# From convexity yield correction for convexity a more

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from convexity Yield Correction for Convexity ¡ A more exact approximation of price changes is a second-order approximation which involves a measure of convexity ( C ): where, ¦ » ¼ º « ¬ ª . . . ¡ n t t t t t y CF y B C 1 2 2 ) ( ) 1 ( ) 1 ( 1 2 * ) ( 2 1 y C y D B B ' ¡ ¡ . ' ¡ 0 # ' 13

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Example – Calculating Convexity Time (in half years) CF PV of CF (@5%) Weight (W) (t 2 +t)xW 1 40 38.095 .0395 .0790 2 40 36.281 .0376 .2257 3 4 40 1040 sum 34.553 855.611 .0358 .8871 .4299 17.7413 Consider a bond with: F =\$1,000, c =8% ( paid semiannually ), n =2 years, and YTM =10%. Use the following table: ¡ Convexity is computed like duration, as a weighted average of the terms (t 2 +t) (rather than t) divided by (1+y) 2 ¡ Thus, in the above example, it is equal to in semiannual terms, or Example – Calculating Convexity (Continued) 14
Convexity of Two Bonds 0 Change in yield to maturity (%) P e r ce n t a g e ch a n g e i n b o n d p r i ce Bond A Bond B ¡ Bonds A and B have the same duration, but bond A is more convex Duration and Convexity of Callable Bonds 0 Yield Region of positive convexity Region of negative convexity Price-yield curve is below tangent 5% 10% Bond Price Call Price 15

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¡ Passive strategy ¢ Control risk (immunization) ¢ Balance risk and return (Bond-Index Funds) ¡ Active strategy ¢ Trade on interest rate predictions ¢ Trade on market inefficiencies Managing Fixed Income Securities: Basic Strategies ¡ Managers believe that bond prices are fairly set ¡ They do not attempt to beat the market ¡ Bond-Index Funds Track a bond market index to obtain its risk-reward profile ¡ Immunization of interest rate risk ¢ Control (eliminate) bond portfolio risk ¢ Target date immunization ¢ Net worth immunization ¢ Cash flow matching and dedication Passive Management 16
Duration of a Bond Portfolio ¡ The duration of a bond portfolio is equal to the weighted average of the durations of the N bonds in the portfolio: where: D p = portfolio Duration D i = bond i ’s Duration X i = investment weight in bond i ¦ u N i i i p X D D 1 ¦ u . . ' ¡ 0 # ' N i i i p X B P y y D P P 1 : where , 1 ) 1 ( ¡ Let P denote the value of the bond portfolio, then: ¡ We face a market condition, which allows us to achieve a given return over a given investment horizon. ¡ For Example: ¡ we have \$100M to invest ¡ current YTM=9% ¡ investment horizon: H =3 years ¡ Investing in the bond market, we can increase the value of a bond portfolio in 3 years to: Target Date Immunization 17

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¡ Problem - if the yield curve will change tomorrow, we may not be able to achieve our goal (\$129.5M) ¡ Solution - an increase (decrease) in the YTM will generate two opposite elements: ¡ an immediate capital loss (gain) ¡ a future gain (loss) from reinvesting the coupons ¡ We can offset the two , to insure the goal of \$129.5M: ¡ Our bond portfolio will be immunized if the duration of the portfolio ( D p ) is equal to investment horizon ( H =3): Target Date Immunization (Continued) y y H P P .
• Fall '11
• brodt
• Fina 385, duration, Managing Bond Portfolio

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