Computing Bond Yields Yield Measure Purpose Nominal Yield Measures the coupon rate Current yield Measures current income rate Promised yield to maturity Measures expected rate of return for bond held to maturity Promised yield to call Measures expected rate of return for bond held to first call date Measures expected rate of return for a bond likely to be sold prior to maturity. It considers specified reinvestment assumptions and an estimated sales price. It can also measure the actual rate of return on a bond during some past period of time. Realized (horizon) yield

Nominal Yield Measures the coupon rate that a bond investor receives as a percent of the bond’s par value

Current Yield Similar to dividend yield for stocks Important to income oriented investors CY = Ci/Pm where: CY = the current yield on a bond Ci = the annual coupon payment of bondi Pm = the current market price of the bond

Promised Yield to Maturity • Widely used bond yield figure • Assumes • Investor holds bond to maturity • All the bond’s cash flow is reinvested at the computed yield to maturity Solve for i that will equate the current price to all cash flows from the bond to maturity

Promised Yield to CallPresent-Value Method Where: Pm= market price of the bond Ci = annual coupon payment nc = number of years to first call Pc = call price of the bond

Example

Calculating Future Bond Prices Where: Pf= estimated future price of the bond Ci = annual coupon payment n = number of years to maturity hp = holding period of the bond in years i = expected semiannual rate at the end of the holding period

Example • Coupon rate 10%, 25-year bond with promised YTM = 12%, face value $1000 • Current price is • But you expect market YTM decline to 8% in 5 years, therefore you want to calculate its future price Pf at the end of year 5 to estimate your expected YTM • Pf = ? • What’s the realized YTM over this 5 years (assuming all cash flows are reinvested at the computed YTM)

The Duration Measure • Since price volatility of a bond varies inversely with its coupon and directly with its term to maturity, it is necessary to determine the best combination of these two variables to achieve your objective • A composite measure considering both coupon and maturity would be beneficial

The Duration Measure Developed by Frederick R. Macaulay, 1938 Where: t = time period in which the coupon or principal payment occurs, typically every 6 month Ct= interest or principal payment that occurs in period t i = semi-annual yield to maturity on the bond

Example • Calculate the Macaulay duration for a 5-year bond with 4% coupon rate, the ytm is 8%.

Characteristics of 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 relation between duration and coupon • There is a positive relation between term to maturity and duration, but duration increases at a decreasing rate with maturity • There is an inverse relation between YTM and duration

Modified Duration and Bond Price Volatility An adjusted measure of duration can be used to approximate the interest-rate sensitivity of a bond Where: m = number of payments a year YTM = nominal YTM

Duration and Bond Price Volatility • Bond price movements will vary proportionally with modified duration for small changes in yields • An estimate of the percentage change in bond prices equals the change in yield time modified duration Where: P = change in price for the bond P = beginning price for the bond Dmod = the modified duration of the bond i = yield change in basis points

Duration • Duration approximation (the straight line) always understates the value of the bond; it underestimates the increase in bond price when the yield falls, and it overestimates the decline in price when the yield rises.

Bond Convexity • Modified duration is a linear approximation of bond price change for small changes in market yields • Price changes are not linear, but a curvilinear (convex) function

Duration and Convexity Price Pricing Error from convexity Duration Yield

Correction for Convexity Correction for 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 and always a positive number

Convexity - example • 30-year maturity, 8% coupon bond, and sells at an initial YTM of 8%, so it is a par bond. The modified duration of the bond at its initial yield is 11.26 years, and its convexity is 212.4. If the yield increases from 8% to 10%, the bond price will fall to: • 1) The duration rule: • 2) The duration with convexity rule:

Convexity - example • 1) The duration rule: • 2) The duration with convexity rule:

Limitations of Macaulay and Modified Duration • Percentage change estimates using modified duration only are good for small-yield changes • 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

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