Lecture07[1]

Lecture07[1] - STAT 2820 Chapter 7 Interest Rate Forwards...

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STAT 2820 Chapter 7 Interest Rate Forwards by K.C. Cheung 7.1 Zero-coupon bonds 7.1.1 A zero-coupon bond is a bond that makes only a single payment at its maturity. The price of a zero-coupon bond quoted at time t , with par value (or maturity value, principal value) 1 and to be purchased at time t 1 and maturing at t 2 , is denoted as P t ( t 1 ,t 2 ). When t = t 1 , we simply write P ( t 1 ,t 2 ). If t < t 1 , then P t ( t 1 ,t 2 ) is the price of the bond to be purchased at time t 1 you are lock in at time t . This is like a forward contract: actual transaction takes place at a future time point, but the price is ﬁxed now. So when t < t 1 we may call P t ( t 1 ,t 2 ) a forward bond price . 7.1.2 We let r t ( t 1 ,t 2 ) be the annual interest rate (compounded annually) from time t 1 to t 2 , prevailing on date t . If t = t 1 , we simply write r ( t 1 ,t 2 ). Remark that buying zero-coupon bond is the same as lending money, selling zero-coupon bond is the same as borrowing money. So r t ( t 1 ,t 2 ) is the lending or borrowing rate for the period [ t 1 ,t 2 ] that one can lock in at time t . 7.1.3 In general, we have P (0 ,n ) = 1 (1 + r (0 ,n )) n 7.1.4 For example, for a 1-year zero-coupon bond with par value 1, if the bond price today ( t = 0) is 0.943396, then P (0 , 1) = 0 . 943396 and r (0 , 1) = 1 P (0 , 1) - 1 = 0 . 06 7.1.5 For a 2-year zero-coupon bond with par value 1, if the bond price today ( t = 0) is 0.881659, then P (0 , 2) = 0 . 881659 and P (0 , 2) = 1 (1 + r (0 , 2)) 2 = r (0 , 2) = 0 . 065 7.1.6 If at time t = 0, we put \$ A in a risk-free n -year deposit, the annual interest rate we earn is r (0 ,n ), then after n years we receive A (1 + r (0 ,n )) n . | {z } r (0 ,n ) per year A (1 + r (0 ,n )) n A 0 n

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STAT 2820 Chapter 7 2 7.1.7 If there is a cash-ﬂow C n at time n , the present value at time 0 is Present value = C n (1 + r (0 ,n )) n = C n · P (0 ,n ) Therefore, P (0 ,n ) can also be thought of as the discount factor for cashﬂow at time n . | {z } r (0 ,n ) per year C n C n · P (0 ,n ) 0 n 7.2 Implied forward rates 7.2.1 Consider the following situation: At time t = 0 we plan to put \$ A at time t 1 in a time deposit for t 2 - t 1 years (i.e. so that principal and interest will be withdrawn at time t 2 ). To most obvious way to do so is: (a) (Unhedged) Do not do anything today. Just wait till time t 1 and deposit \$ A in a time deposit for t 2 - t 1 years. The annual interest rate earned is r t 1 ( t 1 ,t 2 ). So at time t 2 , we receive A (1 + r t 1 ( t 2 ,t 1 )) t 2 - t 1 . The problem of approach (a) is that the interest rate
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This note was uploaded on 07/30/2011 for the course STAT 3801 taught by Professor Kc during the Fall '11 term at HKU.

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Lecture07[1] - STAT 2820 Chapter 7 Interest Rate Forwards...

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