You have a gathered data on a series of ten different $1,000 par value Treasury bonds maturing at different times.All of these bonds pay interest semiannually, and all of them have the same annual coupon rate of 8% (i.e., bonds pay $40 in interest twice per year).Using information below on the price and maturity of each bond, calculate each bond's YTM and calculate the spot interest rate for each maturity date.As before, use the (1+r/2) convention when dealing with semiannual interest payments (i.e., refer to either Equation 1 or Equation 2 in my "Notes for Week 2" file, depending on whether you are solving for a ytm or a spot rate).When you report your answers, you should always report interest rates as annual rates.For example, because the first bond sells at par, its coupon rate and its ytm should be the same, 8%.Hint: To find spot rates, you have to start with the first bond and then work iteratively through the others, using your answers for the prior bonds to find more and more distant spot rates.

MaturityBond priceYTMSpot rate

6-mo$1,000.00

12-mo$ 997.69

18-mo$ 995.52

24-mo$ 991.25

30-mo$ 984.17

36-mo$ 975.51

42-mo$ 965.47

48-mo$ 958.91

54-mo$ 949.47

60-mo$ 929.35

What is the general relationship that you observe between YTMs and spot rates?

What is the forward interest rate from month 24 to month 42?

Notes for Week 2 - Bond Pricing and the Term Structure of Interest Rates

Elementary Bond Pricing

This week we will introduce the notion of the term structure of interest rates and see how investors use the term structure to price bonds.However, the information in the term structure can be used, in principle, to price any stream of cash flows paid over time, not just cash flows from bonds.Furthermore, the uses of the term structure extend beyond the domain of asset pricing as we will see.

Let's start with the simplest possible bond-pricing problem.Suppose a risk-free Treasury bond matures in N years.It pays a coupon of $C in semiannual installments, it has a par value of $1,000, and investors require a rate of return of r% on the bond.The elementary approach to pricing this bond is to solve the following equation:

Price = Equation 1

In this expression, the value "r" is called the yield-to-maturity (YTM) or sometimes the bond-equivalent yield, and it is a measure of the average compound annual return over the life of the bond.Notice that the value "r" is divided by 2 in the denominator of each term in the equation.By dividing by 2, the formula takes an annual value of "r" and converts it to a semiannual figure.Any time we convert from annual to semiannual by dividing the annual rate by 2 (or conversely, when we convert from semiannual to annual by multiplying by 2), we are using simple-interest calculations.That is, in Equation 1, "r" is a simple-interest yield-to-maturity.If we wanted to state the YTM in terms of an effective annual rate, we could do the following:

Or, given the effective YTM, we convert back to a simple-interest form by reversing the expression above:

Let's illustrate this with an example.Suppose a bond matures in 2 years, has a $1,000 par value, offers a coupon rate of 4%, and sells for $1,039.02.What is the simple-interest YTM?Just solve for "r" using Equation 1:

You can solve for "r" using the "=IRR" function in Excel.In cell A1, enter the bond's price as a negative number.In cells A2-A5, enter the bond's cash flows in chronological order.In any empty cell, type the formula "=IRR(A1:A5,)" and Excel will return the value 0.01 (be sure you have the cell formatted to show you several places to the right of the decimal point).In this case, 0.01 represents the value of r/2, so r itself equals 0.02 or 2%.In other words, given the price of $1,039.02, investors require a simple-interest yield-to-maturity on this bond of 2%.The effective YTM on the bond is 2.01% as given below:

Here's another example.Suppose a bond pays a $50 coupon, matures in 3 years, and offers an effective annual YTM of 7%.What is the bond's simple-interest YTM and what is its price?First, calculate the equivalent simple-interest YTM, then use Equation 1 to obtain the price:

Price

Notice in the first example the bond sold at a premium (i.e., above par value), whereas in this example the bond sells at a discount (i.e., below par value).In the first case, the rate of return required by investors was 2% (or 2.01% effective), well below the bond's annual coupon rate of 4%.In the second case, the required return on the bond was 6.88% (7% effective), well above the bond's coupon rate of 5%.The inverse relationship between bond prices and yields is an important one to which we will return frequently.

A More Sophisticated Approach

Almost everyone is introduced to bond pricing through a variant of Equation 1.However, the equation makes an important assumption that is almost always violated in the real world.That assumption is that investors discount all of the bond's cash flows at the same interest rate, no matter when the cash flows arrive.A more realistic bond pricing equation would allow investors to require a different rate of return on each cash flow that arrives at a different point in time, even if the bond is risk-free, like this:

Equation 2

Note that the interest rates in this equation, r₁, r₂, and so on, are not rates that are changing through time.These are rates that are available in the market today. Each of these rates applies to a risk-free cash flow that arrives at a specific time in the future.This distinction is subtle, but important.Bodie, Kane, and Marcus use the term short rates to describe a sequence of interest rates that evolves over time.For example, if the interest rate on a one-year bond today is 5%, the interest rate on a one-year bond one year from today is 6%, and the rate on a one-year bond two years from now is 7%, then we are describing a series of short rates.In the real world, short rates (past the first year) are not observable.We don't know for sure that the one-year interest rate next year will be 6% or that the same rate two years from now will be 7%.

On the other hand, if today's rate on a one-year bond is 5%, today's rate on a two-year bond is 6%, and today's rate on a three-year bond is 7%, then we are talking about spot ratesrates observable in the market right now that apply to instruments having different maturities. In Equation 2, the terms r₁, r₂, and so on are called spot interest rates. The key distinction between spot rates and short rates is that spot rates are observable while short rates are not.(Aside: notice that Equation 2, like Equation 1, simply divides the annual rate by two to get a semiannual rate, a convention that we will consistently follow.)

Definition:The N-year spot rate of interest is the YTM on a zero-coupon, risk-free bond that matures in N years.The term structure of interest rates is the relationship between yield-to-maturity and time-to-maturity for a group of zero-coupon bonds with identical risks.The yield curve is simply a plot of the term structure.

Notice two key pieces of the definition of spot interest rates.First, a spot rate is the YTM on a risk-free bond.The reason we focus on risk-free (Treasury) bonds here is simply to provide a baseline.We can certainly define spot rates for risky bonds, but these will generally equal the spot rate on a risk-free bond plus a risk spread.Therefore, if we understand the concept of a spot rate as applied to risk-free bonds, making the leap to risky bonds only requires adding a premium.

The second key part of this definition is that the spot rate is the YTM on a zero-coupon bond.A zero-coupon bond is simply a bond that pays no interest.An investor pays $X to acquire the bond today and receives $Y at some future date.The dollar return on the bond is simply $Y - $X.

You may wonder why it is necessary to define spot rates in terms of zero-coupon bonds.For example, wouldn't the YTM on a 1-year Treasury bond that makes semi-annual interest payments be the same as the YTM on a 1-year, zero-coupon Treasury bond?Both are 1-year instruments and both are risk-free, so it seems as though the market would price them to provide equal YTMs.This intuition is almost correct, but not quite.There are three reasons why spot rats refer to the YTM on a zero-coupon bond and NOT the YTM on a coupon-bearing bond.The first reason has to do with reinvestment risk.The second is related to the no arbitrage condition.The third is that when different bonds have the same maturity but different coupon rates, their YTMs will be different (even if both bonds are issued by the Treasury and are therefore free of default risk).

Reinvestment Risk

Remember that our objective in talking about spot rates is to create a baseline from which future cash flows can be discounted.We want the baseline to be as pure as it can possibly be, meaning that we want the baseline to be as close as possible to being truly risk-free.A coupon-paying Treasury bond is riskier than a zero-coupon bond due to reinvestment risk.An investor who holds a coupon-paying bond does not know the rate at which the coupon payments can be reinvested.With a zero-coupon bond, there are no coupons to reinvest, so buying and holding the bond to maturity provides a fixed, known return.

Let me illustrate how reinvestment risk comes into play.Suppose we have a 2-year Treasury bond, and for simplicity suppose it makes annual $100 interest payments.If the YTM on the bond is 10%, the bond will sell at par:

What does this equation mean in plain English?It says that an investor is indifferent between a cash payment of $1,000 today or a cash payment of $100 in 1 year plus another payment of $1,100 in 2 years.The two options are equivalent because one is just the present value of another.Now, take both sides of this equation and multiply by 1.10²:

The interpretation of this equation is similar, but now it is in future value terms.On the left, we see the return that an investor would earn today by investing $1,000 for two years at 10% compound interest.That option is just as valuable as the one on the right side of the equation.But the right side of the equation shows what an investor will have at the end of two years if he receives $100 in one year, then reinvests that for one year at 10%, and then receives the remaining $1,100 at the end of the 2nd year.If the investor cannot reinvest the first $100 coupon at 10%, then the equation doesn't hold and the investor is not indifferent between the lump sum on the left and the sequence of cash flows on the right. In other words, the bond-pricing equation implicitly assumes that the investor can reinvest the $100 interest payment received in 1 year at 10% for the 2nd year.That assumption may or may not turn out to be right, and that uncertainty is what we mean by the term reinvestment risk.With a zero-coupon bond, this risk is absent.

No arbitrage pricing

There is a second reason why we define spot rates in terms of YTMs on zero-coupon bonds rather than using the YTMs on coupon bonds.Recall on the syllabus I stated that the fundamental concept underlying this course was the principal of "no arbitrage" which simply means that identical assets must sell for identical prices to prevent arbitrage opportunities.The connection between this concept and the definition of spot rates is as follows.Any coupon-paying bond can be "mimicked" by a series of zero-coupon bonds packaged together in a way that replicates the cash flows of the coupon-paying bond.If that is true, then the total cost (or price) of the sequence of zero-coupon bonds must equal the cost (price) of the coupon bond.But to determine the cost of the sequence of zero-coupon bonds, we must discount each individual bond at the appropriate spot interest rate (remember, by definition the spot rate is the YTM on a zero-coupon bond).That means we should also be able to price a coupon-paying bond by discounting its cash flows using the spot rates.Let's see how this works with an example.For now, imagine that there are no transactions costs or other market frictions associated with buying and selling Treasury securities.

Suppose we have a 4-year Treasury bond.Again for simplicity we will assume annual interest payments.Suppose the coupon rate on this bond is 15%.That is obviously high by today's standards, but perhaps this bond was originally issued a long time ago, and now it just has 4 years of life remaining.At the time that it was first issued, interest rates were much higher than they are now.Finally, suppose that the market price of this bond is $1,244.09.Using Excel's IRR function we can ascertain that the YTM of the bond is 7.6824% (going out to 4 digits to be more precise for reasons that will become clearer in a moment).

Of course, by today's standard an interest rate of 7.6824% is also quite high. I'm using an example with relatively high interest rates to make a key quantitative point a little more obvious. Conceptually, everything I'm talking about is correct whether interest rates are high or low.

There is a part of the government debt market with which you may not be familiar.This is called the Treasury strips market, and it is a market in which investors buy and sell pieces of Treasury bonds that have been "stripped" off existing bonds.This market was launched many years ago when an investment bank decided to buy coupon-paying Treasury bonds and strip off the individual payments, selling each one as a separate zero-coupon security.The investment bank that came up with this idea initially made a lot of money, because the parts of a bond (the strips) were somehow worth more than the whole (the original bond).But competition is a powerful thing.Soon, other investment banks copied the idea, and most of the profits from the undertaking quickly dried up.That's just another way of saying that today, if you buy a bond, strip it, and sell the pieces, the revenue you will get will just about equal the cost of buying the bond in the first place.

The important thing to keep in mind is that a Treasury strip is just a risk-free, zero-coupon bond.Therefore, the YTM on a strip with a given maturity is, by definition, the spot interest rate for the maturity.Suppose an investor looks in the Treasury strips market and finds the current YTMs (i.e., spot rates) on strips maturing at different dates:

Maturity DateYTM

1 year5%

2 years6%

3 years7%

4 years 8%

The investor decides to purchase a sequence of strips to mimic the cash flows from the 4-year, 15% Treasury bond above.The cost of this sequence will be:

Face Value of Spot rate

Strip PurchasedMaturing(i.e., YTM)Cost (PV) of Strip

$1501-year5%$1501.05 = $142.86

$1502-years6%$1501.06² = $133.50

$1503-years7%$1501.07³ = $122.44

$1,1504-years8%$1,1501.08⁴ = $845.28

Total cost of all strips $1,244.08

Notice that the cost of these strips is within $0.01 of the cost of the 4-year bond, and even that difference is simply a rounding error.Of course, if the total cost of the strips did not equal the price of the bond, then there would be an arbitrage opportunity.For example, if the total cost of the strips was $1,000, then the investor could purchase the strips and sell short the bond, realizing an immediate profit of $244.09.This position would be risk-free because the cash flows of the long strips position and the short bond position exactly offset each other.

In the real world there are transactions costs, which we have assumed away here.If transactions costs are not zero, then the actual price of a security can be different from its "no arbitrage" theoretically correct price without giving rise to an arbitrage opportunity, as long as the difference between the actual price and the no arbitrage price is smaller than the transactions costs required to execute an arbitrage trade.The market for Treasury securities is one of the largest, most liquid markets in the world, and transactions costs are quite low in that market.Thus, we would expect bond prices to satisfy the no arbitrage condition to a very close approximation.

The bottom line of this example is that when we define spot rates as the YTMs on zero-coupon bonds, we can see through the principle of no arbitrage that Equation 2 for bond pricing makes sense.It has to work, or an arbitrage opportunity arises.In the context of this example, we can write the bond-pricing equation this way:

Again I want to emphasize that the sequence of rates here is a series of spot rates, not short rates.We can look in the strips market and observe that the rate on a one-year zero-coupon bond is 5%, the rate on a two-year zero is 6%, and so on.We cannot know with certainty what the one-year interest rate will be next year.That would be the short rate.

Thus far, we have seen two reasons why we define spot rates in terms of zero-coupon bonds.First, we want to remove the reinvestment risk issue.Second, it is possible to price coupon-paying bonds using YTMs on zero-coupon bonds via the concept of no arbitrage.That means that discounting the cash flows of a bond using spot interest rates should yield a correct price.The third reason that we focus on zero-coupon bonds is because for bonds that pay coupons, the coupon rate itself affects the YTM.

Pricing new bonds using YTMs on existing bonds

When you first learn about bonds you may hear that bonds that have identical risks and identical maturities will have the same YTM.In the example we've been discussing over the last few pages, is it true that a 4-year, zero-coupon bond should have the same YTM as a 4-year Treasury bond that pays a 15% coupon?Intuitively it might seem that the yield on these two bonds should be identical since they have the same default risk and the same maturity. The answer is no, the bonds will not have the same YTM, and you can see that immediately if you look at the numbers.Remember we said that the four-year spot rate, the YTM on the four-year, zero-coupon bond, was 8%, but we also calculated the YTM on the 15% bond and found that it was 7.6824%.The difference is small, but large enough to matter to an institutional trader investing a large sum in bonds.This discrepancy leads to a general principle.The YTM on one coupon-paying bond cannot be used to determine the price of another coupon-paying bond, even if both bonds have identical risks and identical maturities.This is the third reason why we define spot rates in terms of yields on zero-coupon bonds.YTMs on coupon-paying bonds are unreliable guides to price other bonds.As usual, an example helps clarify this point.

The YTM on the 4-year Treasury bond with a 15% coupon rate was 7.6824%.Suppose a brand new 4-year Treasury bond offers a coupon rate of just 2%.If we use the YTM on the old bond to price the new one, we get $810.45:

But we also know that an investor can easily replicate the cash flows of this bond by purchasing the appropriate sequence of Treasury strips.Furthermore, we know what the cost of each strip will be because we have the YTMs for each maturity date: 5%, 6%, 7%, and 8%.Therefore, the total cost of buying a sequence of strips with the same cash flows as the 2% coupon bond is:

Thought Question:In the last few paragraphs we have assumed that the YTM on a one-year strip is 5%, regardless of whether the face value of the strip is $20 (as in the last example) or $150 (as in the first example).Is this a reasonable assumption?Should the YTM on a strip depend on the strip's face value?

If the price of the sequence of strips is $802.90, then the price of the bond cannot be $810.45.Identical assets cannot sell at different prices.This must mean that our approach for calculating the bond price, discounting the cash flows of the 2% Treasury bond using the YTM of the 15% coupon Treasury bond, is incorrect.But what mistake did we make?If the price of the bond really should be $802.90, then we can calculate the bond's true YTM:

Notice that the YTM on the 2% coupon bond is considerably higher than the YTM on the outstanding 15% coupon bond.Why?The YTM calculation provides a measure of the average return that an investor earns over the life of a bond.The YTM is essentially a weighted average of the individual spot rates, where the weights depend on the timing and the magnitude of the bond's cash flows.Notice that the YTM on the 2% coupon bond, 7.9434%, is very close to the 4-year spot rate of interest, 8%.That's because almost all of this bond's cash flow arrives at the end of the 4^th year.In contrast, the YTM of the 15% coupon bond, 7.6826%, is somewhat lower.The lower YTM results from the fact that the 15% coupon bond distributes more cash flow in earlier periods ($150 each year rather than $20), when spot interest rates are lower.

A bond is a portfolio of cash flows

It may be helpful here to start thinking of a bond in a new light.Rather than viewing a bond as a single investment instrument, think of it as a portfolio of different investments, each one a zero-coupon bond that matures at the end of each year (or at the end of each six-month period for semiannual payments).With that mental frame in place, you can think of the bond's YTM as the average return on this portfolio of zero-coupon investments.For instance, the 2% coupon bond is really a portfolio of four distinct assets.Three of these assets are zero-coupon bonds with a face value of $20, and the 4th asset in the portfolio is a zero-coupon bond with a face value of $1,020.Clearly the average return on this "portfolio" depends mostly on the return on the 4^th component because it is so much larger than the others.By definition, the return on the $1,020 cash flow is 8%, the 4-year spot rate.Therefore, the YTM on the bond must be very close to 8%.In contrast, the 15% coupon bond consists of three, $150 zero-coupon bonds and one, $1,150 bond.The average return on this portfolio will still be close to 8%, because that's the return on the portfolio's largest piece, the $1,150 payment.However, the other three $150 pieces are very large in their own right.Each one of them carries a return less than 8%, so they bring down the overall average.This is why the YTM on the 15% bond is so much less than the YTM on the 2% bond.

Thought Question:Suppose the yield curve is upward sloping, meaning that r₁ < r₂ < r₃....In this environment, what general relationship will obtain between a bond's YTM and its coupon rate, holding maturity constant?That is, if we are looking at two 4-year bonds, one with a high coupon and one with a low coupon, which bond will have a higher YTM?How will this relationship change if the yield curve is downward sloping?

Thought Question:We have said that the YTM on an existing N-year coupon-paying bond cannot be used to price a new N-year bond even if both bonds have identical risks.There are actually two special cases when this is not true.Under what special circumstances would the YTM of an outstanding bond provide a reliable means to calculate the price for a new bond (holding maturity and risk of the two bonds constant)?

Forward Interest Rates

One of the more useful bits of information that we can obtain from the term structure is something called a forward interest rate.A forward interest rate is a kind of economic breakeven point in the bond market that equates the rate of return on one N-year, zero-coupon bond to the return offered by a sequence of two shorter-term bonds.A forward interest rate is defined by two parameters: (1) at what point in the future does the forward rate begin and (2) to what maturity does the forward rate apply.When we write the term _sf_t, we are denoting a forward interest rate that begins in "s" years and lasts for "t" years.(This is different notation than your text uses).For example, the symbol ₃f₂ defines an interest rate that starts 3 years in the future and lasts for 2 years.

For example, suppose an individual wants to invest money for 5 years.The current 5-year spot rate is 6%.This means that someone could invest $1 in a 5-year, zero-coupon bond, and at maturity they would have:

$1(1.06)⁵ = $1.34

Of course there are other ways that an investor could choose to put money away for 5 years.For example, the person might buy a 3-year bond and then reinvest in a 2-year bond afterwards.Suppose the 3-year spot interest rate is 5.5%.The forward rate, ₃f₂, is defined as the rate on a 2-year bond, 3 years from now_, that would make the investor just as well off as the strategy of buying the 5-year bond.Mathematically, we can express this idea as follows:

$1(1.06)⁵ = $1.34 = $1(1.055)³(1+ ₃f₂)²

₃f₂ = .06754 = 6.754%

The intuition behind this example is probably easier than the math.If the investor buys the 5-year bond, he earns 6% for 5 years, or 30% ignoring compounding.If the investor instead buys the 3-year bond, he earns 5.5% for 3 years or 16.5%, again ignoring compounding.The investor will then purchase a 2-year bond.How much would this 2-year bond have to pay to allow the investor to reach the same 30%, 5-year return that he could get on the 5-year bond?The 2-year bond would have to pay roughly 13.5% over the next two years, or 6.75% per year.That's just the rate we get when we do the forward rate computation, except for the extra decimal place that comes into play with compound interest....6.754%.

We can derive a general expression for the forward interest rate beginning in "s" years and lasting for "t" years, _sf_t:

where r_n and r_s are the n-year and s-year spot rates respectively. Let's calculate several forward rates using the sequence of spot interest rates from a previous example.

The sequence of spot interest rates looks like this in September 2017:

MaturitySpot Rate

1-year1.2%

2-year1.3%

3-year1.5%

4-year1.7%

Rates are low today, but the yield curve does slope upward, meaning that longer term rates are higher than shorter term rates. What series of forward rates can we calculate given this information?First, we can calculate a series of one-year forward rates, each occurring at a different point in the future.

The interpretation of these numbers is as follows. The value ₁f₁ refers to an interest rate that starts one year from today and lasts for one year, whereas the value ₂f₁ refers to an interest rate that starts two years from today and lasts for one year. In other words, if today is September 1, 2017, the value 1.400% is the forward rate that starts in September 2018 and runs through September 2019.2.302% is the forward rate that starts in September 2020 and runs through September 2021, and so on.

Can we make intuitive sense out of these numbers?Consider the calculation for the 1-year forward rate beginning 2 years from now (the 2^nd of the 3 equations above).This calculation implicitly equates the return on one 3-year bond with the returns on a sequence of a 2-year bond followed by a 1-year bond.Today an investor can earn 4.5% on a 3-year bond (1.5% per year for 3 years, ignoring compounding).The 2-year bond would pay 2.6% (1.3% per year for 2 years, ignoring compounding), so the investor would have to earn 1.9% in the third year to breakeven (i.e, 4.5% - 2.6% = 1.9%).That's the intuition behind why we get 1.901% for the forward rate, ₂f₁.

Thought Question.How many other forward rates can we calculate given the sequence of 4 spot rates above?Do the calculations.

There are a couple of reasons why we go to the trouble of calculating forward rates.First, many derivative contracts, like interest rate futures, are priced using forward rates.That's a topic to which we will return much later in the course.Second, forward rates may be useful in predicting future interest rate movements or changes in overall macroeconomic activity. The relationship between forward rates and future interest rates is a topic that will occupy our attention next.