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48 9.8 Duration 22. Show tiiat lim d is equal to die point in time at which die furst payment is made.; /.00 23. You are given the following series of payments: (0 $100 at times f= 1,3, 5,..., 19 (ii) $200 at times / = 2,4,6.....20 You are asked to determine die time / * such diat die present value of die series of payments is equal to die present value of a single payment of $3000 made at 5 time / * . Derive an exact expression for / * assuming J > 0. [ 24. Find tfie duration of a common stock which pays dividends at the end of each year, if it is assumed that each dividend is 4% greater dian the prior dividend and die effective rate of interest is 8%. 25. Find an expression for die modified duration of . 26. Show thai die modified duration of a perpetuityimmediate and the present value of the same perpetuity are equal. 27. If there is only one payment, then the variance in formula (9.17) would be zero. Verbally interpret the result diat = 0 in this situation. 28. A loan is to be repaid widi payments of $1000 at die end of one year, $20( at die end of two years, and $3000 at die end of duee years. The effective rate of interest is 25%. a) Find the amount of the loan. b) Find the duration. c) Find the modified duration. d) Find the variance 0 in formula (9.17). 9.9 Immunization 29. a) Confirm that the modified duration of the assets obtained in Example 9.8(1) is equal to the modified duration of the liabihty. *) Confirm that die convexity of die assets obtained in Example 9.8(2) is greater dian die convexity of die UabUity. 30. Find the present value at 8% convertible semiannually of a 10year annuityimmediate whose successive semiannual payments are 1,4,9, .. . ,400. Answer to the nearest dollar. 31. Derive die following relationship between modified duration and convexity: v = vc. di 32. The following relationship holds between convexity and the variance (f in formula (9.17): = a{i)a + b(i) . Find tfie fiinctions a{i) and b(J). 33. a) In Example 9.7 assume that the investor puts $600 in the money market fund and $400 in twoyear bonds. Find: (1) P(.09) (2) P(.IO) (3) PC 11) b) Rework (o) assuming die investor puts $400 in die money market fiind and $600 in twoyear bonds. c) Verbally interpret the answers to (o) and ) in comparison widi the answers in Example 9.7. 34. Find the convexity of the following investments, assuming the effective rate of interest is 8%: a) A money market fund. f) Would you expect the difference betv/een dse answers to become larger for longer term bonds? 17. Consider the bonds in Exercise 16 from a different vantage point. Assume an investor has die choice of buying two 5% bonds or one 10% bond. Either choice will produce an idenUcal flow of coupons. a) Find die difference in die amounts originally invested. b) Find the difference in the payouts when the bonds mature. c) Verify diat ) is die accumulated value of (a) at die spot rate of interest for the term of the bonds. 18. Find die value of j using spot rates of interest in which die spot rates in Table 9.2 apply for die first year and die entire yield curve shifts downward by .25% for each year diereafter dirough die end of die fiveyear period. 19. Based on die yield curve given in Table 9.2, find die following expected forwaid rates: a) 1year deferred 2year forward rate. b) 2year deferred 3year forward rate. 20. A 6year bond widi 6% annual coupons has a yield rate of 12% effective. A 6year ; bond widi 10% annual coupons has a yield rate of 8% effective. Find die 6year spot rate. tl 21. An investor has $100,000 to invest for three years. The account balance may be] reinvested at die end of eidier the first or second years. The yield curve given in Table 9.2 applies for die first year. The entire yield curve shifts upward by 2% for die second year and anodier 2% for die diird year. Considering all possible patterns , of investment, find die minimum and maximum accumulation at die end of the ttiree: year investment period. Answer to die nearest dollar.
Stochastic approaches to interest 10.1 INTRODUCTION The prior chapters in this book have presented the theory of interest on largely a deterministic basis. Chapter 10 introduces the reader to somei stochastic approaches to interest. Achially, there were two areas in the prior chapters where probabilities] were explicitly inhoduced. One area was the consideration of default! probabilities in Section 9.5, and die second was the assumption regarding? withdrawals from the certificate of deposit illustration in Section 9.10. Also,! formula (9.17) for tiie derivative of duration was inte reted in terms of a; variance. Finally, there was implicit recognition of risk and uncertainty in such j areas as the exercise of the call provision on callable bonds and the prepayment! rates on mortgage loans. Neverdieless, the overall approach taken previously] in the book has been largely deterministic. In Chapter 10 we first consider die rate of interest directly as a randoml variable. We then provide a basic introduction to several models widi a stochastic basis which have important applications in practice. Finally, wej examine the widely used approach of scenario testing as a means of dealing wiflij fiiture uncertainty. We assume that die reader has a basic background in mathematical statistics.! The level of statistics used in diis chapter is not advanced. Readers who would! like a condensed refresher of die results from statistics that are used in diis chapter are referred to Appendix FV. 10.2 INDEPENDENT RATES OF INTEREST We now consider the rate of interest to be a random variable. In Section 10.2 we consider the case in which the rate of interest in one period is independent of the rate of interest in any other period. In Section 10.3 we will remove the independence assumption and examine some results when successive rates are correlated in some fashion. A preliminary illustration We will first demonstrate that expected accumulated and present values are not necessarily equal to accumulated and present values at the expected rate of <. interest. In order to demonstrate this possibility, consider the investment of one I unit for ten years at an effective rate of interest which is unknown, but is equally likely to be 7%, 8%, or 9%. From elementary statistics the expected rate of interest is given by E[i] = [.07 + .08 + .09j = .08. The expected accumulated value is given by E[(l + i) (1.07)° + (1.08)° + (1.09)°] = 2.16448. However, the accumulated value at the expected rate of interest is (1.08)° = 2.15892. Thus, in this illustration the expected accumulated value is not equal to the accumulated value at the expected rate of interest. The rate of interest which would produce the expected accumulated value can be determined from (1 +/)° = 2.16448 I Which can be solved to give / = .08028, or 8.028%. t A similar result can also occur for present values. Assume we wish to I invest an amount today sufficient to accumulate to one unit at die end of ten [years and assume the same conditions described above concerning the unknown i rate of interest. The expected present value is given by E[(1 + 0"°] = l[(1.07)° + (1.08)° + (1.09)°] = .46465 while the present value at the expected rate of interest is (1.08)°= .46319. The rate of interest which would produce the expected present value can be (1 +0"° = .46465 determined from
s , s yar[. ] = Li "l s s 41 s s W2  2 .. (10.8) Present values Parallel results to those above for accumulated values can also be develoned for present values. However, we must be careful in the choice of interest rates to use, since in general 1 +1 E[l + it) Thus, when working with present values we must define i by £[(1 +(",)"] = (1 +0. It must be stressed that this value of is different than the value of i used above for accumulated values for which E[i,] = i. Our first result is to develop a formula for the mean of the present value of a single payment analogous to formula (10.1). We have £[ (/ )] = (1+0". (10.9) The derivation of formula (10.9) is similar to formula (10.1) and is left as an exercise. For the variance of the present value we have var[a(n)] = E[a\n)]  {E[aHn)]} = (1 +A:)"(l +0" (10.10) where (1 +fc)" = £[(1 + i)]. Unfortunately, this is as far as we can carry the development without knowing how ij is distributed. The approach used above for finding the second moment for accumulated values will not work for present values. Evaluation of formula (10.10) will require computing the second moment based on a particular probability density fiinction. Now turning to the present value of an period annuityimmediate we have from Section 3.9 = (1 +/,)+ (1 +/,)(! +/2)" + • • • n t +(i+j,)(i+/2)"(i+g= E Ild+P" (334) /=1 s=l The mean of  is the anticipated result [ ]= . (10.11) mf = £[(1 +/,)] = (1+0" = £[(1 +itr] = (1 +A:). Again, applying the results from Appendix IX, we have var[a] = a , a mt + Wj  my mj  m Finding numerical solutions Thus, if we know the first and second moments of 1 + /, and (1 + /,)", we can find the mean and variance of the accumulated value or the present value of either a single payment or a level annuity. We may choose to make an assumption about the probability density function for j,. Unfortunately, the above formulas do not generally result in a known probability density function, even if we know the mean and the variance. The standard approach to handle this situation in practice is to use simulation. The procedure is as follows: 1. Make an appropriate assumption about the probability density function for ij and its parameters. 2. Generate a series of enough random numbers to run as many trials as desired. If m trials are desired, then mn random numbers will be necessary. 3. Using standard simulation techniques, use these random numbers to compute m sets of values for ij, 4. For each of the m sets f2> • • • > compute the required financial function, i.e. a{n), a~(n), s, , or some other function. 5. The m outcomes can be used to develop an approximate probability density function for that financial function. Estimated probabilities for various possible outcomes can be calculated from these m outcomes. We will not discuss the technique of simulation further, since we assume that the reader is familiar with simulation. Readers who are unfamiliar with the technique are referred to standard textbooks in mathematical statistics. The derivation of formula (10.11) is similar to formula (10.5) and is left as an exercise. Finally, we consider the variance of . Define Wj and mj to be the first and second moments of (1 + /,)" about the origin, respectively, i.e. (10.12) (10.13)
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