Chapter 7: Interest Rates and Bond Valuation

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1 Chapter 7: Interest Rates and Bond Valuation Faculty of Business Administration Lakehead University Spring 2003 May 13, Bonds and Bond Valuation 7.2 More on Bond Features 7A On Duration 7C Callable Bonds Outline of the Lecture 1

2 7.1 Bonds and Bond Valuation A bond is normally an interest-only loan. If, for example, a firm wants to borrow $1,000 for 30 years and the actual interest rate for similar corporations is 12%, then the firm will pay $120 in interest each year for 30 years and repay the $1,000 loan after 30 years. The security that guarantees these payments is called a bond. A bond may involve more than one interest payment during a year Bonds and Bond Valuation In the above example, interest payments could be as follows: one payment of $120 per year; two payments of $60 per year; four payments of $30 per year; any arrangement such that a total of $120 in interest is paid each year. 3

3 7.1 Bonds and Bond Valuation With a single interest payment per year, the timing of cash flows to the lender would be as follows: Interest Principal $120 $120 $120 $120 $120 $1,000 With semiannual payments, the timing would be: Interest Principal $60 $60 $60 $60 $60 $60 $60 $60 $60 $1,000 4 Bond Features and Prices A bond is basically characterized by the following items: Face Value (F ): The amount of principal to be repaid at the bond s maturity date. Coupon Rate (i): The fraction of F paid in interest each year. Maturity (T ): The number of years until the face amount is repaid. Number of Payments (m): Number of interest payments during a year. 5

4 Bond Features and Prices Using the above parameters, we can compute the bond s coupon payment per period, which is i F m. Bonds may have other characteristics such as Seniority Repayment Call Provision Protective covenants 6 Bond Values and Yields The above values are set by the firm itself, although very often dictated by market conditions. The bond price and/or the bond s yield to maturity are entirely determined by market conditions. The yield to maturity of a bond is its APR. 7

5 Bond Values and Yields Consider a bond with coupon rate i, face value F, time to maturity T and number of payments m. What is the price of such a bond? This bond s annual coupon payment is C = i F. Hence, this bond makes m payments of C m = C m = if m per year, and this for the next T years. 8 Bond Values and Yields Let y denote the yield to maturity, or APR, of bonds with similar risk characteristics. Then the APR of our bond must also be y. The period interest rate for the bond is then r m = y/m and there are T m periods until maturity. 9

6 Bond Values and Yields The value of the bond, i.e. its price (P), is then ( P = C ( ) ) m 1 T m F 1 + r m 1 + r m (1 + r m ) T m = if/m y/m = if y ( ( 1 1 ( 1 ) ) T m 1 + y/m ( 1 ) ) T m 1 + y/m + + F (1 + y/m) T m F (1 + y/m) T m 10 Bond Values and Yields Expressing the bond price as ( ( ( ) ) i 1 T m P = 1 y 1 + y/m + ( ) ) 1 T m F, 1 + y/m We can see that P > F if i > y, = F if i = y, < F if i < y. 11

7 Bond Values and Yields A bond is said to sell at a premium when P > F; sell at par when P = F; sell at a discount when P < F. 12 Finding the Yield Bond prices are often listed in the newspaper. From these prices, we can compute a bond s yields to maturity as long as we know F, i, T and m. We cannot solve for y analytically, and thus a computer or the trial-and-error method must be used to this end. Note, however, that bond prices are determined by their yield to maturity. Looking at the yield to maturity of bonds with maturities may help understand the trend in interest rates in general. 13

8 Interest Rate Risk The risk arising from a change in market interest rates (i.e. a change in y) is called interest rate risk. Changes in market interest rates affect bond prices. Let P 0 denote the bond price when y = y 0, and suppose that y changes to y 1, inducing a new bond price P 1. The bond price sensitivity to a change in interest rate will be defined as P 1 P 0 P 0 = P P 0, i.e. the proportional increase or decrease in the bond price. 14 Interest Rate Risk The sensitivity of a bond price to changes in y depends on the bond s characteristics. With respect to i, T remember the following results: 1. All other things being equal, the lower i, the greater the interest rate risk; 2. All other things being equal, the greater T, the greater the interest rate risk; 15

9 Interest Rate Risk Example 1 Take, for instance, a zero coupon bond with T years to maturity. The price of such a bond is P = F (1 + y) T. If y is initially y 0 and then decreases to y 1, the relative change in the bond price is (note that P > 0) P P 0 = F (1+y 1 ) T F (1+y 0 ) T = F (1+y 0 ) T ( ) 1 + T y0 1, 1 + y 1 16 Interest Rate Risk Since, in this example, y 1 < y 0, the term ( ) 1 + T y y 1 increases with T and thus P P 0 increases with T. Therefore, the greater T, the greater the interest rate risk of a zero-coupon bond. Does this result hold with coupon bonds? 17

10 Interest Rate Risk Example 2 Consider two bonds, bond 1 and bond 2, with F = $1,000, i = 10% and m = 1. Bond 1 has 5 years to maturity while bond 2 has 10 years to maturity. If, initialls y = 10%, then the price of each bond is $1, Interest Rate Risk Suppose that y suddenly decreases to 5%. Then the price of Bond 1 becomes $1,216, an increase of 1,216 1,000 1,000 = 21.6%, and the price of Bond 2 becomes $1,386, an increase of 1,386 1,000 1,000 = 38.6%. The increase is proportionally greater for the longer-term bond. 19

11 Interest Rate Risk Suppose now that y increases to 15%. Then the price of Bond 1 becomes $832, a decrease of 1, ,000 = 16.8%, and the price of Bond 2 becomes $749, a decrease of 1, ,000 = 25.1%. Again, the decrease is proportionally greater for the longer-term bond. 20 Example 3 Interest Rate Risk Consider now two bonds with the same characteristics except for the coupon rates. More specifically, F = $1,000, T = 10 and m = 1 for both bonds, but Bond 1 has a 5% coupon rate while Bond 2 has a 10% coupon rate. If, initialls y = 10%, then the price of Bond 1 is $693; the price of Bond 2 is $1,

12 Suppose that y decreases to 5%. Interest Rate Risk Then the price of Bond 1 becomes $1,000, an increase of 1, = 44.3%, and the price of Bond 2 becomes $1,386, an increase of 1,386 1,000 1,000 = 38.6%. The increase is proportionally greater for the bond with a lower coupon rate. 22 Interest Rate Risk Suppose now that y increases to 15%. Then the price of Bond 1 becomes $498, a decrease of = 28.1%, and the price of Bond 2 becomes $749, a decrease of 1, ,000 = 25.1%. Again, the decrease is proportionally greater for the bond with a lower coupon rate. 23

13 7A Duration How to compare bonds with different coupon rates and different time to maturity? Starting point: It is easy to compare zero-coupon bonds, in which case interest rate risk increases with time to maturity. If it were possible to find the equivalent zero-coupon bond to any coupon bond, it would then be easy to compare different bonds: We would just have to compare the time to maturity of their equivalent zero-coupon counterparts. 24 7A Duration The time to maturity of a bond s equivalent zero-coupon counterpart is called duration. How to calculate duration? Note that a coupon bond can be viewed as a set of T m zero-coupon bonds. 1. A zero-coupon bond paying if m at time m 1 (in years); 2. A zero-coupon bond paying if m at time m 2 ;. T m. A zero-coupon bond paying F + if m at time T. 25

14 7A Duration Take a 3-year bond with a 10% coupon rate, F = $1,000 and m = 1. This bond can be viewed as a set of three zero-coupon bonds: Bond z 1 ; A zero-coupon bond paying $100 in one year; Bond z 2 ; A zero-coupon bond paying $100 in two years; Bond z 3 ; A zero-coupon bond paying $1,100 in three years. 26 7A Duration If y = 8%, then the price of the coupon bond is P = 100 }{{} 1.08 P z (1.08) 2 }{{} P z2 + 1,100 (1.08) 3 }{{} P z3 = $1,052, i.e. the sum of its zero-coupon bond prices. 27

15 7A Duration Duration, D, is a weighted average of the time to maturity of the zero-coupon bonds composing a bond: D = P z 1 P 1 + P z 2 P 2 + P z 3 P 3 = 100/1.08 1, /(1.08)2 1, ,100/(1.08)3 1,052 3 = 2.74 years. 28 7A Duration More generally, duration is calculated as follows D = 1 m T m n=1 nx n (1 + y/m) n where X n is the cash flow in period n (not necessarily a year here), including the principal when n = T m. The term m 1 in front of T m allows to have a measure in years. n=1 Duration approximates interest rate risk: The greater duration, the greater should interest rate risk be. P, 29

16 7A Duration D = T m n=1[ X n P(1 + y/m) n n m ] where X n P(1 + y/m) n weight on the n th zero-coupon bond, n m time to maturity (in years) of the nth zero-coupon bond. 30 Note that 7A Duration T m n=1 X n P(1 + y/m) n = = T m n=1 T m n=1 T m n=1 X n (1 + y/m) n P X n (1 + y/m) n X n (1 + y/m) n = 1. 31

17 7A Duration Note that the duration of a zero-coupon bond is D = 1 m T m n=1 nx n (1 + y/m) n P = T F (1 + y/m) T P = T. 32 Duration: Example 1 F = $1,000, i = 4%, T = 3, m = 2, y = 8%. (1) (2) (3) (4) (5) Year Cash Flow PV of Flow (PV of Flow)/P (1) (4) ,

18 Duration: Example 2 F = $1,000, i = 6%, T = 3, m = 2, y = 8%. (1) (2) (3) (4) (5) Year Cash Flow PV of Flow (PV of Flow)/P (1) (4) , Duration Hedging A firm can hedge against interest rate risk by matching liabilities with assets. What do we match? Either Duration of assets = Duration of liabilities or Duration of assets Market value of assets = Duration of liabilities Market value of liabilities Using this procedure, a firm can immunize itself against interest rate risk. 35

19 Duration Hedging The IWG Bank Market Value Balance Sheet (in millions of $) Assets Liabilities and Owners Equity Value Duration Value Duration Overnight money 35 0 C&S accounts A/R-backed loans months CD year Inventory loans months LT financing years Industrial loans 40 2 years Equity 100 Mortgages years Total 1,000 Total 1, Duration of IWG s assets: Duration Hedging ,000 = 2.56 years. Duration of IWG s liabilities: = 2.56 years. 37

20 Duration Hedging Is IWG immunized against interest rate risk? ,000 > Both durations being equal, the percentage change in the value of assets following a small change in interest rates is close to the percentage change in the value of liabilities but, since there are more assets than liabilities, the overall change in assets will be greater than the overall change in liabilities. 38 What should the firm do? Duration Hedging 1. Increase the duration of liabilities without changing the duration of assets: Duration of liabilities = , = 2.84 years. 2. Decrease the duration of assets without changing the duration of liabilities: Duration of assets = ,000 = 2.30 years. 39

21 7C Callable Bonds The process of replacing all or part of an issue of outstanding bonds is called bond refunding. It is possible to do so when the issue is callable. The call price is the price at which the firm can repurchase the bond. Why would a firm want to refund its debt issue? To take advantage of lower interest rates. 40 7C Callable Bonds Consider a bond with T years to maturity, a face value F and a coupon rate i. The bond makes annual coupon payments. The actual interest rate is y 0, but is expected to change one year from now. No further changes in y are expected until T. More specifically, the YTM one year from now will be y with probability π, y 1 = y with probability 1 π, where y > y and 0 π 1. 41

22 7C Callable Bonds Let P 1 denote the value of the bond at the beginnig of year 1. Then P 1 = P l = if y P h = if y ( ( y ( ( y ) T 1 ) ) T 1 ) + F ( 1 1+y) T 1 if y = y, + F ( 1 1+y) T 1 if y = y. Note that P h > P l 42 7C Callable Bonds If the bond is not callable, its price as of time 0 is P n = if + πp l + (1 π)p h 1 + y 0. 43

23 7C Callable Bonds Suppose now that the firm can call the bond in period 1 only and at a at a call price CP. The firm will call the bond only if CP < P h, and thus there is no point in setting CP > P h. 44 7C Callable Bonds The value as of time 0 of a callable bond with call price CP is P c = if + πp l + (1 π)cp 1 + y 0. 45

24 7C Callable Bonds Let s compare two bonds, one callable and one non-callable, where i n coupon rate of the non-callable bond; i c coupon rate of the callable bond; CP call price; P n price of the non-callable bond; P c price of the callable bond; y = 10%, y = 6%, and y 0 = 8%. 46 7C Callable Bonds Suppose F = $100 for the two bonds, T = 20, π = 0.5 and CP = $110. Then P n = $100 when i n 7.75%; P c = $100 when i c 8.65%; P l = $88.71 < $110;P h = $ > $110; Thus a = 0.9% premium is needed for the callable bond to be issued at par. 47

Bond Valuation. Lakehead University. Fall 2004

Bond Valuation. Lakehead University. Fall 2004 Bond Valuation Lakehead University Fall 2004 Outline of the Lecture Bonds and Bond Valuation Interest Rate Risk Duration The Call Provision 2 Bonds and Bond Valuation A corporation s long-term debt is