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Ch 9 fully revised 2011

# Ch 9 fully revised 2011 - 1 Chapter 9 Interest Rate Risk II...

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Chapter 9 Interest Rate Risk II 1

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Overview This chapter discusses a market value-based model for assessing and managing interest rate risk: Duration Computation of duration Economic interpretation Immunization using duration Repricing model (discussed in the previous chapter) is used by small banks, whereas the duration model is used by large banks and is a much better valuation model. 2
Macaulay Duration Duration is the weighted average time to maturity using the relative present values of the cash flows as weights. Duration is a more comprehensive measure than maturity since it takes into account the time of arrival of all cash flows as well as maturity. Takes into account all the coupon payments as well as the repayment of principals at maturity. The units of duration are years. 3

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Some Properties of Macaulay Duration As we will also show the duration is equal to the elasticity of bond price with respect to the interest rate. Also equal to the times takes to recover the initial investment on a bond. 4
Duration The duration of any fixed-income security that pays interest annually is given by: D = Σ n t=1 [CF t • t/(1+R) t ] / Σ n t=1 [CF t /(1+R) t ] => D = Σ n t=1 [PV t • t] / Σ n t=1 [PV t ] D = duration measured in years t = number of periods in the future CF t = cash flow received at the end of period t n = last period in which cash-flow is received R = is the annual yield or current level of interest rates in the market PV t = present value of the cash flow from period t 5

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Duration The duration of any fixed-income security that pays interest semi-annually is given by: D = Σ n t=1/2 [CF t • t/(1+R/2) 2t ] / Σ n t=1/2 [CF t /(1+R/2) 2t ] => D = Σ n t=1/2 [PV t • t] / Σ n t=1/2 [PV t ], D = duration measured in years t = number of periods in the future CF t = cash flow received at the end of period t n = last period in which cash-flow is received R = is the annual yield or current level of interest rates in the market 6
Duration Since in an efficient market the price of a bond must equal the present value of all its cash flows, we can state the annual duration formula as follows: D =[Σ n t=1 (t × Present Value of CF t )]/ Price The numerator is equal to the PV of each cash flow multiplied or weighted by the length of time required to receive the cash flow. The denominator is the sum of the present value of all payments which should equal Price in an efficient market. 7

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Examples 1. Zero-coupon Bond For a zero coupon bond, duration equals maturity since 100% of its present value is generated by the payment of the face value at maturity. Derivation:
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Ch 9 fully revised 2011 - 1 Chapter 9 Interest Rate Risk II...

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