Lecture 11[1]

# Lecture 11[1] - Financial Economics for Insurance and...

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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. Fubini’s 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, yield-to-maturity 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 yield-to-maturity, ρ ( 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 ρ ( t , T ) satisfies P ( t , T ) e ρ ( t , T )( T- t ) = 1 = ⇒ ρ ( 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 ) = ρ ( t , t ) . (1.3) By letting t → T in (1.2) 1 r ( T ) = P t ( T , T ) . (1.4) 1 We apply L’ Hopital’s rule since setting t = T in (1.2) yields a meaningless ratio. Thus by L’ Hopital’s lim t → T ρ ( 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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## This note was uploaded on 08/07/2011 for the course ACTL 3004 at University of New South Wales.

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Lecture 11[1] - Financial Economics for Insurance and...

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