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Duration

# Duration - Duration and Portfolio Immunization Macaulay...

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Duration and Portfolio Immunization

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Macaulay duration The duration of a fixed income instrument is a weighted average of the times that payments (cash flows) are made. The weighting coefficients are the present values of the individual cash flows. where PV ( t ) denotes the present value of the cash flow that occurs at time t . If the present value calculations are based on the bond’s yield, then it is called the Macaulay duration . PV t t PV t t PV t t PV D n n ) ( ) ( ) ( 1 1 0 0 + + + =
Let P denote the price of a bond with m coupon payments per year; also, let y : yield per each coupon payment period, n : number of coupon payment periods F : par value paid at maturity C ~ : coupon amount in each coupon payment Now, n n y F y C y C y C P ) 1 ( ) 1 ( ~ ) 1 ( ~ 1 ~ 2 + + + + + + + + = then P y nF y C n y C y C m y d dP P n n 1 ) 1 ( ) 1 ( ~ ) 1 ( ~ 2 1 ~ 1 1 1 1 1 2 + + + + + + + + + - = λ Note that λ = my .

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modified duration = λ d dP P 1 Macaulay duration = P m n k y C k y nF k n = + + + 1 ) 1 ( ~ ) 1 ( 1 The negativity of indicates that bond price drops as yield increases. Prices of bonds with longer maturities drop more steeply with increase of yield. This is because bonds of longer maturity have longer Macaulay duration: λ d dP P 1 . 1 λ + - y D P P Mac
Example Consider a 7% bond with 3 years to maturity. Assume that the bond is selling at 8% yield. × × A B C D E Year Payment Discount factor 8% Present value = B C Weight = D/Price A E 0.5 3.5 0.962 3.365 0.035 0.017 1.0 3.5 0.925 3.236 0.033 0.033 1.5 3.5 0.889 3.111 0.032 0.048 2.0 3.5 0.855 2.992 0.031 0.061 2.5 3.5 0.822 2.877 0.030 0.074 3.0 103.5 0.79 81.798 0.840 2.520 Sum Price = 97.379 Duration = 2.753 Here, λ = 0.08, m = 2, y = 0.04, n = 6, C = 3.5, F = 100.

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