Finance 300 Financial Markets Lecture 12 Fall, 2001© Professor J. Petry

2 Bond Pricing The Term Structure of Interest Rates 1.Term to maturity. Life of the bond contract 2.Coupon Rate. Interest rate specified by the bond contract 3.Call provisions. Contractual provisions whereby the bond can be paid off early. 4.Liquidity. Ability to buy or sell quickly without affecting price. 5.Risk of default. Risk that the issuer will not pay coupon and/or principal when it is due. 6.Tax Status. How income and capital gain are treated under the tax law. These six factors combine to determine the Yield Structure.

3 Bond Pricing The Term Structure of Interest Rates (cont’d) –These six factors combine to determine the Yield Structure. (All facets and characteristics of existing yields.) –The “Term Structure of Interest Rates” looks at the Yield Structure across time periods—yield to maturity and “Term to Maturity” for like bonds. The only difference between the bonds is the maturity. –When we derive the “Term Structure” we are going to hold all things constant except the relationship between term to maturity and yield to maturity. Therefore we are assuming liquidity, call provisions, coupon rates, etc are equivalent. –The “Yield Curve” is the graphic representation of the “Term Structure”.

4 Bond Pricing The Term Structure of Interest Rates (cont’d) –Interest rates over long durations can be thought of as nothing more than a series of short term interest rates spanning the same time period. –So what should the interest rate for 5 years be? It should be equivalent to what you could get if you invested your money in 1 year instruments for 5 consecutive years. –To develop this approach, we have to assume for a moment that there is perfect foresight, or certainty, regarding these future 1 year interest rates. –If I know that 1 year rates today are 6%, and that 1 year rates, 1 year from now are going to be 4.5%, I also know what the two year rate today should be. [(1.06)*(1.045)] 1/2 -1 = = %

5 Bond Pricing The Term Structure of Interest Rates (cont’d) –TERMINOLOGY: The interest rate from now to any time in the future is the “spot- rate”, so we talk about the 1 year spot rate, or the 5 year spot rate. The interest rates which take effect sometime in the future are “forward rates”. We have for instance, the “one year rate two years forward”, which is the one year rate, which takes effect two years from today. The ”two year rate, 5 years forward”, is the two year interest rate which takes effect 5 years from today. What we have concluded thus far is that the long term spot rate, is equivalent to the series of short-term forward rates. –Using this approach, we can easily derive the interest rates for whatever period we are interested in, provided we know one year rates for each year from now to the ultimate maturity we are interested in.

6 Bond Pricing The Term Structure of Interest Rates (cont’d) Things to Do: IV-12 –My crystal ball tells me that the one year interest rates over the next five years will be: this year3.0% in one year (1 year rate, 1 year forward)5.0% in two years7.0% in three years8.0% in four years9.0% Calculate and plot the yield curve.

7 Bond Pricing The Term Structure of Interest Rates (cont’d) –The difficulty here is that for this to happen we have to assume that we know one year rates, in one year, in two years, in three years, etc.—obviously not realistic. We have assumed perfect certainty about future interest rates to obtain these results. –The relationship between long term spot rates and short term forward rates across time however can be modified slightly, allowing us to relax this condition of certainty, and get the same result under conditions of uncertainty. –Instead of going from short-term forward rates (not observable until the time they come into effect) to long term spot rates (observable now), we can go in the other direction. Take long term spot rates, and back into what the market is expecting short term forward rates to be in each period.

8 Bond Pricing The Term Structure of Interest Rates (cont’d) Things to Do: IV-11 –My Bridge Terminal gives me the following simple interest rates: one year3.0% two year4.0% three year4.6% four year5.0% five year5.2% 1.If the yield curve is formed solely from expectations, what does the market expect the one year rate to be in one, two, three, and four years? 2.Plot the yield curve.

9 Bond Pricing The Term Structure of Interest Rates (cont’d) –Though we have added one layer of realism, we are not yet ready to go to the WSJ and take current spot rates and calculate all applicable forward rates, or even construct a yield curve of our own. Why not? –To construct an accurate yield curve, we need like instruments, which are free from other issues, like default risk, liquidity risk, etc. To do this, we generally rely on US Government securities, but... T-Bill: Zero coupon Treasury security of less than one year T-Note: Treasury security of between two and ten years bearing semi-annual coupons and maturing on the 15 th of the specified month and year. T-Bond: Treasury security of more than ten years bearing semi- annual coupons and maturing on the 15 th of the specified month and year.

10 Bond Pricing The Term Structure of Interest Rates (cont’d) –We go to the WSJ and find the yield on the six month T-bill is 8%, the yield on the 12 month T-Bill is 8.3%, and when looking for the 18 month T-Bill find there isn’t one! All we find is an 18 month T-Note, yielding 8.9%. –But the T-Note is a coupon bond, not a zero coupon bond. We are no longer talking about similar bonds. –How to convert the T-Note into the equivalent of a T-Bill???

11 Bond Pricing The Term Structure of Interest Rates (cont’d) –Over the last few class periods, we have been discussing the fact that bonds are nothing but a series of cash-flows. This holds for zero coupon bonds as well as coupon bonds. –We have also concluded that you can break down these cash flows and value them separately, then add them up to value the entire income stream. –We can use this same reasoning to find the equivalent of an 18 month zero coupon rate by combining what we know about the 6 and 12 month zero coupon rates, with what we know about the 18 month coupon rate. How?? –A 1.5 year Treasury Note that generates $425, $425, and $10,425 over the next three six month periods must have the same price and yield as a portfolio of three zero coupon bonds that generates $425, $425 and $10,425 over the next three six month periods.

12 Bond Pricing The Term Structure of Interest Rates (cont’d) –This means that our 18 month coupon bond, can be valued as if it were a series of zero coupon bonds, at the appropriate zero coupon bond yields, and the total worth of these zero coupon bonds, should be equivalent to the value of our 18 month coupon bond. We know what the value of our 18 month coupon bond (10,000 face, 8.5% coupon) is worth by using our now familiar formula: What is our first “zero coupon” bond worth when evaluated at the yield for a six month zero coupon bond? Our six month zero yield from the WSJ is 8%, the one year zero yield is 8.3%... –Price = 425/(1+.08 /2) = $408.65; Our second zero is worth... –Price = 425/( /2) 2 = $391.81; Our final zero is worth... –Price = 10,425/(1+ yld /2) 3 = $9, =9,144.51;

13 Bond Pricing The Term Structure of Interest Rates (cont’d) In the case of the 18 month zero, we do not have a yield, but we do have a price! –Price = 10,425/(1+ yld /2) 3 = $9, = $9, ; Rearranging, we can then solve for the yield: Thus, the rate on a zero coupon 1.5 year T-Note should be 8.93%. Gosh, what a blast!!! Now, you try!

14 Bond Pricing The Term Structure of Interest Rates (cont’d) Things to Do: IV-13 –The WSJ quotes a 9% T-Note with 24 months to maturity at a yield of 8.92%. Using the 6 month rate of 8.0%, the 12 month rate of 8.3%, and the 18 month rate of 8.93% from the examples above, calculate the 24 month zero-coupon rate and graph the yield curve.