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Introduction to Bond Valuation Valuing a bond is very similar to valuing an annuity. As before, we care about the size, timing, and risk of the cash flows. The basic idea in bond valuation is discounting a stream of level cash flows with return of principal at the end of the bond’s life. Most of what is new is terminology. 1 What is a Bond A bond is a loan, typically made by investors to a corporation or government. The indenture spells out the terms of the loan: Coupon Maturity Seniority A corporation can deduct the interest payments on bonds (dividends paid on stock are not deductible). 2 US Treasury Securities The US Treasury is the largest security issuer in the world, with £5.2 trillion in debt in 1996. It issues three basic kinds of securities: Bills (maturities less than one year at issuance) Notes Bonds (up to 30 year maturity) There are other hybrid securities such as STRIPS and inflation-indexed bonds. Treasury securities are important “benchmark” instruments (e.g., “riskless” interest rate). 3 Corporate Bonds Generally have £1000 par value, semiannual coupons. Default risk is rated by agencies such as Moody’s and S&P. Other features may include: call provision convertibility floating rate option features 4 Junk Bonds Junk bonds are high-yield instruments with a significant probability of default. Made popular by Michael Milken of Drexel Burnham Lambert in the 1980’s. Often used in “leveraged” transactions such as LBOs, mergers, acquisitions. 5 Characteristics of Bond Prices The cash flows on a bond are constant (“fixed income”). A bond’s market price changes in response to the market interest rate. When market rates increase, the fixed payments from the bond are worth less so the price falls. If rates decrease, the fixed payments are now worth more. [A bond’s price also changes in response to changes in the risk of the cash flows, but we are not quite ready for that discussion.] 6 Basics of Bond Valuation The bond pricing equation consists of two components PV of Coupons PV of Face Value The price of a bond (these PVs) depends on: Discount Rate (r) Number of Periods (N) Size of Cash Flows (C and P N) 7 Yield to Maturity The yield to maturity is an important number in bond valuation. It is the rate which equates the market price of the bond with the value of the discounted cash flows. That is, YTM is the r such that the bond equation holds. Finding the YTM requires a financial calculator, a goal-seeking solver, or trial and error. 8 Example - Annual Coupon £1000 10 year bond paying a 10% annual coupon What is the value when the interest rate is 10%? If r = 11%? If r = 9%? 9 Example - Semiannual Coupon Now the coupon is split semiannually At 10% At 11% At 9% 10 YTM and the Coupon Rate Relationship between YTM and Coupon Rate YTM = Coupon bond is selling at par (P0 = PN). YTM > Coupon bond is at a discount (P0 < PN). YTM < Coupon bond is at a premium (P0 > PN). Why does the YTM differ from the coupon? The coupon is set when the bond is issued. The YTM is the market’s required interest rate. It may change as economic fundamentals shift. 11 Remembering the YTM-Coupon Relationship Zero Coupon Bonds Pays no coupon so interest comes in the form of a discount from the repayment (P0 < PN). Since Coupon = 0, YTM must be greater than Coupon. Putting these pieces together gives the answer. Capital Gains If the YTM is greater than the Coupon, the extra return must be coming from somewhere. The extra return comes from capital gains (P0 < PN). 12 Example - Solving for YTM Consider a £1000 5 year bond with a 8% coupon What is the YTM if it is selling for £1000? If it is selling for £900? If it is priced at £1100? 13 Duration As we have seen, bonds have value from two sources: coupons and return of principal. Intuitively, bonds with high coupon rates or short maturities will return value more quickly than those with low coupons or long maturities. At the extreme is a zero coupon bond, which returns all value at maturity. Duration is a measure of how quickly the (present) value of a bond is returned. 14 Duration To calculate duration: Find the present value of each cash flow individually Sum these to get the present value of all cash flows (price) Calculate the proportion of the total value from each individual cash flow Multiply each proportion by the corresponding number of periods and sum The answer will give a measure of the average life of the bond in a present value sense. A bonds with a low duration gets most of its value from cash flows occurring early. 15 Bond Theorems Price and interest rates move inversely. A decrease in interest rates raises bond prices by more than a corresponding increase in rates lowers price. Price volatility is inversely related to coupon. Price volatility is directly related to maturity. Price volatility increases at a diminishing rate as maturity increases. 16 Illustration of Bond Theorems A decrease in interest rates raises bond prices by more than a corresponding increase in rates lowers price. This is known as convexity. 17 Illustration of Bond Theorems Price volatility is inversely related to coupon. 18 Illustration of Bond Theorems Price volatility is directly related to maturity. Price volatility increases at a diminishing rate as maturity increases. 19 Illustration of Bond Theorems Price volatility is directly related to maturity. Price volatility increases at a diminishing rate as maturity increases. 20 Risk and Bond Valuation So far we have ignored risk in valuing the bonds. We will now discuss qualitatively the types of risk a bondholder faces. Quantification of the price impact due to risk is still coming. In all cases, adding risk to a security increases the required, or expected, return. This implies that an increase in the risk of a bond will lower its current price. 21 Types of Risk Bondholders Face Interest Rate Risk The risk of a bond changing in value when interest rates change. This affects all bonds regardless of credit quality, but is more severe for longer maturity bonds. Reinvestment Risk The risk that investors will be unable to reinvest the coupon payments at the coupon rate. This is more important for high coupon bonds. Default (Credit) Risk The risk that the firm will go bankrupt and not make all payments to bondholders. Other Risks: Inflation, Call, Liquidity 22 Inflation and Interest Rates Inflation is the increase in the nominal (or cash) cost of goods and services over time. Put differently, it is the decrease in purchasing power over time. In the end, we are generally concerned with consumption in finance (and in life). The amount of dollars you have is really much less important than their purchasing power. Nominal rates are the rates observed in the market and quoted in contracts. Real rates are actually very illusive since measuring inflation accurately is difficult. 23 Term Structure of Interest Rates A graph of interest rates on securities of various maturities. Generally constructed using riskless zero coupon bonds (i.e., Treasuries). Serves as a measure of the Time Value of Money. Generally upward sloping, but can also be downward sloping, inverted, or humped. 24 % 40% 20% 0% -20% -40% 4% 6% 8% 10% 12% 14% 16% Interest Rate 18 Illustration of Bond Theorems Price volatility is directly related to maturity. Price volatility increases at a diminishing rate as maturity increases. 90% 30 yr, 10% 20 yr, 10% 80% 10 yr, 10% Price Volatility (|% Change from par|) 70% 60% 50% 40% 30% 20% 10% 0% 4% 6% 8% 10% 12% 14% 16% Interest Rate 19 Illustration of Bond Theorems Price volatility is directly related to maturity. Price volatility increases at a diminishing rate as maturity increases. 200% Illustration of Bond Theorems 5% Interest Rate 10% Interest Rate 180% 15% Interest Rate 160% Percentage Price Change 140% 120% 100% 80% 60% 40% 20% 0% 0 5 10 15 20 25 30 Years to Maturiy 20 Risk and Bond Valuation So far we have ignored risk in valuing the bonds. We will now discuss qualitatively the types of risk a bondholder faces. Quantification of the price impact due to risk is still coming. In all cases, adding risk to a security increases the required, or expected, return. This implies that an increase in the risk of a bond will lower its current price. 21 Types of Risk Bondholders Face Interest Rate Risk The risk of a bond changing in value when interest rates change. This affects all bonds regardless of credit quality, but is more severe for longer maturity bonds. Reinvestment Risk The risk that investors will be unable to reinvest the coupon payments at the coupon rate. This is more important for high coupon bonds. Default (Credit) Risk The risk that the firm will go bankrupt and not make all payments to bondholders. Other Risks: Inflation, Call, Liquidity 22 Inflation and Interest Rates Inflation is the increase in the nominal (or cash) cost of goods and services over time. Put differently, it is the decrease in purchasing power over time. In the end, we are generally concerned with consumption in finance (and in life). The amount of dollars you have is really much less important than their purchasing power. Nominal rates are the rates observed in the market and quoted in contracts. Real rates are actually very illusive since measuring inflation accurately is difficult. 23 Term Structure of Interest Rates A graph of interest rates on securities of various maturities. Generally constructed using riskless zero coupon bonds (i.e., Treasuries). Serves as a measure of the Time Value of Money. Generally upward sloping, but can also be downward sloping, inverted, or humped. 24

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posted: | 5/26/2011 |

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