Economics 135. Bond Pricing and Interest Rates. Professor Kevin D. Salyer. UC Davis. Fall 2009
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1 Economics 135 Bond Pricing and Interest Rates Professor Kevin D. Salyer UC Davis Fall 2009 Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
2 Bond Pricing Formulas - Interest Rates and Bond Prices A bond is a contract - the price is determined by the present discounted values of the promised payments. Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
3 Bond Pricing Formulas - Interest Rates and Bond Prices A bond is a contract - the price is determined by the present discounted values of the promised payments. Some key terms Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
4 Bond Pricing Formulas - Interest Rates and Bond Prices A bond is a contract - the price is determined by the present discounted values of the promised payments. Some key terms Face Value (F ) the amount that the bond pays out at maturity Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
5 Bond Pricing Formulas - Interest Rates and Bond Prices A bond is a contract - the price is determined by the present discounted values of the promised payments. Some key terms Face Value (F ) the amount that the bond pays out at maturity Coupon payment (C t ): the amount received in period t. Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
6 Bond Pricing Formulas - Interest Rates and Bond Prices A bond is a contract - the price is determined by the present discounted values of the promised payments. Some key terms Face Value (F ) the amount that the bond pays out at maturity Coupon payment (C t ): the amount received in period t. Coupon rate (r): the coupon payment expressed as a fraction of the face value: r = C /F Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
7 Bond Pricing Formulas - Interest Rates and Bond Prices A bond is a contract - the price is determined by the present discounted values of the promised payments. Some key terms Face Value (F ) the amount that the bond pays out at maturity Coupon payment (C t ): the amount received in period t. Coupon rate (r): the coupon payment expressed as a fraction of the face value: r = C /F Maturity (N): The length of the bond contract. Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
8 Bond Pricing Formulas - Interest Rates and Bond Prices A bond is a contract - the price is determined by the present discounted values of the promised payments. Some key terms Face Value (F ) the amount that the bond pays out at maturity Coupon payment (C t ): the amount received in period t. Coupon rate (r): the coupon payment expressed as a fraction of the face value: r = C /F Maturity (N): The length of the bond contract. Price (P b ): the price of the bond. Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
9 Bond Pricing Formulas - Interest Rates and Bond Prices A bond is a contract - the price is determined by the present discounted values of the promised payments. Some key terms Face Value (F ) the amount that the bond pays out at maturity Coupon payment (C t ): the amount received in period t. Coupon rate (r): the coupon payment expressed as a fraction of the face value: r = C /F Maturity (N): The length of the bond contract. Price (P b ): the price of the bond. Yield to Maturity (i): the interest rate associated with the bond (always expressed as an annual rate). Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
10 Bond Pricing Formulas - Interest Rates and Bond Prices A bond is a contract - the price is determined by the present discounted values of the promised payments. Some key terms Face Value (F ) the amount that the bond pays out at maturity Coupon payment (C t ): the amount received in period t. Coupon rate (r): the coupon payment expressed as a fraction of the face value: r = C /F Maturity (N): The length of the bond contract. Price (P b ): the price of the bond. Yield to Maturity (i): the interest rate associated with the bond (always expressed as an annual rate). A bond de nes (F, C, N). Given the market price, P b, (and the frequency of coupon payments), this determines i by the present discounted formula. Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
11 Basic Bond Pricing Formulas If the coupon payments are received annually, the bond pricing formula is: P b = = C 1 (1 + i) + C 2 (1 + i) C N (1 + i) N + F (1 + i) N N t=1 C t (1 + i) t + F (1 + i) N Typically, the coupon payment is made semi-annually. With a constant annual coupon payment, C, the formula then becomes P b = or P b = C /2 (1 + i/2) + C /2 (1 + i/2) C /2 (1 + i/2) 2N + F (1 + i/2) 2N 2N t=1 C /2 (1 + i/2) t + F (1 + i/2) 2N A key relationship: Bond prices and interest rates are inversely related!! Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
12 The e ective annual yield in this case is de ned by: (1 + i/2) 2 That is, the yield includes interest on interest = (i/2) 2. If the number of compounding periods (de ned as m) in a year grows, then the e ective annual yield is determined by: (1 + i/m) m Suppose i = 1 (100% interest rate). What is the e ective yield as m!? This is continuous compounding and yields the mysterious number e: e = lim m! (1 + 1/m)m = Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
13 As the number of compounding periods grows, the formula changes accordingly: Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
14 As the number of compounding periods grows, the formula changes accordingly: P b = mn t=1 C /m (1 + i/m) t + F (1 + i/m) mn Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
15 Common types of bonds Coupon bond (as above). Pure discount bond: C t = 0 for all t. The entire amount is received at maturity. Example: 1 year Treasury Bill. P b = F (1 + i) Amortizing Bond: F = 0, C t = C. The face value (or principal) is included in the coupon payment. Example: 4 year Car Loan with monthly payments. P b = 48 t=1 Some more examples on the board. C (1 + i/12) t Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
16 Two Critical Relationships between Bond Prices and Interest Rates 1 Bond Prices and Interest Rates are inversely related. Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
17 Two Critical Relationships between Bond Prices and Interest Rates 1 Bond Prices and Interest Rates are inversely related. 1 This makes sense since bond prices are determined by the PDV of cash ows. Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
18 Two Critical Relationships between Bond Prices and Interest Rates 1 Bond Prices and Interest Rates are inversely related. 1 This makes sense since bond prices are determined by the PDV of cash ows. 2 The greater the maturity, the greater the change in P b for a given change in i. Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
19 Two Critical Relationships between Bond Prices and Interest Rates 1 Bond Prices and Interest Rates are inversely related. 1 This makes sense since bond prices are determined by the PDV of cash ows. 2 The greater the maturity, the greater the change in P b for a given change in i. 1 Cash received in the future is discounted at a greater rate. So a change in i is compounded more times. Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
20 The elasticity of bond prices with respect to interest rate changes Calculating the Duration of a Bond First - a little review of elasticity. Suppose we have a function y = f (x) Recall that the elasticity of y with respect to x is de ned as dy % y % x = y dx x = dy x dx y But note that this is the derivative of the logs 1 d ln y d ln x = y dy dy = dx dx 1 x So - the easy way to calculate elasticities is to take logs (natural) and then take the derivative. x y Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
21 The elasticity of bond prices with respect to interest rate changes Calculating the Duration of a Bond Recall the formula for the price of a pure discount bond: Now take logs: F P b = (1 + i) N ln P b = ln F N ln (1 + i) Now take the derivative with respect to i (note we don t want the percentage change in i but absolute change): d ln P b 1 = N di 1 + i Hence the elasticity of bond prices is appoximately equal to the maturity of the bond. Or, using discrete notation: % P b = N i 1+i Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
22 The elasticity of bond prices with respect to interest rate changes Calculating the Duration of a Bond But for a typical bond, this relationship is not so straightforward. But we can use the following insight: Recall the formula for a coupon bond (with annual payments): P b = C 1 (1 + i) + C 2 (1 + i) C N (1 + i) N + F (1 + i) N Note that each term can be thought of as the price of a pure discount bond for that period.then a coupon bond can be interpreted as a portfolio of pure discount bonds. Furthermore, the elasticity of the coupon bond will be equivalent to a weighted average of the elasticities of the underlying coupon bonds. (Suppose y = x + z. Then y = x + z. Divide both sides by y and rewrite as: y y This elasticity is de ned as Duration. = x y x x + z y z z ) Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
23 The elasticity of bond prices with respect to interest rate changes Calculating the Duration of a Bond The formula for duration of a coupon bond is: D = C (1+i) P b (1) + C (1+i) 2 P b (2) C (1+i) N P b (N) + F (1+i) N P b (N) Then, once we have D calculated, the elasticity of bond prices is given by the direct equivalent to a pure discount bond: i % P b = D 1 + i Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
24 Duration Example Consider a 3-year coupon bond with face value of $700 and coupon rate of 14%. Suppose also that i = 14%. 1 Calculate the duration of the bond. Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
25 Duration Example Consider a 3-year coupon bond with face value of $700 and coupon rate of 14%. Suppose also that i = 14%. 1 Calculate the duration of the bond. 2 Calculate the change in the price of the bond if interest rates go to 15%. Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
26 Duration Example Consider a 3-year coupon bond with face value of $700 and coupon rate of 14%. Suppose also that i = 14%. 1 Calculate the duration of the bond. 2 Calculate the change in the price of the bond if interest rates go to 15%. 1 First, since the interest rate and coupon rate are the same, this bond is selling at par so P b = $700. Duration is given by Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
27 Duration Example Consider a 3-year coupon bond with face value of $700 and coupon rate of 14%. Suppose also that i = 14%. 1 Calculate the duration of the bond. 2 Calculate the change in the price of the bond if interest rates go to 15%. 1 First, since the interest rate and coupon rate are the same, this bond is selling at par so P b = $700. Duration is given by 2 D = 98 (1.14) 700 (1) + 98 (1.14) (2) (1.14) (3) = 2.65 Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
28 Duration Example Consider a 3-year coupon bond with face value of $700 and coupon rate of 14%. Suppose also that i = 14%. 1 Calculate the duration of the bond. 2 Calculate the change in the price of the bond if interest rates go to 15%. 1 First, since the interest rate and coupon rate are the same, this bond is selling at par so P b = $700. Duration is given by 2 D = 98 (1.14) 700 (1) + 98 (1.14) (2) + 3 The percentage change in the price is: % P b = 2.65 (0.01/1.14) = (1.14) (3) = 2.65 Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
29 Duration Example Consider a 3-year coupon bond with face value of $700 and coupon rate of 14%. Suppose also that i = 14%. 1 Calculate the duration of the bond. 2 Calculate the change in the price of the bond if interest rates go to 15%. 1 First, since the interest rate and coupon rate are the same, this bond is selling at par so P b = $700. Duration is given by 2 D = 98 (1.14) 700 (1) + 98 (1.14) (2) + 3 The percentage change in the price is: % P b = 2.65 (0.01/1.14) = (1.14) (3) = So the change in the price is P b = ($700) = $ (If you recalculate P b = $ so not bad). Professor Kevin D. Salyer (UC Davis) Money and Banking Fall / 12
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