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Bond Price Elasticity

. . Bond Price Elasticity. The sensitivity of bond prices (BP) to changes in the required rate of return (I) is commonly measured by bond price elasticity (BPe), which is estimated as. Example of Elasticity. If the required rate of return changes from 10 percent to 8 percent, the bond price of a zero coupon bond will rise from $386 to $463. Thus the bond price elasticity is.

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Bond Price Elasticity

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    1. Bond Price Elasticity Business 4179

    2. Bond Price Elasticity The sensitivity of bond prices (BP) to changes in the required rate of return (I) is commonly measured by bond price elasticity (BPe), which is estimated as

    3. Example of Elasticity If the required rate of return changes from 10 percent to 8 percent, the bond price of a zero coupon bond will rise from $386 to $463. Thus the bond price elasticity is

    4. Example of Elasticity

    5. Bond Price Elasticity and Bond Price Theorums The following table demonstrates how bond price elasticity measures the effects of a given change in interest rates on bonds with different coupon rates. Zero coupon or stripped bonds have the longest durations because there are no intermediate cash flows, hence they exhibit the greatest elasticity. The higher the coupon rate, the lower the elasticity all other things being equal.

    6. Sensitivity of 10-year bonds with different coupon rates to interest rate changes

    7. Bond Price Sensitivity and Term to Maturity The following chart explores the impact of the term to maturity on bond price sensitivity clearly, the longer the term to maturity, the greater the bond price elasticity. When interest rates rise, the bond price will rise by a greater percentage, than the fall in bond price in response to an equal but opposite increase in interest rates.

    8. Sensitivity of 10-year bonds with different coupon rates to interest rate changes

    9. Bond Prices and Term to Maturity

    10. Duration An alternative measure of bond price sensitivity is the bond’s duration. Duration measures the life of the bond on a present value basis. Duration can also be thought of as the average time to receipt of the bond’s cashflows. The longer the bond’s duration, the greater is its sensitivity to interest rate changes.

    11. Duration and Coupon Rates A bond’s duration is affected by the size of the coupon rate offered by the bond. The duration of a zero coupon bond is equal to the bond’s term to maturity. Therefore, the longest durations are found in stripped bonds or zero coupon bonds. These are bonds with the greatest interest rate elasticity. The higher the coupon rate, the shorter the bond’s duration. Hence the greater the coupon rate, the shorter the duration, and the lower the interest rate elasticity of the bond price.

    12. Duration The numerator of the duration formula represents the present value of future payments, weighted by the time interval until the payments occur. The longer the intervals until payments are made, the larger will be the numerator, and the larger will be the duration. The denominator represents the discounted future cash flows resulting from the bond, which is the bonds present value.

    13. Duration Example As an example, the duration of a bond with $1,000 par value and a 7 percent coupon rate, three years remaining to maturity, and a 9 percent yield to maturity is:

    14. Duration Example ... As an example, the duration of a zero-coupon bond with $1,000 par value and three years remaining to maturity, and a 9 percent yield to maturity is:

    15. Duration is a handy tool because it can encapsule interest rate exposure in a single number. rather than focus on the formula...think of the duration calculation as a process... semi-annual duration calculations simply call for halving the annual coupon payments and discounting every 6 months.

    16. Duration Rules-of-Thumb duration of zero-coupon bond (strip bond) = the term left until maturity. duration of a consol bond (ie. a perpetual bond) = 1 + (1/R) where: R = required yield to maturity duration of an FRN (floating rate note) = 1/2 year

    17. Other Duration Rules-of-Thumb Duration and Maturity duration increases with maturity of a fixed-income asset, but at a decreasing rate. Duration and Yield duration decreases as yield increases. Duration and Coupon Interest the higher the coupon or promised interest payment on the security, the lower its duration.

    18. Economic Meaning of Duration duration is a direct measure of the interest rate sensitivity or elasticity of an asset or liability. (ie. what impact will a change in YTM have on the price of the particular fixed-income security?) interest rate sensitivity is equal to: dP = - D [ dR/(1+R)] P Where: P = Price of bond C = Coupon (annual) R = YTM N = Number of periods F = Face value of bond

    19. Interest Rate Elasticity the percent change in the bond’s price caused by a given change in interest rates (change in YTM)

    20. Economic Meaning of Duration interest rate sensitivity is equal to: dP = - D [ dR/(1+R)] P dP/P = change in bond price [ dR/(1+R)] = change in interest rate Obviously, the relationship is an inverse function of Duration (D)

    21. Example of Calculation of Interest Rate Sensitivity given: n = 6 years (Eurobond ... annual coupon payments) 8 percent coupon 8 YTM if yields are expected to rise by 10%, what impact will that have on the price of the bond? the first step is to calculate the duration of the bond. If there were no coupon payments the duration would be = 6. since there are coupon payments the duration must be less than 6 years. D = 4.993 years the second step is to calculate the % change in price for the bond. = -(4.993)(.1/1.08) = - 0.4623 = - 46.23%

    22. Immunization fully protecting or hedging an FI’s equity holders against interest rate risk. elimination of interest rate risk by matching the duration of both assets and liabilities. (not their average lives or final maturities). when immunized: the gains or losses on reinvestment income that result from an interest rate change are exactly offset by losses or gains from the bond proceeds on sale of the bond.

    23. Example of Bond Price The Canada 10.25 1 Feb 04 is quoted at 123.95 yielding 5.27%. This means that for a $1,000 par value bond, these bonds are trading a premium price of $1,239.50 The figure represents bond prices as of June 17, 1998. This bond has 5 years and 8 months (approximately) until maturity = 5+(8/12) = 5.7 years Bond Price = $102.50(PVIFAn=5.7 ,r=5.27%) + $1,000 / (1.0527)5.7 = $102.50(PVIFAr=5.27%%, n= 5.7) + $746.21 = $102.50(4.8156653) + $746.21 = $493.61 + $743.42 = $1,237.03 Can you explain why the quoted price might differ from your answer?

    24. Example of a Duration Calculation

    25. Sensitivity Analysis of Bonds

    26. Prices over time

    27. Duration of a Portfolio Bond portfolio mangers commonly attempt to immunize their portfolio, or insulate their portfolio from the effects of interest rate movements. For example, a life insurance company knows that they need $100 million 30 years from now cover actuarially-determined claims against a group of life insurance policies just no sold to a group of 30 year olds. The insurance company has invested the premiums into 30-year government bonds. Therefore there is no default risk to worry about. The company expects that if the realized rate of return on this bond portfolio equals the yield-to-maturity of the bond portfolio, there won’t be a problem growing that portfolio to $100 million. The problem is, that the coupon interest payments must be reinvested and there is a chance that rates will fall over the life of the portfolio.

    28. Duration of a Portfolio ... The life insurance company example illustrates a keep risk in fixed-income portfolio management - interest rate risk. The portfolio manager, before-the-fact calculates the bond portfolio’s yield-to-maturity. This is an ex ante calculation. As such, a naïve assumption assumption is made that the coupon interest received each year is reinvested at the yield-to-maturity for the remaining years until the bond matures. Over time, however, interest rates will vary and reinvestment opportunities will vary from that which was forecast.

    29. Duration of a Portfolio ... The insurance company will want to IMMUNIZE their portfolio from this reinvestment risk. The simplest way to do this is to convert the entire bond portfolio to zero-coupon/stripped bonds. Then the ex ante yield-to-maturity will equal ex post (realized) rate of return. (ie. the ex ante YTM is locked in since there are no intermediate cashflows the require reinvestment). If the bond portfolio manager matches the duration of the bond portfolio with the expected time when they will require the $100 m, then interest rate risk will be eliminated.

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