Slide 1

Chapter 4Bond Price Volatility1IntroductionBond volatility is a result of interest rate volatility:When interest rates go up bond prices go down and vice versa. Goals of the chapter:To understand a bonds price volatility characteristics.Quantify price volatility.2Price-Yield Relationship - MaturityConsider two 9% coupon semiannual pay bonds:Bond A: 5 years to maturity.Bond B: 25 years to maturity.Yield5 Years25 Years6 1,127.95 1,385.95 7 1,083.17 1,234.56 8 1,040.55 1,107.41 9 1,000.00 1,000.00 10 961.39 908.72 11 924.62 830.68 12 889.60 763.57 The long-term bond price is more sensitive to interest rate changes than the short-term bond price.3Price-Yield Relationship - Coupon RateConsider three 25 year semiannual pay bonds: 9%, 6%, and 0% coupon bondsNotice what happens as yields increase from 6% to 12%:Yield9%6%0%9%6%0%6% 1,127.95 1,000.00 228.11 0%0%0%7% 1,083.17 882.72 179.05 -4%-12%-22%8% 1,040.55 785.18 140.71 -8%-21%-38%9% 1,000.00 703.57 110.71 -11%-30%-51%10% 961.39 634.88 87.20 -15%-37%-62%11% 924.62 576.71 68.77 -18%-42%-70%12% 889.60 527.14 54.29 -21%-47%-76%4Bond Characteristics That Influence Price VolatilityMaturity:For a given coupon rate and yield, bonds with longer maturity exhibit greater price volatility when interest rates change.

Coupon Rate:For a given maturity and yield, bonds with lower coupon rates exhibit greater price volatility when interest rates change.5Shape of the Price-Yield CurveIf we were to graph price-yield changes for bonds we would get something like this:YieldPriceWhat do you notice about this graph?It isnt linearit is convex.It looks like there is more upside than downside for a given change in yield.6Price Volatility Properties of BondsProperties of option-free bonds:All bond prices move opposite direction of yields, but the percentage price change is different for each bond, depending on maturity and couponFor very small changes in yield, the percentage price change for a given bond is roughly the same whether yields increase or decrease.For large changes in yield, the percentage price increase is greater than a price decrease, for a given yield change.7Measures of Bond Price VolatilityThree measures are commonly used in practice:Price value of a basis point (also called dollar value of an 01)Yield value of a price changeDuration

8Price Value of a Basis PointThe change in the price of a bond if the required yield changes by 1bp.Recall that small changes in yield produce a similar price change regardless of whether yields increase or decrease.Therefore, the Price Value of a Basis Point is the same for yield increases and decreases.9Yield Value of a Price ChangeProcedure:Calculate YTM.Reduce the bond price by X dollars.Calculate the new YTM.The difference between the YTMnew and YTMold is the yield value of an X dollar price change.10DurationThe concept of duration is based on the slope of the price-yield relationship:YieldPriceWhat does slope of a curve tell us?How much the y-axis changes for a small change in the x-axis. Slope = dP/dyDurationtells us how much bond price changes for a given change in yield.Note: there are different types of duration.11Two Types of DurationModified duration:Tells us how much a bonds price changes (in percent) for a given change in yield.Dollar duration:Tells us how much a bonds price changes (in dollars) for a given change in yield.We will start with modified duration.12Duration, contTherefore we get:

Modified duration gives us a bonds approximate percentage price change for a small (100bp) change in yield.

Duration is measured on a per period basis. For semi-annual cash flows, we adjust the duration to an annual figure by dividing by 2:

13Durationduration is less than (coupon bond) or equal to (zero coupon bond) the term to maturityall else equal,the lower the coupon, the larger the durationthe longer the maturity, the larger the durationthe lower the yield, the larger the durationthe longer the duration, the greater the price volatility

14ConvexityDuration is a good approximation of the price yield-relationship for small changes in y.

For large changes in y duration is a poor approximation.Why? Because the tangent line to the curve cant capture the appropriate price change when y is large.15The Value of ConvexitySuppose we have two bonds with the same duration and the same required yield:YieldPriceBond ABond BNotice bond B is more curved (i.e., convex) than bond A. If yields rise, bond B will fall less than bond A.If yields fall, bond B will rise more than bond A.That is, if yields change from y0, bond B will always be worth more than bond A!Convexity has value! Investors will pay for convexity (accept a lower yield) if large interest rate changes are expected.y016Properties of ConvexityAll option-free bonds have the following properties with regard to convexity.Property 1: As bond yield increases, bond convexity decreases (and vice versa). This is called positive convexity.Property 2:For a given yield and maturity, the lower the coupon the greater a bonds convexity.17