Valuing risky debt

1. Valuing risky debt The story teller makes no choice, soon you will not hear his voice. His job is to shed light and not to master. – Garcia, Hunter

2. Debt & Interest Rates Classical Theory of Interest Rates (Economics) developed by Irving Fisher: r Supply Real r Demand $ Qty Real Interest Rate = The theoretical rate (absent inflation) that you pay when you borrow money, as determined by supply and demand.

3. Debt & Interest Rates • Nominal Interest Rate = The rate you actually pay when you borrow money. • Relationship between nominal rate, inflation, and real rate:

4. Global Inflation Rates Averages from 1900-2006

5. The Term Structure of Interest Rates • Shows the relationship between interest rates (spot rates) and time to maturity. • A graph of the term structure is known as the yield curve. • The Term Structure tells us the cost of debt for various maturities.

6. Term Structure YTM (r) 1981 & 1987 Normal Spot Rate - The actual interest rate today (t=0) Forward Rate - The interest rate, fixed today, on a loan made in the future at a fixed time. Future Rate - The spot rate that is expected in the future Yield To Maturity (YTM) - The IRR on an interest bearing instrument 1976 Year 1 5 10 20 30

7. Spot rates • n-year spot rate = rate market uses to value a single payment n years hence • Example: • Value of single payment of 10 in 3 years time: • 10 , where r3 = 3-year spot rate • (1 + r3)3 • Value of 3-year bond with annual coupon of 10: • 10 10 110 • (1 + r1) (1 + r2)2 (1 + r3)3 • Think of a spot rate as the yield on a zero-coupon bond + +

8. What are forward rates? • Forward Rates: are rates from investing additional time periods. • Forward rates are implicit in spot rates: • (1 + r2)2 = (1 + r1)(1 + f2) • The forward rate for year 2 = f2 = (1 + r2)2 (1 + r1)1 - 1

9. Bond Values • Bond prices are found by calculating the present value of the cash flows from the bond at the corresponding spot rate for each cash flow. • Previously, we assumed a flat yield curve (constant spot rates in our bond calculations.)

10. Yield to Maturity • Is the estimated IRR from investing in a bond and holding it to maturity. It is a complex average of the spot rates. Yields measure expected return only if coupons are reinvested to earn yield. • Like IRRs, yields to maturity do not add up. • If know the yield to maturity, you can use it to calculate bond values.

11. Convexity • Convexity refers to the fact that bond price changes are not symmetric with changes in interest rates (yields). • As yields fall, prices rise at an increasing rate. • As yields rise, prices fall at a decreasing rate. • Value • Yield

12. Value of investment in zero-coupon bond

13. Coupon bonds

14. Classical Duration • Classical Duration weighs the percentage of value received by the time it is received. • Where %PVt = PVt / Bond Value • Duration is a measure of Interest Rate Risk.

15. Duration Calculation 1000 Face value 10% coupon bond with 3 years left to maturity and 5% yield.

16. Duration Example (Bond 1) Calculate the duration of our 6 7/8 % bond @ 4.9 % YTM • Year CF PV@YTM % of Total PV % x Year • 1 68.75 65.54 .060 0.060 • 2 68.75 62.48 .058 0.115 • 3 68.75 59.56 .055 0.165 • 4 68.75 56.78 .052 0.209 • 5 1068.75 841.39 .775 3.875 • 1085.74 1.00 Duration 4.424

17. Duration Considers The Magnitude and Timing of Cash Flows • What is the Duration of a zero coupon paying bond? • All else being equal, is Duration larger or smaller for long term versus short term bonds? • All else being equal, is Duration larger or smaller for bonds that pay a high coupon rate versus those that pay a low coupon rate?

18. Modified Duration • Modified Duration is often employed in estimating a change in bond prices for a change in yields. • Where: • Dmodified = DClassical / (1+ y) Change in bond price: This is a linear approximation to actual changes.