11-3 Inverse relationship between price and yield. An increase in a bond’s yield to maturity results in a smaller price decline than the gain associated with a decrease in yield. Long-term bonds tend to be more price sensitive than short-term bonds. Bond Pricing Relationships
11-4 Figure 11.1 Change in Bond Price as a Function of YTM
11-5 As maturity increases, price sensitivity increases at a decreasing rate. Price sensitivity is inversely related to a bond’s coupon rate. Price sensitivity is inversely related to the yield to maturity at which the bond is selling. Bond Pricing Relationships (cont’d)
11-6 Table 16.1 Prices of an 8% Coupon Bond (Coupons Paid Semiannually)
11-7 Table 16.2 Prices of Zero-Coupon Bond (Coupons Paid Semiannually)
11-8 A measure of the effective maturity of a bond. The weighted average of the times until each payment is received, with the weights proportional to the present value of the payment. Duration is shorter than maturity for all bonds except zero coupon bonds. Duration is equal to maturity for zero coupon bonds. Duration
11-9 Figure 11.2 Cash Flows of 8-yr Bond with 9% annual coupon and 10% YTM
11-11 Duration Calculation: Coupon Bond Time Cash Flow CF t PV of CF CF t /(1+r) t wtwtwtwt t ∙w t 18072.727.0765.0765 28066.116.0696.1392 31080811.420.85392.5617 Sum950.2631.00002.7774 P B = 950.26; r = 10%; 3 yrs to maturity; coupon rate = 8%
11-12 Duration Calculation: Zero Coupon Bond Time Cash Flow: CF t PV of CF: CF t / (1+r) t w t : [CF t /(1+r) t ]/P B t ∙w t 10000 20000 31000751.31513 Sum751.31513 P B = 751.315; r = 10%; 3 yrs to maturity; coupon rate = 0%
11-13 Spreadsheet 16.1 Calculating the Duration of Two Bonds
11-14 Price change is proportional to duration and not to maturity. P/P = -D x [ (1+y) / (1+y) D * = modified duration D * = D / (1+y) P/P = - D * x y Modified duration is proportional to the derivative of the bond’s price w.r.t. changes in the bond’s yield. For small changes, D* = (1/P)(dP/dy). Duration/Price Relationship
11-15 Rules for Duration Rule 1 The duration of a zero-coupon bond equals its time to maturity. Rule 2 Holding maturity constant, a bond’s duration is higher when the coupon rate is lower. Rule 3 Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity. Rule 4 Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower. Rules 5 The duration of a level perpetuity is equal to: (1+y) / y
11-16 Figure 11.3 Duration as a Function of Maturity
11-17 Table 16.3 Bond Duration (Initial Bond Yield 8% APR)
11-18 Duration and Convexity Duration calculation is valid for small changes in bond yield. It is less accurate for large changes in bond yield. The duration approximation – a straight line relating change in bond price to change in YTM – always understates the price of the bond; it underestimates the increase in bond price when yields fall, and overestimates the fall in price when yields rise.
11-19 Pricing Error from Convexity Price Yield Duration Pricing Error from Convexity
11-20 Duration and Convexity This effect is due to the convex nature of the relationship between prices and yields. The curvature of the price-yield relationship for bonds is called the convexity of the bond. We can quantify the convexity. Formula given below. Basically it corrects for the second derivative…
11-22 Correction for Convexity Correction for Convexity:
11-23 Convexity Investors generally like convexity. Bonds with more convexity gain more in price when yields fall than they lose when yields rise. See following graph. Bond A is more convex; It enjoys greater price increases for falling yields and small price decreases for rising yields than Bond B. Of course, if convexity is valuable it is not free. Investors pay for it with a lower yield on bonds with greater convexity.
11-26 Immunization Passive management –Net worth immunization: Make net worth today unchanged in response to interest rate movements. Interest rate changes cause changes in bond prices and hence in net worth. Can we eliminate this effect? Yes: match duration of assets and liabilities! –Target date immunization: Make funds available at a target future date unchanged in response to interest rate movements. Interest rate changes alter our future cash position by changing the value of our bonds held as assets and changing the rate of return earned on reinvested coupon payments. Can we achieve a balance between these offsetting effects? Yes: set duration equal to the holding period, the time until the target date.
11-27 Net Worth Immunization A strategy to shield net worth from interest rate movements Problem is mismatch in duration of assets and liabilities; assets may be held to pay an obligation fixed at a particular future date. Interest rate movements affect assets differently depending on their duration; a mismatch of duration between assets and liabilities means that interest rate changes can change net worth, because assets and liabilities may change by different amounts in response to interest rate changes.
11-28 Target Date Immunization If interest rates rise, two offsetting effects: 1) bond price falls; and 2) coupon interest payments can be reinvested at higher rate. First effect is called price risk Second effect is called reinvestment risk For a horizon equal to the portfolio’s duration, price risk and reinvestment risk exactly cancel out; the obligation is immunized
11-29 Net Worth Immunization -- Again When interest rates change, need to worry about value of liabilities and assets. – Pensions – when interest rates change, the PV of their assets change, but also the PV of their liabilities. True for all institutions and individuals with future obligations and future income streams. PV of these both depend on interest rates. Matching durations can be a solution.
11-30 Immunization Example Consider an insurance company that issues a guaranteed investment contract for $10,000. Essentially a GIC is a zero coupon bond issued by an insurance company to its customer. If the GIC has a 5-year maturity and a guaranteed interest rate of 8% per year, the insurance company is obligated to pay $14,693.28 in 5 years time. Suppose the insurance company funds this obligation with $10,000 of 8% annual coupon bonds, selling at par, with 6 years to maturity. As long as the interest rate stays at 8%, the insurance company has fully funded its obligation.
11-31 Immunization Example If interest rates change? Say interest rates rise. Then the bonds will be worth less in 5 years than if the interest rate had remained at 8%. However, the reinvested coupons will now earn more than 8%. Thus there are 2 offsetting risks, price risk and reinvestment rate risk. Increases in interest rates cause capital losses (price risk) but provide higher reinvestment rates (reinvestment risk).
11-32 Immunization Example When duration is chosen appropriately, these two effects will cancel out! When the portfolio duration is set equal to the investor’s horizon date, the accumulated value of the investment fund at the horizon date will be unaffected by interest rate fluctuations. In the example the duration of the 6-year bonds is 5 years. (Show that?) Consider following table:
11-34 Table 16.4 Terminal value of a Bond Portfolio After 5 Years (All Proceeds Reinvested)
11-35 Immunization Example Several points to highlight from the above table –Duration matching balances the difference between the accumulated value of the coupon payments and the sale value of the bond. –This is balancing price risk and reinvestment rate risk. –Portfolios must be rebalanced over time. As time passes, duration will change, so portfolio must be adjusted to maintain immunization.