Chapter 4

1. Chapter 4 Bond Price Volatility

2. Introduction • Bond 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 bond’s price volatility characteristics. • Quantify price volatility.

3. Price-Yield Relationship - Maturity • Consider two 9% coupon semiannual pay bonds: • Bond A: 5 years to maturity. • Bond B: 25 years to maturity. • The long-term bond price is more sensitive to interest rate changes than the short-term bond price.

4. Price-Yield Relationship - Coupon Rate • Consider three 25 year semiannual pay bonds: • 9%, 6%, and 0% coupon bonds • Notice what happens as yields increase from 6% to 12%:

5. Bond Characteristics That Influence Price Volatility • Maturity: • For a given coupon rate and yield, bonds with longer maturity exhibit greater price volatility when interest rates change. • Why? • Coupon Rate: • For a given maturity and yield, bonds with lower coupon rates exhibit greater price volatility when interest rates change.

6. Shape of the Price-Yield Curve • If we were to graph price-yield changes for bonds we would get something like this: • What do you notice about this graph? • It isn’t linear…it is convex. • It looks like there is more “upside” than “downside” for a given change in yield. Price Yield

7. Price Volatility Properties of Bonds • Properties 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 coupon • For 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.

8. Price Volatility Properties of Bonds • Exhibit 4-3 from Fabozzi text, p. 61 (required yield is 9%):

9. Measures of Bond Price Volatility • Three measures are commonly used in practice: • Price value of a basis point (also called dollar value of an 01) • Yield value of a price change • Duration

10. Price Value of a Basis Point • The 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.

11. Price Value of a Basis Point • We examine the price of six bonds assuming yields are 9%. We then assume 1 bp increase in yields (to 9.01%)

12. Yield Value of a Price Change • Procedure: • 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.

13. Duration • The concept of duration is based on the slope of the price-yield relationship: • What does slope of a curve tell us? • How much the y-axis changes for a small change in the x-axis. • Slope = dP/dy • Duration—tells us how much bond price changes for a given change in yield. • Note: there are different types of duration. Price Yield

14. Two Types of Duration • Modified duration: • Tells us how much a bond’s price changes (in percent) for a given change in yield. • Dollar duration: • Tells us how much a bond’s price changes (in dollars) for a given change in yield. • We will start with modified duration.

15. Deriving Duration • The price of an option-free bond is: • P = bond’s price • C = semiannual coupon payment • M = maturity value (Note: we will assume M = $100) • n = number of semiannual payments (#years  2). • y = one-half the required yield • How do we get dP/dy?

16. Duration, con’t • The first derivative of bond price (P) with respect to yield (y) is: • This tells us the approximate dollar price change of the bond for a small change in yield. • To determine the percentage price change in a bond for a given change in yield we need: Macaulay Duration

17. Duration, con’t • Therefore we get: • Modified duration gives us a bond’s 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:

18. Calculating Duration • Recall that the price of a bond can be expressed as: • Taking the first derivative of P with respect to y and multiplying by 1/P we get:

19. Example • Consider a 25-year 6% coupon bond selling at 70.357 (par value is $100) and priced to yield 9%. (in number of semiannual periods) • To get modified duration in years we divide by 2:

20. Duration • duration is less than (coupon bond) or equal to (zero coupon bond) the term to maturity • all else equal, • the lower the coupon, the larger the duration • the longer the maturity, the larger the duration • the lower the yield, the larger the duration • the longer the duration, the greater the price volatility

21. Properties of Duration • Duration and Maturity: • Duration increases with maturity. • Duration and Coupon: • The lower the coupon the greater the duration. • Earlier we showed that holding all else constant: • The longer the maturity the greater the bond’s price volatility (duration). • The lower the coupon the greater the bond’s price volatility (duration).

22. Properties of Duration, con’t • What is the relationship between duration and yield? • The lower the yield the higher the duration. • Therefore, the lower the yield the higher the bond’s price volatility.

23. Approximating Dollar Price Changes • How do we measure dollar price changes for a given change in yield? • We use dollar duration: approximate price change for 100 bp change in yield. • Recall: • Solve for dP/dy: • Solve for dP:

24. Example of Dollar Duration • A 6% 25-year bond priced to yield 9% at 70.3570. • Dollar duration = 747.2009 • What happens to bond price if yield increases by 100 bp? • A 100 bp increase in yield reduces the bond’s price by $7.47 dollars (per $100 in par value) • This is a symmetric measurement.

25. Example, con’t • Suppose yields increased by 300 bps: • A 300 bp increase in yield reduces the bond’s price by $22.4161 dollars (per $100 in par value) • Again, this is symmetric. • How accurate is this approximation? • As with modified duration, the approximation is good for small yield changes, but not good for large yield changes.

26. Portfolio Duration • The duration of a portfolio of bonds is the weighted average of the durations of the bonds in the portfolio. • Example: • Portfolio duration is: • If all the yields affecting the four bonds change by 100 bps, the value of the portfolio will change by about 5.4%.

27. Price-Yield Relationship

28. Accuracy of Duration • Why is duration more accurate for small changes in yield than for large changes? • Because duration is a linear approximation of a curvilinear (or convex) relation: Price • Duration treats the price/yield relationship as a linear. P3, Actual Error • Error is small for smallDy. P3, Estimated • Error is large for largeDy. • The error is larger for yield decreases. P0 • The error occurs because of convexity. P1 P2, Actual Error P2, Estimated y3 y0 y1 y2 Yield

29. Convexity • Duration 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 can’t capture the appropriate price change when ∆y is large.

30. How Do We Measure Convexity? • Recall a Taylor Series Expansion from Calculus: Divide both sides by P to get percentage price change: • Note: • First term on RHS of (1) is the dollar price change for a given change in yield based on dollar duration. • First term on RHS of (2) is the percentage price change for a given change in yield based on modified duration. • The second term on RHS of (1) and (2) includes the second derivative of the price-yield relationship (this measures convexity)

31. Measuring Convexity • The first derivative measures slope (duration). • The second derivatives measures the change in slope (convexity). • As with duration, there are two convexity measures: • Dollar convexity measure – Dollar price change of a bond due to convexity. • Convexity measure – Percentage price change of a bond due to convexity. • The dollar convexity measure of a bond is: • The percentage convexity measure of a bond:

32. Calculating Convexity • How do we actually get a convexity number? • Start with the simple bond price equation: • Take the second derivative of P with respect to y: • Or using the PV of an annuity equation, we get:

33. Convexity Example • Consider a 25-year 6% coupon bond priced at 70.357 (per $100 of par value) to yield 9%. Find convexity. Note: Convexity is measured in time units of the coupons. • To get convexity in years, divide by m2 (typically m = 2)

34. Price Changes Using Both Duration and Convexity • % price change due to duration: • = -(modified duration)(dy) • % price change due to convexity: • = ½(convexity measure)(dy)2 • Therefore, the percentage price change due to both duration and convexity is:

35. Example • A 25-year 6% bond is priced to yield 9%. • Modified duration = 10.62 • Convexity measure = 182.92 • Suppose the required yield increases by 200 bps (from 9% to 11%). What happens to the price of the bond? • Percentage price change due to duration and convexity

36. Important Question: How Accurate is Our Measure? • If yields increase by 200 bps, how much will the bond’s price actually change? • Note: Duration & convexity provides a better approximation than duration alone. • But duration & convexity together is still just an approximation.

37. Why Is It Still an Approximation? • Recall the “error” in the our Taylor Series expansion? • The “error” includes 3rd, 4th, and higher derivatives: • The more derivatives we include in our equation, the more accurate our measure becomes. • Remember, duration is based on the first derivative and convexity is based on the 2nd derivative.

38. Some Notes On Convexity • Convexity refers to the curvature of the price-yield relationship. • The convexity measure is the quantification of this curvature • Duration is easy to interpret: it is the approximate % change in bond price due to a change in yield. • But how do we interpret convexity? • It’s not straightforward like duration, since convexity is based on the square of yield changes.

39. The Value of Convexity • Suppose we have two bonds with the same duration and the same required yield: • Notice 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. Price Bond B Bond A y0 Yield

40. Properties of Convexity • All 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 bond’s convexity.

41. Approximation Methods • We can approximate the duration and convexity for any bond or more complex instrument using the following: • Where: • P– = price of bond after decreasing yield by a small number of bps. • P+ = price of bond after increasing yield by same small number of bps. • P0 = initial price of bond. • ∆y = change in yield in decimal form.

42. Example of Approximation • Consider a 25-year 6% coupon bond priced at 70.357 to yield 9%. • Increase yield by 10 bps (from 9% to 9.1%): P+ = 69.6164 • Decrease yield by 10 bps (from 9% to 8.9%): P- = 71.1105. • How accurate are these approximations? • Actual duration = 10.62 • Actual convexity = 182.92 • These equations do a fine job approximating duration & convexity.