Bond Yield Under Various Assumptions - Essay Example

?Bond Yield Under Various Assumptions I. Overview Bond valuation and yield uses the concept of present values for assessments. Friedlob and Schleifer(2003, p. 36) defined the present value as value of future cash flows discounted to their “present value at an appropriate interest rate.” For example, using slide 14 of our PowerPoint slides, “Bond prices and interest rate risk,” the present value of a bond with zero coupon is the present value of its face value or . In the equation immediately above, P is the present value of the bond while F is the face value of the bond. Further, i is the interest rate, n the maturity while m is the number of compounding per year. Of course, the PowerPoint slide refer to the price of a zero coupon bond. Nevertheless, the concept of the price of a zero coupon bond articulated in the PowerPoint slides and the concept of the present value are similar. One of the more important concepts in bond valuation is term to maturity. Term to maturity “specifies the date or number of years before a bond matures (or expires)” (Reilly and Brown, 2002, p. 697). Another important concept is the coupon of bond which “indicates the income that the bond investor will receive over the life (or holding period) of the issue” of a bond (Reilly and Brown, 2002, p. 697). Other than concepts term to maturity and coupon of bond, the other important concepts include the principal or the par value of the bond but the public is generally familiar with these concepts. II. Measures of Bond Yield Under Various Assumptions (and examples) There are at least five measures of bond yield. Each measure involves a set of assumptions. 1. Yield to Maturity (YTM) As pointed out by our PowerPoint slides, “Bond prices and interest rate risk”, the yield to maturity or YTM “is the yield promised to the bondholder if the bond held up to maturity and all coupons are reinvested at the promised yield” (Slide 17, “Bond prices and interest rate risk”). Our PowerPoint slides, “Bond prices and interest rate risk”, elaborated that “the YTM is also the rate that equates the present value of bond’s cash flows with its price” (Slide 17). Yield to maturity or YTM is also discussed by Breasley and Myers (2003, p. 671-673) and Fabozzi (2008, p. 214-215). Fabozzi (2008, p. 214) confirmed that yield to maturity “is the interest rate that will make the present value of the cash flow from a bond equal to its market price plus accrued interest.” Fabozzi (2008, p. 214) pointed out that “an iterative procedure is used to find the interest rate that will make the present value of the cash flows equal to the market price plus accrued interest.” Following the Fabozzi (2008, p. 214) example, suppose a bond with a face value of $100 promising payments of 7% per annum payable semi-annually or every six months is being sold at $94.17. Based on the parameters defined for the bond, the bond will earn for the bond buyer the value of $3.50 every six months plus $100 at the end of the eight year. Fabozzi (2008, p. 214) pointed out that when the discount rate used to obtain the present value of the payments from the bond is 3.5%, the present value of the bond is $100.00. When the discount rate of 3.6% is used to determine the present value of payments from bond, the present value of the bond is $98.80. When the discount rate of 3.7% is used, the present value of the bond is $97.62. When the discount rate of 3.8% is used, the present value of the bond is $96.45. When the discount rate of 3.9% is used, the present value of the bond is $95.30. Finally, when the discount rate of 4.0% is used, the present value of the bond is $94.17. Thus, based on these, Fabozzi (2008, p. 214) concluded that 4.0% is the price of the bond and “hence, 4.0% is the semi-annual yield to maturity.” All computations came from Fabozzi (2008). Thus, we can consider that the yield to maturity or YTM of the bond as the interest actually paid to the investment of $94.17 made by the buyer of bond and the cash flows of $3.50 semi-annually for eight years plus the $100 payment at the end of the eight year. In this case, the cash flows are equivalent to earning 4.0% compounded semi-annually for 8 years on the invested or $94.17 or the amount spent for buying the bond at the bond price of $94.17. Fabozzi (2008, p. 214) provides the following data on the relationship between the interest rate and present value of the 8-years cash flows from the bond we are discussing (market price plus accrued interests): The YTM is 4.0% because 4.0% is the interest rate that equates the present value of the cash flows from the bond to its price. Although Fabozzi did not provide a formula, the formula implied is P.V. = C/(1+(i/2))^(mn) + FV/(1+i)^n where the problem really is to determine the i or the interest rate that will equalize P.V. with total values on the right. In the formula I have illustrated C is the coupon payments from the bond, m is the compounding per year, n is the number of years until maturity of the bond, PV is the present value of the bond, FV is the final value of the bond, and i is the interest rate. The mathematical procedures to compute for the i are not immediately possible to illustrate over a few pages of work but Microsoft Excel through the internal rate of return function can compute for the YTM as internal rate of return in only a few seconds if not in only one second. The internal rate of return computation command function can be called to service from Excel through the @irr function of Excel. 2. Realised Yield (RY) The realised yield is actual yield realised by the bondholder when the bond is sold before maturity or when the bond issuer defaults. Thus, in the two possible situations and as correctly pointed out by our course PowerPoint slides, “Bond prices and interest rate risk”, the investor’s return will not be equal to YTM we have discussed earlier. Our PowerPoint, “Bond prices and interest rate risk”, clarify that the actual return or the realised yield on a bond actually received by the investor and assuming reinvestment at the coupon rate is called the realised yield. For an analysis, let us assume the first case for the example we have been discussing earlier in Fabozzi (2008, p. 214): the buyer of the bondholder succeeded in selling the bond immediately after the end of the first year at the same price. In this case, after buying the bond at $94.17, our buyer earned $3.50 on the first six months and another $3.50 on the second six months. In addition, suppose the buyer of the bond was able to sell the bond the original purchase price of $94.17. This situation is similar to the case of earning $3.50 semi-annually or every six months for the next 12 months getting back your original investment of $94.17. The effective yield, therefore, is (3.5/94.17) x (2) or 7.43%. In the second case for our example and following Fabozzi (2008, p. 214), let us assume that the bond issuer announced at the end of year 1 or after paying out the second $3.50 that the bond issuer will not be able implement obligations and will buy back the bonds instead at the original purchase price. In this case, the buyer of the bond obtained a yield of 7.43% from the bond even if the issuer bought back the bond at the original purchase price of $94.17. 3. Expected Yield (EY) Our PowerPoint materials, “Bond prices and interest rate risk”, noted that an investor “may wish to compute the yield they would receive if interest rate changed” (Slide 19). Our PowerPoint slides, “Bond prices and interest rate risk”, pointed out that “the predicted or forecast yield is called the expected yield and is based on an expected sales price” (Slide 19). If the expected yield is computed at the current yield, then following Fabozzi (2008, p. 215), a 7% semi-annual payments offered on a bond priced at $94.17 has an expected yield computed based at the current yield equal to 7.43%. This is computed through ($7/$94.17) or 7.43%. However, if the interest rate offered by the bond is increased to 8% then the expected yield computed at the current yield is equal to ($8/$94.17) or 8.495%. If the interest rate offered by the bond is equal to 9%, the expected yield computed at the current yield is equal to ($9/$94.17) or 9.557%. If the offered interest rate by the issuer has been increased to 10% on a bond with a face value of $100 but bought at $94.17, then the expected yield computed at the current yield is $10/$94.17 or 10.619%. All computations on the expected yield based on the current yield follows Fabozzi (2008, p. 214). According to Fabozzi (2008, p. 214), the current yield is computed as: Current Yield = [Annual dollar coupon interest/Bond Price]. Fabozzi (2008, p. 214) pointed out that “the current yield will be greater than the coupon rate when the bond sells a discount; the reverse is true for selling the bond at premium.” Of course, as also pointed out by Fabozzi (2008, p. 214) himself, the drawback of the current yield is that “it considers only the coupon interest and no other source that will impact an investor’s return.” In the current yield estimate, there is no consideration for “the capital gain that the investor will realize when a bond is purchased at a discount and held to maturity” (Fabozzi, 2008, p. 214). Further, in the estimation of expected yield based on the current yield, there is no “recognition of the capital loss that the investor will realize if a bond purchased at a premium is held to maturity.” In this discussion, it is implied that when the coupon interest rate of a bond is increased, the expected yield increases but this considers only the earning from the coupon interest rate of a bond. 4. Holding Period Return of a Bond (HPR) According to Reilly and Brown (2002, p. 699), “the rate of return of a bond is computed in the same way as the rate of return on stock or any asset.” In particular, “it is determined by the beginning and ending price and the cash flows during the holding period” (Reilly and Brown, 2002, p. 699). Reilly and Brown (2002, p. 699) defined the holding period return of a bond or HPR as . In the equation or formula: In the HPR, market forces determine the beginning and ending prices while in-between the beginning and ending prices are payments to the bond determined by the contractual obligation of the issuer of the bond to the bondholders (Reilly and Brown, 2002, p. 699). Thus, if the beginning market price of a bond is $94.17, the accrued interest on the bond is $7 x 8 years or $56, and the end of period market price of the bond is $100, the HPY=((100+56)/94.17)=1.66 (rounded). 5. Holding Period Yield of a Bond (HPY) Meanwhile, according to Reilly and Brown (2002, p. 699), the holding period yield (HPY) of a bond is equal to: . Thus given an HPR of 1.66 in our earlier problem, the applicable HPY is equal to 0.66. III. Interest Rates and Bond Prices Assuming a fixed coupon rate on a bond, bond prices rise when the market rates of interest decline (“Bond prices and interest rate risk”, Slide 22). Based on the same PowerPoint slide, bond prices decrease when the market rates of interest increase. This means that given a fixed coupon rate of payment guaranteed by a bond, a higher market interest rate implies that lower yields are realized from the fixed coupon interest rate of the bond and that, therefore, to attract buyers, bond price must decrease. Otherwise, if the bond price is unchanged given a fixed coupon rate for the bond, the yield of the bond from the fixed coupon rate actually decreases compared to what can be obtained from other market financial instruments like a time deposit account, for example. Thus, bond prices will have to decrease or there will a pressure for bond prices to decrease given the bond’s lower yield rate given a fixed coupon interest rate and an increasing market interest rate. Counting out the principal for simplicity, the price of the bond will be a function of the present value of returns of the cash flows from the bond coupon. Suppose that this bond coupon is $70 annually at 7% coupon rate for a bond with a face value of $1,000. Assuming annual compounding or payments, from Brealey and Meyers (2003, p. 47), the present value of the annual flow of $70 for five years when discounted based on the market interest rate when the market interest rate is 4.8% and bond maturity is five years equals the following quantity in dollars: . If the market interest rate is 5%, the denominator in all the fractions in the equation immediately above becomes 1.05. When the market interest rate becomes 9%, the denominator becomes 1.09 and if the market interest rate becomes 15%, the denominator becomes 1.15%. It follows that the present value of the bond decreases and that, therefore, the price of the bond should decrease. Thus, the price of a bond is inversely related with the market interest rate. IV. Conclusion Based on the foregoing, it is important to know the intricacies related to bond yield valuation and how variables like interest rates can affect bond values. In particular, we have shown how variables like market interest rates and coupon interest rates may affect bond values. For advance topics or details, we can also consult the Australian Securities Exchange (2009), Choudhry (2001), Deardoff (2002), Elan Guides (2002), Investopedia (2010a), Investopedia (2010b), PIMCO (2007), PIMCO (2010), and QFinance (2010). References Australian Securities Exchange. (2009). A guide to the pricing conventions of ASX interest rates products. Available at: http://www.sfe.com.au/content/sfe/products/pricing.pdf (accessed 27 September 2011). Brealey, R. and Myers, S. (2003). Principles of corporate finance. 7th ed. McGraw-Hill Companies. Choudhry, M. (2001). Understanding bond and repo markets. Powerpoint slides. Available in http://janroman.dhis.org (accessed 27 September 2011). Deardorff, A. (2002). Bond prices and interest rates. University of Michigan. Available in: http://www.-personal.umich.edu (accessed 27 September 2011). Elan Guides. (2002). Reading 65: Yield measures, spot rates and forward rates. Available in: www.elanguides.com (accessed 27 September 2011). Fabozzi, F. (2008). Bonds: Investment features and risks. In: F. Fabozzi (ed.), Handbook of finance, Volume 1. John Wiley & Sons, Inc. New Jersey and Canada, 207-220. Friedlob, G. and Schleifer, L. (2003). Essentials of financial analysis. New Jersey and Canada: John Wiley & Sons, Inc. Investopedia. (2010a). Bond basics tutorial. Available in: http://www.investopedia.com/university/bonds/ (accessed 27 September 2011). Investopedia. (2010b). Advanced bond concepts. Available in: http://www.investopedia.com/university/advancedbond/ (accessed 27 September 2011). PIMCO. (2007). Bond basics. Sydney: PIMCO Australia Pty Ltd. Available in www.pimco.com (accessed 27 September 2011). PIMCO. (2010). Investment basics. Sydney: PIMCO Australia Pty Ltd. Available in www.pimco.com (accessed 27 September 2011). QFinance. (2010). Bond Yield. Available in: www.qfinance.com (accessed 27 September 2011). Reilly, F. and Brown, K. (2002). Portfolio management. Cengage. Read More