Part III Financial Markets

6.1 Discount Bonds

Alexander Hamilton, the first Secretary of the U. S. Treasury, brought bonds to the U. S. One of his first acts was to consolidate all debt from the Revolutionary War, resulting in the first U. S. government bonds. Many features of original bonds are the same, even with a more complex bond market.

A bond is a promise to make a series of payments on specific future dates. Bonds create obligations, and are therefore thought of as legal contracts that require the borrower to make payments to the lender, and specify what happens if the borrower fails to do so. Who makes this promise to pay? Governments and firms. Governments at various levels: local (city level/municipality level), subnational (regional level, state level), and national (often called treasury bonds). Firms also make this promise to pay. These are called corporate bonds.

Governments and firms create new bonds (this is called “issuing”) to sell them to the public. An original seller of a bond is called the issuer and the buyer of the bond is called the holder. For example, when the U. S. Treasury issues a new bond, it is selling a claim for a future payment that the Treasury promises to fulfill sometime in the future. Who does it sell it to? Private citizens, foreigners, businesses, other governments, and financial firms. The buyers of the bond are called bond holders. They pay for the bond now in exchange for a future payment. The bond is considered an instrument of debt.

The bond issuer is selling an instrument of debt, so in essence it becomes a borrower. The bond holder is buying an instrument of debt, so the holder becomes a lender.

Bonds come in two types: coupon bonds and discount bonds. Discount bonds are sometimes called zero-coupon bonds, because they promise no interim payments between the date when the bond is issued and the date when the final payment is due. When an issuer of a discount bond sells it, it enters into a legal contract with the holder, whereby the seller promises to make a future payment, the amount of which is called the principal, face value (FV), or par value, of the bond. The contract also specifies when the payment is due, which is called the maturity date or the term to maturity.

What incentivizes the investor/lender to buy the discount bond is that it is sold at a discount—where the price is below the face value. This is because the face value is paid sometime in the future and the value of that future payment must be discounted to the present with a present value calculation.

The U. S. Treasury issues Treasury bills (T-bills), which are a type of discount bond with a maturity of up to one year. If $\mathrm{FV}$ denoted the face value of a discount bond, n is the time to maturity and i is the interest rate, the price of a discount bond (${\mathrm{P}}_{\mathrm{DB}}$) is given by:

Say, we have a T-bill that represents a promise by the U. S. government to pay $1,000 on a fixed future date. If the interest rate were 5%, the price of a one-year T-bill would be $952.38. If the bond had a shorter maturity, we would discount fewer periods, which would increase the price of the bond. So the price of a six-month T-bill with a face value of $1,000 and an interest of 5% would be:

This is slightly higher than $952.38. The following snippet of Python code defines the equation to calculate the value of a discount bond.

Applying this formula, we can see that on a $1,000 bond at an interest rate of 5%, the price value declines the longer we have to wait for the final payment of the face value of the bond.

When the maturity of this bond increases to two or three years, the price of the bond decreases.

Our first takeaway is: The price of the discount bond is inversely related to its maturity. Incidentally, if the discount bond were redeemable immediately, there would be no periods to discount the face value. No discounting means the price of the bond would exactly equal the cash value of the promised principal.

In addition to the term to maturity, another crucial factor that impacts the price of a discount bond is the interest rate. As we have seen in previous chapters, if the prevailing rate increased from 5% to, say, 7% or 10%, the discount bond would be discounted faster.

Keeping the maturity fixed at three periods, what happens to the price of the bond as the interest rate increases?

At any fixed maturity (n), as the interest rate increases, we are discounting the face value faster, so the price decreases. Similarly, if the interest rates decreases, future payments are discounted more slowly, which serves to appreciate the value of the contract promising the payment, thereby raising its price. The following Python snippet shows the price of a three-year bond goes up as the interest rate decreases.

Our second takeaway is: The price of the discount bond is inversely related to its yield or interest rate.

An implicit assumption we are making here is that market interest rates are non-negative. While, technically, there is no upper bound on yields, it is believed there is a zero-lower-bound (ZLB) on (nominal) interest rates below which they cannot go. However, in the late 2000s and early 2010s, some negative yields were reached for a small number of financial instruments in a few countries. If this were the case for bonds, it begs the question: What would be the price of a discount bond with a negative yield?

In that case, the denominator in the price formula would be smaller than one, which would render the price of the bond even larger than its face value. For example, the price of a three-year bond with a face value of $1,000 and a yield of −1% would actually be $1,030, which is higher than its face value.

In other words, if a lender were to pay $1,030.61 today for a $1,000 payment in three years, this bond would not be sold at a discount, but it would actually be sold at a premium. This is clearly an unusual proposition. Why would anyone pay more today for the privilege of receiving a lower cash payment in the future?

There could be some tax implications or some unusual carrying costs of cash to explain this somewhat aberrant behavior. In any case, some yields have, on rare occasions, become slightly negative, but it is highly unlikely that they will ever be largely, or even moderately, negative. This would lead to what is known as a liquidity trap problem with an instrument in infinite supply and zero demand...the market would most likely break!

In summary: 1. The price of the discount bond is inversely related to its maturity. We will see that this rule can be extended to any type of bond. 2. Prices and yields of discount bonds are inversely related. This will hold for any type of bond. 3. Discount bonds are also called zero-coupon bonds (meaning they do not make any coupon payments)–the only payment is the face value at maturity. Most notable examples of discount bonds are issued by government institutions such as the U. S. Treasury.

6.2 Coupon Bonds

A second type of bond is known as coupon bonds. Coupon bonds are somewhat more common than discount bonds. When an issuer of a coupon bond sells it, the issuer enters into a legal contract with the holder, whereby the issuer/borrower is required to make annual payments, called coupon payments. The contract specifies: the annual interest the borrower pays (${\mathit{i}}_{\mathit{c}}$), which is known as the coupon rate; the date on which the payments stop and the loan is repaid (n), which is called the maturity date or term to maturity; and the final payment to be made when the bond matures, which is the principal, face value, or par value of the bond. Therefore, what distinguishes coupon bonds from discount bonds is that the former actually makes periodic payments even before the bond matures—before paying back the face value—while the latter makes a single face value payment at maturity. As with discount bonds, it must also be the case that, say, a coupon bond with a $100 face value payment at a future date is worth less than 100 dollars cash today.

What would be the value of a coupon bond’s $100 principal to be paid in five years at an interest rate of 3%? This would require the exact same calculation we used before for a discount bond. In fact, the price of this coupon bond would be exactly the same as the price of a discount bond (${\mathrm{P}}_{\mathrm{DB}}$) with a five year maturity with the same principal of $100 and the same interest rate of 3%.

So, if we let (${\mathrm{P}}_{\mathrm{DB}}$) denote the price of the bond’s principal, the price of a coupon bond’s $1,000 principal with maturity of five years at an interest rate of 3% would also be $862.61.

So, does this represent the full price of the coupon bond? Not necessarily. We need to remember that the coupon bond promises regular future payments between the date that it is issued and the date that it matures. While these payments are made periodically at regular intervals, they are also made some time in the future. Therefore, we need to discount this stream of future coupon payments to the present. Since these payments add value to the bond, they must also be reflected in its price.

The fact that coupon payments are made on regular intervals allows us to model them as an annuity. Luckily, we have discussed how to calculate the present value of an annuity in previous chapters.

The price of a stream of coupon payments (${\mathrm{P}}_{\mathrm{CP}}$) can be calculated generally as the sum of the present values of each coupon payment as follows:

These resemble fixed loan payments, where C is the fixed amount of a coupon payment. This present value expression gives us a general formula for the string of yearly coupon payments made over n years. This formula can be easily written into Python as follows: … taking as inputs the fixed amount of the coupon payment (C), the number of periods until maturity (n) and the interest rate (i). With this equation defined, we can now call it to calculate the present value of any collection of coupon payments.

For example, what is the PV of a string of five ($\mathit{n}=5$) coupon payments in the amount of $100 at an interest rate of three percent ($\mathit{i}=0.03$)?

If the amount of the coupon payment were to increase, the bond would be more valuable so its price would also increase. In fact, the increase is one-to-one, meaning that—while keeping all other terms unchanged—if the coupon payment were to double, the price of the bond would also double. The snippet below shows that if the interest rate and maturity remained the same, but the coupon payment were to double from $100 to $200, the present value of the coupon bond would double as well, from $457.9 to $915.9.

Keeping everything else constant, the present value of a stream of coupon payments would decrease if the interest rate were to increase. This is because each future coupon payment would be discounted faster at a higher interest rate. The following snippet of Python code shows that the present value of a $200 coupon bond, paid once a year for the next five years, would decrease from $915.9 if the interest rate were to rise from 3% to 7%.

This means that the present value of a stream of coupon bonds decreases as the interest rate increases. This should also make sense. At higher interest rates, we could collect more returns on deposits that we hold now. But if we are holding a coupon bond, we cannot collect returns now on payments that are not made yet. Since we are waiting to be paid that coupon payment, waiting is more costly at higher interest rates, because there are potentially higher returns that we are not collecting while we wait. In other words, there is an opportunity cost of holding a coupon bond and that opportunity cost is higher at higher interest rates. So, a higher interest rate generally makes the coupon bond less valuable, all else being equal.

Finally, what happens to the present value of a coupon bond as it promises more payments (as n increases)?

All else being equal, a coupon bond that promises more payments will be more valuable. For example, if at an interest rate of 7%, the maturity of a $200 coupon bond were to increase from five years to eight years (meaning, it would make eight payments instead of five), the present value of that bond would increase from $820 to $1,194.

Given that a coupon bond provides two sources of revenue—a stream of coupon payments and the bond principal at maturity—the price of the coupon bond must reflect both factors. Specifically, the price of the coupon bond (${\mathrm{P}}_{\mathrm{CB}}$) adds up the price of the coupon payments (${\mathrm{P}}_{\mathrm{cp}}$) and the bond’s principal (${\mathrm{P}}_{\mathrm{BP}}$), or the face value of the bond.

The following lines of code define a Python function to calculate the price of the coupon bond:

This formula allows us to answer questions such as: What is the price of a five-year coupon bond with a face value of $1,000 and $100 coupon payment at an interest rate of 10%?

The price of a coupon bond may not always equal its face value. However, in this case, the price of this particular coupon bond would be $1,000, the same as its face value of $1,000.

Why is it the case here that the price of this bond equals its face value? We can answer this by noticing that this bond pays $100 coupon payments out of $1,000 FV, which means that the return from a coupon payment is 10% (= $100/$1,000). This is called the coupon rate ($\mathrm{cr}$). The coupon rate equals the coupon payment divided by the face value of the bond.

Any time the coupon rate equals the interest rate ($\mathrm{cr}=\mathit{i}$), the price of the coupon bond equals its face value (${\mathrm{P}}_{\mathrm{CB}}=\mathrm{FV}$). This holds whether the bond has a short or a long maturity. This also holds independently of the amount of the face value or coupon payment or interest. We can verify this by applying the formula to a much longer coupon bond with 30 year maturity ($\mathit{n}=30$) that has a much larger face value ($30,000) with coupon payments of $600 at an interest rate of 2%.

The price of this coupon bond also equals its face value of $30,000 even when it has a much longer maturity than five years. This is because its coupon rate of 2% equals the prevailing interest rate of 2%.

Let’s get back to our five-year coupon bond with a $1,000 face value and coupon payments of $100 at an interest rate of 10%.

Some of the same conditions we learned about the discount bond also apply to the coupon bond. For example, if the face value increases, all else being equal, the bond’s price increases. This makes sense, because the bond is more desirable when it promises a higher future payment.

On the other hand, when the amount of the coupon payment declines, so does the value and, therefore, the price of the coupon bond. This also follows the same reasoning, that there is a direct relationship between coupon payments, face value, and the price of a coupon bond.

Finally, when we examined discount bonds before, we learned that higher interest rates lead to lower bond prices, because with higher interest rates, investors incur higher opportunity costs of holding these bonds, so they discount the future faster, making the wait for the future payment at maturity less desirable today. This also holds generally for coupon bonds. However, when we are pricing coupon bonds, we now must contend with not just one but two rates: the coupon rate ($\mathrm{cr}$) and the prevailing market rate (i).

We have seen that any time the coupon rate and the interest rate are equal, the price of the coupon bond equals its face value. What happens to the price of the coupon bond when the interest rate is higher than the promised coupon rate? If the coupon rate is lower than prevailing rates in the market (), this must mean that the opportunity cost of holding the coupon bond outweighs the benefit of coupon payments, which makes the coupon bond less valuable. Therefore, whenever the interest rate is above the coupon rate, the price of the coupon bond is lower than its face value.

In the previous example, if the coupon payments promised decreased from $100 to $40, the coupon rate would decrease from 10% to 4%. Therefore, at an interest rate of 10%, the price of a five-year $1,000 face value coupon bond with a coupon rate of 4% would be $772—which is lower than its $1,000 face value.

Similarly, any time a coupon bond promises a coupon rate above the prevailing interest rate ($\mathrm{cr}\mathrm{>}10\mathrm{\%}$) the price of the bond is above its face value. For example, if we had another coupon bond with the exact same terms as the previous one, but with a coupon rate of 15% instead of 4%, the price of the bond would increase from $772 to $1,189, which is above its $1,000 face value.

Finally, all else being equal, what happens as the maturity increases? Holding everything the same, what if instead of a five-year bond it was a 30-year bond? In this case, we have tension: two forces pulling in opposite directions. The longer we have to wait for maturity, the less the face value means to us today because we are discounting longer. But the more coupons we get in the interim (as we wait), the more valuable the coupon bond is today. However, recall the value of those coupons also declines the longer we have to wait for them.

So as we saw in earlier sections, the price and the time to maturity are inversely related for discount bonds. The longer in the future we have to wait for the face value payment, the lower the price of the discount bond today. The relationship is not as clear for coupon bonds because of this tension between coupons and time. In other words, longer maturities may affect the price of the bond, but this effect is factored by the relationship between the coupon rate and the prevailing interest rate.

For example, as we saw earlier, if the bond has a coupon rate that equals the prevailing interest rate ($\mathrm{cr}=\mathit{i}$), maturity will not impact the price of the bond. Whether it is a five-year, a thirty-year or any other maturity, the price of the bond equals its face value.

On the other hand, we saw that if the coupon rate was above the prevailing interest rate ($\mathrm{cr}\mathrm{>}\mathit{i}$), the price of the coupon bond will be above face value. For this type of bond, as the maturity increases, the return from coupon payments outweigh the opportunity cost at each period, for more periods. Therefore, a coupon bond that is priced above its face value would experience an increase in price at longer maturities.

For example, at an interest rate of 10%, a $1,000 face value coupon bond with a coupon rate of 15% at five years would have a price above $1,000 and at 30 years would have an even higher price.

Conversely, if the coupon rate stood below the market rate (), the price of the bond would be below its face value. In that case, if the maturity were to increase, the opportunity cost of holding that bond would be higher than the expected return from the coupon payment each period… for more periods. This effect would compound at longer maturities, decreasing the price of the coupon bond. For example, at an interest rate of 10%, the price of a coupon bond with a coupon rate of 8% that pays a $1,000 face value in five years would be $924. Holding everything else the same, increasing the maturity would lower the price further below its face value. If instead of five years, the bond had a maturity of 30 years, its price would go down to $811.

In summary: 1. The price of a coupon bond is directly related to its coupon payments and face value. All else being equal, if either the price or the coupon payment increases, so does the price. 2. The price of zero-coupon bonds always lies (at a discount) below its face value. This is not necessarily the case for coupon bonds. 3. Whenever the coupon rate equals the market rate, the price of the coupon bond equals its face value. If the coupon rate is above the market rate, the coupon bond is priced above its face value. And if the coupon rate is below the market rate, the price of the coupon bond lies below its face value.

6.3 Interest Rates, Yields, and Bond Prices

The relationship between bond prices and interest rates is important. Bonds promise fixed payments at future dates, so generally the higher the interest rate, the lower their present value. The value of a bond varies inversely with the interest rate used to calculate the present value of the promised payment.

Suppose a bond has four years until maturity. Its face value is $1,000, which means it pays $1,000 at maturity with annual coupons of $50. If the interest rate were 5%, the price of this bond would be $1,000—the same as its face value.

If the interest rate were to increase from 5% to 6%, the price of the bond would decrease from $1,000 down to $965.34.

This is the case because the interest increased, and at 6% it is now higher than the coupon rate of 5%. It is often the case that interest rates may depart from coupon rates. This is because a coupon rate is generally fixed at origination—when the bond is newly issued. The amount of the coupon payment is guaranteed not to change over the life of the bond. Short of a payment default by the issuer, the coupon bond makes the same payment at the prescribed time. These terms are contractually guaranteed at issuance. So, while coupon rates are fixed at issuance and do not change over the life of bond, interest rates vary daily.

Once a newly minted bond has been sold, it can be resold again at any time before maturity in a secondary market. The terms of the bond, such as the coupon rate and the face value, never change since they are fixed at issuance. But the prevailing rate may certainly fluctuate from day to day. Generally, bonds are traded daily. Every trading day, there are two bond markets: one for “newly minted” bonds and one for “second-hand” bonds, where these bonds are bought and sold. Even if the terms that give fundamental value to the bond like the face value and coupon rate are fixed, the prevailing rate may change daily and, therefore, the price of the bond may change daily.

The current yield ($\mathrm{cy}$) is the measure of the proceeds the bond holder receives for making the loan (purchasing the bond). The current yield measures that part of the return from buying the bond that arises solely from the coupon payments. When we buy a coupon bond today, the coupon payments are agreed upon and fixed at that time. Therefore, as we purchase the bond, we know what it currently yields because we know what the coupon payments will be.

For example, what is the current yield of a one-year coupon bond with a face value of $100 and a coupon rate of 5% selling for $98?

Its current yield would be 5.1%. Notice the current yield of 5.1% is above the coupon rate of 5% because the $98 price of the bond is below its face value of $100.

The most useful measure of the return from holding a bond is called the yield to maturity (YTM). This is the yield bond holders receive if they hold the bond to its maturity when the final principal payment is made. In other words, the YTM equals the coupon rate when the bond is sold. One way to think about this is that when we buy a bond, we need to know the interest rate that prevails currently in the market.

If the interest rate is high today, the price of the bond is low today, which means we get a face value in the future for a relatively low price today. If the interest rate is low today, the price of the bond is relatively high, which means we get a face value in the future for a relatively high price today. Therefore, the market interest rate (i) gives as a sense of what the current bond price should be (e. g. is the bond’s value high or low?) today. Then, comparing the price to the promised coupon rate from the bond gives us insight into what the bond currently yields at purchase. Importantly, if the bond is held to its maturity, then the current yield is what the bond eventually yields at maturity. Therefore, the YTM equals the current yield (at purchase) if the bond is held to its maturity.

Since the YTM is the yield bond holders receive if they hold the bond to its maturity when the final principal payment is made, it must be equal to the interest rate (i) that solves the equation for the present value of the coupon bond. Let’s return to our example. Say we have a one-year coupon bond with a face value of $100 and a coupon rate of 5% selling for $98. We calculated the current yield of this bond is 5.1%. What is its YTM? We need to solve the following equation for the interest rate i

Solving the equation for i reveals the YTM for this bond is 7.07%. Therefore, we find that this bond has a coupon rate of 5%, a current yield of 5.1% and yield to maturity of 7.1%.

We will find that it is always the case that for a bond that is priced below its face value, its coupon rate is below its current yield, which in turn is below its yield to maturity.

Take another example. Say we have a $1,000 face value coupon bond that pays $50 coupon payments, matures in four years, and the prevailing interest rate is 5%. We already know how to calculate the price of this coupon bond. The snippet below shows the price is $1,000.

We know the coupon rate is $\mathrm{cr}=\frac{\mathrm{C}}{\mathrm{FV}}=\frac{50}{1\mathrm{,}000}=5\mathrm{\%}$ and the current yield is the same, since we now know the price and the face value equal each other $\mathrm{cy}=\frac{\mathrm{C}}{\mathrm{P}}=\frac{50}{1\mathrm{,}000}=5\mathrm{\%}$. What is the YTM of this bond? The following Python snippet provides a definition:

Calling this function with the details of this bond reveals that the YTM is also 5%, as shown in the following code snippet

The table below summarizes what we have learned about the relationship between the coupon rate, the current yield, and the yield to maturity.

In summary:

1.

When Bond Price < Face Value: Coupon rate < Current Yield < Yield to Maturity.

2.

When Bond Price = Face Value: Coupon rate = Current Yield = Yield to Maturity.

3.

When Bond Price > Face Value: Coupon rate > Current Yield > Yield to Maturity.

For a fixed coupon rate, the YTM is always inversely related to the price of the bond. shows the YTM of various bonds with 10% coupon rates.

When the YTM equals the cr, the price of the bond equals its face value. As the yield to maturity increases above the coupon rate, the price of the bond decreases. When the coupon bond is priced at its face value, the yield to maturity equals the coupon rate. The yield to maturity is greater than the coupon rate when the bond price is below its face value. Therefore, the price of a coupon bond and the yield to maturity are negatively related.

6.4 Rates of Return

There are two possible sources of earning a return from holding a coupon bond: the coupon payments collected at regular intervals and the face value collected at maturity. For example, if we paid $95 for a one-year coupon bond with a $100 face value and a coupon rate of 10%, we would get two returns. The 10% coupon rate means we would get a $10 coupon payment. This is a dollar return obtained from holding the bond. And if we held this bond to its maturity, we would get $100 when the bond expires.

Since we paid $95 for the bond, we would also be collecting an extra $\mathrm{(}\mathrm{\$}100-\mathrm{\$}95=\mathrm{)}\mathrm{\$}5$. The rise in value from an investment is referred to as a capital gain and it is part of the return on our investment from holding the bond. On the other hand, if the price of the bond was above the face value, the holder would incur a capital loss at maturity. In this particular example, we would be getting a $10 coupon payment plus a $5 capital gain from holding a bond that we paid $95 for. Therefore, the total return from holding this bond to maturity would be $\mathrm{(}\mathrm{\$}15\mathrm{/}\mathrm{\$}95=\mathrm{)}15.8\mathrm{\%}$.

Since we have two sources of returns from holding a coupon bond, its rate of return must reflect both. This is an equation for the rate of return (RoR):

The first term in the sum is what the bond currently yields ($\mathrm{cy}=\frac{\mathrm{C}}{\mathrm{P}}$) and the second term is the capital gain ($\frac{\mathrm{FV}-\mathrm{P}}{\mathrm{P}}\mathrm{>}0$) or capital loss ($\frac{\mathrm{FV}-\mathrm{P}}{\mathrm{P}}\mathrm{<}0$) from selling the bond.

As we discussed in the previous section, if we buy a multi-period coupon bond for a price that exactly equals its face value and hold it to its maturity, then the current yield, the coupon rate, and the yield to maturity we get from the bond are the same. We are now prepared to draw some insight into the relationship between all of these rates (cy, cr, YTM) and the rate of return (RoR).

For example, imagine that with a prevailing interest rate of 6% ($\mathit{i}=0.06$), we buy a 10-year coupon bond that promises a 6% coupon rate ($\mathrm{cr}=0.06$) and a principal payment of $1,000 at maturity. We know that since ($\mathrm{cr}=\mathit{i}$), the price of this coupon bond should be equal to its $1,000 face value.

Since the price of the bond equals its face value, if we hold this bond to maturity, we know its coupon rate is 6% ($\mathrm{cr}=\frac{\mathrm{C}}{\mathrm{FV}}=\frac{\mathrm{\$}60}{\mathrm{\$}1\mathrm{,}000}=6\mathrm{\%}$) and its current yield is also 6% ($\mathrm{cy}=\frac{\mathrm{C}}{\mathrm{P}}=\frac{\mathrm{\$}60}{\mathrm{\$}1\mathrm{,}000}=6\mathrm{\%}$), which also equals to a $\mathrm{YTM}=6\mathrm{\%}$.

So if we hold this coupon bond to its maturity, what return do we get? Since we are holding to maturity, we should be able to sell it back to the issuer for its face value, and since the face value exactly equals what we originally paid for it, we will incur neither a capital gain nor a capital loss. The rate of return equation tells us we also get a 6% rate of return from holding this bond.

Since we are holding this bond to its maturity, what we really get from this bond is its coupon rate ($\mathrm{cr}=6\mathrm{\%}$), which is equal to what the bond currently yields at any given time ($\mathrm{cy}=6\mathrm{\%}$), which equals its yield to maturity since we are holding it to its maturity ($\mathrm{YTM}=6\mathrm{\%}$), which is ultimately the return we get from holding it ($\mathrm{RoR}=6\mathrm{\%}$).

The following snippet of Python code calls the various equations we defined and applies the terms of this coupon bond:

We can conclude that if we buy a coupon bond with the intention of holding it to its maturity as an investment vehicle, the rate of return from holding the bond to its maturity equals its yield to maturity, or its current yield, or its coupon rate. Remember that while we hold this bond for the 10 years, interest rates, prices, and coupon rates promised by other bonds may fluctuate daily. But this will not have any bearing on the fundamentals of the bond we are holding, since the coupon rate and face value are locked in and guaranteed (so long as the issuer does not default on its promise to pay them).

This often leads to the conclusion that high interest rates are good for financial investors because they imply high rates of return. But this is not always the case. If we buy coupon bonds as a long-term investment vehicle and we hold them to their maturity, then yes… the highest the yield ($\mathrm{cr}$, $\mathrm{cy}$, or $\mathrm{YTM}$), the highest the rate of return from holding the bond, all else being equal.

However, even if we have all the intention of a “buy-and-hold” strategy, we may not be able to hold on to a particular coupon bond until it matures. In that case, we will need to consider the holding period return as the return to holding a bond and selling it before maturity. In the cases of selling ahead of maturity, the holding period return can differ from the yield to maturity.

For example, let’s say that we buy the previous bond we discussed for $1,000 today, planning to hold it for its maturity of 10 years. Every year we will collect a coupon payment of $60 and in 10 years we will collect the $1,000 face value by selling it back to the issuer. We know that we should be able to receive a 6% rate of return from holding this bond to its maturity, which is what the bond yields to its maturity. However, let’s say that shortly after collecting the first annual coupon payment of $60 next year, we get into a small car accident leading to some unforeseen medical expenses. Unfortunately, we are not insured, we don’t have credit cards, and the hospital will not accept bonds as payment. We know we have a $1,000 principal payment coming, but the hospital is not willing to wait nine years to be paid.

This bond we are holding is a financial instrument that has value. It has already made a $60 payment, but the bond certainly has residual value, because it should make nine future payments of $60 over the next nine years. Moreover, in nine years’ time, it will make a 1,000 payment to whomever is holding it. So we are going to re-enter the bond market looking to sell this bond in a process that is called “liquidation.” We would have liked to keep the bond to its maturity and collect the 6% yield to maturity, but we are going to convert this bond into cash now so that we can pay our medical expenses.

We want to calculate the holding period return. What is the return from holding the bond for one year and selling it with nine years left before its maturity? We know we have already been enjoying a current yield of 6% ($60/$1,000), but we do not know what the holding period return is until we find out what is the price we can sell our bond for. Since we are re-entering the bond market with a 10-year bond that has nine years until it matures, we now have to compete with other bonds that may promise other coupon rates in a market where prevailing interest rates may be different from when we originally purchased our bond. All of this will likely impact the price that our bond will fetch.

Let’s say that at the time we want to sell the bond, interest rates have gone down. This sounds like it’s a bad turn for us because we concluded earlier that maybe lower interest rates mean lower rates of return. But this is actually the opposite. If yields are lower now than when we first bought the bond, this is very good news for us!

Let’s assume that the prevailing interest rate today is 5% and brand-new coupon bonds with various maturities (of 10 years, nine years, five years…) now promise lower coupon payments. If the average 10-year coupon bond today promises a coupon rate of 5% ($50 coupon payments for the next 10 years and $1,000 at the end), the 10-year bond we are holding is more valuable than the competition (since it promises a 6% coupon rate and the $1,000 principal back in nine years). Since we are holding a more valuable bond than the competition, the price of the bond should be higher.

In fact in our example, the price will be higher than what we bought it for. Since it promises a 6% coupon rate when the prevailing interest rate is 5%, the bond’s future payments (higher than the competition) will be discounted at a slower pace, which leads to an increased price today. The price of our bond that we sell under these conditions would be $1,071 today.

This is the price when we bought the bond:

And this is the price if we sell today with nine years left in the life of the bond:

Here is the calculation in Python code:

If we sold the bond for $1,071, we would be incurring a capital gain. Since we originally bought the bond for $1,000, selling the bond would net us a 7.1% capital gain. This would also be a return from the bond above and beyond the 6% we were getting the first year we held it. Therefore, the holding period return from buying this bond and selling it with nine years left of its maturity for a higher price would be

In this example, we had to liquidate our bond because of our emergency, and we were lucky that interest rates were lower at the time we had to sell. This led to an appreciation in value of the bond we were holding and we were able to collect a capital gain. The holding period return from our bond is higher than the return we would have derived had we kept the bond to maturity. In other words, by selling early, the rate of return is higher than the yield to maturity of the bond.

While we may have invested in bonds as a long-term investment, this turn of events resulted in a profitable proposition. This brings up another motive for investing in the bond market, namely speculation. Some investors enter the bond market with little intention of holding bonds to their maturity. Instead, they might chase higher returns than what the bonds yield to their maturity by speculating on future interest rate movements that might propitiate capital gains from selling early. Specifically, if interest rates decline after a bond has been purchased, the value of the bond increases, generating a higher holding period return by selling ahead of maturity. However, speculation is risky, and it requires taking a calculated risk because interest rates could increase rather than decline, and this would generate capital losses when selling the bond ahead of maturity.

Here is a different scenario facing the same bond. Three years ago, we bought a 10-year bond promising $60 coupon payments when the interest rate stood at 6% ($\mathit{i}=0.06$). We held the bond for three years and now we are contemplating selling it. Does it make sense to sell it?

To answer the question, we may want to calculate the current yield, the yield to maturity, the holding period rate of return, and the price the bond might fetch if we sell. However, to make all of these calculations, we need to know one crucial bit of information. What is the prevailing market rate today? Let’s say the interest rate increased from the 6% we bought at up to now 7%. The following bit of Python code defines a function that calculates the rate of return of a coupon bond:

Given the price (${\mathrm{P}}_{\mathrm{buy}}$) and interest rate (${\mathit{i}}_{\mathrm{buy}}$) at the time of the purchase, the coupon payment specified at the bond’s issuance, the prevailing market interest rate at the time we want to sell the bond (${\mathit{i}}_{\mathrm{sell}}$), and the time left until the bond matures ($\mathrm{tim}2\mathrm{mat}$), we can now make all the necessary calculations. Calling these functions, we defined in Python, with the appropriate information reveals the rate of return from selling this bond early when the interest increased from 6% to 7% would be a low 1.2%, which is much lower than the 6% return we would get if we held the bond to its maturity.

The interest rate increasing from 6% to 7% in our example leads to a reduction in the price of the bond so that, if sold, it incurs a reduction in the rate of return. Our example considered a modest increase in the interest rate. However, had the interest rate increased more dramatically, the rate of return might turn out to be negative if the bond sold incurs a capital loss rather than a gain.

Interest rate changes matter greatly for the ultimate period returns on bonds. However, how far in the future we need to wait for the bond to mature will also affect holding period returns. shows what happens to the rate of return of various bonds that promise the same yield (10%) and have the same face value ($1,000), but differ in their maturities, if the prevailing market interest rate increases.

shows that if the interest rate rose from 10% to 20% at the end of year one, the price of those bonds that have not matured yet would all decrease.

Looking at the first row of the table shows the treasury with one year maturity would come due, returning the face value of $1,000 to the holder. The holder in this case would have held the one-year treasury to its maturity so she would have collected the coupon rate of 10%, which equals the current yield and the yield to maturity. The price of the bond would be $1,000 at the end of the year. Purchasing the bond for $1,000 and collecting the $1,000 face value back when it matures affords the holder neither a gain nor a loss. In this case, the rate of return on this one-year bond is the sum of the 10% current yield in column (b) plus the 0% capital gain in column (e). This is a case, where $\mathrm{cr}=\mathrm{cy}=\mathrm{YTM}=\mathrm{RoR}$.

However, the interest rate increase at the end of year one will affect the rates of return of all those bonds that have not yet matured by then. These are all shown in the second to the sixth row of the table.

As we discussed earlier, interest rates and bond prices are inversely related. We can see that as the interest rate doubles, the price of all these bonds from the two-year to the 30-year decline. However, the longer we have to wait for the bond to mature, the more heavily that bond’s price is discounted.

As the interest rate doubles, we can see on the second row that the price of the two-year bond decreases from $1,000 down to $917. This is because this bond has only one year left before it matures, so it requires a relatively short wait.

Comparatively, the thirty-year bond on the last row experiences a much larger decline from $1,000 down to $503. This is because there are a lot more periods left (29) to discount the $1,000 principal to its present value. Therefore, the longer we have to wait for the bond to mature, the larger the capital loss as a result of an interest rate increase, or the larger the capital gain as a result of an interest rate reduction.

Generally, there is no interest rate risk for any bond whose time to maturity matches the holding period. However, since bonds are not always held to their maturity, prices and returns for long-term bonds are more volatile than those for shorter-term bonds. This introduces the concept of risk in the bond market, which we discuss in the next section.

6.5 Bond Risk

Bonds are promises to pay a certain pre-specified dollar amount sometime in the future. How can that be risky?

Essentially, risk arises if the promised payment were to be reneged on. Alternatively, risk could arise if the value of the payment were eroded even if the promised payment itself were fulfilled. Finally, unforeseen movements in interest rates might bring about risk if a bond is sold before maturity. Therefore, the risk of holding bonds falls under three main categories: 1) default risk, 2) inflation risk, and 3) interest rate risk.

Default risk refers to the probability that the bond’s issuer may not fulfill the payment. Inflation risk refers to uncertainty in future inflation, which might make uncertain the real return of the payments made. Interest rate risk refers to the chance that unforeseen increases in the interest rate could mean capital losses.

Some bonds have low or near zero default risk. They fulfill payments with such high levels of certainty that the interest rate they pay is called the risk-free rate. These are typically treasury bonds of financially sound economies such as Treasury bonds of the U. S. or Sterling Bonds of the U. K. They promise a safe or risk-free interest rate, which is the rate that savers can receive with certainty.

But other types of bonds may not be so safe. Other types of bonds, most notably corporate bonds, may carry some default risk. We can use the risk-free rate from treasuries as a reference (a benchmark) to understand how risky other corporate bonds might be. Since interest rates are used to determine the present value of future payments, they should reflect an asset’s riskiness. If future payments are not certain, an asset is risky, and that risk should reduce the present value of future income.

The risk premium (ψ) is a payment on an asset that compensates the owner for taking risk, and it is typically measured as a percentage.

The present value of a certain future payment is determined using the risk-free interest rate (${\mathit{i}}_{\mathrm{safe}}$). But the present value of a riskier future payment is determined using an interest rate that includes a risk premium (ψ) reflecting the riskiness of the asset. This risk premium can be added to the YTM that the bond would typically promise if the bond were held to its maturity (i. e., no interest rate risk) and the face value were redeemed with certainty (i. e., no default risk).

For example, imagine the risk-free rate treasury rate offered today is 5%. Let’s say the Microsoft corporation offers its own one-year corporate bond that also promises 5% with a face value of $100. If Microsoft’s bond were free of risk, the price of the bond would be the present value of $105 in one year. This means if we bought Microsoft’s bond today at a price of $100, we would collect a 5% coupon rate, which is what the bond currently yields (${\mathit{i}}_{\mathit{c}}=5\mathrm{\%}$), and we would redeem it at maturity (selling it back to Microsoft) for its $100 face value, thereby collecting no capital gain/loss. Our return from this investment would have been 5%, which is what the bond yielded to its maturity. Without risk, the YTM for this bond would be 5%.

Now imagine there is a 10% chance that Microsoft would go bankrupt before paying back the bond, bringing about some degree of default risk. So, there are two possible payoffs: $105 or $0. Now that the value of the bond at its maturity is not certain, we must formulate an expectation of its value. Rather than the certain $105 payment in a year, now the expected payment in a year is $94.50.

Why? Because in the presence of future uncertainty, formulating an expected value involves multiplying the probability and the amount of each expected payoff, and summing up all the outcomes. In our example, the expected value of our bond would be:

But this is the expected value of the bond in one year, so it needs to be discounted to the present. The price of the one-year coupon bond today would be:

Recall that in the absence of risk, the YTM of this one-year Microsoft bond was 5%. But now with some risk of default, the YTM will increase.

How? Well, since there is some risk of default now, the price of that bond today is discounted to $90 (when it would have been $100 absent risk). This means that while the coupon rate is still guaranteed at 5% ($\mathrm{cr}=\frac{\mathrm{\$}5}{\mathrm{\$}100}=5\mathrm{\%}$), the bond’s current yield has now increased to 5.6% ($\mathrm{cy}=\frac{\mathrm{\$}5}{\mathrm{\$}90}=5.6\mathrm{\%}$). Moreover, the yield to maturity would be much higher because—were Microsoft not to default on its payment—the bond would also render a capital gain:

Notice that as we contend with some probability of default, the price of the bond would decline from $100 to $90 and its yield would increase from 5% to 15.8%. With higher risk, investors would demand a higher risk premium, which would lead to further yield increases and a deeper erosion in the price of bonds.

6.6 Glossary

Bonds

A financial instrument representing a promise to make a series of payments on specific future dates, creating a legal contract between the borrower (issuer) and lender (holder).

Capital Gain/Loss

The difference between the selling price and purchase price of a bond, realized when a bond is sold before maturity.

Corporate Bonds

Bonds issued by companies, which typically carry more default risk than government bonds.

Coupon Bond

A bond that makes regular coupon payments throughout its life and returns the face value at maturity.

Coupon Payment

Regular, fixed payments made by the bond issuer to the bond holder at specified intervals, typically annually or semi-annually, calculated as a percentage of the face value.

Coupon Rate (cr)

The annual coupon payment expressed as a percentage of the bond’s face value, fixed at issuance.

Current Yield (cy)

The annual coupon payment divided by the bond’s current market price, representing the return from coupon payments relative to the bond’s price.

Default Risk

The probability that a bond issuer will fail to make promised coupon payments or repay the face value at maturity.

Discount Bond (Zero-Coupon Bond)

A type of bond that makes no interim payments between issuance and maturity, sold at a price below its face value and paying the full face value at maturity.

Face Value (FV) / Par Value / Principal

The final payment amount promised by the bond at maturity, representing the principal amount that will be repaid to the bond holder.

Holder

The buyer of a bond who becomes a lender by purchasing the debt instrument.

Holding Period Return

The total return earned from holding a bond for a specific period, including both coupon payments received and any capital gains or losses.

Interest Rate Risk

The risk that changes in market interest rates will affect the value of a bond, particularly relevant when selling before maturity.

Issuer

The original seller of a bond, typically a government or firm that creates and sells the bond to raise funds.

Liquidity Trap

A market situation where an instrument has infinite supply and zero demand, typically associated with negative interest rates.

Maturity Date

The specific future date when the bond expires and the issuer must repay the face value to the bond holder.

Risk-Free Rate

The interest rate that can be earned with certainty and no risk of default, typically represented by government treasury securities.

Risk Premium ($\mathit{\psi }$)

The additional return required by investors to compensate for taking on risk, measured as a percentage above the risk-free rate.

Term to Maturity

The time remaining until a bond’s maturity date.

Treasury Bills (T-bills)

A type of discount bond issued by the U. S. Treasury with maturity up to one year.

Yield to Maturity (YTM)

The total return an investor would receive if they held the bond to maturity, accounting for coupon payments and any difference between purchase price and face value.

Zero-Lower-Bound (ZLB)

The theoretical floor below which nominal interest rates cannot go, traditionally thought to be zero.

7.1 Bond Supply and Bond Demand

The forces of supply and demand determine bond prices and bond yields. A bond supply curve is the relationship between the price and the quantity of bonds people are willing to sell, all else being equal. So it is a schedule of prices and quantities. We typically express bond supply as a curve in a two-dimensional space, with bond prices on the vertical axis and bond quantities on the horizontal axis. In that two-dimensional space, the bond supply curve slopes upward. This means that, ceteris paribus, the higher the price of a bond, the larger the quantity supplied. But why is this the case?

There are two main reasons related to each side of the market. The first is from the point of view of the investor (or lender or holder) of the bond and the second is from the issuer (or borrower or seller). Consider the first point of view of a typical bond investor who holds a portfolio of bonds. All else being equal, the higher the price of those bonds, the more tempting it is to sell those bonds ahead of maturity and possibly collect higher capital gains. So, at higher prices, the quantity supplied of second-hand bonds (those originally issued in the past but that are yet to mature) should also increase.

Now consider the second point view of some corporation or government. The higher the price at which they can sell the bond, the lower the yield they have to promise on it. This means that if, say, a company is seeking to finance a project, issuing brand new bonds at high prices is a way to cheaply borrow to finance capital expenditures. So, at higher prices, the quantity supplied of newly issued bonds should also increase. All else being equal, for a $1,000 one-year discount bond, the quantity of bonds supplied will be higher at $950 than it will be at $900. See .

The bond demand curve is the relationship between the price and the quantity of bonds that investors demand, all else being equal.

Let’s now discuss the demand side. Recall from previous chapters that bond prices are inversely related to their yield. Therefore, the higher the yield of a given bond, the lower its price, all else being equal. The lower the price bond holders must pay for a fixed-dollar payment on a future date, the more likely they are to buy a bond.

There are two reasons for this. Since lower prices come hand-in-hand with higher yields, buying inexpensive bonds typically means receiving higher current yields and, importantly, higher YTM were the bonds to be held until maturity (all else remaining the same, particularly risk remaining the same).

Another reason why at lower prices investors might be motivated to buy more is that if prices are low, there may be more room for upward movement in bond prices in the future, which would translate to capital gains were those cheap bonds to be sold before maturity. All else being equal, for a $1,000 one-year discount bond, the quantity of bonds demanded will be lower at $950 than it will be at $900. See .

When the quantity of bonds supplied (${\mathit{Q}}_{\mathit{S}}$) and the quantity of bonds demanded (${\mathit{Q}}_{\mathit{D}}$) equal each other, we say the bond market is in equilibrium, and this point is reflected where both curves meet. In below, (${\mathit{P}}^{\ast }$) denotes the equilibrium price. What if bonds prices were below equilibrium? Say current bonds prices (${\mathit{P}}_1$) were unusually low. Then, many investors who wish to buy bonds cannot generally get them at a price below equilibrium. This would generate a shortage (an excess demand) of bonds in the market.

Bonds are clearly a desirable product. Whenever there is a shortage of a given good that is typically desirable, we tend to see a bidding up of its price.

Therefore, if bond investors cannot find the bonds they wish to hold at the current price ${\mathit{P}}_1$, they may begin to bid up the prices they are willing to buy them for. This is reflected by the arrows alongside the demand curve in .

Bond issuers would be all too willing to supply more bonds at higher prices, which is reflected by the arrows alongside the supply curve. Both of these forces drive up the price of the bond from ${\mathit{P}}_1$ toward equilibrium.

In a process called market clearing, this bidding up of prices will shrink the shortage regions (the difference between ${\mathit{Q}}_{\mathit{S}}$ and ${\mathit{Q}}_{\mathit{D}}$) until the bond market returns to its equilibrium condition, where ${\mathit{Q}}_{\mathit{S}}={\mathit{Q}}_{\mathit{D}}$ when the price reaches ${\mathit{P}}^{\ast }$.

Conversely, what if bond prices were above equilibrium? Say current bonds prices (${\mathit{P}}_2$) were unusually high. In this case, we would have the opposite scenario, where at a high price bond issuers would be all too happy to supply more bonds. However, it would be difficult to find investors to be willing to pay such a high price. This would generate a surplus (or excess supply) of bonds.

Since at the high price of ${\mathit{P}}_2$ suppliers cannot find buyers, bond issuers/sellers will begin to cut prices. Excess supply puts downward pressures on the price (reflected on the downward arrows alongside the two curves in ) until once again markets clear and the quantity supplied equals the quantity demanded.

From this analysis, we can see that any change in the price (and the quantity demanded or supplied) of the bond causes movements alongside the supply and demand curves for bonds. Essentially, any market surprise that impacts the variables in the axes (price and quantity) can be reflected by movements along the curves. In other words, the curves themselves do not change (they do not shift or tilt) in response to shocks to the bond price (or quantity).

This begs the question: what factors will impact (shift) the supply and demand curves? Whatever those factors might be, we know they cannot be the two variables (P and Q) shown on the axes. They must be something else.

Let’s first discuss factors that may potentially affect the demand (curve) for bonds. Since demand typically describes the willingness and ability to purchase, if a given factor is to have an impact on bond demand it must be because it affects the willingness or ability to buy it.

One factor is the personal (or institutional) wealth of the investor. Insofar as bonds are generally desirable goods (what economists call normal goods), bond demand should increase when the ability to purchase (wealth) increases. On the other hand, if an investor experiences a sharp reduction in wealth, she can now afford fewer bonds, so, all else being equal, her demand for bonds will decrease.

These shocks affecting the ability to purchase bonds are typically called an income or wealth effect, which is often determined by current forces in the market.

While the ability to purchase bonds is generally tied to the current financial health of the investor and current market conditions, the willingness to buy bonds is typically shaped by expectations of the future.

Since inflation often erodes real returns, changes in expected inflation may impact bond demand. For example, if investors expected inflation to increase in the future, the real value of the bond’s promised payments would decrease, making it less desirable to purchase today. On the other hand, with declines in inflation expectations, investors would expect an appreciation in the real value of the bond’s future payments, making the bonds more desirable to purchase today.

Expectations of future interest rates will also affect the demand curve for bonds. Recall from previous chapters that prices and interest rates are inversely related. Since changes in prices are on the vertical axis of our graphs, they do not affect the demand (curve) for bonds, they only affect the quantity demanded through movements alongside the curve. So, this must also be the case for interest rates or yields.

Changes in current interest rates will not impact (shift) the demand for bonds, but changes in expected future interest rates will. For example, if interest rates were expected to fall, investors would want to buy more bonds now while the yield is higher. Because expected declines in interest rates would foretell expected increases in the price of the bonds—opening opportunities for higher capital gains—bonds would be more desirable to buy now if we expected interest rates to decline.

Conversely, if interest rates are expected to increase, investors might buy fewer bonds now and wait until those expectations materialize to higher yields.

Expected returns work in a similar fashion to expected interest rates in their effect on bond demand. But, while expected interest rate changes affect the willingness to buy the bond, expected returns affect the willingness to buy the bond relative to alternative assets. This is typically called a substitution effect.

For example, if relative to alternative assets (e. g., money markets) bond returns were expected to increase in the future, then even on a fixed budget, the investor might sell off (substitute away from) other assets and buy (substitute into) more bonds.

Other examples of factors that determine bond demand through a substitution effect are risk and liquidity. As we discussed in previous chapters, risk is not an absolute, but a relative concept.

For example, if bonds became less risky relative to alternative investment, all else being equal, they would become more desirable, thereby increasing demand for bonds. Furthermore, investors like liquidity, all else being equal. The easier it is to convert a particular bond into cash, the more liquid it is. The more liquid the bond is, the more desirable it is, thereby increasing its demand.

An increase in bond demand means that, independently of prices, more quantity is being demanded. This suggests that whether prices are high, moderate, or low, some factor that serves to increase bond demand would shift the bond demand curve to the right, guaranteeing more quantity demanded at any given price. This is shown in . Say we begin in the equilibrium point 1, where the S and D curves meet at price ${\mathit{P}}^{\ast }$, as the market clears to ${\mathit{Q}}_{\mathit{S}}={\mathit{Q}}_{\mathit{D}}={\mathit{Q}}^{\ast }$.

The demand for bonds would increase for any given price if investors: 1) experienced an increase in wealth, 2) expected lower levels of future inflation, 3) expected future interest rates to decrease, 4) expected lower returns in money markets (a substitute for bond markets), or 5) expected more uncertainty/risk in the stock market—all else being equal.

This would lead to a wholesale shift to the right of the demand curve to a new demand curve ${\mathit{D}}_2$.

After the rightward shift in , the old equilibrium point, (Point 1 in the graph) would no longer hold. That is because, while ${\mathit{P}}^{\ast }$ used to imply ${\mathit{Q}}_{\mathit{S}}={\mathit{Q}}_{\mathit{D}}={\mathit{Q}}^{\ast }$, after the curve shifts right, ${\mathit{P}}^{\ast }$ pins down the same quantity supplied (${\mathit{Q}}_{\mathit{S}}$) from the supply curve as before.

However, ${\mathit{Q}}_{\mathit{D}}$ now falls in the new ${\mathit{D}}_2$ curve generating a shortage of bonds at the old price ${\mathit{P}}^{\ast }$. Market forces would bid up the price of bonds to a new equilibrium condition (point 2 in the graph) at a higher equilibrium price ${\mathit{P}}^{\ast \ast }$ and higher quantity of bonds sold ${\mathit{Q}}_{\mathit{S}}^{\mathrm{new}}={\mathit{Q}}_{\mathit{D}}^{\mathrm{new}}={\mathit{Q}}^{\ast \ast }$. summarizes the various factors that would result in the rightward demand shift displayed in .

In conclusion, all else being equal, any factor that would serve to increase the demand for bonds would eventually increase the quantity sold of bonds, as well as raise equilibrium prices, and lower equilibrium yields in the bond market.

Table 7.1Factors that affect bond demand and shift the demand curve.

Surprise change in	Effect on bond demand
An increase in wealth increases demand for all normal (non-inferior) assets, including bonds	Bond Demand shifts
	Bond Prices
	Bond Yields
An increase in expected inflation makes bonds with fixed nominal payments less desirable	Bond Demand shifts
	Bond Prices
	Bond Yields
An increase in expected returns on bonds relative to substitute financial instruments makes purchasing bonds more attractive now	Bond Demand shifts
	Bond Prices
	Bond Yields
An increase in the risk of bonds relative to substitute assets makes purchasing bonds less attractive now	Bond Demand shifts
	Bond Prices
	Bond Yields
An increase in the liquidity of bonds relative to the liquidity of alternative assets makes purchasing bonds more attractive now	Bond Demand shifts
	Bond Prices
	Bond Yields

Let’s now discuss factors that may potentially affect the supply (curve) for bonds. Recall, the bond supply curve is the relationship between the price and the quantity of bonds that institutions supply, all else being equal. Factors that may shift the supply curve of bonds may fall under two main categories: those that affect the institution that issues the bonds (idiosyncratic factors) and those that affect the economy (systemic or economy-wide factors).

Institutionally, any change in a government’s borrowing needs will impact the supply of bonds outstanding, all else being equal. Often, local governments finance expenditures by issuing municipal bonds. When local governments want to increase expenditures on public works (such infrastructure projects as parks, roads, government buildings like City Hall, etc.) or expand on services (as those offered by parks and recreational facilities), new bond issuance will increase.

When central governments want to increase spending (e. g., on national defense) above and beyond what they appropriate in taxes, they may issue treasury bonds. Increases in a central government’s spending (Federal spending in the U. S.) lead to new treasury issuance. Any increase in municipal or treasury bond issuance serves to increase the supply of bonds at any price. This shifts the supply curve of bonds to the right.

Another institutional source of bond supply is the issuance of corporate bonds. When firms and corporations want to increase capital spending (higher expenditures on property, plant, and equipment) or invest in research and development, they may finance these expenditures by issuing new corporate bonds. Any increase in corporate bond issuance serves to increase the supply of bonds at any price. This shifts the supply curve of bonds to the right.

There are also economy-wide factors that may also affect the bond supply. For example, a general improvement in business conditions or an increase in confidence about economic conditions may lead firms to become more optimistic about the future and increase investment. This would serve to shift the supply of bonds to the right.

Recall we concluded earlier that changes in expected inflation affect the demand for bonds. They may also affect supply. For example, even if nominal face values and coupon payments may remain unaffected by inflation expectations, when expected inflation rises, the real cost of borrowing falls. Cheaper borrowing costs may spur more borrowing by way of higher bond issuance, which shifts the supply curve of bonds to the right.

An increase in bond supply means that, independently of prices, more quantity is being supplied. This suggests that whether prices are high, moderate, or low, some factor that serves to increase bond supply would shift the bond supply curve to the right, guaranteeing more quantity supplied at any given price. This is shown in . Say we begin in the equilibrium point 1, where the S and D curves meet at price ${\mathit{P}}^{\ast }$ as the market clears to ${\mathit{Q}}_{\mathit{S}}={\mathit{Q}}_{\mathit{D}}={\mathit{Q}}^{\ast }$.

An increase in government deficits resulting from higher expenditures relative to government revenues and/or an improvement in general business conditions and/or an increase in inflation expectations would all lead to an increase in the supply of bonds for any given price, leading to a wholesale shift to the right of the supply curve to a new supply curve ${\mathit{S}}_2$. After the rightward shift in , the old equilibrium point, (point 1 in the graph) would no longer hold.

That is because, while ${\mathit{P}}^{\ast }$ used to imply ${\mathit{Q}}_{\mathit{S}}={\mathit{Q}}_{\mathit{D}}={\mathit{Q}}^{\ast }$, after the curve shifts right, ${\mathit{P}}^{\ast }$ pins down the same quantity demanded (${\mathit{Q}}_{\mathit{D}}$) from the demand curve (D) as before, but the quantity supplied (${\mathit{Q}}_{\mathit{S}}$) now falls on the new supply curve (${\mathit{S}}_2$) generating a surplus of bonds at the old price ${\mathit{P}}^{\ast }$.

Market forces would bid down the price of bonds to a new equilibrium condition (point 2 in the graph) at a lower equilibrium price ${\mathit{P}}^{\ast \ast }$ and a higher quantity of bonds sold ${\mathit{Q}}_{\mathit{S}}^{\mathrm{new}}={\mathit{Q}}_{\mathit{D}}^{\mathrm{new}}={\mathit{Q}}^{\ast \ast }$. summarizes the various factors that would result in the rightward shift of the supply curve displayed in . In conclusion, all else being equal, any factor that would serve to increase the supply of bonds would eventually increase the quantity sold of bonds, reduce equilibrium prices, and raise equilibrium yields in the bond market.

Table 7.2Factors that affect bond supply and shift the supply curve.

Surprise change in	Effect on bond supply
An increase in government deficits where government expenditures outpace government revenues	Bond Supply shifts
	Bond Prices
	Bond Yields
An increase in expected inflation that reduces the real cost of debt repayment	Bond Supply shifts
	Bond Prices
	Bond Yields
An increase in the demand for financing capital expenditures by corporations	Bond Supply shifts
	Bond Prices
	Bond Yields
An increase in market confidence or an improvement in business conditions	Bond Supply shifts
	Bond Prices
	Bond Yields

So far, we have discussed different factors that affect the supply or the demand curves for bonds and their ultimate effect on bond yields.

Sometimes, a given factor may affect both supply and demand. In those cases, we need to ascertain whether the effect on both curves lead yields to shift in the same direction or not. If both curves shift in the same direction in response to a given change in a factor, then we have what we call an augmenting effect.

An augmenting effect occurs when the effect of a particular shock is compounded due to the similar responses of supply and demand. On the other hand, if the supply and demand for bonds lead yields to move in different directions, then we have what we call an offsetting effect.

An offsetting effect occurs when two shocks counteract each other. In these cases, we need to determine which is the dominant shift to effectively understand what the ultimate effect on the bond’s yield will be.

For example, an economic downturn would impact both curves. Such a downturn typically erodes consumer confidence. Firms would expect a reduction of consumer demand for the goods they sell. They may respond by reducing their own capital investment, so their demand for financing added capital expenditures would go down and they would issue fewer bonds. An economic downturn may lead to a reduction in the supply of bonds (regardless of price), essentially shifting the bond supply curve to the left.

The other side of this coin is the effect on bond investors. An economic downturn often comes with job separation and reductions of income. So, it often means the economy is experiencing a negative income/wealth effect. When there is an erosion in income, wages, and wealth, investors and households will demand fewer goods, and this may include bonds.

An economic downturn may lead to a reduction in the demand for bonds (regardless of price), essentially shifting the bond demand curve to the left. Both curves shifting left lead to an offsetting effect on bond yields. Therefore, the effect would be ambiguous unless we knew which was the dominant force.

shows a scenario where investors are more responsive than bond issuers to an economic downturn. If the demand curve for bonds were the dominant force, then the demand curve would shift to the left more than the leftward shift of the supply curve. Therefore, the new equilibrium price of the bond would go down from point 1 to point 2 on the graph. In this case, bond prices would go down and bond yields would increase in an economic downturn.

shows the opposite scenario to the same economic downturn. In this case, if bond issuers responded more than investors during an economic downturn, then it would be the supply curve for bonds that would dominate. In this case the bond supply curve would shift to the left more than the leftward shift of the demand curve. Therefore, the new equilibrium price of the bond would increase from point 1 to point 2 on the graph, and we would conclude that bond prices would increase and bond yields would go down in an economic downturn.

If we were not clear on whether supply or demand dominates the effect, both curves shifting could give an ambiguous answer in terms of equilibrium bond prices and yields. Even if the theory were ambiguous, the real world is not. The data suggests that in economic recessions, interest rates tend to fall, meaning that bond prices should rise.

7.2 Interest Rate Spreads and Bond Ratings

Changes in bond prices and the associated changes in interest rates can have a pronounced effect on the borrowing costs corporations face. The financial market often experiences simultaneous increases in some interest rates and reductions in others. These gaps—the percent differences among various rates—are called interest rate spreads, and economists often look for meaningful market signals arising from interest rate spreads.

For example, an increase in the perceived risk of companies’ bonds often leads to declines in prices, which results in interest rate increases and higher corporate borrowing costs. If treasuries remain insulated from this higher risk, this serves to increase the interest rate spread (or simply “spread”) between corporate and treasury yields.

We must be able to distinguish between many different types of bonds that are traded in financial markets. In previous chapters, we have already discussed, many factors affecting the price of a bond, such as time to maturity, prevailing interest rates, inflation expectations, perceptions of risk, and so on.

Because default is one of the most important risks a bond holder faces, independent companies (rating agencies) regularly evaluate the creditworthiness of potential borrowers (bond issuers). These companies analyze the financial well-being of bond issuers and estimate the likelihood that a corporate or government borrower will make a bond’s promised payments.

The U. S. government has acknowledged a few firms as “nationally recognized statistical rating organizations” (NRSROs). The best known bond rating services are: Moody’s , Standard and Poor (S&P) , and Fitch . They monitor the status of individual bond issuers and assess the likelihood a lender will be repaid by the bond issuer. A high rating suggests that a bond issuer will have little problem meeting a bond’s payment obligations.

Firms or governments with an exceptionally strong financial position carry the highest ratings, typically referred to as triple A. Since these highly rated institutions are deemed relatively safe from default risk, they can issue the highest-rated bonds with the promise of relatively low yields (e. g. Johnson & Johnson, Microsoft, and the government of Norway).

shows the various grades that are available for bonds. The top four categories are considered investment-grade bonds. These are reserved for bonds with little risk of default, typically selected for most government issuers and corporations that are among the most financially sound.

Anything below Baa for Moody’s or BBB for Standard & Poor’s is considered below investment grade. They are often referred to as speculative or junk bonds, because they carry significantly higher risk of default.

The distinction between investment-grade and speculative, non-investment-grade bonds is important. For example, a number of regulated institutional investors are not allowed to invest in bonds rated below investment grade, which is Baa on Moody’s scale or BBB on Standard & Poor’s scale. This means fewer potential investors can demand these bonds, so companies or governments that are rated below investment grade must promise higher yields to attract demand.

There are two types of junk bonds. Fallen angels are bonds that were once investment-grade, but their issuers fell on hard times. There are also junk bonds that begin trading at a junk rating because they may be sold by issuers about which there is little known. Ratings are not fixed. Material changes in a firm’s or government’s financial conditions may precipitate changes in its debt ratings, leading to rating downgrades—which lower an issuer’s bond rating—or rating upgrades—which raise an issuer’s bond rating.

In addition to corporate and treasury bonds, rating agencies also rate other financial instruments, such as commercial paper.

Commercial paper is a short-term version of a bond, where the borrower offers no collateral, so the debt is unsecured. Commercial paper is issued on a discount basis, as a zero-coupon bond specifying a single future payment with no promise for coupon payments. While commercial paper has typical maturities of less than 270 days, most of it is issued with a maturity of 5 to 45 days and is used, almost exclusively, for short-term financing. A large portion of commercial paper is held by money-market mutual funds.

The rating agencies rate the creditworthiness of commercial paper issuers in the same way they do bond issuers. Almost all commercial paper instruments carry Moody’s $\mathit{P}-1$ or $\mathit{P}-2$ rating. P stands for prime grade commercial paper. Speculative-grade commercial paper does exist, but not because it was issued as such.

Ratings agencies have enormous power to affect interest rates and yields. In 2010, the government issued financial reform legislation called the Dodd-Frank Wall Street Reform and Consumer Protection Act, partially to reduce reliance on ratings agencies.

Bond ratings are designed to reflect default risk. The lower the rating, the higher the risk of default. This drives the price of the bond down and the associated yield higher. Because risk is a relative concept, to understand quantitative ratings, it is easier to compare them to a benchmark. Since the U. S. Treasury issued bonds are viewed as having little default risk, they are often used as a benchmark, even globally.

Since U. S. treasuries are a benchmark for many other bonds—and the yield of any given bond is explained in part by the yield of U. S. Treasury—when Treasury yields move, all other yields tend to move along with them in a process that is called co-movement.

Note: Figure provided by the Federal Reserve Economic Database (FRED) of the Federal Reserve Bank of St. Louis.

shows a plot of the risk structure of interest rates. For the vast majority of the postwar experience, the 10-year U. S. Treasury rate, which acts as a reference rate for many corporate bonds, lies below investment-grade corporate yields. Changes in the U. S. Treasury yields account for most of the movement in the Aaa and Baa bond yields.

From 1965 to 2021, the 10-year U. S. Treasury bond yield has averaged 6.27%, about a full percentage point below the average yield on Aaa bonds (7.34%) and two percentage points below the average yield on Baa bonds (8.38%).

Ratings are crucial to corporations’ ability to raise financing. A lower rate increases the costs of funding. Since investors clearly must be compensated for assuming risk, the lower the rating of the corporate bond, the higher the risk premium, and, thus, the bigger the spread with the 10-year rate. This explains why the spread between the Aaa and the 10-year U. S. Treasury is smaller than the spread between the Baa and the 10-year U. S. Treasury.

7.3 Term Structure of Interest Rates

In addition to risk, other factors separate different types of bonds. On the other hand, bonds often have similar characteristics. Bonds with the same default rate and tax status but different maturity dates often have different yields. We can think of longer-term bonds as a composite of a series of shorter-term bonds, and their yields depend on what people expect to happen in the future.

The relationship among bonds with the same risk characteristics but different maturities is called the term structure of interest rates.

Note: Figure provided by the Federal Reserve Economic Database (FRED) of the Federal Reserve Bank of St. Louis.

compares the three-month and 10-year U. S. Treasury yields in the postwar period. We can see that, while these two treasuries have different maturities, their interest rates tend to move together. We can also see that yields on short-term bonds are somewhat more volatile than yields on long-term bonds. Moreover, long-term yields tend to be higher than short-term yields, as the 10-year treasury rate, generally, tends to be above the three-month treasury yield. Importantly, the gap between the two rates is not fixed. The spread between the 10-year treasury bond and the three-month treasury bill is sometimes wider and sometimes narrower.

A yield curve is a popular chart in financial analysis that summarizes the yields of treasuries of various maturities at a specific point in time. Yield curves are a graphical representation of the term structure of interest rates. shows the yield curve on May 13, 2018 (source: Wikipedia). The yield curve is typically a snapshot in time where the maturities are shown on the horizontal axis, with short-term bonds on the left and longer-term bonds on the right-hand side.

This yield curve is upward sloping, which means that the short-term bonds had a lower yield than longer-term bonds on that date. While the slope of the yield curve can vary from day to day—and over the months and years—most of the time the yield curve tends to slope upward, as it is shown in .

Many theories have been proposed to explain the term structure of interest rates. The first theory we will examine is called the expectations hypothesis. This theory focuses on the risk-free interest rate. The risk-free interest rate can be computed, assuming there is no uncertainty about the future. This is an unrealistic assumption, which other theories will relax. For now, we make the assumption of future certainty. Certainty means that bonds of different maturities are perfect substitutes for each other.

For example, if there is no uncertainty, then an investor should be indifferent between holding a single two-year bond or a series of two one-year bonds. The expectations hypothesis implies that the current two-year interest rate should equal the average of the current one-year rate and the one-year interest rate one year in the future. For example, if the current interest rate on the one-year bond is $1\mathrm{\%}$ and the future interest rate on the one-year bond is $3\mathrm{\%}$, then the current interest rate on the two-year bond is $\mathrm{(}1+3\mathrm{)}\mathrm{/}2=2\mathrm{\%}$.

When interest rates are expected to rise, long-term interest rates will be higher than short-term interest rates. In these cases, the yield curve will slope up. On the other hand, if interest rates are expected to fall, the yield curve will slope down, and if interest rates are expected to stay the same, the yield curve will be flat. This is shown in .

The expectations hypothesis suggests that if bonds of different maturities are perfect substitutes for each other, then we can construct investment strategies that must have the same yields.

For example, Susan, a bond investor, could buy a three-year treasury bond and hold it to maturity. At period t (today), this bond offers a current yield of ${\mathit{i}}_{3\mathit{t}}$. If Susan buys this bond and holds it to maturity, one dollar of this bond today yields $\mathrm{(}1+{\mathit{i}}_{3\mathit{t}}\mathrm{)}\mathrm{(}1+{\mathit{i}}_{3\mathit{t}}\mathrm{)}\mathrm{(}1+{\mathit{i}}_{3\mathit{t}}\mathrm{)}$ three years later.

But Susan has another option. She could buy a one-year treasury with a current yield of ${\mathit{i}}_{1\mathit{t}}$ and, at the end of the year, collect the proceeds and use them to buy the one-year bond again (this is called rollover). Then, at the end of year two she could collect the proceeds from the one-year bond and, again, roll it over into a third year.

Susan can read financial news and gather quotes from many financial assets on offer today. So, she can see the current yields of the one-year and three-year treasury bonds today (at period t). However, today she cannot see what the one-year treasury bond will pay one year from today (at period $\mathit{t}+1$), or what the one-year treasury bond will pay two years from today (at period $\mathit{t}+2$).

So, today, Susan can see what the one-year treasury yields ${\mathit{i}}_{1\mathit{t}}$ trades at, but she cannot see what it will pay a year from today—she cannot see ${\mathit{i}}_{1\mathit{t}+1}$ today. The best she can do today is formulate an expectation (her own educated guess or forecast) of what the one-year bond will pay a year from today. We will denote this with an superscript e to denote expectation (${\mathit{i}}_{1\mathit{t}+1}^{\mathit{e}}$).

Similarly, she needs to formulate an expectation today of what the one-year bond will yield in two years (${\mathit{i}}_{1\mathit{t}+2}^{\mathit{e}}$). If Susan buys this one-year bond with a plan to roll it over twice, one dollar of this bond today yields $\mathrm{(}1+{\mathit{i}}_{1\mathit{t}}\mathrm{)}\mathrm{(}1+{\mathit{i}}_{1\mathit{t}+1}^{\mathit{e}}\mathrm{)}\mathrm{(}1+{\mathit{i}}_{1\mathit{t}+2}^{\mathit{e}}\mathrm{)}$ three years later, given a rollover plan.

Therefore, Susan, has these two options. She could buy a short-term bond (a one-year treasury) three times in three years; or could buy a single long-term bond (a three-year treasury) and hold it to its maturity. The expectations hypothesis suggests Susan should be indifferent between the two options, which means the returns from the two options should equal each other as follows:

Rearranging this equation, we can now write the three-year yield as the average of the current and expected future one-year rates:

More generally, the expectations hypothesis allows us to write the yield of a long-term bond of arbitrary maturity (n) as an average of current and expected future yields of shorter maturity bonds, so that:

This equation seems consistent with the co-movement we see in treasury rates, where interest rates of different maturities will move together.

For example, if the current yield on the one-year treasury (${\mathit{i}}_{1\mathit{t}}$) rises, it is clear from the equation that the long-term yield (${\mathit{i}}_{\mathit{n}\mathit{t}}$) also rises, all else being equal.

This equation also suggests that the yields on short-term bonds will be more volatile than yields on long-term bonds. Long-term rates are averages of short-term rates, so changing one short-term rate has little effect on the long-term rate. For example, if the one-year treasury bonds’ interest increased by 100 basis points, the thirty-year treasury yield would only go up by 3.3 basis points (100/30).

Similarly, if ${\mathit{i}}_{1\mathit{t}}$ decreased by 100 basis points, ${\mathit{i}}_{\mathit{n}\mathit{t}}$ would decrease by 3.3 basis points. This means that a 2% swing in the short-term interest rate comes hand-in-hand with a much smaller swing of 0.07% in the long-term bond.

However, looking at the equation above, the expectations hypothesis cannot explain why long-term yields tend to be higher than short-term yields most of the time. The equation suggests that the yield curve will slope upward only when interest rates are expected to rise.

Since the yield curve normally slopes upward, the expectations hypothesis would suggest that interest rates are expected to rise most of the time. But this is not the case with the data we see. As we saw in Figures and above, interest rates are just as likely to go up as they are to go down over time.

Therefore, the expectations hypothesis provides a good explanation for two features we see in interest rate data: co-movement and higher volatility in the short term. But it does not explain well the third feature that—though interest rates fluctuate over time—they are not always expected to continuously increase over time.

The expectations hypothesis is quite useful but it falls just shy of explaining all three aspects of interest rate behavior. So, we look for a way to augment the hypothesis. So far, we have abstracted from risk in interest rate determination. We need to extend the expectations hypothesis to include risk.

The yield curve provides a summary of the term structure of interest rates as it is populated by treasuries of different maturities. Even treasuries that are free of default risk, such as U. S. treasuries, carry other types of risk. As we discussed in previous chapters, bond holders face both inflation and interest rate risk. The longer the term of the bond, the greater both types of risk.

Inflation risk refers to the impact that inflation will have on the purchasing power of the nominal returns the investor derives from the bond. Therefore, investing in bonds involves successfully computing real returns from nominal returns, which requires a forecast of expected future inflation. The further we look into the future, the greater the uncertainty of where inflation will be, and, thus, the greater the uncertainty that surrounds the real return of the bond. Therefore, a bond’s inflation risk increases with its time to maturity.

As we discussed in previous chapters, there is no interest rate risk for bonds that are held to maturity. Bond holders face interest rate risk when they sell their bonds ahead of maturity. Interest rate risk arises from the mismatch between the investor’s investment horizon and a bond’s time to maturity. If a bond holder plans to sell a bond prior to maturity, changes in the interest rate generate capital gains or losses. Those gains or losses are magnified with time to maturity. The longer the term of the bond, or the more distant the time to maturity, the greater the price changes for a given change in interest rates and the larger the potential for capital losses. Thus, a bond’s interest rate risk increases with its time to maturity.

Since risk increases at longer maturities, investors will require compensation for the increase in risk they take for buying longer-term bonds. So, we can think about bond yields as having two components: one that is risk free—explained by the expectations hypothesis—and one that is a risk premium—explained by inflation and interest rate risk.

These two components are reflected in what is known as the Liquidity Premium Theory (LPT) of the term structure of interest rates. The LPT essentially augments the expectations hypothesis with an extra term that accounts for the risk premium that investors typically demand in compensation for increased risk.

Just like the expectations hypothesis, the LPT allows us to write the yield of a long-term bond of arbitrary maturity (n) as an average of current and expected future yields of shorter maturity bonds. But unlike the expectations hypothesis, the LPT does not require the long-term yield to exactly offset the average of current and expected future shorter-term yields.

Rearranging the expectations hypothesis equation shows the exact offset as follows:

However, the LPT suggests that the two yields for the long and short terms may not offset each other. If the long-term yield is higher than the current and expected short-term yields, then that difference may arise from a higher risk of holding the long treasury bond.

Therefore, any positive difference between the long-term yield and the average short-term yields must be explained by a risk premium (${\mathrm{rp}}_{\mathit{n}}$) that compensates the investor for holding the typically riskier long-term bond. The LPT equation becomes:

This equation shows that risk is the key to understanding the slope of the yield curve. If the yield curve is upward sloping, then it must be the case that the risk premium is positive, which compensates the investor for holding the riskier long-term bond. The fact that risk premia on long-term bonds are typically positive explains why the yield curve is most often upward sloping.

A historical inspection of the average slope of the term structure over a long period can give us some idea of the size of the risk premium. shows that from 1985 to 2023, the difference between the interest rate on a 10-year Treasury bond (5.2%) and that on a three-month Treasury bill (3.4%) has averaged nearly 2 percentage points. The risk premium varies over time, but it has mostly been positive around 2%.

Note: Figure provided by the Federal Reserve Economic Database (FRED) of the Federal Reserve Bank of St. Louis.

We can also see that, quite infrequently, the three-month treasury bill has risen above the 10-year treasury. This would suggest an inversion of the yield curve, where the longer end of the yield curve sits lower than the short end. This indicates that the risk of holding long-term bonds is dominated by the risk of holding short-term bonds—which means the risk premium turns negative. The figure shows the negative risk premium is most evident during 2023. A negative risk premium suggests that long-term treasuries trade at a discount. Over the years, economists have identified that these periods of long-term treasuries trading at risk discounts occur when there is a lot of risk on the short-term of the yield curve, which seems to be followed by recessionary pressures roughly about one year after the yield curve initially inverts.

Risk spreads provide one type of information and the term structure another. For example, an economic slowdown or recession does not affect the risk of holding government bonds that are held to maturity. However, the immediate impact of a pending recession is to raise the risk premium on privately issued bonds. And—while the impact of a recession on companies with high bond ratings is usually quite small—the lower the initial grade of the bond, the more the default-risk premium rises as general economic conditions deteriorate.

Note: Figure provided by the Federal Reserve Economic Database (FRED) of the Federal Reserve Bank of St. Louis.

shows the annual GDP growth over four decades, superimposed on shading that shows the dates of recessions. During these shaded periods, GDP growth is usually negative. On the right axis, the figure shows the spread between yields on Baa-rated bonds and U. S. Treasury bonds. We can clearly see that when risk spread rises, output growth falls.

Note: Figure provided by the Federal Reserve Economic Database (FRED) of the Federal Reserve Bank of St. Louis.

shows annual GDP growth and the slope of the yield curve, measured as the difference between the 10-year and three-month yields. We can see that when the term spread falls, GDP growth tends to fall about one year later, more or less. When this term spread falls below zero, this is consistent with a yield curve inversion. The figure shows the term spread became flat roughly one year before the 1991 recession. And for the 2001 and 2008 recessions and the COVID downturn, the yield curve inverted roughly one year prior to the onset of those economic recessions.

Overall, the graph shows the yield curve is a solid leading indicator of recessions. It shows GDP growth in the current year is strongly correlated with the slope of the yield curve one year earlier: and if we shifted GDP growth one year earlier, then the troughs in the two lines clearly move together. This shows that the yield curve is a valuable forecasting tool for timing recession. However, the yield curve is less useful for predicting the severity of a recession and it did not predict the depth or duration of the Great Recession of 2007–2009.

7.4 Glossary

Benchmark Bond

Reference security (typically U. S. Treasury) used to price other bonds.

Bond Demand Curve

The relationship between price and quantity of bonds investors demand, typically sloping downward.

Bond Rating

Assessment of creditworthiness by rating agencies that estimates the likelihood a borrower will make promised payments.

Bond Supply Curve

The relationship between price and quantity of bonds people are willing to sell, typically sloping upward.

Co-movement

Tendency of different bond yields to move together when U. S. Treasury yields change.

Commercial Paper

Short-term, unsecured debt instrument issued on a discount basis with maturity under 270 days.

Expectations Hypothesis

Theory stating long-term interest rates equal the average of current and expected future short-term rates.

Fallen Angels

Bonds that were once investment-grade but were downgraded to junk status.

Inflation Risk

Uncertainty about how inflation will affect the purchasing power of bond returns.

Interest Rate Risk

Potential for capital losses when selling bonds before maturity due to interest rate changes.

Interest Rate Spreads

The percentage differences between various interest rates, often used as market signals.

Junk Bonds

Bonds rated below investment grade, carrying higher default risk and yields.

Liquidity Premium Theory (LPT)

Enhanced version of expectations hypothesis that includes risk premium compensation.

Market Clearing

Process where prices adjust until quantity supplied equals quantity demanded.

Rating Downgrade

Lowering of an issuer’s bond rating due to deteriorating financial conditions.

Risk Premium

Additional yield required by investors to compensate for increased risk of longer-term bonds.

Term Structure of Interest Rates

Relationship among bonds with same risk characteristics but different maturities.

Yield Curve

Graphical representation of interest rates across different maturities at a specific point in time.

Yield Curve Inversion

When short-term interest rates exceed long-term rates, often predicting recession.

8.1 Background on Common Stock

The idea of common stock is one of shared ownership and shared risk. Essentially, a group of investors would pool resources to collectively own the company and divide up the profits while spreading the risk of the venture. Historically, this facilitated an increase in international trade in the 16th and 17th centuries bringing about the age of mercantilism.

A powerful idea that solidified the viability of the stock market is the concept of ‘apportionment’ of ownership. Ownership of a profitable venture was often deemed valuable, and apportionment implies that an investor could sell her portion of ownership in the venture independently—not needing the permission of the other co-owners. The number of portions, or shares, of ownership that could be sold could be made arbitrarily large by selling them in ever smaller denominations.

Since shares were issued in small denominations, and were also made transferable, an owner could sell them to someone else and investors could buy as little or as much as they wanted. This increased participation, which thickened the market of stocks.

In the past, investors used to receive paper stock certificates—most stockholders no longer do. Information is all computerized, which is safer to store and makes it easier to transfer. Computerization reduced transaction costs of trading stocks, increasing participation again.

The ownership of common stock entitles the stockholder to participate in the profits of the enterprise. It also entitles the stockholder to have a say in the management of the firm, since stockholders are able to vote at the firm’s annual meeting. Another advantage of stock ownership is that investors have limited liability in the firm. Even if the company goes bankrupt, the maximum amount a shareholder can lose is their initial investment. However, in the event of liquidation, as the firm experiences a financial strain, a stockholder is a residual claimant. This means the stockholder is paid last, only after all other creditors have been paid.

To sum up, the market for stocks is thick with wide participation because of the many advantages it offers: 1) an individual share is only a small fraction of the company’s value and is highly transferable, 2) a large number of shares are outstanding, 3) prices of individual stocks are often low, 4) stockholders are residual claimants, 5) stockholders have limited liability, and 6) shareholders can have some influence in the management of the firm.

8.2 Measuring the Stock Market

Many thousands of stocks are being traded every day around the world. While investors typically concentrate their holdings on a smaller subset of all the stocks currently trading, it is useful to measure the overall market. Investors need to understand the dynamics of the stock market. They typically evaluate the connections between stock values and economic conditions. To do this, they need to be able to measure the level of fluctuation in all stock values (even if they do not hold them all). This concept is the value of the stock market. We refer to this overall measurement as a stock market index.

There are many stock indexes around the world. They tell us how much the value of an average stock has changed, and how much total wealth (or equity) has gone up or down. They also facilitate comparisons by providing benchmarks for investor performance. One can tell whether a money manager has done better or worse than “the market” as a whole.

The Dow Jones Industrial Average (DJIA) is the first and most widely known stock market index. It is based on the stock prices of 30 of the largest companies in the U. S. It tracks the value of purchasing a single share of each of the stocks in the index. The percentage change in the DJIA over time is the percentage change in the sum of the 30 prices. The DJIA is a price-weighted average, which gives greater weight to shares with higher prices. Therefore, the behavior of higher priced stocks dominates the movement of this price-weighted index.

Another popular index is the Standard & Poor’s 500 Index. The S&P500 is constructed from the prices of the 500 largest firms in the U. S. economy. It tracks the total value of owning a single share of all 500 firms. It uses a value-weighted index where larger value firms carry more weight. For example, if a firm is priced at $100 and has 10 million shares outstanding, its total market value—or market capitalization—is worth $1 billion. A price-weighted index like the DJIA gives more importance to stocks that have high prices. But a value-weighted index like the S&P 500 gives more importance to companies with a high market value—their price per share is not necessarily relevant.

Both types—price-weighted and value-weighted indexes—are useful because they answer different questions. Changes in price-weighted indexes give information on the performance of a typical stock, whereas changes in value-weighted indexes more closely resemble changes in the economy’s overall wealth.

Other important indexes in the U. S. include the Nasdaq Composite Index and the Wilshire 5,000. The Nasdaq is a value-weighted index of over 3,000 companies traded on the over-the-counter (OTC) market. It is mainly composed of smaller, newer firms and has typically been dominated by technology and Internet companies. The Wilshire 5000 is the most broadly based index in use; it covers all publicly traded stocks in the U. S. with readily available prices. As a value-weighted index, the Wilshire is probably the most comprehensive measure of overall market wealth.

Stock market indexes can be pulled directly from the web into Python via the Pandas package with the Yahoo Finance overlay as follows:

The following snippet of Python code produces a chart of daily values of the four major U. S. stock market indexes between January 2, 2020, and September 10, 2023. The code graphs four indexes each with a different scale in the vertical (y) axis—two for the main chart and one in a sub-graph or inset.

This code produces , which clearly shows all three indexes are highly correlated. All three indexes experienced a substantial drop following March of 2020, because of COVID, and finally bottomed out by the end of 2022. While most of 2022 was a rocky year for stock markets, coinciding with a major increase in headline inflation, 2023 marked a moderate recovery, with the DJIA mostly moving sideways and the S&P500 index and the Nasdaq seeming to grow faster, buttressed by bullish expected returns on superconductors, hardware, and software connected with an expansion of artificial intelligence.

While over the long run, all these indexes show strong co-movement, at shorter windows, differences often arise. Many countries around the world have a stock market and each has an index. Many of them are value-weighted indexes. Investors view global stock markets as a means to diversify risk away from domestic markets. However, global stock markets are typically strongly correlated.

8.3 Valuing Stocks

Investors in equities often differ on how stocks should be valued. Some believe they can predict changes by examining graphical patterns or past movements in particular stocks. These types of investors are known as chartists.

Others estimate the value of stocks based on their perceptions of investor psychology and behavior. They monitor other investors’ trading patterns and follow suit—which is called free riding—or, conversely, bet against them. They also monitor trading chatter on social media and move to invest with the herd. These types of investors are collectively known as behavioralists.

Yet other types of investors estimate the value of stocks based on both their current assets and on estimates of future profitability. They pore over financial statements, they attend earnings meetings, and they forecast macroeconomic activity. These types of investors trade based on the fundamentals of a company. The fundamental value of a stock is based on the timing and uncertainty of the returns it brings.

Chartists and behavioralists question the usefulness of fundamentals in understanding the level and movement of stock prices. Instead, they are more concerned with internalizing correctly and responding quickly to financial news.

We can use the information we have already studied to compute the fundamental value of stocks. A stock represents a promise to make monetary payments on future dates, under uncertain circumstances. These payments are usually made in the form of dividends. These dividends are typically issued from the proceeds of a company when they have a good quarter. In other words, dividends are distributed to the owners of a company when the company makes a profit. When a company is sold, the stockholders receive a final distribution that represents their share of the purchase price.

8.4 The Dividend Discount Model

There are two potential returns from holding a stock. One is future dividends and the other is the potential for future capital gains, which can be earned any time the share value appreciates above the price that the share was originally bought at. Since both dividends and share prices are incurred in the future, they must be discounted to the present to evaluate the desirability of the underlying stock.

According to the Classical Theory of Asset Prices, the price of an asset equals the present-value discounted expected future incomes or returns. Following this definition, the price of a share of stock must equal the discounted future dividend payment and, if the stock is sold, the discounted future price. For example, the dividend discount model suggests that today’s price (${\mathit{P}}_{\mathit{t}}$) of a share of stock that issues a dividend one year from today (${\mathit{D}}_{\mathit{t}+1}$) must be equal to:

which means that for a given interest rate (${\mathit{i}}_{\mathit{t}}$), the higher we expect the dividend payment or the price of the stock to be next year, the more valuable it is to hold this stock today, and, therefore, the higher its price today (${\mathit{P}}_{\mathit{t}}$). This equation can easily be extended to an arbitrary investment horizon of n years:

Dividend payments are not typically guaranteed. A firm might have a bad quarter and turn in no profits to distribute to the owners. At other times, the firm’s management might decide to reinvest the profits in, for example, property, plant, or equipment for the firm, rather than issue out dividend payments to the owners. Alternatively, they might buy back some of the firm’s own stock with quarterly proceeds. This makes it more difficult to apply the equations above to price a share of stock.

Case in point, if a company currently pays no dividends, investors must estimate when the company will begin paying dividends and use the present-value framework. This requires investors to have some extra information about annual dividend payments.

For example, an investor expects some dividend payment (${\mathit{P}}_{\mathit{t}}$) at any given period t. Assume that the investor receives a dividend payment at a specific period t. She does not need to formulate a guess in this case; she sees this payment materialize to the realized value (D), so that (${\mathit{D}}_{\mathit{t}}=\mathit{D}$). If the investor assumes the dividend payments will grow at some constant rate g per year, then the dividend payment next year will be ${\mathit{D}}_{\mathit{t}}=\mathit{D}\mathrm{(}1+\mathit{g}\mathrm{)}$ and the dividend payment in n years will be ${\mathit{D}}_{\mathit{t}}=\mathit{D}{\mathrm{(}1+\mathit{g}\mathrm{)}}^{\mathit{n}}$. With this information, we can rewrite the price equation as follows:

As the planning horizon n gets arbitrarily large, it becomes more difficult to forecast what the price in n years will be. In this case, investors might not attempt to forecast that distant future price (${\mathit{P}}_{\mathit{t}+\mathit{n}}$) at all and consider a buy-and-hold strategy instead. Presumably, holding the stock ‘forever’ would pay dividends forever, which would turn the stock into something like a consol (a bond with no maturity). This would modify the equation above into:

By mathematical properties of geometric progressions, as n grows very large (toward infinity), it can be shown that the equation above reduces to the following equation:

This relationship is the dividend discount model. The model tells us that stock prices should be high when dividend payments are high (D), when dividend growth is rapid (g is large), or when the interest rate (i) is low.

8.5 Assessing Risk in Stocks

The dividend discount model of the previous section abstracted from risk. However, dividend payments are rarely guaranteed. Stockholders are residual claimants, which means they get part of the profits only after everyone else is paid, including bond holders. In addition, when investors buy stocks, they often finance their purchase. Essentially, they often put up some of their wealth to buy shares of the firm and borrow the rest. The borrowing creates leverage, and leverage creates risk. The higher the level of debt used to finance equity purchases, the higher the leverage, and the greater the owners’ risk.

For example, let’s say a firm needs $10,000 for computing equipment. The firm wants to calculate the return on its investment in the first year following the $10K capital expenditure. Once installed, the firm will have an equal probability of earning $900 from the equipment during a bad year of sales or $1,500 during a good one. The $10,000 investment can be financed in equity (with stocks) exclusively, or it can be part equity (stock) and part debt (bonds). Let’s say that equity can be obtained at a 10% interest rate.

shows the risk and expected returns from financing the $10,000 investment with different combinations of equity and debt. The first row shows the case when the investment is fully financed with 100% equity. In other words, the firm does not take on any level of debt to finance the expenditure. Column b in the first row shows zero percent financing through debt, which means that the cost of borrowing 10% of zero means a zero dollar payment (see column c), since no bonds are issued to finance the expenditure. Under this scenario, the payment to equity holders (column d) ranges from $900 in bad times to $1,500 in good times.

Column e, in the first row, shows that in percentage terms, equity returns range from ($\mathrm{\$}900\mathrm{/}\mathrm{\$}10\mathrm{,}000=9\mathrm{\%}$) to ($\mathrm{\$}1\mathrm{,}500\mathrm{/}\mathrm{\$}10\mathrm{,}000=15\mathrm{\%}$). Since there is an equal probability of having a bad year or a good year, the expected equity returns are:

The equity returns can deviate plus or minus 3% from this expected 12% equity return, since they range from 9% to 15%. Column g, in the first row, indicates that at 3%, the standard deviation of these expected returns is quite low when the investment is fully financed with equity.

The second row in the table shows what happens when half of the investment is financed with equity and the other half is financed by issuing bonds. In this scenario, since the firm issues $5,000 worth of bonds, at a 10% interest rate, the bond holders have to be paid ahead of equity holders to the tune of $500 (10% × $5,000). This $500 required payment to bond holders is shown in column c.

Looking at column d in the second row, since equity holders are residual claimants, in the bad times the equity holders would receive $900, but now that bond holders get paid $500 first, the leftover return for equity holders in bad times is $400. Similarly, the equity return in the good times would be ($\mathrm{\$}1\mathrm{,}500-\mathrm{\$}500=\mathrm{\$}1\mathrm{,}000$). This means that the payment to equity holders would range between $400 and $1,000. Column e in the second row shows that, as a share of the $5,000 equity financing, the equity returns range between 8% ($400/$5,000) and 20% ($1,000/$5,000).

The second row shows that when 50% of the investment is financed with debt, the expected equity returns increase from 12% (under zero percent debt shown in the first row) to the 14% that column f shows (under 50% debt in the second row). However, column g in the second row shows that, at 6%, the standard deviation is now much higher when some of the investment is financed with debt. Therefore, debt financing has increased the risk of the investment.

The third row shows what happens to the risk and expected returns of the investment when $3,000 is financed with equity and the firm issues $7,000 worth of bonds. The fourth row shows the case when the computing equipment is purchased with $9,000 of capital raised though bond issuance and a mere $1,000 of equity is used. We can see that the larger the position of the investment that is financed with bonds, the higher the bond payment obligation that has to be made ahead of issuing returns to the equity holders (the payment amount increases as we climb down column c). As the debt issuance gets larger, the dollar amount of equity returns (column d) gets smaller.

However, the equity returns are larger (column e) because the equity used is lower at higher debt levels. As we take on more debt to finance the $10K investment, expected returns increase from 12% to 20% (as we climb down the rows in column f). This may lead to the conclusion that financing investment with debt could be desirable, because expected equity returns increase at higher levels of debt. Importantly, these higher expected returns carry much higher degrees of risk—shown in column g, where standard deviations rise from 3% to 15%—because of what is known as leverage. In other words, a higher level of debt is used to chase higher returns, and this carries higher risk because debt claimants are paid ahead of equity holders.

8.6 Portfolio Risk

The dividend-discount model is useful when looking at the price and risk of holding a single stock. Many investors, however, collect a basket of various stocks into a portfolio. The returns from a portfolio can be simply calculated as the weighted sum of the stocks the portfolio contains.

Financial analysts often normalize share price data so that it can be more easily compared across companies. Normalization involves converting both stock prices to an index so we can plot them as if they were both starting at the same $100 share price. Then, the period-by-period return of a stock can be calculated as the one-period change in the share price:

Or, alternatively, it can be calculated with a log transformation, which is known as log returns, as follows:

And then compute their returns with a log transformation, which can be easily done in a single line of Python code.

We can calculate mean returns over an arbitrary period of time and annualize the returns by assuming 250 trading days in a year. (We also multiply by 100 to express in percentages.)

This analysis shows that from 2007 to 2023, the annualized returns for Microsoft were considerable higher than those of Apple. If we are interested in calculating the portfolio return from holding these two stocks, we simply need to know how we distribute our portfolio among the various returns. In other words, we need a set of weights. For example, if 100% of our portfolio consisted of Apple stock exclusively, the portfolio return would be exactly equal to the 16% of Apple’s return. On the other hand, if we only had Microsoft stock in our portfolio, our portfolio’s return would be 25.1%.

The portfolio return is calculated by multiplying each stock times its associated weight and summing all the weighted returns. If the portfolio was equally weighted among the two stocks (50% Apple and 50% Microsoft), the portfolio’s return would be 20.6%:

If the portfolio were more heavily weighted to Apple (say 80%), its return would be pulled down by the lower return on Apple:

And similarly, if most of the portfolio were heavily weighted on Microsoft (say 80%), the portfolio’s return would be higher:

Each stock has a return, and each stock has its own risk. As we saw in previous chapters, the risk of a given stock can be measured by the standard deviation of its return (what we call “return volatility”, or simply “volatility”) over some period.

The previous code snippets show we can append .mean() to a Python object (in this case our numpy array containing the two stocks) to calculate the mean values. We can do the same for variances by simply appending .var(). Then taking the square root and multiplying by 100 obtains the standard deviation in percentages for each stock.

We can see that Microsoft has a slightly higher standard deviation, suggesting that it is a somewhat riskier stock to hold than Apple. But how risky is a portfolio that holds both stocks?

Calculating portfolio risk is a more difficult proposition than simply combining the weighted returns of each stock to calculate the overall portfolio return. This is because, while both stocks carry their own level of risk, their risk may not be independent of each other. The risk of holding Microsoft may be somewhat related to the risk of holding Apple.

We need to account for that relationship. It is not enough to measure how each stock return varies over time. We also need to determine how both stocks co-vary with each other over time. This is accomplished with a covariance matrix. shows the general form of a covariance matrix between two stocks: a and b. The northwest quadrant shows the covariance of stock a with itself. This is the same as the variance of stock a.

We call the line that cuts through the northwest to the southeast quadrants, the main diagonal. The main diagonal of a covariance matrix holds the variances of each stock. Therefore, the off-diagonals show the covariance among the various stocks in our portfolio. The northeast quadrant shows the covariance of stock a and stock b, while the southwest quadrant shows how stock b co-varies with stock a. These two quadrants should show the same value, because the order in which we place the stocks does not affect their covariance. The covariance of x and z must be the same as the covariance of z and x.

From this we learn two properties of covariance matrices: One, their main diagonal contains variances, so the main diagonal must only contain positive values. Two, covariance matrices are symmetric, meaning their values are mirror images of each other across the main diagonal. The following snippet of Python code shows the covariance matrix of Microsoft and Apple’s returns for a period that spans 2007 to 2023:

We can see the positive values of the main diagonal of the covariance matrix and their symmetry across the main diagonal. The covariance matrix suggests that Microsoft returns show a larger variance over the period, indicating that they are somewhat riskier than Apple. The off-diagonal shows that Microsoft and Apple co-vary positively. However, the strength of that relationship in the risk of both stocks can be more easily interpreted with a correlation matrix.

A correlation matrix essentially divides the covariance matrix through by the standard deviations of the stock in each quadrant. shows a generic correlation matrix for two stock returns.

Dividing the main diagonal elements of the covariance matrix by the square of the standard deviation of each stock is tantamount to dividing the variance of each stock by itself. This should make sense. Each stock should be perfectly correlated with itself.

The off-diagonal elements show the correlation coefficient between the stocks. The correlation values can range from −1 (or −100%) for a perfectly negative correlation to +1 (or 100%) for a perfectly positive correlation. The closer the value it is to zero, the weaker the strength of the relationship between the stocks and the more independent their risk is from each other.

The following Python snippet shows that Microsoft and Apple stock are positively correlated with a 57.1% correlation coefficient. (Note that we never need to annualize a correlation matrix since it does not contain average daily values, so no need to multiply times 250).

Now that we have some measure of the volatility of each stock, as well as how both stocks co-vary, we can now determine the portfolio’s risk. A simple equation for the portfolio’s variance (Ω) that contains two stocks is given by:

where the weights must add up to one (${\mathit{\omega }}_1+{\mathit{\omega }}_2=1$) and (${\mathit{\sigma }}_1$) and (${\mathit{\sigma }}_2$) are the standard deviations of each stock. Applying the quadratic formula, we have the following equation:

where ${\mathit{\rho }}_{12}$ is the correlation coefficient between the two stocks. The following snippet of Python code calculates our two-stock portfolio’s variance:

Again, it is easier to interpret a standard deviation than a variance. The portfolio’s volatility is simply given by the following equation:

which can be easily coded in Python:

Combining all this together, we find that if, since 2007, we have had an equally weighted portfolio with 50% stock in Apple and 50% stock in Microsoft, the portfolio return would have been 20.6% and the portfolio volatility would have been 26.7%.

If more weight in our portfolio had leaned toward Microsoft (say 20% Apple and 80% Microsoft), we have already shown the portfolio return would be higher (than 20.6% ) at 23.3% (pulled up by the higher performing returns of Microsoft during the period). But the code below shows the portfolio volatility would also be somewhat higher (at 29.1% up from 26.7%) since a larger portion is held in the riskier Microsoft stock:

As we discussed in previous chapters, risk can be characterized as either idiosyncratic or systemic. As we saw earlier, idiosyncratic risk can be mitigated by diversification or hedging strategies. On the other hand, systemic risk is, to some extent, “un-diversifiable.”

We now set out to disaggregate our calculated portfolio’s risk into the diversifiable versus the systemic risk.

First, we need the annualized variances of each stock, recalling that Microsoft looks slightly riskier. Then, applying our equation above shows the idiosyncratic risk is 0.026:

So subtracting this idiosyncratic risk from the portfolio’s variance will give us a way to quantify systemic risk.

Showing that risk associated with the economy-wide factors is almost twice as large as the idiosyncratic risk. We can alternatively calculate systemic risk and check that both alternatives find the same number.

8.7 Monte Carlo

Monte Carlo simulation (MCS) methods were invented by Ulam, Neumann, and Metropolis in the 1940s as a set of numerical techniques that use random sampling for computing approximate estimates or for simulating uncertain outcomes. MCS is useful for analyzing both deterministic and probabilistic systems with numerical approximations when an analytical solution may be difficult to reach.

MCS converts a complex, multi-dimensional, difficult problem in integral calculus into an easier problem of descriptive statistics. It can also be useful when best/worst/base case scenarios may not be enough for decision-making and, instead, we need to quantify the probability and impact of all outcomes.

MCS may help when we need a better understanding of risk of complex financial models so long as we have some answers to the following questions:

• 1. What are the independent variables/risk factors/features that drive the model?

• 2. What are the correlations across time and among these probabilistic/stochastic variables?

• 3. What theoretical probability distribution best approximates their real-life behavior?

MCS methods are based on two of the most important theorems in probability and statistics: the law of large numbers and the central limit theorem (CLT). The law of large numbers shows that when we simulate/estimate data samples, convergence to the true mean is often guaranteed as the sample size gets larger. The CLT proves the sampling distribution of the mean approaches a normal/Gaussian distribution as the sample size gets large. See Figure .

Sampling error is independent of the dimension of the variable and goes to zero asymptotically as the square root of the sample size increases. Therefore, sampling errors/fluctuations from the true estimate are normally distributed due to CLT. MCS requires a large number of iterations; in order to increase accuracy by a single digit, MCS iterations need to increase by a factor of 100.

Implementing a sound framework for building an MCS requires a few steps:

Step 1. We must formulate how dependent variables are affected by independent variables. Let’s say that those independent variables denote risk factors for a given investment strategy.

Step 2. We must specify the probability distribution of each risk factor. This requires: (i) assuming some initial values for the risk factors and deciding the frequency of the measurement; (ii) specifying how each risk factor changes over time; and (iii) specifying how each risk factor is affected by other risk factors.

Step 3. We need to draw a random value from the probability distribution of each independent risk factor.

Step 4. We want to compute the value of each risk factor based on that random value.

Step 5. We need to compute target/dependent variables based on the computed value of all risk factors.

Step 6. Repeat Step 3 through Step 5 as many times as necessary in a process called iteration.

Step 7. Collect and analyze descriptive statistics of all iterations.

There are some challenges to conducting a MCS successfully. First, we often need to specify how each independent variable changes over time, but the degree with which it varies (its serial correlation) usually changes over time. We also need to specify how each variable is affected by other variables of the model, but the relationships among input variables/risk factors also usually change over time.

MCS typically requires fitting a theoretical probability distribution to the actual outcomes. However, probability distributions of variables usually change over time, which likely means nonlinear dynamics. Therefore, convergence to the best estimate is often nonlinear, making it slow and costly, so it may not occur quickly enough to be of any practical value to trading or investing.

8.8 Forecasting Stock Prices with MCS

Let’s say we want a daily forecast of a given stock. We begin with the growth rate between a stock price today and its price yesterday. Let g denote the daily growth rate:

This growth rate (g) describes the return from holding the asset between yesterday and today. This return can also be approximated by a log return (r). Therefore, the daily stock return can be calculated as follows:

Now, we can begin with an identity:

Multiplying and dividing by ${\mathit{P}}_{\mathrm{yesterday}}$ does not change the identity:

We can now leverage the following condition: ${\mathit{e}}^{ln\mathrm{(}\mathit{x}\mathrm{)}}=\mathit{x}$, which always holds, so that we have the following:

Substituting our definition for the daily return gives us the following equation of the price of a stock, which allows us to describe today’s price as a function of yesterday’s price as well as the daily return between yesterday and today, as follows:

We now make two main assumptions that allow us to make predictions about the stock price with MCS. First, we assume we know yesterday’s price, but we do not know the daily return (${\mathit{r}}_{\mathit{t}}$). Second, we assume the daily return is a random variable.

There are several ways to model a random variable. One way to model returns as a random variable is to focus on two typical behaviors of returns. Past returns often have a certain degree of persistence. This means that while they may tend to revert to their long-term averages at relatively high frequency, these averages themselves might move (often more slowly). This is typically referred to as drift. On the other hand, daily returns typically fluctuate at high frequencies. The spreads of those daily fluctuations are typically referred to as volatility. This means that past returns often exhibit both drifts and volatilities at various degrees.

Therefore, we decompose daily returns into their drift plus their volatility. The drift of the daily return of a stock is a rough approximation of the future’s return of the stock given by:

where μ is the historical mean and the ${\mathit{\sigma }}^2$ is the variance of the returns. As we discussed in previous chapters, the volatility is given by the standard deviation (σ) of the data. However, given our assumption of randomness in daily returns, we randomize the volatility with a Z-value from a standard normal distribution so that our volatility becomes $\mathit{\sigma }\ast \mathit{Z}$. So, we decompose daily returns in the following way:

We can combine our stock price equation with the decomposition of the daily return from earlier to arrive an equation for the share price at a given period, (say, today):

This equation gives us a single value for the price forecast today. But there is randomness involved. To deal with that randomness, we will want to repeat the calculation many 1,000s of times with an MCS.

Moreover, we can predict future prices by projecting this equation into the future in a recursive manner. This means that if we start from the previous price (recall that we assume we know this initial price), and we calculate the daily return between yesterday and today, we can calculate the price for today. Subsequently, we can repeat the analysis. Now that we know today’s price, we can calculate the return between today and tomorrow and predict tomorrow’s price. We can then repeat this process on an on. For example, if we wanted to predict the daily price of a share of stock for the next 364 days, we could begin with an arbitrary initial price at period 0 say (${\mathit{P}}_0$) and proceed as follows:

The following snippet of Python code enters various stock prices into Jupyter lab:

We then select the share price of Netflix (NFLX) to predict and calculate the mean and the variance of its log returns, which allows us to calculate the drift:

We now calculate the drift and the standard deviation. The drift of the log returns is a rough approximation of the future returns of the stock. Recalling the equation, we can code it into Python as follows:

Now that we have values for the drift and the variance for the stock, we can simulate as many paths for as many periods as we want. Each path corresponds to an independent simulated history of the price we are modeling with MCS. That path can be as long or short as we want:

This Python snippet sets up a simulation for six different paths of a share price of Netflix for 252 trading days; so our simulation will look like the following:

The question now is which price (${\mathit{P}}_0$) to begin from? Since we want to forecast the price, we can begin with the most recent price available.

The last two lines of the Python snippet chart the six simulated paths. See .

We can append these six simulated paths to the historical price of Netflix splicing where the price ends and our simulations begin (connecting where the last value of the price is available with the first value of the simulated paths). The following snippet of Python code accomplishes this. See .

As we covered before, there is a substantial amount of randomness introduced in MCS; therefore, we need to run a very large number of simulated paths. We can simulate a much larger set of paths, then we can compute the mean (μ) and standard deviation (σ) across all paths for each period. The following snippet of Python code simulates 500 paths of the price of Netflix. See .

We can use the following formula to construct a 90% confidence interval (with a Z-value of 1.64) such that $\mathit{\mu }\pm 1.64\ast \mathit{\sigma }$ creates an upper and lower bound where 90% of the simulated paths fall within, as follows:

We can then graph the price of Netflix and append it to the forecast (from the mean across all paths) and the 90% confidence interval. The following snippet of code accomplishes this. See .

We interpret this (dotted red) interval as follows: If we were to repeat this simulation, say 100 times, the forecast would lie within this interval 90 of those 100 times. The width of this interval suggests the accuracy of our forecast. The narrower this band, the more confident we can be about our forecast in the dashed red line.

8.9 Glossary

Apportionment

The foundational concept that allows ownership in a company to be divided into independently transferable portions (shares), enabling partial ownership to be sold without requiring permission from other owners.

Behavioralists

Investors who base their trading decisions on perceptions of market psychology and investor behavioral patterns, often monitoring social media sentiment and trading patterns to inform their decisions.

Buy-and-Hold Strategy

An investment approach where stocks are purchased with the intention of keeping them for an extended period, treating them similarly to perpetual bonds that pay regular dividends.

Chartists

Investment analysts who attempt to predict stock price changes by examining graphical patterns and historical price movements, focusing on technical analysis rather than company fundamentals.

Common Stock

Shares in a firm’s ownership that entitle stockholders to participate in profits and vote at company meetings, while maintaining limited liability for losses.

Dividend Discount Model

A valuation method that determines a stock’s fundamental value based on the present value of its expected future dividend payments.

Drift

In stock price modeling, the general directional trend of returns, calculated as the historical mean return minus half the variance.

Free Riding

A trading behavior where investors monitor and copy other investors’ trading patterns, essentially following the market rather than conducting independent analysis.

Fundamental Value

The estimated worth of a stock based on both current assets and projected future profitability, determined through analysis of financial statements and macroeconomic conditions.

Historical Mean

The average return of a stock over a specified past period, used as a key component in calculating drift for price forecasting models. This represents the central tendency of returns over time.

Joint-Stock Company

An early form of business organization that issued stock to finance ventures, spreading risk among multiple investors who shared in potential profits.

Leverage

The use of borrowed money to finance stock purchases, which can amplify both potential returns and risks in investment positions. The text notes this as a significant factor in determining overall investment risk.

Limited Liability

A key feature of stock ownership where the maximum amount an investor can lose is their initial investment, even if the company goes bankrupt.

Market Capitalization

The total value of a company calculated by multiplying the current stock price by the number of outstanding shares.

Monte Carlo Simulation

A numerical technique using random sampling to simulate multiple possible future paths of stock prices and estimate probable outcomes.

Over-the-Counter (OTC) Market

A decentralized market where stocks, particularly of smaller and newer companies, are traded directly between parties rather than on a centralized exchange.

Portfolio Risk

The combined risk of multiple stocks held together, which accounts for both individual stock volatility and the correlations between stocks.

Residual Claimant

The position of stockholders in receiving payment last, after all other creditors (including bond holders), during company liquidation.

Return Volatility

The degree of variation in a stock’s returns over time, typically measured by the standard deviation of returns.

Stock Market Index

A measurement tool that tracks the overall performance of a group of stocks, providing a benchmark for market performance.

Value-Weighted Index

A stock market index, like the S&P 500, where companies with a larger market capitalization have a greater influence on the movement of the index.