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46 One final observation is to note that we have not dealt with reinvestment rate risk in this section. This is a very important risk, but it is addressed elsewhere in this book. Example 9.2 The prevailing yield rate for riskfree 10year bonds is 9% effective. A $1000 10year bond with annual coupons is issued on which the coupon rate is 9%. (1) Find the price an investor would be willing to pay if the probability of default each year is .005 and if the investor requires a yield rate of 11% in compensation for the risk of default. (2) Find the risk premium in the interest rate in this transaction. 1. If die bond were riskfree, it would obviously sell at par, i.e. die price would be $1000. We can utilize formula (9,10) to find die price  .995  .995  +   .995   1 1000  .995   1.11  1.11    1.11   1.11 
= $852,825 upon summing the geometric progression. We have the following equation of value to determine the yield rate on the bond 852.825 = 90 1 + 1000(1 + /)°. If we solve diis equation by iteration using the approach developed in Section 7.6, we obtain die yield rate / = 11.56%. Thus, die risk premium in die interest rate is 11.56%  9% = 2.56%. It is instructive to note diat die risk premium is approximately equal to die difference between die two interest rates, i.e. 11 % minus 9%, plus die default rate, 0.5%. Example 9.3 Consider the illustration in this section in which a $1000 oneyear 8% bond selling for $940 bought at a yield rate of 11% has an implicit probability of default equal to .0339. Assume that a diversified portfolio of 500 such bonds is purchased. Use the normal distribution to find a 95% confidence interval for the average probability of default on the entire portfolio. The standard deviation for die probabihty of default is equal to El = (.0339)(.9661) oogj 500 Thus, die 95% confidence interval using die normal distribution is tip ± 1.96 Op = .0339 ± 1.96(.0081) or (.0180, .0498). When franslated into yield rates, die confidence interval becomes (9.17%, 12.83%). Readers desiring a brief statistical review are referred to Appendix IV. 9,6 YIELD CUR\TES Another factor that affects the level of interest rates is the length of the investment period. Typically, at any point in time shortterm and longterm rates of interest are different. This phenomenon is called the term structure of interest rates. Table 9.2 is a hypothetical table of interest rates illustrating the term structure. Table 9.2 Hypoflietical Term Structure of Interest Rates Length of investment  Interest rate  1 year  7.00%  2 years  8.00  3 years  8.75  4 years  9.25  5 years  9.50 
Table 9.2 illustrates a situation in which the rate of interest increases as the length of the investment period increases. In other words, longterm interest rates are higher than shortterm interest rates. This pattern is usually found in practice. Periods beyond five years are of concern as well, but Table 9.2 was kept short for simplicity in illustrating the concept. Another phrase used to describe the term structure of interest rates is the yield curve. Figure 9.1 graphically illustrates the yield curve for die rates of interest contained in Table 9.2. 9.00 "S 8.00 7.00  12 3 4 Length of investment in years Figure 9.1 DIustrative yield curve
and this will be reflected when the computation is based on different spot rates from the yield curve. An analytical approach to measuring this difference in average bond terms is given in Section 9.8. Another type of interest rate is the forward rate. This is a spot rate that will come into play in future. In essence, a forward rate can be considered as a future reinvestment rate. In order to illustrate this concept, consider a business firm which needs to borrow a sizable amount of money for two years. The firm is presented with die yield curve illustrated in Table 9.2. The firm has two options. The first option is to borrow for two years at the 2year spot rate of 8%. The second option is to borrow for one year at the 1year spot rate of 7% and then borrow for the second year at the 1year spot rate in effect a year later. This 1year spot rate for the second year is called a "forward rate." We will now analyze the two options. Let the forward rate be denoted by /. The firm will then be indifferent between the two options if (1.08)2 = (1.07)(1 +/) which is solved to give / = .0901, or 9.01%. Thus, if the firm expects the forward rate to be greater than 9.01%, it should use the first option to borrow. However, if die firm expects the forward rate to be less than 9.01%, it should use the second option. The reader should note that the symbol/was also used in Section 5.9 to represent the project financing rate. Note that in referring to forward rates we need to be precise in specifying both die period of deferral and the period for which die rate applies. For example, a projected forward rate of interest for the interval between 3 and 8 years from the present is, in essence, an estimated 3year deferred 5year spot rate. The forward rate/used in the above illustration is an estimated 1year deferred 1year spot rate. The use of spot rates and forward rates in the term structure of interest rates introduces a more sophisticated way of analyzing alternatives for investing and borrowing dian we have considered previously. It has become die basis for many complex modern financial transactions, including extensive hedging and arbitrage strategies. This section is not a complete treatment of the subject but it does introduce the basic concepts. The reader is encouraged to pursue the subject in more depth in the references listed in the bibliography. Example 9.4 Find the present value of payments of $1000 at the end of each year for five years using the spot rates given in Table 9.2. What level yield rate would produce an equivalent value? The present value of the payments is 1000[(1.07) + (1.08)2 + (1.0875)" + (1.0925)" + (1.095)] = $3906.63. The equivalent level yield rate is found by solving aj,, = 3.90663 . If we solve this by iteration using the techniques discussed in Section 3.8, we obtain i = .0883, or Example 9.5 Find the present value of the remaining payments in the annuity given in Example 9.4 immediately after two payments have been made. The forward rates at that time are expected to be 1% higher for all periods than the current spot rates. The comparison date for diis calculation is at die end of two years. At diat time there are diree remaining annuity payments to be made. The expected forward rates are die current spot rates for 1, 2, and 3 years (not for 3, 4, and 5 years) each increased by 1%. Thus, the remaining present value is 1000[(1.08) + (1.09) + (1.0975)"] = $2524.07. 9,7 INTEREST RATE ASSUMPTIONS Within the realm of standard borrowing and lending transactions the rate of interest involved is typically specified or implied by the provisions of the transaction. For example, loan rates on mortgages and yield to maturity rates on bonds are either known or can readily be determined. The same can be said of other borrowing and lending transactions we have discussed in this book. However, as we have also seen, the use of discounted cash flow analysis is a powerful analytical and decisionmaking tool with applicability going well beyond borrowing and lending transactions. In particular, the discussion of capital budgeting, net present value, and internal rate of return in Chapter 5 was motivated by the need to analyze more general business and financial transactions involving estimated fuhire receipts and/or disbursements at various points of time. This type of analysis is widely used in die field of actuarial science in which present value analysis of future contingent events is a fundamental activity. Another area in which this type of analysis is becoming increasingly important is in the field of accounting. An increasing number of values appearing on financial statements are based on present value calculations. Many other examples could also be cited illustrating the importance of discounted cash flow analysis in financial analysis and decision making. In using discounted cash flow analysis for these types of more complex situations, the decision maker must resolve three basic ingredients listed in Section 9.5:
Let «2, .., „ be a series of payments made at times 1, 2... ., „ method of equated time was defined in formula (2.9). If we apply thisfor with the required change in notation, we obtain t R, 7= IzL For example, consider the two bonds cited above. If the coupons are annually, then the average term to maturity of the 5% bond is (9.13 1 • 5 + 2 • 5 + 5 + 5 + + 10 • 5 + 10 • 100 + 100 = 8.50 Similarly, the average term to maturity of the 10% bond is 1 • 10 + 2 • 10 + 10 + 10 + + 10 • 10 + 10 • 100 = 7.75 The 5% bond has an average payment date of 8.50 years, while the 10% bon has an average payment date of 7.75 years. Thus, we could say that the 5*, bond is a longer term bond than the 10% bond. This confirms the statement made comparing Bonds A and in Section 9.6. An even better index is given by duration. The concept behind duration similar to the method of equated time, except that the present value of each payment is used as the weight instead of the payment itself. Then duration, denoted by d, is given by (9.13J Note that d is a function of /. Several special cases of formula (9.13) are of importance: 1. If i = 0, then d  t. This is obvious upon inspection, since formula (9.13) immediately simplifies to formula (9.12). Thus, the method equated time is, in essence, a special case of duration which ignores interest. 2. The duration <f is a decreasing fiinction of i. The proof of this result wiU come later in this section. However, the result does have an interesting verbal inte retation. As the rate of interest increases, the terms in the numerator of formula (9.13) with the higher values of / are discounted relatively more than the terms with the iower valuesof t. This serves to reduce the overall weighted average value of i.e. d. If there is only one future payment, then d is the point in time at which that payment is made. This intuitively appealing result is obvious upon examining formula (9.13), since the summations in the numerator and denominator have only one term each and everything cancels except the time of payment. It is also instructive to examine the rate of change in the present value of 1 series of future payments as the rate of interest changes. Let this present lvalue be denoted by i.e. P{i) = E vJ?, = a +0%. (9.14) we now define the volatility of this present value, denoted by v, as Pii) (9.15) I where v is a function of /. The reasoning behind formula (9.15) is analogous to the reasoning behind the definition of the force of discount given in Section 1.9. The term Pii) measures the instantaneous rate of change in the present value of the f payments as the rate of interest changes. Dividing by P{i) expresses this instantaneous rate of change in units independent of the size of the present value itself The minus sign is necessary to make v positive, since Pii) is negative. The word "volatility" seems appropriate to describe v, since v is a , measure of how rapidly the present value of a series of future payments changes as the rate of interest changes. We obtain a very interesting result if we substitute formula (9.14) into formula (9.15) Pii) j.Ei+irt i: i\ + i)~R, t=i ti\ +i)R, E (1 + iyt
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