Bond Portfolio Management Strategies • What are theoretical spot rates and forward rates and how do we compute them? • When the bond’s yield changes, what characteristics of a bond cause differential price changes for individual bonds? • What is modified duration and what is the relationship between a bond’s modified duration and its volatility?
Bond Portfolio Management Strategies • What is the convexity for a bond, how do you compute it, and what factors affect it? • Under what conditions is it necessary to consider both modified duration and convexity when estimating a bond’s price volatility?
Theoretical spot rates • We have seen that using STRIPS we can determine the spot rate for a particular maturity. • However, the theoretical spot rates may be slightly different from those observed in STRIPS because the stripped securities are not as liquid as the current Treasury issues.
Theoretical spot rates • We can compute a set of theoretical spot rates through a process referred to as boot-strapping. • With this process, we assume that the value of a Treasury coupon security should equal the value of a package of zero coupon securities that duplicates the coupon bond’s cash flows.
Forward rates • Forward rates represent the market’s expectation of future short-term rates. • For example, the yield on a 6-month Treasury bill six months from now would be a forward rate. • Given the current rate for the 6-month and 1-year T-bills, we can extrapolate this forward rate.
Interest Rate Sensitivity • Interest rate sensitivity is the amount of bond price change for a given change in yield. • This sensitivity is a function of: • Coupon rate • Maturity • Direction and level of yield change.
Trading strategies based on interest rate sensitivity • If you expect a decline (increase) in interest rates, you want a portfolio of bonds with maximum (minimum) interest rate sensitivity. • Duration measures provide composite measures of interest rate sensitivity based on coupon and maturity.
Macaulay Duration Measure • The Macaulay Duration can be calculated as: • Where t =time period in which the coupon or principal payment occurs Ct= interest or principal payment that occurs in period t
Characteristics of Macaulay Duration • Duration of a bond with coupons is always less than its term to maturity because duration gives weight to these interim payments • A zero-coupon bond’s duration equals its maturity • There is an inverse relationship between duration and coupon • There is a positive relationship between term to maturity and duration, but duration increases at a decreasing rate with maturity • There is an inverse relationship between YTM and duration
Determining interest rate sensitivity • An adjustment of Macaulay duration called modified duration can be used to approximate the bond price change to changes in yield. • Where: m = number of payments a year i = yield to maturity (YTM)
Modified Duration and Bond Price Volatility • Bond price movements will vary proportionally with modified duration for small changes in yields. • We can estimate the change in bond prices as: Where: P = beginning price for the bond Dmod = the modified duration of the bond i = yield change
Trading Strategies Using Modified Duration • Longest-duration security provides the maximum price variation • If you expect a decline in interest rates, increase the average modified duration of your bond portfolio to experience maximum price volatility • If you expect an increase in interest rates, reduce the average modified duration to minimize your price decline • Note that the modified duration of your portfolio is the market-value-weighted average of the modified durations of the individual bonds in the portfolio
Bond Convexity • Modified duration is a linear approximation of bond price change for small changes in market yields • However, price changes are not linear, but a curvilinear (convex) function.
Determinants of Convexity The convexity is the measure of the curvature and can be calculated as: The change in price due to convexity is then:
Determinants of Convexity • There exists a(n): • Inverse relationship between coupon and convexity • Direct relationship between maturity and convexity • Inverse relationship between yield and convexity
Modified Duration-Convexity Effects • Changes in a bond’s price resulting from a change in yield are due to: • Bond’s modified duration • Bond’s convexity • Relative effect of these two factors depends on the characteristics of the bond (its convexity) and the size of the yield change • Convexity is desirable
Limitations of Macaulay and Modified Duration • Percentage change estimates using modified duration only are good for small-yield changes. • It is difficult to determine the interest-rate sensitivity of a portfolio of bonds when there is a change in interest rates and the yield curve experiences a nonparallel shift. • Initial assumption that cash flows from the bond are not affected by yield changes. This may not be true for bonds with options attached.
Effective Duration • Effective duration is also a measure of the interest rate sensitivity of an asset but adjusts for limitations of modified duration. • It uses a pricing model to estimate the market prices surrounding a change in interest rates. • Many practitioners use this direct measure to estimate interest rate sensitivity of bonds.
Readings • RB 18 (pgs. 704 – 711, 716-730, 733-734)