Time Value of Money Problems (Dr. A. Tezel) Tezel)
Formulas: PV=FVt [ 1/(1+R)t ] Present Value of Annuity: PV = C [ 1/R – 1/(R(1+R) N) ] Growing Annuity PV = [ C / (R – g) ][ 1 – ((1+g)/(1+R)) N ] Future Value of Annuity: FV = [(1+R) N – 1 ] /R
Which amount is worth more at 14%, compounded annually; $1,000 in hand today or $2,000 due in 6 years?
2000(1/1.14)^6=2000(0.4556)=911.20. 1000 today is worth worth more.
How would you find the compound annual return from a stock whose price rose from $10 to $25 over 5 years? (Assume the stock stock paid no dividends)
25=10(1+R)^5 2.5=(1+R)^5 2.5^(1/5) – 1= R
R=20.11%
Find the amount to which $500 will grow under each of the following conditions: 12 % compounded annually for 5 years 12% compounded semiannually for 5 years 12% compounded quarterly for 5 years 12% compounded monthly for 5 years
881.15, 895.40, 903.05, and 908.35
A 20-year ordinary annuity pays an annual payment of 12,000. What is this annuity’s present value for a 9 percent interest rate?
PV=12000(PVIFA(9%,20 years) = 109,543
In the previous question, each annual payment is 4 percent larger than the previous one. Find the present value of this growing annuity?
PV=12000/(0.9-0.04) [1- (1.04/1.09)^20] = 240000 {1-0.39096] = 146,168
Your car loan is for 10,000 to be paid over 5 years. If the interest rate is 12% per year, what are your monthly payments?
10000= PMT(PVIFA(1%,60 yrs) PMT = 10000/ PVIFA = 222.44
What is the price of a b bond ond with a 10 percent yield to maturity if the coupon payments are $40 every six months and the bond matures in 5 years?
Price = 40 PVIFA (5%, 10 periods) + 1000 PVIF (5%, 10 yrs) = 308.87 + 613.91 = 922.78
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While you were a student in college, you borrowed $12,000 in student loans at an interest rate of 9% per year, year, compounded annually. If you repay $1,500 per year, year, how long to the nearest year, will it take you to repay the loan?
12000 = 1500 PVIFA (9%, N)
8=PVIFA(9%,N) N=6.29 years
You need to accumulate $10,000. To do so, you plan to make deposits of $1,250 per year, with the first payment being made a year from today, in a bank account acco unt which pays 12 % interest, compounded annually. Your last deposit will be less than $1,250 if less is needed to round out to $10,000. How many years will will it take you to reach reach your $10,000 goal, and how large will the last deposit be?
N=5.94 years. years. After 5 years, years, 1250(6.3528) = 7,941 7941(1.12) = 8893.92 and 10000 - 8893.92=1106.92 Can you suggest another method for solving this problem?
Your client is 40 years old and wants to begin saving for retirement. You advise the client to put $5,000 a year into the the stock market. You estimate that the market’s market’s return will be on average of 12% a year. Assume the investment will be made at the end of the year. How much money will will she have by age 65? By age 70?
FV= 5000 FVIFA(12%, 25 years) = 666,669 FV= 5000 FVIFA(12%, FVIFA(12%, 30 years) years) = 1,206,663
You just started your first job, and you you want to buy a house within within 3 years. You are currently saving for the the down payment. You plan to save $5,000 the first year. You also anticipate that the amount you save each year will rise by 10 percent a year as your salary increases over time. Interest rates are assumed assumed to be 7 percent, and all savings savings occur at year end. How much money will you you have for a down payment in 3 years?
5000(0.07-0.10) [1-(1.1/1.07)^3 ] =14,415.41
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14415.41(1.07)^3 =17,659.50
A father is planning a savings savings program to put his daughter through college. His daughter is now 13 years old. She plans to enroll at the university in 5 years, years, and it should take her 4 years to complete her education. education. Currently, the cost per year is $12,500, but a 5% annual inflation rate in the cost is expected. The daughter recently received $7,500 from her grandfather’s estate; this money, which is invested in a bank account paying 8% interest, compounded annually, will be used to help meet the costs of the daughter’s education. The rest of the costs will be met by money the father deposits in the savings account. He will make 6 equal deposits to the account, one deposit in each year from noe until his daughter starts college. These deposits will will begin today and will also also earn 8 % on interest, compounded annually. How large must each deposit to to be for him to be able to put his daughter through college? 14
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16
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18 -15954
19 -16751 2
20 -17589
21 -18468
PV of these payments at age 18 is 61,203 minus 7500(1.09)^5 =61203-11,020=50,183
50183= PMT FVIFA(8%,6 yrs)
PMT=6,841
You are 65 years old and an d are considering whether it pays to buy bu y an annuity from an insurance company. For a cost of $10,000 the insurance insurance company will pay you $1,000 per year for the rest of your life. If you can earn 8% per year on your money in a bank account and expect to live until age 80, is is it worth buying the the annuity? What implied interest rate is the insurance company paying you? How long must you live live for the annuity to be worthwhile?
PV=8,559.48. Implied rate=5.56%. If you live 21 years you make 8%. Inflation questions: At age 20 you save $100 and invest it at a nominal interest rate of 8%. At age 65, you will have $3,192. Given the expected inflation is 5% per year over these years, years, what is the real future value?
Method 1: Find the real rate 1.08/1.05 – 1 = 0.02857 100 (1.02857)^45 = $355 or Method 2 : 100(1.08)^35 /1.05^45 =3,192 / 8.985 =$355
You plan to buy a car four years from now and want to invest enough money now to pay for it. Your desired model now costs $10,000, and the interest rate you you can earn on your money is 8% per year. If inflation in car prices is 5% per year, how much do you have to invest now?
Answer is= $8,934
You take out a three-year $10,000 $ 10,000 mortgage with two points at a stated APR of 12% with monthly payments. What is the true APR on the loan?
PMT= 332.14
9800=332.14(PVIFA(R%, 36 mo).
R=1.1175% times 12 = 13.41%
You must pay a creditor $6,000 one year from now, $5,000 two years from now, $4,000 three years from now, $2,000 four years from now, and a final $1,000 five years from now. You would like to restructure restructure the loan into five equal annual payments payments due at the end of each year. If agreed interest rate rate is 6% compounded annually, what is your payment?
PV = 15,800.28
15800.28 = PMT PVIFA(6%,5 yrs)
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PMT = $3,750
Sam charged $1,000 worth of merchandise one year ago on his MasterCard, which has a stated interest rate of 18% APR compounded monthly. He made 12 regular monthly payments of $50, at the end of each month, and refrained from using the card for the past year. How much does she still owe?
1.5% $50 a month 12 payments PV=545.38 454.62(1.015)^12= 543.55
1000-545.38 =454.62
You have decided to acquire a new car that costs costs $30,000. You are considering whether to lease it for the years or to purchase it and finance the purchase with a three-year installment loan. loan. The lease requires no down payment and lasts for three years. Lease payments are $400 monthly starting next month, whereas the installment loan will require monthly payments starting next month at an annual percentage rate (APR) of 8%. If you expect the resale value of the car to be $20,000 three years from now, should you buy or lease it? What is the break-even resale resale price of the car three years years from now?
PV of 20,000 at 8%/12 8%/12 per month for 36 months = 15,745.09 Net loan is 30000-15745.09 = 14,254.09 at 0.08/15 % for 36, monthly payments are 446.67. Lease is better. PV of 400 for 36 months is 12,764.72. 30000 – 12764.28 =17,235.28 PV of resale value or 21,892.89 resale break-even. You are planning to buy a car worth $20,000. Which of the two deals described below would you choose: The dealer offers to take 10% off the price and lend you the balance for 5 years at the regular financing rate (APR of 9%). The dealer offers to lend you $20,000 $ 20,000 (with no discount) for 5 years at a special financing rate of 3%.
10% of the price at 9% interest PMT= 373.65 Full price at 3% interest, PMT=359.37
You bought a house a year ago for $250,000, borrowing$200,000 at 10% on a 30-year term loan(with monthly payments). Interest rates rates have since come down to 9%. You can refinance your loan at this rate with a closing cost cost that will be 3% of the loan. You opportunity cost is 8%. Ignore tax effects.
How much are your monthly payments on your current loan (10%)? How much would your monthly payments be if you could refinance your remaining mortgage plus 3% points at 9% (with a 30-year term loan)? You plan to stay in in this house for next five five years. Given the refinancing cost, would you you refinance this loan? 200,000 = PMT PVIFA(0.1/12%, PVIFA(0.1/12%, 360 months) PMT = 1,755.14 Next year’s loan balance= 1,755.14 PVIFA(0.1/12, 348 months) = 198,887.89
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New loan 198,887.89 (1.03) =204,854.52 PMT on a 30-yr loan at 9% = 1,648.31 Balance at the end of 5 years : Original mortgage 1755.14 PVIFA( 0.1/12, 300) PV= 193,148.30 New loan 1648.31 PVIFA(0.09/12, 300) PV= 196,415.29 1755.14-1648.31 = 106.83 per month savings for 60 months but you owe 3,266.99 more at the end of 5 years. PV of annuity of 106.83 at 8% is 5,268.69 PV of 3,266.99 is 2,194.85 5268.69-2192.85>0, refinance. •
Your client has $250,000 in savings in in the bank. He is 55 years old old and expects to work for 10 more years, making making $100,000 a year. Ignore taxes and assume he can make 5 % on his investments. Once he retires 10 years from now, he h e would like to be able to withdraw $80,000 a year for the following 25 years. How much would he need in the bank 10 years years from now to be able to to do this? How much of his income would he need to save each year for the next 10 years to be able to afford these planned withdrawals after the tenth year?
PV=80,000 PVIFA(5%, 25 years) = 1,159,014.82 FV of 250,000 = 250,000 (1.05) ^10 = 407,223.66 The difference = 679,791.16. 679,791.16 = 88,036.07 PVIFA(5%, 10 years)
Your client is 55 years old with $1 million invested in the stock market and wants wan ts to retire in 5 years and desires to withdraw 200,000 per year for 30 years starting in in 6 years. She calculates that at an 8% annual return earned on investments, she needs to save $133,336 per year for each of the next 5 years before before retirement. [Your client needs $2,251,557 at the time time of the retirement to be able to to withdraw $200,000 per year for 30 years. Her current investment investment of $1 million would be worth $1,469,328 in 5 years. Thus, she needs to save and invest only 133,336 per year for 5 years to accumulate the difference between 2,251,557 and 1,469,328=782,229] You suggest that the client invests at least 25% of her wealth in fixed-income securities. securities. But this recommendation reduces the expected returns from 8% to 6%. Find the required annual investments for the next 5 years under the return assumption of 6%.
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