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Unformatted text preview: Financial Economics for Insurance and Superannuation: Week 11 Interest Rate Term Structure Models October 6, 2010 1 / 18 Summary of Lecture Snapshot of forward rate curves. Interest rate dynamics. Fubinis Theorem HJM  forward rate dynamics. From forward rate dynamics to spot rate dynamics. From forward rate dynamics to bond price dynamics. ACTL3004: Week 11 2 Modelling Interest Rates Traditional Approach So far we have used constant interest rates eg r = 4 % in modelling. Problems: 1 This implies that interest rates do not change. 2 This implies that the term structure is flat. Both are clearly unrealistic assumptions. ACTL3004: Week 11 3 Types of Yield Curves Upward sloping yield curve rates of longer periods tend to be greater than those of shorter periods. Downward slopping yield curve rates of longer periods are smaller than those of shorter periods. Flat yield curve relatively small differences in the rates for differing investment periods. ACTL3004: Week 11 4 Modelling Interest Rate Dynamics We shall clarify the relationship between interest rates, bond prices, yieldtomaturity and forward rates. Let: 1 P ( t , T ) denote the price at time t of a pure discount bond paying $ 1 at time T . 2 r ( t ) be the instantaneous interest rate agreed at time t for borrowing starting at t . 3 f ( t , T ) the instantaneous interest agreed at t for borrowing starting at T . The yieldtomaturity, ( t , T ) is the continously compounded rate of return causing the bond price to satisfy the maturity condition P ( T , T ) = 1 , (1.1) that is ( t , T ) satisfies P ( t , T ) e ( t , T )( T t ) = 1 = ( t , T ) = ln P ( t , T ) T t (1.2) ACTL3004: Week 11 5 The Instantaneous Spot Rate The instantaneous spot rate of interest, r ( t ) , is the yield on the currently maturing bond, that is r ( t ) = ( t , t ) . (1.3) By letting t T in (1.2) 1 r ( T ) = P t ( T , T ) . (1.4) 1 We apply L Hopitals rule since setting t = T in (1.2) yields a meaningless ratio. Thus by L Hopitals lim t T ( t , T ) = lim t T t ln P ( t , T ) t ( T t ) = lim t T 1 P ( t , T ) P t ( T t ) 1 = lim t T P t ( t , T ) P ( t , T ) = P t ( t , T ) ACTL3004: Week 11 6 The Forward Rate The forward rate arises when we consider an investor who holds a bond maturing at T 1 and asking: What return he/she would earn between T 1 and T 2 ( > T 1 ) , if he/she contracted now at time t ....
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