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# Chapter5 - FIXED-INCOME SECURITIES Chapter 5 Hedging...

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Chapter 5 Hedging Interest-Rate Risk with Duration FIXED-INCOME SECURITIES

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Outline Pricing and Hedging Pricing certain cash-flows Interest rate risk Hedging principles Duration-Based Hedging Techniques Definition of duration Properties of duration Hedging with duration
Pricing and Hedging Motivation Fixed-income products can pay either Fixed cash-flows (e.g., fixed-rate Treasury coupon bond) Random cash-flows: depend on the future evolution of interest rates (e.g., floating rate note) or other variables (prepayment rate on a mortgage pool) Objective for this chapter Hedge the value of a portfolio of fixed cash-flows Valuation and hedging of random cash-flow is a somewhat more complex task Leave it for later

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Pricing and Hedging Notation B(t,T) : price at date t of a unit discount bond paying off \$1 at date T (« discount factor ») R a (t, θ ) : zero coupon rate or pure discount rate, or yield-to-maturity on a zero-coupon bond with maturity date t + θ θ a θ t R θ t t B )) , ( 1 ( 1 ) , ( + = + ( 29 ) , ( ln 1 ) , ( θ t t B θ θ t R + - = ( 29 ) , ( exp ) , ( θ t R θ θ t t B × - = + R(t, θ ) : continuously compounded pure discount rate with maturity t + θ : Equivalently,
The value at date t ( V t ) of a bond paying cash-flows F(i) is given by: = + × = + = = × = = 105 100 100 % 5 5 100 % 5 N cN F cN F m i Example: \$100 bond with a 5% coupon Therefore, the value is a function of time and interest rates Value changes as interest rates fluctuate [ ] = = + = + = m i i a i m i i i t R F i t t B F t V 1 1 ) , ( 1 ) , ( ) ( Pricing and Hedging Pricing Certain Cash-Flows

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Example Assume today a flat structure of interest rates R a ( 0 , θ ) = 10% for all θ Bond with 10 years maturity, coupon rate = 10% Price: \$100 If the term structure shifts up to 12% (parallel shift) Bond price : \$88.7 Capital loss: \$11.3, or 11.3%
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Chapter5 - FIXED-INCOME SECURITIES Chapter 5 Hedging...

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