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Chapter 4

Chapter 4. Bond Price Volatility Chapter Pages 58-85,89-91. 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.

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Chapter 4

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  1. Chapter 4 Bond Price Volatility Chapter Pages 58-85,89-91

  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. Review of Price-Yield Relationships • 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. Review of Price-Yield Relationships • 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 • Coupon Rate: • For a given maturity and yield, bonds with lower coupon rates exhibit greater price volatility when interest rates change. • Why? • Maturity: • For a given coupon rate and yield, bonds with longer maturity exhibit greater price volatility when interest rates change. • Why? • Note: The higher the yield on the bond, the lower its volatility.

  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. Quick Review: Why Do Yields Change? • The required return on any security equals: • r = rreal + Expected Inflation + RP • Yields can change for three reasons: • Change in the real rate—compensation for deferring consumption. • Change in expected inflation—i.e., erosion of purchasing power (important). • Change in risk—e.g., credit risk, liquidity risk, etc.

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

  9. 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.

  10. 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

  11. Price Value of a Basis Point • Change in the dollar price of the bond if the required yield changes by 1 bp. • 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.

  12. Price Value of a Basis Point - pg 63 • We examine the price of six bonds assuming yields are 9%. We then assume 1 bp increase in yields (to 9.01%)

  13. 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 YTMinitial is the yield value of an X dollar price change.

  14. 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

  15. 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.

  16. 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?

  17. 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 (called modified duration) we need: Macaulay Duration

  18. Duration, con’t • Therefore we get: • Modified duration gives us a bond’s approximate percentage price change for a small change in yield. • The negative sign reflects the inverse relation between bond price and yield. • Duration is measured over the time horizon of the m periodic CFs that occur during a year (typically m = 2). To get an annual duration:

  19. 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:

  20. 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: (what is Macaulay duration?)

  21. Properties of Duration • Duration and Maturity: • Duration increases with maturity. • Coupon bonds: duration < maturity. • Zeros: Macaulay duration = maturity Modified duration < maturity. • Duration and Coupon: • The lower the coupon the greater the duration (exception is long-maturity deep-discount bonds) • Earlier we showed that holding all else constant: • The longer the maturity the greater the bond’s price volatility. • The lower the coupon the greater the bond’s price volatility. • So, the greater a bond’s duration, the greater its volatility: • So duration is a measure of a bond’s volatility.

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

  23. Duration In Action! • Recall: • Solve for dP/P (the % price change): Formula 4.11 • We can use this to approximate the % price change in a bond for a given change in yield. • Example: Consider the 25-year 6% bond priced at 70.3570 to yield 9%. Modified duration = 10.62. • By how much will the bond price change (in percentage terms) if yields increase from 9% to 9.10%?

  24. Solution • Using our formula: • Here, y is changing from 0.09 to 0.091 so dy = +0.001: • Thus, a 10 bp increase in yield will result (approximately) in a 1.06% decline in bond price. • Note this effect is symmetric: • A 10 bp decline in yield (from 9% to 8.90%) result in a 1.06% price increase.

  25. One More Example • Assume the yield increases by 300 bps. (or –31.86%) • Likewise, a 300 bps decline in yield will change the bond’s price by +31.86% • Are these approximations accurate?

  26. Accuracy of Duration (Exhibit 4-3) • Problems with duration: • It assumes symmetric changes in bond price (not true in reality). • The greater the yield change the larger the approximation error. • Duration works well for small yield changes but is problematic for large yield changes.

  27. Approximating Dollar Changes • How do we measure dollar price changes for a given change in yield? • Recall: • Solve for dP/dy: (This is called Dollar Duration) • Solve for dP:

  28. Consider Previous Example • A 6% 25-year bond priced to yield 9% at 70.3570. • Dollar duration = 747.2009 (= 10.62 x 70.3570) • What happens to bond price if yield increases by 1 bp? • A 1 bp increase in yield reduces the bond’s price by $0.0747 dollars (per $100 of face value) • If an investor had $1,000,000 in face value of the bond, a 1 bp increase in yield would reduce the value of the holdings by $747. • This is a symmetric measurement.

  29. Example, con’t • Suppose yields increased by 300 bps: • A 300 bp increase in yield reduces the bond’s price by $22.42 dollars (per $100 in par value) • A $1,000,000 face value in bond holding would decline in value by $224,161 if the yield were to increase by 300 bps. • 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.

  30. 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

  31. 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 yields affecting all bonds change by 100 bps, the value of the portfolio will change by about 5.4%.

  32. 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. • Also keep in mind that there is a different duration for every different yield for a bond. • This means each time we get a new yield, we need to calculate a new duration.

  33. Measuring Convexity • The first derivative measures slope (duration). • The second derivative 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 convexity measure of a bond:

  34. Measuring Convexity • Now we can measure the dollar price change of a bond due to convexity: • The percentage price change of a bond due to convexity:

  35. 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:

  36. 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)

  37. 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:

  38. 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?

  39. 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.

  40. 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.

  41. 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

  42. 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. • Property 3: • For a given yield and modified duration, the lower the coupon the smaller the convexity (I disagree with this property – possible error)

  43. Additional Concerns with Duration • We know duration ignores convexity and may not be appropriate when measuring price volatility. • However, there are other concerns to address. • Notice that duration is based on the simple bond pricing formula: • This formula assumes that yields for all maturities are the same (i.e., flat yield curve) and that all yield curve shifts are parallel. • This is not true in general! • Recall we can view a bond as a package of zeros, each with it’s own yield. • We also know that the yield curve usually does not shift in a parallel fashion. • Our discussion of duration applies only to option-free bonds.

  44. Duration as an Alternative Measure of “Maturity” • It is popular to interpret duration as the “weighted average” life of a bond. • This is true only with very simple bonds and is not true in general…be careful. • For example, there are 20 year bonds with durations greater than 20 years! • Obviously the interpretation as weighted average life does not hold.

  45. Approximation Methods • We can the approximate 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.

  46. 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.

  47. Additional Series Of Slides: • Additional Slides on Duration & Convexity • Used by Dr. Shaffer in MBA derivatives class • Additional explanations & clarifications • B is used for P for notation

  48. What is Duration? • Measures how long, on average, it takes to receive the cash flows from a bond: • Is a weighted average of the “maturities” of a bond’s cash flows. • Recall, a bond is a package of zero-coupon bonds: • Duration is the weighted average maturity of all of those zero-coupon bonds. Weight given to the ith cash flow (ith zero coupon bond) Maturity of the ith cash flow (ith zero-coupon bond)

  49. Duration • What is the weight, wi, given to each cash flow? • The percentage contribution that each cash flow makes to the value of the bond. • The greater the impact a cash flow has on a bond’s value, the greater the weight assigned to that cash flow: • Example: • Consider two 5-year bonds, identical in every respect except the order in which the cash flows are received. • (for familiarity, we use discrete compounding).

  50. Duration • The bond cash flows look like: Both have the same maturity • But, which bond has less interest rate risk? • Why?

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