Measures of Yield of Bonds - Essay Example

Measures of Yield of bonds Student’s Name: Institutional affiliation: University: Yield to maturity is the same as Internal Rate of Return for bonds, which is at the rate of interest that is equal to Yield to Maturity the present value of payments of interest from a bond plus the discount terminal value of that bond equals to the bond’s market price. Yield to maturity offers a bond investor with a rate that he can apply to determine whether to invest in that particular bond. (Crabbe & Fabozzi, 2002). Normally Yield to Maturity is the same as the coupon rate at the time the bon is originally issued , nevertheless, this may not be true since the risks of defaults on the bon payments or its value of maturity consequently investors would demand yield to maturity that is high than the bond’s coupon interest rate (Livingston, 1999). A bond may be issued for specified period of time. If the bond has a finite maturity, to determine its present value, annual payment plus its maturity or terminal value are considered. By the use of present value concept, the discounted value of these values of these flows can be determined through calculation. Through comparison of the present value of a bond and its current market value, it can be determined if the bond is undervalued or overvalued (Subramani, 2011). Suppose an investor is considering the purchase of a five year, $1000 at par value bond, bearing a nominal (coupon) rate of interest of 7 percent. The investor’s required rate of return is 8%. What should he be willing to pay now to purchase the bond if it matures at par? The investor will receive $70 as interest each year for 5 year and $ 1000 on maturity (i.e. at the yield of the fifth year) Bo = 70/ (1.08)1+ 70/ (1.08)2 + 70/ (1.08)3+ 70/ (1.08)4 + 70/ (1.08)5+ 1000/ (1.08)5 It will be noted that $70 is an annuity for five years and $ 1000 is received as a lump sum at the end of the fifth year. Using the present value tables, the present value of the bond is: Bo = $70 ×3.993 + $ 1000×0.681 = $ 960.51 This means that $1000 bond is worth $960.51 today if the required rate of interest is 8%. $960.51 is a composite value of the present value of interest payments, $279.51 and the present value of the maturity value, $681 (Choudhry, 2001). The following formula is applied to determine the present of a bond: Bo = INT1/(1+kd) + INT2/(1+kd)2 + ………+ INTn +Bn/(1+kd)n Bo = Summation INTt/ (1+kd) + Bn/ (1+kd) n Where; Bo = present value of a bond INTt = amount of interest in period t Kd = required rate of return on bond (%) Bn = terminal, or maturity value in period n N = number of years to maturity A debenture or a bond may be amortized every year. In this case, the principal will decline with annual payments and interest rate will be calculated on the outstanding amount. For example, the government is proposing to sell a five-year bond of $1000 at 8 per cent rate of interest per annum. The bond amount will be amortized equally over its life. If an investor has a minimum required rate of return of 7%, what is the bond present value for him? (Anson & Fabozzi, 2007). The amount of interest will go on reducing because the outstanding amount of the bond will be decreasing owing to amortization. The amount of interest for five years will be: $1000 × 0.08 = $80 for the first year; ($1000-$200) ×0.08 = $64 for the second year and; ($800-$200) ×0.08 = $48 for the third year; ($600-$200) ×0.08 = $32 for the fourth year and ($400-$200) × 0.08 = $16 for the fifth year (Anson & Fabozzi, 2007). Since the government will have to return $200 every year, the outflows every year will be $200+$80 = $280; $200+$64 = $264; $200+$48 = $248; $200+$32 = $232; and $200+$16 = $ 216 respectively from the first through five years. Referring to the present value table, the value of the bond can be calculated as follows: Bo = 280/ (1.07)1 + 264/(1.07)2 + 248/(1.07)3 + 232/(1.07)4 + 216/(1.07)5 = 280×0.935+264×0.873+248×0.816+232×0.763+216×0.713 = $261.80 + $230.47+$202.37+$177.02+$154.00 = $ 1025.25 Semi-annual interest payment In practice, it is quite common to pay interest on bonds/debentures semi-annually. The formula for bond valuation can be modified in terms of half-yearly interest payments and compounding periods as given below: B0 = Summation ½(INTt)/(1+kd/2)t + Bn/(1+kd/2)2n A ten year bond of $1000 has an annual rate of interest of 12%. The interest is paid semi annually. If the required rate of return is 16%, what is the value of the bond? The equation above can be used as follows: B0 = Summation ½(INTt)/(1+kd/2)t + Bn/(1+kd/2)2n = summation (2×10, t = 10), ½(120)/(1+0.16/2)t + 1000/(1+0.16/2)2×10 = summation (20, t = 1), 60/(1.08)t + 1000/(1.08)20 = 60×9.818+1000×0.215 = $804.08 If the interest is paid annually, then the value of the bond would be: Bo = 120/(1.16)t + 1000/(1.16)10 = 120×4.833 + 0.227×1000 = 580+227 = $807 Perpetual bonds Perpetual bonds are the types of bonds which will never mature. Perpetual bonds are rarely found in practice. After the Napoleonic War, England issued three types of bonds to pay off many smaller issues that had been floated in prior year years to pay for the war. In case of perpetual bonds, as there is no maturity, or terminal value, the value of the bonds would simply be discounted value of the infinite streams of interest flows. If a bond pays $70 interest annually into perpetuity, what will be its value if the current yield is 85 per annum? The value of the bond will be determined as follows: Bo = INT/kd = 70/0.08 = $875 If the going interest rate is 7%, the value of the bond will be $1000 and it is a 9% the value will be $777.78. Consequently, the value of the bond will decrease as the interest rate increases. They are inversely proportional. The table below gives the present value of perpetual bond paying annual interest of $70 at different discount rates. % Discount rate Value of Bonds ($) 4 1750.00 5 1400.00 6 1166.67 7 1000 8 875.00 9 777.78 10 700.00 When the price of the bond and the cash flows are known the required rate of return which is the yield to maturity of the bond can be calculated. If the price of the bond (BO) AND ITS CASH FLOWS (INTt and Bn), then the required rate of return can be calculated (kd). Suppose the price market of the bond is $883.40 (face value being $1000). The bond will pay interest at 6% per annum for five years, after which it will be retired at par. The bond yield to maturity will be determined as follows: 883.4 = 60/(1+kd)1 + 60/ (1+kd)2 + 60/(1+kd)3 + 60/(1+kd)4 + 60/(1+1060/(1+kd)5 Kd is obtained through using 10% by trial and error. It is simple to calculate yield to maturity of perpetual bonds. It is normally equal to interest amount divided by the price of the bond. If an investor purchases a bond in the secondary market and pays a price that varies from par value, then not only will the current yield varies from the nominal yield, but there will be a loss or gain when the bond matures and the individual holding the bond receives the par value of the bond. If the investor remains with the bond until maturity, he will lose his money if he if at all he paid a premium for the bond or he may earn money if it was bought at a discount. The required rate of return of a bond that will have to account for the loss or gain that happens when the par value is repaid (Reilly & Norton, 2003). When a bond is paid at a discount, required rate of return will normally be greater than the current yield since there will be some gain when the bond is held to maturity, and the person holding the bond receives back the par value, therefore, raising the true yield, when the bond is bought at a premium, the required rate of return will normally be less than the current yield since there will be some loss when the par value is reduced, which brings down the true yield. Yield to call bonds are the ones which are callable. The date of call is normally substituted for the maturity date and the call premium or the call price (Steiner & Heinke 2001) When a bond is issued at a premium, the yield to call will register the lowest yield of that particular bond. Some bonds which are redeemed periodically by a sinking fund, in this case, the issuer establishes to retire debt at different periods at regular intervals at dates of sinking fund which are specified in the schedule of redemption on such dates. Consequently yield to sinker is determined as if the bond will be subsequently retired at the next sinking fund date. If the bond is retired as specified, then the person holding the bond will simply receive the price of the sinking fund, and yield to sinker is determined just like the sinking fund price for the par value of the bond (Wansley, Glascock & Clauretie, 1992). The yield obtained by reinvesting all payments on the coupon for additional interest income is knows as realized compound yield. It will also be determined by the bond price if it is disposed before it reaches maturity. Whatever will be realized in the end will depend on the manner in which interest rates will change of the period of holding the bond. Horizon analysis is sometimes used to forecast interest rates and prices of bonds over a pre-determined period of time to yield an expectation of compound yield realized (Mann & Fabozzi, 2010). How changes in interest rates would affect bond prices The value of the bond depends upon the interest rate. As interest rate changes, the value of the bond also varies. There is an inverse relationship between the value of a bond and the interest rate. The value will decline when the interest rate rises and vice-versa. For example, the value 0f a five year bond in the example discussed previously comes down to $960.51 from $1000 when interest rate is assumed to rise from 7% to 8% resulting in a loss of $39.49 to the bond holder. Interest rates have the tendency of falling or rising in practice. Consequently, investors investing their funds in bonds are exposed to risk from increasing or falling interest rates (Fabozzi, 2005). The intensity of interest rate risk would be high on bonds with longer maturities than those in short periods. Interest rate (%) Value of 5-year bond ($) Value of 10-year bond ($) Value of perpetual bonds ($) 4 1134 1244 1750 5 1087 1155 1400 6 1042 1073 1167 7 1000 1000 1000 8 961 933 875 9 922 871 778 10 886 861 700 When interest rate rise to, say, 8% 5-year bond fall to $961, 10-year bond to $933 and perpetual bond still further to $875. Similarly, the value of long-term bond will fluctuate (increase) more when the rates fall below 7%. The differential value response to interest rates changes between short and long term bonds will always be true. Thus, two bonds of same quality (in terms of quality of default) would have different exposure to interest rate risk-the one with longer maturity is exposed to greater degree of risk from increasing interest rate (Pinches & Singleton, 1978). The reason for differential responsiveness is not difficult to understand. For example, in the 10-year bond situation, one would get just $700 even if interest rate rises to, say, 10 per cent. In case of the 5-year bond, one can, at least, sell the bond after five years, and reinvest money to receive $100 for the next five year. The interest rate is very important when it comes to investment and yield of any give security. References Choudhry, M. (2001). The bond and money markets: strategy, trading, analysis. Melbourne: Butterworth-Heinemann. Subramani, R.V. (2011). Accounting for Investments: Volume 2 - Fixed Income and Interest Rate Derivatives: A Practitioner's Handbook. London: John Wiley and Sons. Livingston, M. (1999). Bonds and bond derivatives. Sidney: Wiley-Blackwell. Crabbe, L.E, Fabozzi , F.J. (2002). Corporate bond portfolio management. London: John Wiley and Sons. Fabozzi, F. (2005). The Handbook of Fixed Income Securities, Chapter 5 - Bond Pricing, Yield Measures, and Total Return, Part 5. Canberra: McGraw-Hill Professional. Anson, M.J.P. & Fabozzi, F.J. (2007). Fixed income analysis. London: John Wiley and Sons. Mann, V. S. & Fabozzi, F. J. (2010). Introduction to Fixed Income Analytics: Relative Value Analysis, Risk Measures and Valuation. London: John Wiley and Sons. Reilly, F. K. & Norton, E. (2003). Investments. New Jersey: Thomson/South-Western. Steiner, M. & Heinke, V.G. (2001). Event Study Concerning International Bond Price Effects of Credit Rating Actions. International Journal of Finance and Economics, 6(4) 139-157. Wansley, J. W., Glascock, J. L., & Clauretie, T. M. (1992). Institutional Bond Pricing andInformation Arrival: The Case of Bond Rating Changes. Journal of BusinessFinance & Accounting, 19(5), 733-750. Pinches, G. E. & Singleton J. C. (1978). The Adjustment of Stock Prices to Bond Rating Changes. Journal of Finance, 33, 1, 29-44. Read More