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 usually involves interest-only loans. If, for example, a firm wants to borrow $1,000 for 30 years and the actual interest rate on loans with similar risk characteristics is 12%, then the firm will pay a total of $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. 3
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. 4
Bonds and Bond Valuation With a single interest payment per year, the timing of cash flows to the lender is as follows: Year Interest Principal 0 1 2 3... 29 30 $120 $120 $120 $120 $120 $1,000 With semiannual payments, the timing is: Year Interest Principal 0 0.5 1 1.5 2 2.5 3... 29 29.5 30 $60 $60 $60 $60 $60 $60 $60 $60 $60 $1,000 5
Bonds and Bond Valuation A bond is 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 value is repaid. Number of Payments (m): Number of interest payments in a year. 6
Bonds and Bond Valuation The annual coupon payment of a bond is then C = i F. If the bond makes m payments per year, each coupon payment is C m = C m = if m and there are m T of these payments over the life of the bond. 7
Bonds and Bond Valuation A bond is usually issued at par, i.e. it sells for $F when issued. If, for example, the face value of a bond is $1,000, an investor pays $1,000 for the bond when issued. As time evolves, the return required by buyers of bonds with similar characteristics changes. This required return depends on the market interest rates. Market interest rates determine the yield to maturity of a bond, which is the annual return to an individual buying the bond at its market price and holding it until maturity. 8
Bonds and Bond Valuation The yield to maturity of a bond is an APR, not an EAR. Let y denote the yield to maturity of a bond, which is also the yield to maturity of bonds with similar risk characteristics. The market interest rate of a bond between each coupon payment is then r m = y/m. 9
Bonds and Bond Valuation What is the price of a bond with a face value F = $1,000, a coupon rate i = 12% and a time to maturity T = 30 years if the bond makes annual interest payments and the rate of return on securities with similar characteristics (yield to maturity) is 10%? Year Interest Principal 0 1 2 3... 29 30 $120 $120 $120 $120 $120 $1,000 10
Bonds and Bond Valuation The price of this bond is ( P = 120 1.10 ( ) ) 1 30 1.10 + 1,000 (1.10) 30 = 120 9.4269 + 1,000 0.0573 = $1, 188.53 11
Bonds and Bond Valuation What if the bond makes semiannual coupon payments? Each coupon payment is then 120/2 = $60, the bond makes 2 30 = 60 coupon payments and the rate of return between payments is 10%/2 = 5%. Year Interest Principal 0 0.5 1 1.5 2 2.5 3... 29 29.5 30 $60 $60 $60 $60 $60 $60 $60 $60 $60 $1,000 12
Bonds and Bond Valuation The price of this bond is ( P = 60 1.05 ( ) ) 1 60 1.05 = 1,135.75 + 53.54 = $1, 189.29 + 1,000 (1.05) 60 13
Bonds and Bond Valuation More generally, the price of a bond making m coupon payments per year over T years when the yield to maturity is y is P = C m r m = if/m y/m = if y ( ( ) ) 1 T m 1 1 + r m ( ( 1 1 + ( 1 ) ) T m 1 + y/m ( 1 ) ) T m 1 + y/m + F (1 + r m ) T m + F (1 + y/m) T m F (1 + y/m) T m 14
Bonds and Bond Valuation Note that P > F if i > y, = F if i = y, < F if i < y. 15
Bonds and Bond Valuation 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. 16
Bonds and Bond Valuation Zero-Coupon Bonds A zero-coupon bond is a bond that does not make coupon payments, i.e. it is a bond with a 0% coupon rate. The price of a zero-coupon bond with $1,000 face value and 22 years to maturity when the return on similar bonds is 6% is P = 1,000 = $277.51. (1.06) 22 Clearly, a zero-coupon bond always sells at a discount. 17
Bonds and Bond Valuation A bond is normally issued at par, i.e. bonds with a coupon rate of 12% are issued when the yield to maturity of similar bonds is 12%. If the face value of the bond is $1,000, its price at the time it is issued is $1,000. If, later on, the yield to maturity of similar bonds increases above (decreases below) 12%, then the bond price will be less (more) than $1,000. 18
Bonds and Bond Valuation Knowing the face value F, the coupon rate i, the time to maturity T and the number of coupon payments per year m, we can compute P when y is known and vice versa. Most often, we know the price at which a bond trades and we use the bond features (F, i and T ) to determine its yield to maturity. Most Canadian bonds make semiannual coupon payments. 19
Bonds and Bond Valuation Bond prices are usually quoted as a percentage of face value. If a bond with a face value of $5,000 is quoted at 97.02, this means that the bond price is 97.02% 5,000 = $4,851. If this bond has a coupon rate of 8%, makes semiannual payments and has 21 years to maturity, can we find its yield to maturity? 20
Bonds and Bond Valuation The price of the bond is ( 4,851 = 200 1 y/2 ( 1 ) ) 42 1 + y/2 + 5,000 (1 + y/2) 42. What do we know about y? It has to be more than 8% since the bond is selling at a discount. The yield is 8.29%. 21
Interest Rate Risk A bond trader may be interested in two types of gains: Interest income Capital gain arising from an increase in the bond price The risk associated with bond price changes arising from changes in market interest rates is called interest rate risk. 22
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 sensitivity of a bond price to a change in interest rate is defined as P 1 P 0 P 0 = P P 0, i.e. the percentage increase or decrease in the bond price. 23
Interest Rate Risk The sensitivity of a bond price to changes in y depends on the bond s characteristics. With respect to i and 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; 24
Interest Rate Risk Take, for instance, a zero-coupon bond with a face value F and T years to maturity. The price of such a bond is P = F (1 + y) T. If y is initially y 0 and then cahnges to y 1, the relative change in the bond price is (note that P > 0) F P F (1+y 1 ) = T (1+y 0 ) T P 0 = F (1+y 0 ) T ( ) 1 + T y0 1 1 + y 1, 25
Interest Rate Risk If y 1 < y 0, then P P 0 = ( ) 1 + T y0 1 1 + y 1 = ( ) 1 + T y0 1, 1 + y 1 which increases as T increases. If y 1 > y 0, then P P 0 = ( ) 1 + T y0 1 1 + y 1 = 1 ( ) 1 + T y0, 1 + y 1 which also increases as T increases. 26
Interest Rate Risk Therefore, the greater T, the greater the interest rate risk of a zero-coupon bond. Does this result hold with coupon bonds? 27
Interest Rate Risk 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, initially y = 10%, then the price of each bond is $1,000. 28
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 percentage change is greater for the longer-term bond. 29
Interest Rate Risk Suppose now that y increases to 15%. Then the price of Bond 1 becomes $832, a decrease of 1,000 832 1,000 = 16.8%, and the price of Bond 2 becomes $749, a decrease of 1,000 749 1,000 = 25.1%. Again, the percentage change is greater for the longer-term bond. 30
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, initially y = 10%, then the price of Bond 1 is $693; the price of Bond 2 is $1,000. 31
Interest Rate Risk Suppose that y decreases to 5%. Then the price of Bond 1 becomes $1,000, an increase of 1,000 693 693 = 44.3%, and the price of Bond 2 becomes $1,386, an increase of 1,386 1,000 1,000 = 38.6%. The percentage change is greater for the bond with a lower coupon rate. 32
Interest Rate Risk Suppose now that y increases to 15%. Then the price of Bond 1 becomes $498, a decrease of 693 498 693 = 28.1%, and the price of Bond 2 becomes $749, a decrease of 1,000 749 1,000 = 25.1%. Again, the percentage change is greater for the bond with a lower coupon rate. 33
Duration How to compare bonds with different coupon rates and different time to maturity? It is easy to compare zero-coupon bonds: Interest rate risk of a zero-coupon bond is simply its time to maturity. Thus translating coupon bonds into equivalent zero-coupon bonds should facilitate interest rate risk comparisons. 34
Duration Duration is a measure of the interest rate sensitivity of a bond s market price taking into consideration its coupon rate and time to maturity. The duration of a bond can be seen as the the time to maturity of the bond s equivalent zero-coupon bond. Duration is measured in units of time. 35
Duration How to calculate 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. 36
Duration If y = 8%, then the price of the coupon bond is P = 100 }{{} 1.08 P z1 + 100 (1.08) 2 }{{} P z2 + 1,100 (1.08) 3 }{{} P z3 = $1,052, i.e. the sum of its zero-coupon bond prices. 37
Duration Duration, D, is the average time to maturity of all the zero-coupon bonds of a bond: D = P z 1 P 1 + P z 2 P 2 + P z 3 P 3 = 100/1.08 1,052 1 + 100/(1.08)2 1,052 = 2.74 years. 2 + 1,100/(1.08)3 1,052 3 This is a weighted average, the weight on each payment time being given by the present value of the payment divided by the bond price. 38
Duration More generally, a coupon bond can be viewed as a set of T m zero-coupon bonds: A zero-coupon bond paying if m at time 1 m (in years) A zero-coupon bond paying if m at time m 2. A zero-coupon bond paying F + if m at time T 39
Duration The general equation for duration is T m D = n=1[ Xn /(1 + y/m) n P n m ] where X n is the cash flow at time n, including the principal when n = T m. Duration approximates interest rate risk: The greater duration, the greater interest rate risk should be. 40
Duration Note that T m n=1 X n /(1 + y/m) n P = = 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. 41
Duration The duration of a zero-coupon bond is then T m D = n=1[ Xn /(1 + y/m) n n ] F/(1 + y)t = P m F/(1 + y) T T = T, and the duration of a coupon bond is T m (if/m)/(1 + y/m) n D = n=1[ n P m ] + F/(1 + y/m)t m P T. 42
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) 0.5 20 19.23 0.0215 0.0107 1.0 20 18.49 0.0207 0.0207 1.5 20 17.78 0.0199 0.0298 2.0 20 17.10 0.0191 0.0382 2.5 20 16.44 0.0184 0.0459 3.0 1,020 806.12 0.9005 2.7016 895.16 2.8469 43
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) 0.5 30 28.85 0.0304 0.0152 1.0 30 27.74 0.0293 0.0293 1.5 30 26.67 0.0281 0.0422 2.0 30 25.64 0.0271 0.0541 2.5 30 24.66 0.0260 0.0651 3.0 1,030 814.02 0.8591 2.5772 947.58 2.7831 44
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. 45
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 400 0 A/R-backed loans 500 3 months CD 300 1 year Inventory loans 275 6 months LT financing 200 10 years Industrial loans 40 2 years Equity 100 Mortgages 150 14.8 years Total 1,000 Total 1,000 46
Duration Hedging Duration of IWG s assets: 35 0 + 500 1 4 + 275 1 2 + 40 2 + 150 14.8 1,000 = 2.56 years. Duration of IWG s liabilities: 400 0 + 300 1 + 200 10 900 = 2.56 years. 47
Duration Hedging Is IWG immunized against interest rate risk? 2.56 1,000 > 2.56 900. 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. 48
Duration Hedging What should the firm do? 1. Increase the duration of liabilities without changing the duration of assets: Duration of liabilities = 2.56 1,000 900 = 2.84 years. 2. Decrease the duration of assets without changing the duration of liabilities: Duration of assets = 2.56 900 1,000 = 2.30 years. 49
The Call Provision A bond may be issued with a call provision. That is, the issuer has the right to repurchase the bond issue before its expiration. The price at which the bonds can be repurchased is called the call price. The call price is normally given by the bond s face value plus a call premium. 50
The Call Provision Suppose, for example, that Company XYZ has a bond issue with 22 years until maturity, $1,000 face value and a coupon rate of 8%. The bonds make semiannual payments. If the yield to maturity is 6%, the price of each of these bonds is ( P = 80 ( ) ) 1 44 1 + 1,000 = $1,242.54..06 1.03 (1.03) 44 51
The Call Provision If the firm had the right to repurchase the bonds for $1,150 each, should it do it (bondholders would be obliged to sell at that price)? What if it had the right to repurchase each bond for $1,300? A call provision on a bond is an option that gives the issuer (the firm) the right to repurchase the issue at a pre-specified price (the call price), usually after a pre-specified date. When the issuer exercises its right, bondholders have the obligation to sell at the pre-specified price. It is a contractual arrangement. 52
The Call Provision Clearly, a call provision and the size of the call price on a bond will affect its yield to maturity or the coupon rate needed for the bond to be sold at par. The idea behind a call provision is that, if interest rates fall in the future, the issuer will be able take advantage of the low rates by repurchasing its high-interest-rate bonds and replacing them with low-rate bonds. 53
The Call Provision Example Company XYZ wants to raise $1M by issuing $1,000-bonds with 25 years to maturity. The current yield to maturity for similar bonds is 10%. The bonds will make annual coupon payments. Without a call provision, the coupon rate has to be 10% for these bonds to be issued at par (i.e. $1,000 each). 54
The Call Provision Example XYZ, however, wants to be able to replace the bonds with low-rate ones if interest rates fall before the bonds mature. Suppose then that the bonds are issued with a call provision allowing XYZ to repurchase each bond for $1,200 one year from now. If XYZ does not exercise its next year, then it won t able to repurchase the bonds at any other time in the future. This is not realistic, it is for expositional simplicity only. 55
The Call Provision Example If the bonds have a coupon rate of 10% and the yield to maturity falls to 7% in one year, will XYZ repurchase the issue? If y = 7% one year from now, the value of each bond will be ( P = 100 ( ) ) 1 24 1 + 1,000 = $1,344.08..07 1.07 (1.07) 24 56
The Call Provision Example If the firm repurchases each bond for $1,200, its profit per bond is 1,344.08 1,200.00 = $144.08 and thus it should do it. 57
The Call Provision Example Put differently, suppose that, if repurchasing the issue, XYZ replaces it by a non-callable issue with a coupon rate equal to the market rate. If, for instance, y = 7% in one year, replacing the 10% callable bonds with 7% non-callable bonds would save the company 100 70 = $30 per bond in interest each year. 58
The Call Provision Example The present value of the interest savings is ( ( ) ) 30 1 24 1 = $344.08..07 1.07 This $344.08 is the cash inflow from the repurchase. 59
The Call Provision Example Repurchasing the bonds, however, creates a cash outflow of 1,200 1,000 = $200 per bond. That is, $1,200 has to be spent to repurchase an old bond and each new bond provides $1,000. The net cash flow from the operation is then 344.08 200.00 = $144.08. 60
The Call Provision Example How much would you pay for XYZ s callable bond today if you believed that next year s yield to maturity had 50% chance to be 7% and 50% chance to be 13%? Today s yield is 10%. If y = 7%, XYZ repurchases the bond and you get $1,200. If y = 13% XYZ does not repurchase and the bond is worth ( ( ) ) 100 1 24 1 + 1,000 = $781.51..13 1.13 (1.13) 24 In each case, you also receive $100 after one year. 61
The Call Provision Example What you would be willing to pay for a callable bond is the present value of its expected cash flow in one year, which is PV =.5 1,200.00 +.5 781.51 + 100.00 1.10 = $991.60. That is, you would pay less than $1,000 even though the coupon rate is equal to the market rate. Why? Because the call feature penalizes you: It prevents you from receiving high interest payments when interest rates are low. 62
The Call Provision Example At what coupon rate would XYZ be able to issue its callable bonds at par? Assume that the yield to maturity one year from now has 50% chance of being 7% and 50% chance of being 13% and each bond can be repurchased at a call price of $1,200 one year from. If not repurchased in one year, the bonds become regular bonds (they cannot be repurchased at a later date). 63
The Call Provision Example Let C denote the coupon payment on XYZ s callable bonds. The expected payoff (EP) of such a bond in one year, assuming it is repurchased if the market rate falls to 7%, is EP =.5 1,200 +.5 ( C.13 ( 1 = 600 + 3.6414C + 26.6126 + C = 626.6126 + 4.6414C. ( ) ) 1 24 + 1,000 1.13 (1.13) 24 ) + C 64
The Call Provision Example If the market rate is currently 10%, C has to be such that 1,000 = 626.6126 + 4.6414C 1.10 which means a coupon rate of 10.199%. C = 101.99, 65
The Call Provision Example To be issued at par, a callable bond s coupon rate must be higher than the actual return on non-callable bonds with similar characteristics. Purchasers of callable bonds have to be compensated for the possible loss in interest payments. 66