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

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

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

Global Inflation Rates Averages from

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. The Term Structure tells us the cost of debt for various maturities.

Term Structure 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 YTM (r) Year 1981 & 1987 Normal

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 r 3 = 3-year spot rate (1 + r 3 ) 3 Value of 3-year bond with annual coupon of 10: (1 + r 1 ) (1 + r 2 ) 2 (1 + r 3 ) 3 Think of a spot rate as the yield on a zero-coupon bond ++

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

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. 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.)

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.

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

Value of investment in zero-coupon bond

Coupon bonds

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

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

Duration of Total PV% x Year Duration Example (Bond 1) Calculate the duration of our 6 7/8 % 4.9 % YTM

Duration Considers The Magnitude and Timing of Cash Flows What is the Duration of a zero coupon paying bond? 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 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? 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?

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