# f515&415p05 - Spot Forward and Par Rates Finance...

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1 Spot, Forward, and Par Rates Finance 515/415

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2 Pricing coupon bonds based on the discount function The yield curve, spot curve, or zero-(coupon bond) curve Forward prices that preclude arbitrage Forward rates and their relationship to spot rates Par rates and their relationship to spot rates Outline
3 Suppose the current date is 0 and a bond pays m cashflows at dates t 1 , t 2 , …, t m , where the cashflows equal C 1 , C 2 , …, C m . This bond can be viewed as a portfolio of m zero-coupon bonds, where the i th zero-coupon bond pays C i at date t i . Moreover, this i th zero-coupon bond equals C i times the value of a zero-coupon bond that pays 1 at date t i . Suppose that we observed the current date 0 prices of these zero-coupon bonds with unit face value, where P (0, t i ) is the date 0 price of the zero-coupon bond that pays 1 at date t i . Then the multiple cashflow bond would be worth Multiple Cashflow Bonds and the Discount Function ( ) 0 1 0, m i i i B C P t = =

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4 The date 0 prices of zero-coupon bonds that pay 1 at maturity: Consider a coupon bond that pays the following cashflows Then the value of this coupon bond equals Example ( ) ( ) ( ) 1 2 3 Maturity Date Current Price 0.5 0,0.5 0.95 1.0 0,1.0 0.93 1.5 0,1.5 0.84 = = = = = = t P t P t P 1 1 2 2 3 3 Payment Date Cashflow 0.5 50 1.0 50 1.5 1050 = = = = = = t C t C t C ( ) 3 0 1 0, 50 0.95 50 0.93 1050 0.84 976 i i i B C P t = = = × + × + × =
5 The prices of zero-coupon bonds with face value of 1, which we denoted as P (0, t ), is referred to as the discount function . Once we know the discount function, we can price multiple cashflow bonds. A discount function must start at 1 and monotonically decline. A typical shape is Discount Function 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 P (0, t ) t , years until maturity

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6 From the discount function, one can compute the yields-to- maturity of the zero-coupon bonds maturing at each future date. It is convenient to work with continuously-compounded yields, so let y (0, t ) be the date 0 yield on the bond that pays 1 at date t : so that Therefore, an equivalent formula for the value of a multiple cashflow bond is Yields-to-Maturity ( ) ( ) 0, 0, y t t P t e = ( ) ( ) 1 0, ln 0, y t P t t = − ª º ¬ ¼ ( ) 0, 0 1 i i m y t t i i B C e = =
7 The current or “spot” yield curve, y (0, t ) = -(1/ t )ln[ P (0, t )], for the previous discount function is The Spot Rate (Yield) Curve 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t , years until maturity y (0, t )

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8 The actual return from holding a bond over a finite period may be distinct from its yield to maturity. Consider the following two strategies for investing \$100 at date 0. Strategy 1: Invest in a two-year bond. At the end of two years, the investment is worth Strategy 2: Invest in a one-year bond. At the the year, reinvest the proceeds in another one-year bond. At the end of two years, the investment is worth Yields and Holding Period Returns ( ) 0,2 2 100 y e × ( ) ( ) 0,1 1 1,2 1 100 × × y y e e
9 Over a two-year horizon, Strategy 1 produces a riskless annualized, continuously-compounded, rate of return of y (0,2).

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