1 Interest Rate Risk Part 2, Convexity

2 Convexity Empirical evidence shows that duration works well in estimating the percent change in value of relatively short- term bonds with relatively small changes in interest rates. The process can be improved by including convexity as a correction factor in the ΔB/B formula

3 Total Percentage Price Change Total percentage change in the price of an asset is equal to Price effect of duration (the first derivative of price yield relationship) + Effect of Convexity(the second derivative) + Higher order effects(higher order derivatives) Higher order effects are relatively small so the entire difference between the total price change and duration can be ascribed to convexity

4 Effect of Convexity on % Chg in Price Two bonds With identical calculated duration for a given level of rates Will experience different actual price changes if the convexity differs Why does convexity differ? Longer the maturity, the greater the convexity Holding yield and duration constant constant, the higher the coupon, the greater is the security's convexity

Total Price Change for a 2 Percentage Point Change in Rates Coupon Bond: 10-Year Maturity 8% Coupon YTM: 6% Calculated modified duration: 7.18 Suppose its YTM increases by 2 percentage points Total % change in price: % Effect of Duration % Effect of Convexity+ 1.25% This is an example of positive convexity where effect of rise in rates is not as large as effect of fall in rates on the percent change in the price of a security What if rates fell 2 percentage points Total price change:+15.81% Effect of Duration:+14.36% Effect of Convexity:+ 1.45% 5 Note: this ignores effects of all higher order derivatives on the price of the security

6 Convexity Positive Convexity Positive convexity: The increase in price stemming from a given fall in a security’s yield is greater than the fall in price stemming from a rise in the rate of the same magnitude Positive convexity has value If two securities have all the same characteristics except for convexity, the one will the greater convexity will have the greater value

7 Positive Convexity Illustrating Effects as YTM Increases

8 Positive Convexity Illustrating Effects as YTM Decreases

9 Convexity – the Formula Note: This formula is from the text. B is the price of the bond and the convexity formula is the second derivative of the bond’s price with respect to interest rate.

Convexity Formula Continued 10 This can be simplified to: C = dollar value of periodic coupon per $100 of face value r = periodic YTM or initial YTM / 2 for semi-annual compounding n = number of periods to maturity (n=2 x years to maturity for semi-annual compounding

Using Convexity In accounting for the effects of convexity on price, the dollar convexity measure must first be converted to a convexity measure, or This is a periodic convexity measure measured in m period per year, which must be converted to an annual measure (measured in years) by 11 Bringing it together: the convexity correction factor changes the ΔB/B formula to: Note: This formula is captured in lines28 and 34 in the accompanying excel sheet

Approximation The annualized convexity measure is approximated by Where VH is the security price at the lower rate VL is the security price at the higher rate V0 is the original price  r is the change in rates 12

Duration, Convexity, and Higher Moments 13 The estimate of the effect of an interest rate change is improved. It is not exactly correct. So, so why go through this trouble to estimate the percent change in the value of a bond? The answer is that in the financial markets duration and convexity are tools to manage bond portfolios. Te math is the same for one bond or one portfolio, so we showed all our examples with one bond to keep the numbers simpler.

14 Convexity Negative Convexity The gain is price from a given fall in the security’s yield is less than the price loss stemming of a rise in rates of the same magnitude Negative convexity stems from options embedded in the security These amount to call options sold by the buyer of the security to the issuer of the security These call options could be Explicit: e.g., call feature of a callable corporate bond Implicit: e.g., ability of individuals to prepay their mortgage or other types of amortizing debt

Graphic Illustration of Negative Convexity 15 Negative Convexity Range Positive Convexity Range