Finance 300 Financial Markets Lecture 10 Professor J. Petry, Fall, 2002©

2 Housekeeping Equity projects due next class (October 3 rd ). Bring to class. Fill out Investment Challenge Team Forms today so you can get a UISES account. Class on October 10 th may have to be cancelled. I will keep you posted. Next exam is October 15 th, will cover chapters IV and V. We’ll start today’s class by doing the example we ended last class doing...

3 Bond Pricing The Term Structure of Interest Rates We are trying to derive and understand the term structure of interest rates by deriving the yield curve. We start with an unrealistic assumption: we know all future one-year interest rates. If this assumption holds, we can derive the yield curve... Things to Do: IV-14 –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.

4 Bond Pricing The Term Structure of Interest Rates (cont’d) –The point of this exercise it to demonstrate that long-term rates are nothing more than the product of short-term rates. –Armed with this knowledge, we can relax our assumption that we have perfect foresight into future interest rates— obviously unrealistic--and go from current long-term spot rates, to find forward rates. Essentially reversing the process we just undertook. –We found that a 2-year spot rate is equal to: –[(current 1-yr spot) x (1 yr forward in one year)] 1/2 =current 2-yr spot –[(1.03)*(1.05)] 1/2 = ; or [(1.03)(1.05)]= –Similarly, the 3-yr spot was found... –[(1.03)(1.05)(1.07)] 1/3 = ; or alternatively, [( ) 2 (1.07)] 1/3 –What if instead of not knowing the 2-yr spot, I didn’t know the 1 yr forward rate in one year? –[(1.03)(1+ r)]= ; (1+ r)=( ) 2 /1.03; 1+ r = 1.05

5 Bond Pricing The Term Structure of Interest Rates (cont’d) Things to Do: IV-13 –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.

6 Bond Pricing The Term Structure of Interest Rates (cont’d) –We can now go from forward rates to spot rates and back again. We are still 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.

7 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???

8 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 8.5% Coupon 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 each of the next three six month periods.

9 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, 8.9% yld) is worth by using our formula/calculator*: 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;

10 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!

11 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.

12 Yield Curve The Shape of the Yield Curve –There yield curve can generally be described by one of the following four classifications: Normal—yield curve is upward sloping. This is the most common form of the yield curve, and is typically the case when growth is neither too fast nor too slow. Inverted—short maturity yields are higher than long maturity yields. Often precipitated by Central Bank monetary tightening, usually because of persistent rapid economic growth, or increasing inflation concerns. Oftentimes precedes recession. Flat—short and long maturities have about the same yields. Most common as a temporary condition as yields adjust from either inverted to normal or normal to inverted. Humped—middle maturity yields are highest, usually followed by long maturities, with short maturity yields lowest. Also frequently transitory, though expectations, liquidity and segmented market theories can explain this remaining in place for some time.

13 Yield Curve The Shape of the Yield Curve –Which are usually explained by one of three theories: Expectations Hypothesis—yields reflect expectations now and in the future for interest rates. If rates are expected to go up, the yield curve will be upward sloping. If they are expected to go up, then fall further down the road, it would be humped. Expectations are formed explicitly about interest rates, but implicitly about inflation. They are also incorporating monetary policy responses. Liquidity Preference—investors prefer liquidity so that they can adjust their holdings as conditions warrant. The future is uncertain, and the further into the future you are interested in, the more uncertain it becomes. To the extent you have to commit now to specific yields in the future, those yields should be higher than shorter term commitments. Segmented Market Hypothesis—(a.k.a. preferred habitat theory, institutional theory, hedging pressure theory) yield curve is a function of supply & demand for funds as dictated by liability matching, tax liability concerns and other specific needs of investors/depositors.

14 Footnote to slide 9 *FYI: You should be able to do using the CF keys as well as the bond keys and the formula to find this answer. Using the CF keys: Using the Bond function, you should be able to do using 100 RV and 10,000 RV: