financial management

ALTERNATIVE PROBLEMS AND SOLUTIONS

ALTERNATIVE PROBLEMS

5-1A. (Compound Interest) To what amount will the following investments accumulate?
a. $4,000 invested for 11 years at 9% compounded annually
b. $8,000 invested for 10 years at 8% compounded annually
c. $800 invested for 12 years at 12% compounded annually
d. $21,000 invested for 6 years at 5% compounded annually
5-2A. (Compound Value Solving for n) How many years will the following take?
a. $550 to grow to $1,043.90 if invested at 6% compounded annually
b. $40 to grow to $88.44 if invested at 12% compounded annually
c. $110 to grow to $614.79 if invested at 24% compounded annually
d. $60 to grow to $73.80 if invested at 3% compounded annually
5-3A. (Compound Value Solving for i) At what annual rate would the following have to be invested?
a. $550 to grow to $1,898.60 in 13 years
b. $275 to grow to $406.18 in 8 years
c. $60 to grow to $279.66 in 20 years
d. $180 to grow to $486.00 in 6 years
5-4A. (Present Value) What is the present value of the following future amounts?
a. $800 to be received 10 years from now discounted back to present at 10%
b. $400 to be received 6 years from now discounted back to present at 6%
c. $1,000 to be received 8 years from now discounted back to present at 5%
d. $900 to be received 9 years from now discounted back to present at 20%
5-5A. (Compound Annuity) What is the accumulated sum of each of the following streams of payments?
a. $500 a year for 10 years compounded annually at 6%
b. $150 a year for 5 years compounded annually at 11%
c. $35 a year for 8 years compounded annually at 7%
d. $25 a year for 3 years compounded annually at 2%
5-6A. (Present Value of an Annuity) What is the present value of the following annuities?
a. $3,000 a year for 10 years discounted back to the present at 8%
b. $50 a year for 3 years discounted back to the present at 3%
c. $280 a year for 8 years discounted back to the present at 7%
d. $600 a year for 10 years discounted back to the present at 10%
5-7A. (Compound Value) Trish Nealon, who recently sold her Porsche, placed $20,000 in a savings account paying annual compound interest of 7%.
a. Calculate the amount of money that will have accrued if she leaves the money in the bank for 1, 5, and 15 years.
b. If she moves her money into an account that pays 9% or one that pays 11%, rework part a using these new interest rates.
c. What conclusions can you draw about the relationships between interest rates, time, and future sums from the calculations you have done above?
5-8A. (Compound Interest with Nonannual Periods) Calculate the amount of money that will be in each of the following accounts at the end of the given deposit period:

Compounding Period Annual (Compounded Deposit Amount Interest Every Period
Account Deposited Rate Month) (Years)
Korey Stringer $2,000 12 % 2 2
Eric Moss 50,000 12 1 1
Ty Howard 7,000 18 2 2
Rob Kelly 130,000 12 3 2
Matt Christopher 20,000 14 6 4
Juan Porter 15,000 15 4 3
5-9A. (Compound Interest with Nonannual Periods)
a. Calculate the future sum of $6,000, given that it will be held in the bank 5 years at an annual interest rate of 6%.
b. Recalculate part a using a compounding period that is (1) semiannual and (2) bimonthly.
c. Recalculate parts a and b for a 12% annual interest rate.
d. Recalculate part a using a time horizon of 12 years (annual interest rate is still 6%).
e. With respect to the effect of changes in the stated interest rate and holding periods on future sums in parts c and d, what conclusions do you draw when you compare these figures with the answers found in parts a and b?
5-10A. (Solving for i in Annuities) Ellen Denis, a sophomore mechanical engineering student, receives a call from an insurance agent, who believes that Ellen is an older woman ready to retire from teaching. He talks to her about several annuities that she could buy that would guarantee her an annual fixed income. The annuities are as follows: Initial Payment into Duration Annuity Amount of Money of Annuity
Annuity (at t = 0) Received per year (Years)
A $50,000 $8500 12
B $60,000 $7000 25
C $70,000 $8000 20
If Ellen could earn 12% on her money by placing it in a savings account, should she place it instead in any of the annuities? Which ones, if any? Why?
5-11A. (Future Value) Sales of a new marketing book were 10,000 copies this year and were expected to increase by 15% per year. What are expected sales during each of the next three years? Graph this sales trend and explain.
5-12A. (Future Value) Reggie Jackson, formerly of the New York Yankees, hit 41 home runs in 1980. If his home-run output grew at a rate of 12% per year, what would it have been over the following 5 years?
5-13A. (Loan Amortization) Sefani Moore purchased a new house for $150,000. She paid $30,000 down and agreed to pay the rest over the next 25 years in 25 equal annual payments that included principal payments plus 10% compound interest on the unpaid balance. What will these equal payments be?
5-14A. (Solving for PMT in an Annuity) To pay for your child’s education, you wish to have accumulated $25,000 at the end of 15 years. To do this, you plan on depositing an equal amount in the bank at the end of each year. If the bank is willing to pay 7% compounded annually, how much must you deposit each year to obtain your goal?
5-15A. (Solving for i in Compound Interest) If you were offered $2,376.50 ten years from now in return for an investment of $700 currently, what annual rate of interest would you earn if you took the offer?
5-16A. (Present Value and Future Value of an Annuity) In 10 years, you plan to retire and buy a house in Marco Island, Florida. The house you are looking at currently costs $125,000 and is expected to increase in value each year at a rate of 5%. Assuming you can earn 10% annually on your investment, how much must you invest at the end of each of the next 10 years to be able to buy your dream home when you retire?
5-17A. (Compound Value) The Knutson Corporation needs to save $15 million to retire a $15 million mortgage that matures on December 31, 2002. To retire this mortgage, the company plans to put a fixed amount into an account at the end of each year for 10 years, with the first payment occurring on December 31, 1993. The Knutson Corporation expects to earn 10% annually on the money in this account. What annual contribution must it make to this account to accumulate the $15 million by December 31, 2002?
5-18A. (Compound Interest with Nonannual Periods) After examining the various personal loan rates available to you, you find that you can borrow funds from a finance company at 24% compounded monthly or 26% compounded annually. Which alternative is the most attractive?
5-19A. (Present Value of an Uneven Stream of Payments) You are given three investment alternatives to analyze. The cash flows from these three investments are as follows: Investment
End of
Year A B C
1 $15,000 $20,000
2 15,000
3 15,000
4 15,000
5 15,000 $15,000
6 15,000 60,000
7 15,000
8 15,000
9 15,000
10 15,000 20,000
Assuming a 20% discount rate, find the present value of each investment.
5-20A. (Present Value) The Shin Corporation is planning to issue bonds that pay no interest but can be converted into $1,000 at maturity, 8 years from their purchase. To price these bonds competitively with other bonds of equal risk, it is determined that they should yield 9%, compounded annually. At what price should the Shin Corporation sell these bonds?
5-21A. (Perpetuities) What is the present value of the following?
a. A $400 perpetuity discounted back to the present at 9%
b. A $1,500 perpetuity discounted back to the present at 13%
c. A $150 perpetuity discounted back to the present at 10%
d. A $100 perpetuity discounted back to the present at 6%
5-22A. (Solving for n with Nonannual Periods) About how many years would it take for your investment to grow sevenfold if it were invested at 10% compounded annually?
5-23A. (Complex Present Values) How much do you have to deposit today so that beginning 11 years from now you can withdraw $10,000 a year for the next 5 years (periods 11 through 15) plus an additional amount of $15,000 in that last year (period 15)? Assume an interest rate of 7%.
5-24A. (Loan Amortization) On December 31, Loren Billingsley bought a yacht for $60,000, paying $15,000 down and agreeing to pay the balance in 10 equal annual installments that include both principal and 9% interest on the declining balance. How big would the annual payments be?
5-25A. (Solving for i in an Annuity) You lend a friend $45,000, which your friend will repay in 5 equal annual payments of $9,000 with the first payment to be received one year from now. What rate of return does your loan receive?
5-26A. (Solving for i in Compound Interest) You lend a friend $15,000, for which your friend will repay you $37,313 at the end of 5 years. What interest rate are you charging your “friend”?
5-27A. (Loan Amortization) A firm borrows $30,000 from the bank at 13% compounded annually to purchase some new machinery. This loan is to be repaid in equal annual installments at the end of each year over the next 4 years. How much will each annual payment be?
5-28A. (Present Value Comparison) You are offered $1,000 today, $10,000 in 12 years, or $25,000 in 25 years. Assuming that you can earn 11% on your money, which should you choose?
5-29A. (Compound Annuity) You plan to buy some property in Florida five years from today. To do this, you estimate that you will need $30,000 at that time for the purchase. You would like to accumulate these funds by making equal annual deposits in your savings account, which pays 10% annually. If you make your first deposit at the end of this year and you would like your account to reach $30,000 when the final deposit is made, what will be the amount of your deposit?
5-30A. (Complex Present Value) You would like to have $75,000 in 15 years. To accumulate this amount, you plan to deposit each year an equal sum in the bank, which will earn 8% interest compounded annually. Your first payment will be made at the end of the year.
a. How much must you deposit annually to accumulate this amount?
b. If you decide to make a lump-sum deposit today instead of the annual deposits, how large should this lump-sum deposit be? (Assume you can earn 8% on this deposit.)
c. At the end of 5 years you will receive $20,000 and deposit this in the bank toward your goal of $75,000 at the end of 15 years. In addition to this deposit, how much must you deposit in equal annual deposits to reach your goal? (Again, assume that you can earn 8% on this deposit.)
5-31A. (Comprehensive Present Value) You are trying to plan for retirement in 10 years, and currently you have $150,000 in a savings account and $250,000 in stocks. In addition, you plan to add to your savings by depositing $8,000 per year in your savings account at the end of each of the next 5 years and then $10,000 per year at the end of each year for the final 5 years until retirement.
a. Assuming your savings account returns 8% compounded annually while your investments in stocks will return 12% compounded annually, how much will you have at the end of 10 years? (Ignore taxes.)
b. If you expect to live for 20 years after you retire, and at retirement you deposit all of your savings in a bank account paying 11 percent, how much can you withdraw each year after retirement (20 equal withdrawals beginning one year after you retire) to end up with a zero balance at death?
5-32A. (Loan Amortization) On December 31, Eugene Chung borrowed $200,000, agreeing to repay this sum in 20 equal annual installments that included both the principal and 10% interest on the declining balance. How large will the annual payments be?
5-33A. (Loan Amortization) To buy a new house, you must borrow $250,000. To do this, you take out a $250,000, 30-year, 9% mortgage. Your mortgage payments, which are made at the end of each year (one payment each year), include both principal and 9% interest on the declining balance. How large will your annual payments be?
5-34A. (Present Value) The state lottery’s million-dollar payout provides for one million dollars to be paid over 24 years in $40,000 amounts. The first $40,000 payment is made immediately with the remaining 24 payments occurring at the end of each of the next 24 years. If 10% is the appropriate discount rate, what is the present value of this stream of cash flows? If 20% is the appropriate discount rate, what is the present value of the cash flows?
5-35A. (Solving for i in Compound Interest—Financial Calculator Needed) In March 1963, issue number 39 of Tales of Suspense was issued. The original price for that issue was 12 cents. By March of 1997, 34 years later, the value of this comic book had risen to $2,000. What annual rate of interest would you have earned if you had bought the comic in 1963 and sold it in 1997?
5-36A. (Comprehensive Present Value) You have just inherited a large sum of money and you are trying to determine how much you should save for retirement and how much you can spend now. For retirement, you will deposit today (January 1, 1997) a lump sum in a bank account paying 10% compounded annually. You do not plan to touch this deposit until you retire in 5 years (January 1, 2002), and you plan to live for 20 additional years and then to drop dead on December 31, 2021. During your retirement, you would like to receive income of $60,000 per year to be received on the first day of each year, with the first payment on January 1, 2002, and the last payment on January 1, 2021. Complicating this objective is your desire to have one final 3-year fling during which time you’d like to track down all the original members of the “Mr. Ed Show” and “The Monkeys” and get their autographs. To finance this you want to receive $300,000 on January 1, 2017 and nothing on January 1, 2018, and January 1, 2019, as you will be on the road. In addition, after you pass on (January 1, 2022), you would like to have a total of $100,000 to leave to your children.
a. How much must you deposit in the bank at 10% on January 1, 1997 in order to achieve your goal? (Use a time line in order to answer this question.)
b. What kinds of problems are associated with this analysis and its assumptions?
SOLUTIONS TO ALTERNATIVE PROBLEMS

5-1A. a. FVn = PV (1 + i)n FV11 = $4,000(1 + 0.09)11 FV11 = $4,000 (2.580) FV11 = $10,320 b. FVn = PV (1 + i)n FV10 = $8,000 (1 + 0.08)10 FV10 = $8,000 (2.159) FV10 = $17,272 c. FVn = PV (1 + i)n FV12 = $800 (1 + 0.12)12 FV12 = $800 (3.896) FV12 = $3,117 d. FVn = PV (1 + i)n FV6 = $21,000 (1 + 0.05)6 FV6 = $21,000 (1.340) FV6 = $28,140
5-2A. a. FVn = PV (1 + i)n $1,043.90 = $550 (1 + 0.06)n 1.898 = FVIF6%, n yr. Thus, n = 11 years (because the value of 1.898 occurs in the 11-year row of the 6% column of Appendix B). b. FVn = PV (1 + i)n $88.44 = $40 (1 + .12)n 2.211 = FVIF12%, n yr. Thus, n = 7 years c. FVn = PV (1 + i)n $614.79 = $110 (1 + 0.24)n 5.589 = FVIF24%, n yr. Thus, n = 8 years d. FVn = PV (1 + i)n $78.30 = $60 (1 + 0.03)n 1.305 = FVIF3%, n yr. Thus, n = 9 years
5-3A a. FVn = PV (1 + i)n $1,898.60 = $550 (1 + i)13 3.452 = FVIFi%, 13 yr. Thus, i = 10% (because the Appendix B value of 3.452 occurs in the 12-year row in the 10% column) b. FVn = PV (1 + i)n $406.18 = $275 (1 + i)8 1.477 = FVIFi%, 8 yr. Thus, i = 5% c. FVn = PV (1 + i)n $279.66 = $60 (1 + i)20 4.661 = FVIFi%, 20 yr. Thus, i = 8% d. FVn = PV (1 + i)n $486.00 = $180 (1 + i)6 2.700 = FVIFi%, 6 yr. Thus, i = 18%
5-4A. a. PV = FVn PV = $800 PV = $800 (0.386) PV = $308.80 b. PV = FVn PV = $400 PV = $400 (0.705) PV = $282.00 c. PV = FVn PV = $1,000 PV = $1,000 (0.677) PV = $677 d. PV = FVn PV = $900 PV = $900 (0.194) PV = $174.60
5-5A. a. FVn = PMT FV = $500 FV10 = $500 (13.181) FV10 = $6,590.50 b. FVn = PMT FV5 = $150 FV5 = $150 (6.228) FV5 = $934.20 c. FVn = PMT FV7 = $35 FV7 = $35 (10.260) FV7 = $359.10 d. FVn = PMT FV3 = $25 FV3 = $25 (3.060) FV3 = $76.50
5-6A. a. PV = PMT PV = $3,000 PV = $3,000 (6.710) PV = $20,130 b. PV = PMT PV = $50 PV = $50 (2.829) PV = $141.45 c. PV = PMT PV = $280 PV = $280 (5.971) PV = $1,671.88 d. PV = PMT PV = $600 PV = $600 (6.145) PV = $3,687.00
5-7A. a. FVn = PV (1 + i)n compounded for 1 year FV1 = $20,000 (1 + 0.07)1 FV1 = $20,000 (1.07) FV1 = $21,400 compounded for 5 years FV5 = $20,000 (1 + 0.07)5 FV5 = $20,000 (1.403) FV5 = $28,060 compounded for 15 years FV15 = $20,000 (1 + 0.07)15 FV15 = $20,000 (2.759) FV15 = $55,180 b. FVn = PV (1 + i)n compounded for 1 year at 9% FV1 = $20,000 (1 + 0.09)1 FV1 = $20,000 (1.090) FV1 = $21,800 compounded for 5 years at 9% FV5 = $20,000 (1 + 0.09)5 FV5 = $20,000 (1.539) FV5 = $30,780 compounded for 15 years at 9% FV15 = $20,000 (1 + 0.09)15 FV15 = $20,000 (3.642) FV15 = $72,840 compounded for 1 year at 11% FV1 = $20,000 (1 + 0.11)1 FV1 = $20,000 (1.11) FV1 = $22,200 compounded for 5 years at 11% FV5 = $20,000 (1 + 0.11)5 FV5 = $20,000 (1.685) FV5 = $33,700 compounded for 15 years at 11% FV15 = $20,000 (1 + 0.11)15 FV15 = $20,000 (4.785) FV15 = $95,700
c. 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.
5-8A. FVn = PV (1 + )mn

Account PV i m n (1 + )mn PV(1 + )mn Korey Stringer 2,000 12% 6 2 1.268 $2,536 Eric Moss 50,000 12 12 1 1.127 56,350 Ty Howard 7,000 18 6 2 1.426 9,982 Rob Kelly 130,000 12 4 2 1.267 164,710 Matt Christopher 20,000 14 2 4 1.718 34,360 Juan Porter 15,000 15 3 3 1.551 23,265

5-9A. a. FVn = PV (1 + i)n FV5 = $6,000 (1 + 0.06)5 FV5 = $6,000 (1.338) FV5 = $8,028 b. FVn = PV (1 + )mn FV5 = $6,000 (1 + )2 x 5 FV5 = $6,000 (1 + 0.03)10 FV5 = $6,000 (1.344) FV5 = $8,064 FVn = PV (1 + )mn FV5 = $6,000 FV5 = $6,000 (1 + 0.01)30 FV5 = $6,000 (1.348) FV5 = $8,088 c. FVn = PV (1 + i)n FV5 = $6,000 (1 + 0.12)5 FV5 = $6,000 (1.762) FV5 = $10,572 FV5 = PV mn 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 d. FVn = PV (1 + i)n FV12 = $6,000 (1 + 0.06)12 FV12 = $6,000 (2.012) FV12 = $12,072
e. 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.
5-10A. Annuity A: PV = PMT PV = $8,500 PV = $8,500 (6.194) PV = $52,649
Since the cost of this annuity is $50,000 and its present value is $52,649, given a 12% opportunity cost, this annuity has value and should be accepted. Annuity B: PV = PMT PV = $7,000 PV = $7,000 (7.843) PV = $54,901
Since the cost of this annuity is $60,000 and its present value is only $54,901 given a 12% opportunity cost, this annuity should not be accepted. Annuity C: PV = PMT PV = $8,000 PV = $8,000 (7.469) PV = $59,752
Since the cost of this annuity is $70,000 and its present value is only $59,752, given a 12% opportunity cost, this annuity should not be accepted.
5-11A. Year 1: FVn = PV (1 + i)n FV1 = 10,000(1 + 0.15)1 FV1 = 10,000(1.15) FV1 = 11,500 books Year 2: FVn = PV (1 + i)n FV2 = 10,000(1 + 0.15)2 FV2 = 10,000(1.322) FV2 = 13,220 books Year 3: FVn = PV (1 + i)n FV3 = 10,000(1 + 0.15)3 FV3 = 10,000(1.521) FV3 = 15,210 books

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% of 10,000 books, the second year 15% of 11,500 books, and the third year 15% of 13,220 books.
5-12A. 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. FV3 = 41(1 + 0.12)4 FV4 = 41(1.574) FV4 = 64.534 Home Runs in 1984 (for what at that time would have been a new major league record). FV5 = 41(1 + 0.12)5 FV5 = 41(1.762)
FV5 = 72.242 Home Runs in 1985 (again for a new major league record, but not up to Barry Bonds’ 73 homers in 2001). Actually, Reggie never hit more than 41 home runs in a year. In 1982, he hit 15 only; 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.
5-13A. PV = PMT $120,000 = PMT $120,000 = PMT(9.077) Thus, PMT = $13.220.23 per year for 25 years
5-14A. FVn = PMT $25,000 = PMT $25,000 = PMT(25.129) Thus, PMT = $994.87
5-15A. FVn = PV (1 + i)n $2,376.50 = $700 (FVIFi%, 10 yr.) 3.395 = FVIFi%, 10 yr Thus, i = 13%
5-16A. 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 = PMT $203,625 = PMT(15.937) PMT = $12,776.87
5-17A. FVn = PMT $15,000,000 = PMT $15,000,000 = PMT(15.937) Thus, PMT = $941,206
5-18A. One dollar at 24.0% compounded monthly for one year FVn = PV (1 + i)n FV12 = $1(1 + .02)12 = $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 monthly is more attractive.
5-19A. Investment A PV = PMT = $15,000 = $15,000(2.991) = $44,865

Investment B First, discount the annuity back to the beginning of year 5, which is the end of year 4. Then, discount this equivalent sum to present. PV = PMT = $15,000 = $15,000(3.326) = $49,890—then discount the equivalent sum back to present. PV = FVn = $49,890 = $49,890(.482) = $24,046.98 Investment C PV = FVn = $20,000 + $60,000 + $20,000 = $20,000(.833) + $60,000(.335) + $20,000(.162) = $16,660 + $20,100 + $3,240 = $40,000
5-20A. PV = FVn PV = $1,000 = $1,000(.502) = $502
5-21A. a. PV = PV = PV = $4,444 b. PV = PV = PV = $11,538 c. PV = PV = PV = $1,500 d. PV = PV = PV = $1,667
5-22A. FVn = PV (1 + )m x n 7 = 1(1 + )2n 7 = (1 + 0.05)2n 7 = FVIF5%, 2n yr. A value of 7.040 occurs in the 5% column and 40-year row of the table in Appendix B. Therefore, 2n = 40 years, and n = approximately 20 years.
5-23A. The Present value of the $10,000 annuity over years 11-15. PV = PMT = $10,000(9.108 - 7.024) = $10,000(2.084) = $20,840 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.
5-24A. PV = PMT $45,000 = PMT(6.418) PMT = $7,012
5-25A. PV = PMT $45,000 = $9,000 (PVIFAi%, 5 yr.) 5.0 = PVIFAi%, 5 yr. i = 0%
5-26A. PV = FVn $15,000 = $37,313 (PVIFi%, 5 yr.) .402 = PVIF20%, 5 yr. Thus, i = 20%
5-27A. PV = PMT $30,000 = PMT $30,000 = PMT(2.974) PMT = $10,087
5-28A. The present value of $10,000 in 12 years at 11% is: PV = FVn () PV = $10,000 () PV = $10,000 (.286) PV = $2,860 The present value of $25,000 in 25 years at 11% is: PV = $25,000 () = $25,000 (.074) = $1,850 Thus, take the $10,000 in 12 years.
5-29A. FVn = PMT $30,000 = PMT $30,000 = PMT(6.105) PMT = $4,914
5-30A. a. FV = $75,000 = $75,000 = PMT (FVIFA8%, 15 yr.) $75,000 = PMT(27.152) PMT = $2,762.23 per year b. PV = FVn PV = $75,000 (PVIF8%, 15 yr.) PV = $75,000(.315) PV = $23,625 deposited today c. The contribution of the $20,000 deposit toward the $75,000 goal is FVn = PV (1 + i)n FVn = $20,000 (FVIF8%. 10 yr.) FV10 = $20,000(2.159) = $43,180 Thus, only $31,820 need be accumulated by annual deposit. FV = PMT $31,820 = PMT (FVIFA8%, 15 yr.) $31,820 = PMT [27.152] PMT = $1,171.92 per year
5-31A. 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 c and d.) a. Future value of $150,000 FV10 = $150,000 (1 + .08)10 FV10 = $150,000 (2.159) FV10 = $323,850 b. Future value of $250,000 FV10 = $250,000 (1 + .12)10 FV10 = $250,000 (3.106) FV10 = $776,500 c. Compound annuity of $8,000, 10 years FV10 = PMT = $8,000 = $8,000 (14.487) = $115,896 d. Compound annuity of $2,000 (years 6-10) FV5 = $2,000 = $2,000 (5.867) = $11,734 At the end of 10 years, you will have $323,850 + $776,500 + $115,896 + $11,734 = $1,227,980. PV = PMT $1,227,980 = PMT (7.963) PMT = $154,210.72
5-32A. PV = PMT (PVIFAi%, n yr.) $200,000 = PMT (PVIFA10%, 20 yr.) $200,000 = PMT(8.514) PMT = $23,491
5-33A. PV = PMT (PVIFAi%, n yr.) $250,000 = PMT (PVIFA9%, 30 yr.) $250,000 = PMT(10.274) PMT = $24,333
5-34A. At 10%: PV = $40,000 + $40,000 (PVIFA10%, 24 yr.) PV = $40,000 + $40,000 (8.985) PV = $40,000 + $359,400 PV = $399,400 At 20%: PV = $40,000 + $40,000 (PVIFA20%, 24 yr.) PV = $40,000 + $40,000 (4.938) PV = $40,000 + $197,520 PV = $237,520
5-35A. FV = PMT (FVIFi%, n yr.) $2,000 = .12(FVIFi%, 35 yr.) solving using a financial calculator: i = 32.70%
5-36A. a.

There are a number of equivalent ways to discount these cash flows back to present, one of which is as follows (in equation form): PV = $60,000 (PVIFA10%, 19 yr. - PVIFA10%, 4 yr.) + $300,000 (PVIF10%, 20 yr.) + $60,000 (PVIF10%, 23 yr. + PVIF10%, 24 yr.) + $100,000 (PVIF10%, 25 yr.) = $60,000 (8.365-3.170) + $300,000 (.149) + $60,000 (0.112 + .102) + $100,000 (.092) = $311,700 + $44,700 + $12,840 + $9,200 = $378,440
b. If you live longer than expected, you could end up with no money later on in life.

5-37. rate (i) = 8% number of periods (n) = 7 payment (PMT) = $0 present value (PV) = $900 type (0 = at end of period) = 0

Future value = $1,542.44

Excel formula: =FV(rate,number of periods,payment,present value,type)

Notice that present value ($900) took on a negative value.

5-38. In 20 years you’d like to have $250,000 to buy a home, but you only have $30,000. At what rate must your $30,000 be compounded annually for it to grow to $250,000 in 20 years? number of periods (n) = 20 payment (PMT) = $0 present value (PV) = $30,000 future value (FV) = $250,000 type (0 = at end of period) = 0 guess = i = 11.18%
Excel formula: =RATE(number of periods,payment,present value,future value,type,guess)
Notice that present value ($30,000) took on a negative value.
5-39. To buy a new house you take out a 25 year mortgage for $300,000. What will your monthly interest rate payments be if the interest rate on your mortgage is 8 percent?

Two things to keep in mind when you're working this problem: first, you'll have to convert the annual rate of 8 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.

Excel formula: =PMT(rate,number of periods,present value,future value,type)

rate (i) = 8%/12 number of periods (n) = 300 present value (PV) = $300,000 future value (FV) = $0 type (0 = at end of period) = 0

monthly mortgage payment = ($2,315.45)

Notice that monthly payments take on a negative value because you pay them.
You can also use Excel to calculate the interest and principal portion of any loan amortization payment. You can do this using the following Excel functions:

Calculation: Formula:

Interest portion of payment =IPMT(rate,period,number of periods,present value,future value,type)
Principal portion of payment =PPMT(rate,period,number of periods,present value,future value,type)
Where period refers to the number of an individual periodic payment.
Thus, if you would like to determine how much of the 48th monthly payment went toward interest and principal you would solve as follows:
Interest portion of payment 48: ($1,884.37)
The principal portion of payment 48: ($431.08)

5-40. a. N = 379
I/Y = 6
PV = -24
PMT = 0
CPT FV = 93.57 billion dollars
b. For this problem, first make P/Y = 12
N = 379 X 12 = 4,548
I/Y = 6
PV = -24
PMT = 0
CPT FV = 170.40 billion dollars
c. N = 10
I/Y = 10
CPT PV = -77.108 billion dollars
PMT = 0
FV = 200. billion
d. N = 10 CPT I/Y = 14.87%
PV = -15 billion
PMT = 0
FV = 60. billion
e. N = 40
I/Y = 7
PV = -30. billion
CPT PMT = 2.25 billion dollars
FV = 0
5-41. What will the car cost in the future?
N = 6
I/Y = 3
PV = -15,000
PMT = 0
CPT FV = 17,910.78 dollars
How much must Bart put in an account today in order to have $17,910.78 in 6 years?
N = 6
I/Y = 7.5
CPT PV = -11,605.50 dollars
PMT = 0
FV = 17,910.78
5-42. N = 45
I/Y = 8.75
PV = 0
CPT PMT = -2,054.81 dollars
FV = 1,000,000
5-43. First, we must calculate what Mr. Burns will need in 20 years, then we will know what he needs in 20 years and we can then calculate how much he needs to deposit each year in order to come up with that amount (note: once you calculate the present value, you must multiply your answer, in this case -$4.192 billion times (1 + i) because this is an annuity due):
N = 10
I/Y = 20
CPT PV = -4.1925 billion  1.20 = -5.031 billion dollars
PMT = 1 billion
FV = 0
Next, we will determine how much Mr. Burns needs to deposit each year for 20 years to reach this goal of accumulating $5.031 billion at the end of the 20 years:
N = 20
I/Y = 20
PV = 0
CPT PMT = -26.9 million dollars
FV = 5.031 billion
5-44. What’s the $100,000 worth in 25 years (keep in mind that Homer invested the money 5 years ago and we want to know what it will be worth in 20 years)?
N = 25
I/Y = 7.5
PV = -100,000
PMT = 0
CPT FV = 609,833.96 dollars
Now we determine what the additional $1,500 per year will grow to (note that since Homer will be making these investments at the beginning of each year for 20 years we have an annuity due, thus, once you calculate the present value, you must multiply your answer, in this case $64,957.02 times (1 + i)):
N = 20
I/Y = 7.5
PV = 0
PMT = -1,500
CPT FV = 64,957.02  1.075 = 69,828.80 dollars
Finally, we must add the two values together:
$609,833.96 + $69,828.80 = $679,662.76
5-45. Since this problem involves monthly payments we must first, make P/Y = 12. Then, N becomes the number of months or compounding periods,
N = 60
I/Y = 6.2
PV = -25,000
CPT PMT = 485.65 dollars
FV = 0
5-46A. Since this problem involves monthly payments we must first, make P/Y = 12. Then, N becomes the number of months or compounding periods,
N = 36
CPT I/Y = 11.62%
PV = -999
PMT = 33
FV = 0
5-47. First, what will be the monthly payments if Suzie goes for the 4.9 percent financing? Since this problem involves monthly payments we must first, make P/Y = 12. Then, N becomes the number of months or compounding periods,
N = 60
I/Y = 4.9
PV = -25,000
CPT PMT = 470.64 dollars
FV = 0
Now, calculate how much the monthly payments would be if Suzie took the $1,000 cash back and reduced the amount owed from $25,000 to $24,000. Again, since this problem involves monthly payments we must first, make P/Y = 12.
N = 60
I/Y = 6.9
PV = -24,000
CPT PMT = 474.10 dollars
FV = 0
5-53A. Since this problem involves quarterly compounding we must first, make P/Y = 4. Then, N becomes the number of quarters or compounding periods,
N = 16
I/Y = 6.4%
PV = 0
PMT = -1000
CPT FV = 18,071.11 dollars
5-49. Since this problem involves monthly payments we must first, make P/Y = 12. Then, N becomes the number of months or compounding periods,
CPT N = 41.49 (rounded up to 42 months)
I/Y = 12.9
PV = -5000
PMT = 150
FV = 0
5-50. a. Since this problem begins using annual payments, make sure your calculator is set to P/Y=1.
N = 12
CPT I/Y = 8.37%
PV = -160,000
PMT = 0
FV = 420,000
b. Again, since this problem begins using annual payments, make sure your calculator is set to P/Y=1
N = 10
CPT I/Y = 11.6123%
PV = -140,000
PMT = 0
FV = 420,000
c. Since this problem now involves monthly payments we must first, make P/Y = 12. Then, N becomes the number of months or compounding periods,
N = 120
I/Y = 6
PV = -140,000
CPT PMT = -1,008.57 dollars
FV = 420,000
d. Since this problem now involves monthly payments we must first, make P/Y = 12. Then, N becomes the number of months or compounding periods. Also, since Professor ME will be depositing both the $140,000 (immediately) and $500 (monthly), they must have the same sign,
N = 120
CPT I/Y = 8.48%
PV = -140,000
PMT = -500
FV = 420,000