1 1 1 1 w t duration modified duration if we want a

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1 ) 1 ( 1 1 w t
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Duration – Modified Duration If we want a more direct measure of the relationship between changes in bond prices and interest rates, we can use Modified Duration, defined as So D * measures the sensitivity of the % change in bond price to changes in yield ) 1 ( * y D D y D P P *
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Calculating the Duration of Two Bonds
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Duration of Bonds with Different Features
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Duration Rules 1. Duration is shorter than maturity for all bonds except zero coupon bonds. Duration of a zero-coupon bond equals maturity. 2. Holding time to maturity and YTM constant, duration is higher when coupons are lower. 3. Holding coupon and YTM constant, duration generally (but not always) increases with time to maturity. 4. Holding coupon and time to maturity constant, duration is higher when YTM is lower. 5. Duration of a perpetuity is (1+y)/y.
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Bond Durations (Yield to Maturity = 8% APR; Semiannual Coupons)
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Duration is Additive The duration of a portfolio of securities is the weighted average of the durations of the individual securities with the weights reflecting the proportion invested in each. Example: Let 25% of a portfolio be invested in a bond with a duration of 5 and let 75% of the portfolio be invested in a bond with a duration of 10. D p = (0.25 * 5) + (0.75 * 10) = 8.75 years
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Why is Duration a Big Deal? Simple summary statistic of effective average maturity Measures sensitivity of bond price to interest rate changes Measure of bond price volatility Measure of interest-rate risk Useful in the management of risk You can match the duration of assets and liabilities Or hedge the interest rate sensitivity of an investment
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Example Consider a 3-year 10% coupon bond selling at $1078.7 to yield 7%. Coupon payments are made annually. What’s the duration of this bond? If yields increase to 7.10%, how does the bond price change (duration rule & PV formula)?
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Example Modified duration of this bond: If yields increase to 7.10%, how does the bond price change? The percentage price change of this bond is given by:
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Example What is the predicted change in dollar terms? New predicted price: $ Actual dollar price (using PV formula): N=3; PMT=100; FV=1000; I/Y =7.1; PV=$1075.966
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Duration and Interest Rate Sensitivity Example: A 30-year bond that pays an annual coupon of 8%. The current YTM is 9%. Then the duration should be 11.37 years. The current bond price is $897.26. If YTM changes to 9.1%, we predict that the bond price should change by: P = D P y/(1+y) = 11.37     /1.09 = $9.36 (Decrease) When using the annuity formula, we find the bond price equals $887.98. The difference of $9.28 is again quite close to the one predicted by duration formula.
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Duration is a Local Concept Suppose that in the last example the YTM changed to 10%. What’s the price change predicted by the duration formula? P = What’s the price change predicted by the annuity formula?
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