CHAPTER The Time Value of Money CHAPTER ORIENTATION In this chapter the concept of a time value of money is introduced, that is, a dollar today is worth more than a dollar received a year from now Thus if we are to logically compare projects and financial strategies, we must either move all dollar flows back to the present or out to some common future date CHAPTER OUTLINE I Compound interest results when the interest paid on the investment during the first period is added to the principal and during the second period the interest is earned on the original principal plus the interest earned during the first period A Mathematically, the future value of an investment if compounded annually at a rate of i for n years will be FVn = where n i PV FVn PV (l + i)n = the number of years during which the compounding occurs = the annual interest (or discount) rate = the present value or original amount invested at the beginning of the first period = the future value of the investment at the end of n years The future value of an investment can be increased by either increasing the number of years we let it compound or by compounding it at a higher rate If the compounded period is less than one year, the future value of an investment can be determined as follows: FVn where m = PV mn = the number of times compounding occurs during the year 91 II Determining the present value, that is, the value in today's dollars of a sum of money to be received in the future, involves nothing other than inverse compounding The differences in these techniques come about merely from the investor's point of view A Mathematically, the present value of a sum of money to be received in the future can be determined with the following equation: PV where: III = FVn n = the number of years until payment will be received, i = the interest rate or discount rate PV = the present value of the future sum of money FVn = the future value of the investment at the end of n years The present value of a future sum of money is inversely related to both the number of years until the payment will be received and the interest rate An annuity is a series of equal dollar payments for a specified number of years Because annuities occur frequently in finance, for example, bond interest payments, we treat them specially A A compound annuity involves depositing or investing an equal sum of money at the end of each year for a certain number of years and allowing it to grow This can be done by using our compounding equation, and compounding each one of the individual deposits to the future or by using the following compound annuity equation: FVn where: = PMT i n FVn B  n −1  PMT  ∑ (1 + i) t   t =0  = the annuity value deposited at the end of each year = the annual interest (or discount) rate = the number of years for which the annuity will last = the future value of the annuity at the end of the nth year Pension funds, insurance obligations, and interest received from bonds all involve annuities To compare these financial instruments we would like to know the present value of each of these annuities This can be done by using our present value equation and discounting each one of the individual cash flows back to the present or by using the following present value of an annuity equation: 92 PV = where: PMT IV This procedure of solving for PMT, the annuity value when i, n, and PV are known, is also the procedure used to determine what payments are associated with paying off a loan in equal installments Loans paid off in this way, in periodic payments, are called amortized loans Here again we know three of the four values in the annuity equation and are solving for a value of PMT, the annual annuity Annuities due are really just ordinary annuities where all the annuity payments have been shifted forward by one year Compounding them and determining their present value is actually quite simple Because an annuity, due merely shifts the payments from the end of the year to the beginning of the year, we now compound the cash flows for one additional year Therefore, the compound sum of an annuity due is FVn(annuity due) A VI = PMT (FVIFA i,n) (1 + i) Likewise, with the present value of an annuity due, we simply receive each cash flow one year earlier – that is, we receive it at the beginning of each year rather than at the end of each year Thus the present value of an annuity due is PV(annuity due) V   (1 + i) t  = the annuity deposited or withdrawn at the end of each year = the annual interest or discount rate = the present value of the future annuity = the number of years for which the annuity will last i PV n C  n PMT  ∑  t =1 = PMT (PVIFA i,n) (1 + i) A perpetuity is an annuity that continues forever, that is every year from now on this investment pays the same dollar amount A An example of a perpetuity is preferred stock which yields a constant dollar dividend infinitely B The following equation can be used to determine the present value of a perpetuity: PV = where: PV = the present value of the perpetuity pp = the constant dollar amount provided by the perpetuity i = the annual interest or discount rate To aid in the calculations of present and future values, tables are provided at the back of Financial Management (FM) A To aid in determining the value of FV n in the compounding formula FVn = PV (1 + i)n = PV (FVIF i,n) 93 tables have been compiled for values of FVIF i,n or (i + 1)n in Appendix B, "Compound Sum of $1," in FM 94 B To aid in the computation of present values PV = FVn = FVn (PVIFi,n) tables have been compiled for values of or PVIF i,n and appear in Appendix C in the back of FM C Because of the time-consuming nature of compounding an annuity, FVn = PMT n −1 ∑ t=0 (1 + i) t = PMT (FVIFA i,n) Tables are provided in Appendix D of FM for n −1 ∑ t=0 (1 + i) t or FVIFA i,n for various combinations of n and i D To simplify the process of determining the present value of an annuity  n PV = PMT  ∑  t =1   (1 + i)  t = PMT (PVIFA i,n) tables are provided in Appendix E of FM for various combinations of n and i for the value n t =1 (1 + i) t ∑ V or PVIFA i,n Spreadsheets and the Time Value of Money A While there are several competing spreadsheets, the most popular one is Microsoft Excel Just as with the keystroke calculations on a financial calculator, a spreadsheet can make easy work of most common financial calculations Listed below are some of the most common functions used with Excel when moving money through time: Calculation: Formula: Present Value Future Value Payment = PV(rate, number of periods, payment, future value, type) = FV(rate, number of periods, payment, present value, type) = PMT(rate, number of periods, present value, future value, type) Number of Periods = NPER(rate, payment, present value, future value, type) Interest Rate = RATE(number of periods, payment, present value, future value, type, guess) 95 where: rate = i, the interest rate or discount rate number of periods = n, the number of years or periods payment = PMT, the annuity payment deposited or received at the end of each period future value = FV, the future value of the investment at the end of n periods or years present value = PV, the present value of the future sum of money type = when the payment is made, (0 if omitted) = at end of period = at beginning of period guess = a starting point when calculating the interest rate, if omitted, the calculations begin with a value of 0.1 or 10% ANSWERS TO END-OF-CHAPTER QUESTIONS 5-1 The concept of time value of money is recognition that a dollar received today is worth more than a dollar received a year from now or at any future date It exists because there are investment opportunities on money, that is, we can place our dollar received today in a savings account and one year from now have more than a dollar 5-2 Compounding and discounting are inverse processes of each other In compounding, money is moved forward in time, while in discounting money is moved back in time This can be shown mathematically in the compounding equation: FVn = PV (1 + i)n We can derive the discounting equation by multiplying each side of this equation by and we get: PV 5-3 = FVn = PV(1 + i)n We know that FVn Thus, an increase in i will increase FV n and a decrease in n will decrease FVn 5-4 Bank C which compounds daily pays the highest interest This occurs because, while all banks pay the same interest, percent, bank C compounds the percent daily Daily compounding allows interest to be earned more frequently than the other compounding periods 5-5 The values in the present value of an annuity table (Table 5-8) are actually derived 96 from the values in the present value table (Table 5-4) This can be seen, by examining the values represented in each table The present value table gives values of for various values of i and n, while the present value of an annuity table gives values of n t =1 (1 + i) t ∑ for various values of i and n Thus the value in the present value of annuity table for an n-year annuity for any discount rate i is merely the sum of the first n values 10 in the present value table PVIFA 10%,10yrs = 6.145 ∑ PVIF10%,n = 6.144 = n =1 0.909 + 0.826 + 0.751 + 0.683 + 0.621 + 0.564 + 0.513 + 0.467 + 0.424 + 0.386 5-6 An annuity is a series of equal dollar payments for a specified number of years Examples of annuities include mortgage payments, interest payments on bonds, fixed lease payments, and any fixed contractual payment A perpetuity is an annuity that continues forever, that is, every year from now on this investment pays the same dollar amount The difference between an annuity and a perpetuity is that a perpetuity has no termination date whereas an annuity does SOLUTIONS TO END-OF-CHAPTER PROBLEMS Solutions to Problem Set A 5-1A (a) (b) FVn = PV (1 + i)n FV10 = $5,000(1 + 0.10)10 FV10 = $5,000 (2.594) FV10 = $12,970 FVn = PV (1 + i)n FV7 = $8,000 (1 + 0.08)7 FV7 = $8,000 (1.714) FV7 = $13,712 97 (c) (d) 5-2A (a) (b) (c) (d) 5-3A (a) FV12 = PV (1 + i)n FV12 = $775 (1 + 0.12)12 FV12 = $775 (3.896) FV12 = $3,019.40 FVn = PV (1 + i)n FV5 = $21,000 (1 + 0.05)5 FV5 = $21,000 (1.276) FV5 = $26,796.00 FVn = PV (1 + i)n $1,039.50 = $500 (1 + 0.05)n 2.079 = FVIF 5%, n yr Thus n = 15 years (because the value of 2.079 occurs in the 15 year row of the percent column of Appendix B) FVn = PV (1 + i)n $53.87 = $35 (1 + 09)n 1.539 = FVIF 9%, n yr Thus, n = years FVn = PV (1 + i)n $298.60 = $100 (1 + 0.2)n 2.986 = FVIF 20%, n yr Thus, n = years FVn = PV (1 + i)n $78.76 1.486 = = $53 (1 + 0.02)n FVIF 2%, n yr Thus, n = 20 years FVn = PV (1 + i)n $1,948 3.896 = = $500 (1 + i)12 FVIF i%, 12 yr Thus, i = 12% (because the Appendix B value of 3.896 occurs in the 12 year row in the 12 percent column) 98 (b) (c) (d) 5-4A (a) (b) (c) FVn = PV (1 + i)n $422.10 = $300 (1 + i)7 1.407 = FVIF i%, yr Thus, i = 5% FVn = PV (1 + i)n $280.20 = $50 (1 + i)20 5.604 = FVIF i%, 20 yr Thus, i = 9% FVn = PV (1 + i)n $497.60 = $200 (1 + i)5 = FVIF i%, yr Thus, i = 20% PV = FVn PV = $800 PV = $800 (0.386) PV = $308.80 PV = FVn PV = $300 PV = $300 (0.784) PV = $235.20 PV = FVn PV = $1,000 PV = $1,000 (0.789) PV = $789 99 (d) 5-5A (a) (b) (c) (d) PV = FVn PV = $1,000 PV = $1,000 (0.233) PV = $233 FVn =  n −1  PMT  ∑ (1 + i) t  t=0  FV10 =  10 − t $500  ∑ (1 + 0.05)   t =0  FV10 = $500 (12.578) FV10 = $6,289 FVn =  n −1  PMT  ∑ (1 + i) t  t=0  FV5 =  −1  $100  ∑ (1 + 0.1) t  t =0  FV5 = $100 (6.105) FV5 = 610.50 FVn =  n −1  PMT  ∑ (1 + i) t  t=0  FV7 =  −1 t $35  ∑ (1 + 0.07)  t =0  FV7 = $35 (8.654) FV7 = $302.89 FVn =  n −1  PMT  ∑ (1 + i) t  t=0  FV3 =  −1 t $25  ∑ (1 + 0.02)  t =0  FV3 = $25 (3.060) FV3 = $76.50 100 (c) (d) 5-6B (a) (b) (c) FVn =  n −1  PMT  ∑ (1 + i) t  t=0  FV8 =  −1 t $35  ∑ (1 + 0.07)  t =0  FV8 = $35 (10.260) FV8 = $359.10 FVn =  n −1  PMT  ∑ (1 + i) t  t=0  FV3 =  −1 t $25  ∑ (1 + 0.02)  t =0  FV3 = $25 (3.060) FV3 = PV = $76.50  n PMT  ∑ t  t = (1 + i)     PV =  10 $3,000  ∑ t  t = (1 + 0.08) PV = $3,000 (6.710) PV = $20,130 PV =  n PMT  ∑  t =1   (1 + i) t  PV =  $50  ∑  t =1   t  (1 + 0.03)  PV = $50 (2.829) PV = $141.45 PV =  n PMT  ∑  t =1   (1 + i) t  PV =  $280  ∑  t =1   (1 + 0.07) t  PV = $280 (5.971) PV = $1,671.88 1 129     (d) 5-7B (a) PV =  n PMT  ∑ t  t = (1 + i) PV =  10 $600  ∑  t =1 PV = $600 (6.145) PV = $3,687.00 FVn = PV (1 + i)n       (1 + 0.1) t  compounded for year FV1 = $20,000 (1 + 0.07)1 FV1 = $20,000 (1.07) FV1 = $21,400 compounded for years FV5 = $20,000 (1 + 0.07)5 FV5 = $20,000 (1.403) FV5 = $28,060 compounded for 15 years (b) FV15 = $20,000 (1 + 0.07)15 FV15 = $20,000 (2.759) FV15 = $55,180 FVn = PV (1 + i)n compounded for year at 9% FV1 = $20,000 (1 + 0.09)1 FV1 = $20,000 (1.090) FV1 = $21,800 compounded for years at 9% FV5 = $20,000 (1 + 0.09)5 FV5 = $20,000 (1.539) FV5 = $30,780 compounded for 15 years at 9% FV5 FV5 = $20,000 (1 + 0.09)15 = $20,000 (3.642) 130 FV5 = $72,840 compounded for year at 11% FV1 = $20,000 (1 + 0.11)1 FV1 = $20,000 (1.11) FV1 = $22,200 compounded for years at 11% FV5 = $20,000 (1 + 0.11)5 FV5 = $20,000 (1.685) FV5 = $33,700 compounded for 15 years at 11% (c) 5-8B FV5 = $20,000 (1 + 0.11)15 FV5 = $20,000 (4.785) FV5 = $95,700 There is a positive relationship between both the interest rate used to compound a present sum and the number of years for which the compounding continues and the future value of that sum FVn = Account Korey Stringer Erica Moss Ty Howard Rob Kelly Mary Christopher Juan Diaz 5-9B (a) (b) FVn = PV (1 + )mn PV 2,000 50,000 7,000 130,000 20,000 15,000 i 12% 12% 18% 12% 14% 15% m 12 PV (1 + i)n FV5 = $6,000 (1 + 0.06)5 FV5 = $6,000 (1.338) FV5 = $8,028 FVn = PV (1 + )mn FV5 = $6,000 (1 + )2 x FV5 = $6,000 (1 + 0.03)10 FV5 = $6,000 (1.344) FV5 = $8,064 FVn = PV (1 + )mn 131 n 2 (1 + )mn 1.268 1.127 1.426 1.267 1.718 1.551 PV(1 + )mn $2,536 56,350 9,982 164,710 34,360 23,265 (c) (d) (e) 6X5 FV5 = $6,000 (1 + ) FV5 = $6,000 (1 + 0.01)30 FV5 = $6,000 (1.348) FV5 = $8,088 FVn = PV (1 + i)n FV5 = $6,000 (1 + 0.12)5 FV5 = $6,000 (1.762) FV5 = $10,572 FV5 = PV FV5 = $6,000 FV5 = $6,000 (1 + 0.06)10 FV5 = 6,000 (1.791) FV5 = $10,746 FV5 = PV mn FV5 = $6,000 FV5 = $6,000 (1 + 0.02)30 FV5 = $6,000 (1.811) FV5 = $10,866 FVn = PV (1 + i)n FV12 = $6,000 (1 + 0.06)12 FV12 = $6,000 (2.012) FV12 = $12,072 mn 2X5 6X5 An increase in the stated interest rate will increase the future value of a given sum Likewise, an increase in the length of the holding period will increase the future value of a given sum 5-10B Annuity A: PV =  n PMT  ∑  t =1 PV =  12 $8,500  ∑  t =1 PV = $8,500 (6.194) 132   t  (1 + i)    (1 + 0.12) t  PV = $52,649 Since the cost of this annuity is $50,000 and its present value is $52,649, given a 12 percent opportunity cost, this annuity has value and should be accepted Annuity B: PV =  n PMT  ∑  t =1   t  (1 + i)  PV =  25 $7,000  ∑  t =1 PV = $7,000 (7.843) PV =$54,901   (1 + 0.12) t  Since the cost of this annuity is $60,000 and its present value is only $54,901 given a 12 percent opportunity cost, this annuity should not be accepted Annuity C: PV =  n PMT  ∑ t  t = (1 + i)     PV =  20 $8,000  ∑  t =1 PV = $8,000 (7.469)   (1 + 0.12) t  PV = $59,752 Since the cost of this annuity is $70,000 and its present value is only $59,752, given a 12 percent opportunity cost, this annuity should not be accepted 5-11B Year 1: Year 2: Year 3: FVn = PV (1 + i)n FV1 = 10,000(1 + 0.15)1 FV1 = 10,000(1.15) FV1 = 11,500 books FVn = PV (1 + i)n FV2 = 10,000(1 + 0.15)2 FV2 = 10,000(1.322) FV2 = 13,220 books FVn = PV (1 + i)n FV3 = 10,000(1 + 0.15)3 FV3 = 10,000(1.521) FV3 = 15,210 books 133 Book sales 20,000 15,000 10,000 years The sales trend graph is not linear because this is a compound growth trend Just as compound interest occurs when interest paid on the investment during the first period is added to the principal of the second period, interest is earned on the new sum Book sales growth was compounded; thus, the first year the growth was 15 percent of 10,000 books, the second year 15 percent of 11,500 books, and the third year 15 percent of 13,220 books 5-12B FVn = PV (1 + i)n FV1 = 41(1 + 0.12)1 FV1 = 41(1.12) FV1 = 45.92 Home Runs in 1981 (in spite of the baseball strike) FV2 = 41(1 + 0.12)2 FV2 = 41(1.254) FV2 = 51.414 Home Runs in 1982 FV3 = 41(1 + 0.12)3 FV3 = 41(1.405) FV3 = 57.605 Home Runs in 1983 FV4 = 41(1 + 0.12)4 FV4 = 41(1.574) FV4 = 64.534 Home Runs in 1984 (for a new major league record) 134 FV5 = 41(1 + 0.12)5 FV5 = 41(1.762) FV5 = 72.242 Home Runs in 1985 (again for a new major league record) Actually, Reggie never hit more than 41 home runs in a year In 1982, he only hit 15, in1983 he hit 39, in 1984 he hit 14, in 1985 25 and 26 in 1986 He retired at the end of 1987 with 563 career home runs 5-13B 5-14B 5-15B 5-16B PV =  n PMT  ∑  t =1   (1 + i) t  $120,000 =  25 PMT  ∑  t =1    t  (1 + 0.1)  $120,000 = PMT(9.077) Thus, PMT = $13,220.23 per year for 25 years FVn =  n −1  PMT  ∑ (1 + i) t  t=0  $25,000 =  15 − t PMT  ∑ (1 + 0.07)   t=0  $25,000 = PMT(25.129) Thus, PMT = $994.87 FVn = PV (1 + i)n $2,376.50 = $700 (FVIF i%, 10 yr.) 3.395 = FVIF i%, 10 yr Thus, i = 13% 1 The value of the home in 10 years FV10 = PV (1 + 05)10 = $125,000(1.629) = $203,625 How much must be invested annually to accumulate $203,625? $203,625 =  10 − t PMT  ∑ (1 + 10)   t =0  $203,625 = PMT(15.937) PMT = $12,776.87 135 5-17B 5-18B FVn =  n −1  PMT  ∑ (1 + i) t  t=0  $15,000,000 =  10 − t PMT  ∑ (1 + 10)   t =0  $15,000,000 = PMT(15.937) Thus, PMT = $941,206 One dollar at 24.0% compounded monthly for one year FVn = PV (1 + )nm FV1 = $1(1 + 02)1 = $1(1.268) = $1.268 One dollar at 26.0% compounded annually for one year FVn = PV (1 + i)n FV1 = $1(1 + 26)1 = $1(1.26) = $1.26 The loan at 26% compounded annually is more attractive 5-19B Investment A PV =  n PMT  ∑  t =1   (1 + i) t  =  $15,000  ∑  t =1 = $15,000(2.991) = $44,865 i   (1 + 20)  t Investment B First, discount the annuity back to the beginning of year 5, which is the end of year Then discount this equivalent sum to present PV =  n PMT  ∑  t =1   (1 + i) t  =  $15,000  ∑  t =1 = $15,000(3.326) 136   t  (1 + 20)  PV = $49,890 then discount the equivalent sum back to present = FVn = $49,890 = $49,890(.482) = $24,046.98 = FVn = $20,000 + $60,000 Investment C PV + $20,000 5-20B 5-21B (a) (b) (c) (d) = $20,000(.833) + $60,000(.335) + $20,000(.162) = $16,660 + $20,100 + $3,240 = $40,000 PV = FVn PV = $1,000 = $1,000(.502) = $502 PV = PV = PV PV = = PV = PV = PV = PV = PV = PV = PV = PV = $1,667 = PMT(PVIFA i,n)(l + i) = $1000(3.791)(1 + 10) = $3791(1.1) = $4,170.10 = PV (1 + )m n 5-22B PV(annuity due) 5-23B FVn $4,444 $11,538 $1,500 137 n = 1(1 + )2 = (1 + 0.05)2 = FVIF 5%, 2n yr n A value of 7.040 occurs in the percent column and 40-year row of the table in Appendix B Therefore, 2n = 40 years and n = approximately 20 years 5-24A Investment A: PV = FVn (PVIFi,n) PV = $5,000(PVIF 10%, year 1) + $5,000(PVIF 10%, year 2) + $5,000(PVIF 10%, year 3) - $15,000(PVIF 10%, year 4) + $15,000(PVIF 10%, year 5) = $5,000(.909) + $5,000(.826) + $5,000(.751) - $15,000(.683) + $15,000(.621) = $4,545 + $4,130 + $3,755 - $10,245 + $9,315 = $11,500 138 Investment B: PV = FVn (PVIFi,n) PV = $1,000(PVIF 10%, year 1) + $3,000(PVIF 10%, year 2) + $5,000(PVIF 10%, year 3) + $10,000(PVIF 10%, year 4) $10,000(PVIF 10%, year 5) = $1,000(.909) + $3,000(.826) + $5,000(.751) + $10,000(.683) - $10,000(.621) = $909 + $2,478 + $3,755 + $6,830 - $6,210 = $7,762 PV = FVn (PVIFi,n) PV = $10,000(PVIF 10%, year 1) + $10,000(PVIF 10%, year 2) + $10,000(PVIF 10%, year 3) + $10,000(PVIF 10%, year 4) $40,000(PVIF 10%, year 5) = $10,000(.909) + $10,000(.826) + $10,000(.751) + $10,000(.683) - $40,000(.621) = $9,090 + $8,260 + $7,510 + $6,830 - $24,840 = $6,850 Investment C: 5-25B The Present value of the $10,000 annuity over years 11-15 PV =   15 PMT   ∑   t =1 = $10,000(9.108 - 7.024) = $10,000(2.084) = $20,840   10  − ∑ (1 + 07) t   t =   t  (1 + 07)   The present value of the $15,000 withdrawal at the end of year 15: PV = FV15 = $15,000(.362) = $5,430 Thus, you would have to deposit $20,840 + $5,430 or $26,270 today 139 5-26B 5-27B PV 5-28B PV 5-29B PV PV =  10 PMT  ∑  t =1   t  (1 + 09)  $45,000 = PMT(6.418) PMT = $7,012 =  PMT  ∑  t =1 $45,000 = $9,000 (PVIFA i%, yr.) 5.0 = PVIFA i%, yr i = 0% = FVn $15,000 = $37,313 (PVIF i%, yr.) 402 = PVIF20%, yr Thus, i = 20% =  n PMT  ∑  t =1 $30,000 =  PMT  ∑  t =1 $30,000 = PMT(2.974) PMT = $10,087   t  (1 + i)    (1 + i) t    t  (1 + 13)  5-30B The present value of $10,000 in 12 years at 11 percent is: PV =  FVn  n  (1 + i)    PV = $10,000 () PV = $10,000 (.286) PV = $2,860 The present value of $25,000 in 25 years at 11 percent is: PV = $25,000 () = $25,000 (.074) = $1,850 Thus take the $10,000 in 12 years 5-31B FVn =  n −1  PMT  ∑ (1 + i) t  t=0  140 5-32B (a) (b) (c) $30,000 =  −1 t PMT  ∑ (1 + 10)  t =0  $30,000 = PMT(6.105) PMT =$4,914 FVn =  n −1 t PMT  ∑ (1 + i)  t=0  $75,000 =  15 − t PMT  ∑ (1 + 08)   t=0  $75,000 = PMT (FVIFA 8%, 15 yr.) $75,000 = PMT(27.152) PMT = $2,762.23 per year PV = FVn PV = $75,000 (PVIF 8%, 15 yr.) PV = $75,000(.315) PV = $23,625 deposited today The contribution of the $20,000 deposit toward the $75,000 goal is FVn = PV (1 + i)n FVn = $20,000 (FVIF 8%, 10 yr.) FV10 = $20,000(2.159) = $43,180 Thus only $31,820 need be accumulated by annual deposit FVn =  n −1 t PMT  ∑ (1 + i)  t=0  $31,820 = PMT (FVIFA 8%, 15 yr.) $31,820 = PMT [27.152] PMT = $1,171.92 per year 141 5-33B.(a) This problem can be subdivided into (1) the compound value of the $150,000 in the savings account, (2) the compound value of the $250,000 in stocks, (3) the additional savings due to depositing $8,000 per year in the savings account for 10 years, and (4) the additional savings due to depositing $2,000 per year in the savings account at the end of years 6-10 (Note the $10,000 deposited in years 6-10 is covered in parts (3) and (4).) (1) (2) (3) Future value of $150,000 FV10 = $150,000 (1 + 08)10 FV10 = $150,000 (2.159) FV10 = $323,850 Future value of $250,000 FV10 = $250,000 (1 + 12)10 FV10 = $250,000 (3.106) FV10 = $776,500 Compound annuity of $8,000, 10 years FV10 = (4)  n −1  PMT  ∑ (1 + i) t  t=0  =  10 −  $8,000  ∑ (1 + 08) t   t =0  = $8,000 (14.487) = $115,896 Compound annuity of $2,000 (years 6-10) FV5 =  −1  $2,000  ∑ (1 + 08) t  t =0  = $2,000 (5.867) = $11,734 At the end of ten years you will have $323,850 + $776,500 + $115,896 + $11,734 = $1,227,980 PV =  20 PMT  ∑ t  t = (1 + 11) $1,227,980 = PMT (7.963) (b) PMT = $154,210.72 142     5-34B PV = PMT (PVIFA i%, n yr.) $200,000 = PMT (PVIFA 10%, 20 yr.) $200,000 = PMT(8.514) PMT = $23,491 PV = PMT (PVIFA i%, n yr.) $250,000 = PMT (PVIFA 9%, 30 yr.) $250,000 = PMT(10.274) 5-35B PMT = $24,333 5-36B At 10%: PV = $40,000 + $40,000 (PVIFA 10%, 24 yr.) PV = $40,000 + $40,000 (8.985) PV = $40,000 + $359,400 PV = $399,400 PV = $40,000 + $40,000 (PVIFA 20%, 24 yr.) PV = $40,000 + $40,000 (4.937) PV = $40,000 + $197,480 PV = $237,480 5-37B FVn(annuity due) = PMT(FVIFA i,n)(l + i) = $1000(FVIFA 5%, years )(l + 05) = $1000(5.526)(1.05) = $5802.30 = PMT(FVIFA i,n)(l + i) = $1,000(FVIFA 8%, years )(1 + 08) = $1,000(5.867)(1.08) = $6,336.36 = PMT(PVIFA i,n)(l + i) = $1000 (PVIFA 12%, 15 years )(1 + 12) = $1000(6.811)(1.12) = $7,628.32 At 20%: FVn(annuity due) 5-38B PV(annuity due) 143 ... amount provided by the perpetuity i = the annual interest or discount rate To aid in the calculations of present and future values, tables are provided at the back of Financial Management (FM)... perpetuity is that a perpetuity has no termination date whereas an annuity does SOLUTIONS TO END-OF -CHAPTER PROBLEMS Solutions to Problem Set A 5-1A (a) (b) FVn = PV (1 + i)n FV10 = $5,000(1 + 0.10)10... convert the annual rate of percent into a monthly rate by dividing it by 12, and second, you''ll have to convert the number of periods into months by multiplying 25 times 12 for a total of 300 months