Business Finance Final Exam (Chapter 4)
The future value of a lump sum investment will increase if you:
increase the time period
The relationship between the present value and the investment time period is best described as:
inverse
Starlite Industries will need $2.2 million 4.5 years from now to replace some equipment. Currently, the firm has some extra cash and would like to establish a savings account for this purpose. The account pays 3.6 percent interest, compounded annually. How much money must the company deposit today to fully fund the equipment purchase?
$1,876,306.49 Present value = $2,200,000/(1 + .036)4.5= $1,876,306.49
Starlite Industries will need $2.2 million 4.5 years from now to replace some equipment. Currently, the firm has some extra cash and would like to establish a savings account for this purpose. The account pays 3.6 percent interest, compounded annually. How much money must the company deposit today to fully fund the equipment purchase?
$1,876,306.49 Present value = $2,200,000/(1 + .036)^4.5= $1,876,306.49
You are scheduled to receive $7,500 in two years. When you receive it, you will invest it at 4.5 percent per year. How much will your investment be worth ten years from now?
$10,665.75 Future value = $7,500 ×(1 + .045)(10 - 2) = $10,665.75
What is the future value of $1,000 invested for 15 years at a rate of 5%?
$2,079 Financial calculator key strokes: N = 15 I/YR = 5 PV = $1,000 PMT = $0 FV = ??? = $2,079
You have $5,000 you want to invest for the next 45 years. You are offered an investment plan that will pay you 6 percent per year for the next 15 years and 10 percent per year for the last 30 years. How much will you have at the end of the 45 years?
$209,092.54 Future value A = $5,000 ×(1 + .06)15 ×(1 + .10)30 = $209,092.54
Western Bank pays 5 percent simple interest on its savings account balances, whereas Eastern Bank pays 5 percent compounded annually. If you deposited $6,000in each bank, how much more money would you earn from the Eastern Bank account at the end of 3 years?
$45.75 Future value Western = $6,000 + ($6,000 ×.05 ×3) = $6,900 Future value Eastern = $6,000 × (1 + .05)3 = $6,945.75 Difference = $6,945.75- 6,900 = $45.75
Today, you deposit $2,500 in a bank account that pays 3.6 percent simple interest. How much interest will you earn over the next 5 years?
$450.00 Interest = $2,500 ×.036 ×5 = $450
Eleven years ago, you deposited $3,200 into an account. Seven years ago, you added an additional $1,000 to this account. You earned 9.2 percent, compounded annually, for the first 4 years and 5.5 percent, compounded annually, for the last 7 years. How much money do you have in your account today?
$8,073.91 Future value = {[$3,200 ×(1 + .092)4] + $1,000} ×(1 + .055)7 = $8,073.91
How long will it take to double your savings if you earn 6.4 percent interest, compounded annually?
11.17 years 2 = $1 ×(1 + .064)^t t = 11.17 years
Isaac only has $1,090 today but needs $1,979 to buy a new computer. How long will he have to wait to buy the computer if he earns 5.4 percent compounded annually on his savings? Assume the price of the computer remains constant
11.34 years $1,979 = $1,090 ×(1 + .054)^t t = 11.34 years
At 10 percent interest, how long does it take to triple your money?
11.53 years $3 = $1 ×(1 + .10)^t t = 11.53 years
You have been told that you need $32,000 today for each $100,000 you want when you retire 28 years from now. What rate of interest was used in the present value computation? Assume interest is compounded annually
4.15 percent $100,000 = $32,000 ×(1 + r)28 r = 4.15 percent
Jenny needs to borrow $5,500 for four years. The loan will be repaid in one lump sum at the end of the loan term. Which one of the following interest rates is best for Jenny?
6.5 percent simple interest
The variables in a future value of a lump sum problem include all of the following, except:
Annuity Payments
You're trying to save to buy a new car valued at $48,690. You have $38,000 today that can be invested at your bank. The bank pays 3.7 percent annual interest on its accounts. How long will it be before you have enough to buy the car for cash? Assume the price of the car remains constant.
6.82 years $48,690 = $38,000 ×(1 + .037)^t t = 6.82 years
When you were born, your parents opened an investment account in your name and deposited $1,500 into the account. The account has earned an average annual rate of return of 5.3percent. Today, the account is valued at $42,856. How old are you?
64.91 years $42,856 = $1,500 ×(1 + .053)^t t = 64.91 years
You have $1,500 today in your savings account. How long must you wait for your savings to be worth $4,000 if you are earning 1.1 percent interest, compounded annually?
89.66 years $4,000 = $1,500 ×(1 + 0.011)t t = 89.66years
How would a decrease in the interest rate effect the future value of a lump sum, single amount problem (all other variables remain the same)?
Decrease the future value
You can earn .49 percent per month at your bank. If you deposit $3,200, how long must you wait until your account has grown to $6,200?
Explanation: The time line is: 0 t Picture −$3,200 $6,200 To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)^t $6,200 = $3,200(1.0049)^t t = ln($6,200 / $3,200) / ln1.0049 t = 135.31 months
You have $18,000 you want to invest for the next 36 years. You are offered an investment plan that will pay you 8 percent per year for the next 18 years and 12 percent per year for the last 18 years. How much will you have at the end of the 36 years? If the investment plan pays you 12 percent per year for the first 18 years and 8 percent per year for the next 18 years, how much will you have at the end of the 36 years?
Explanation: In this case, we have an investment that earns two different interest rates. We will calculate the value of the investment at the end of the first 18 years, then use this value with the second interest rate to find the final value at the end of 36 years. Using the future value equation, at the end of the first 18 years, the account will be worth: Value in 18 years = PV(1 + r)^t Value in 18 years = $18,000(1.08)^18 Value in 18 years = $71,928.35 Now we can find out how much this will be worth 18 years later at the end of the investment. Using the future value equation, we find: Value in 36 years = PV(1 + r)^t Value in 36 years = $71,928.35(1.12)^18 Value in 36 years = $553,126.56 It is irrelevant which interest rate is offered when as long as each interest rate is offered for 18 years. We can find the value of the initial investment in 36 years with the following: FV = PV(1 + r1)^t(1 + r2)^t FV = $18,000(1.08)^18(1.12)^18 FV = $553,126.56
Assume that in 2015, the first edition of a comic book was sold at auction for $2,157,000. The comic book was originally sold in 1943 for $.10. For this to have been true, what was the annual increase in the value of the comic book?
Explanation: To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)^t Solving for r, we get: r = (FV / PV)^1/t - 1 r = ($2,157,000 / $.10)^1/72 - 1 r = .2643, or 26.43%
Assume the total cost of a college education will be $365,000 when your child enters college in 18 years. You presently have $59,000 to invest. What annual rate of interest must you earn on your investment to cover the cost of your child's college education?
Explanation: To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)^t Solving for r, we get: r = (FV / PV)^1/t - 1 r = ($365,000 / $59,000)^1/18 - 1 r = .1065, or 10.65%
In 1906, the first Green Jacket Golf Championship was held. The winner's prize money was $260. In 2015, the winner's check was $1,600,000. What was the annual percentage increase in the winner's check over this period? If the Winner's prize increases at the same rate, what will it be in 2048?
Explanation: To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)^t Solving for r, we get: r = (FV / PV)^1/t - 1 r = ($1,600,000 / $260)^1/109 - 1 r = .0833, or 8.33% To find the FV of the first prize in 2048, we use: FV = PV(1 + r)^t FV = $1,600,000(1.0833)^33 FV = $22,453,953.40
In 1902, the first Green Jacket Golf Championship was held. The winner's prize money was $320. In 2015, the winner's check was $1,680,000. What was the annual percentage increase in the winner's check over this period? If the winner's prize increases at the same rate, what will it be in 2051?
Explanation: To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)^t Solving for r, we get: r = (FV / PV)^1/t - 1 r = ($1,680,000 / $320)^1/113 - 1 r = .0788, or 7.88% To find the FV of the first prize in 2051, we use: FV = PV(1 + r)^t FV = $1,680,000(1.0788)^36 FV = $25,733,139.64
Assume that in 2014, an 1876 $20 double eagle sold for $16,400. What was the rate of return on this investment?
Explanation: To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)^t Solving for r, we get: r = (FV / PV)^1/t - 1 r = ($16,400 / $20)^1/138 - 1 r = .0498, or 4.98%
You have decided that you want to be a millionaire when you retire in 45 years. If you can earn an annual return of 11.24 percent, how much do you have to invest today? What if you can earn an annual return of 5.62 percent?
Explanation: To find the PV of a lump sum, we use: PV = FV / (1 + r)^t So, if you can earn 11.24 percent, you will need to invest: PV = $1,000,000 / (1.1124)^45 PV = $8,284.30 And if you can earn 5.62 percent, you will need to invest: PV = $1,000,000 / (1.0562)^45 PV = $85,393.08
You have just received notification that you have won the $2.03 million first prize in the Centennial Lottery. However, the prize will be awarded on your 100th birthday (assuming you're around to collect), 77 years from now. What is the present value of your windfall if the appropriate discount rate is 9 percent?
Explanation: To find the PV of a lump sum, we use: PV = FV / (1 + r)^t PV = $2,030,000 / 1.0977 PV = $2,664.75
Imprudential, Inc., has an unfunded pension liability of $765 million that must be paid in 25 years. To assess the value of the firm's stock, financial analysts want to discount this liability back to the present. If the relevant discount rate is 8 percent, what is the present value of this liability?
Explanation: To find the PV of a lump sum, we use: PV = FV / (1 + r)^t PV = $765,000,000 / (1.08)^25 PV = $111,703,697.26
You need $87,000 in 12 years. If you can earn .54 percent per month, how much will you have to deposit today?
Explanation: To find the PV of a lump sum, we use: PV = FV / (1 + r)^t PV = $87,000 / (1.0054)^144 PV = $40,060.87
First City Bank pays 7 percent simple interest on its savings account balances, whereas Second City Bank pays 7 percent interest compounded annually. If you made a deposit of $16,000 in each bank, how much more money would you earn from your Second City Bank account at the end of 11 years?
Explanation: The simple interest per year is: $16,000 × .07 = $1,120 So, after eleven years, you will have: $1,120 × 11 = $12,320 in interest. The total balance will be: Total balance = $16,000 + 12,320 Total balance = $28,320 With compound interest, we use the future value formula: FV = PV(1 + r)^t FV = $16,000(1.07)^11 FV = $33,677.63 The difference is: Difference = $33,677.63 - 28,320 Difference = $5,357.63
Solve for the unknown number of years in each of the following: Present Value Years Interest Rate Future Value 900 ? 12 1,755 2,591 ? 10 4,350 34,105 ? 15 393,620 33,80 ? 22 217,868
Explanation: To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)^t Solving for t, we get: t = ln(FV / PV) / ln(1 + r) FV = $1,755 = $900(1.12)^t t = ln($1,755 / $900) / ln1.12 t = 5.89 years FV = $4,350 = $2,591(1.10)^t t = ln($4,350 / $2,591) / ln1.10 t = 5.44 years FV = $393,620 = $34,105(1.15)^t t = ln($393,620 / $34,105) / ln1.15 t = 17.50 years FV = $217,868 = $33,800(1.22)^t t = ln($217,868 / $33,800) / ln1.22 t = 9.37 years
At 9 percent interest, how long does it take to double your money? At 9 percent interest, how long does it take to quadruple your money?
Explanation: To find the length of time for money to double, triple, etc., the present value and future value are irrelevant as long as the future value is twice the present value for doubling, three times as large for tripling, etc. To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)^t Solving for t, we get: t = ln(FV / PV) / ln(1 + r) The length of time to double your money is: FV = $2 = $1(1.09)^t t = ln2 / ln1.09 t = 8.04 years The length of time to quadruple your money is: FV = $4 = $1(1.09^)t t = ln4 / ln1.09 t = 16.09 years
What is the present value of $1,000 to be received in 12 years invested at a rate of 8%?
Financial calculator key strokes: N = 12 I/YR = 8 PMT = $0 FV = $1,000 PV = ??? = $397
What is the present value of $1,200 to be received in 18 years invested at a rate of 5%?
Financial calculator key strokes: N = 18 I/YR = 5 PMT = $0 FV = $1,200 PV= ??? = $499
What is the future value of $1,200 invested for 20 years at a rate of 6%?
Financial calculator key strokes: N = 20 I/YR = 6 PV = $1,200 PMT = $0 FV = ??? = $3,849
The variables in a present value of a lump sum problem include all of the following, except:
Free Cash Flow
The variable that you are solving for in a future value of a lump sum problem is:
Future value
How would a decrease in the interest rate effect the present value of a lump sum, single amount problem (all other variables remain the same)?
Increase the present value
Stacey deposits $5,000 into an account that pays 2 percent interest, compounded annually. At the same time, Kurt deposits $5,000 into an account paying 3.5 percent interest, compounded annually. At the end of three years:
Kurt will have a larger account value than Stacey will
Which one of the following is the correct formula for the current value of $600 invested today at 5 percent interest for 6 years?
PV = $600 / (1 + .05)^6
The variables in a future value of a lump sum problem include all of the following, except:
Payments
The variables in a present value of a lump sum problem include all of the following, except:
Payments
Katlyn needs to invest $5,318 today in order for her savings account to be worth $8,000 six years from now. Which one of the following terms refers to the $5,318?
Present Value
The variable that you are solving for in a present value of a lump sum problem is:
Present value
Today, Charity wants to invest less than $3,000 with the goal of receiving $3,000 back some time in the future. Which one of the following statements is correct?
The period of time she has to wait decreases as the amount she invests increases
You're trying to save to buy a new $205,000 Ferrari. You have $34,000 today that can be invested at your bank. The bank pays 4.1 percent annual interest on its accounts. How long will it be before you have enough to buy the car?
To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)^t Solving for t, we get: t = ln(FV / PV) / ln(1 + r) FV = $205,000 = $34,000(1.041)^t t = ln($205,000 / $34,000) / ln1.041 t = 44.71 years
For each of the following, compute the future value: Present Value Years Interest Rate Future Value 2,500 9 16 ? 9,753 22 7 ? 102,305 16 9 ? 240,382 32 3 ?
To find the FV of a lump sum, we use: FV = PV(1 + r)^t FV = $2,500(1.16)^9 = $9,507.40 FV = $9,753(1.07)^22 = $43,209.71 FV = $102,305(1.09)^16 = $406,182.14 FV = $240,382(1.03)^32 = $619,003.54
You have $9,100 to deposit. Regency Bank offers 9 percent per year compounded monthly (.75 percent per month), while King Bank offers 9 percent but will only compound annually. How much will your investment be worth in 19 years at each Bank? Investment value Regency Bank ? King Bank ?
To find the FV of a lump sum, we use: FV = PV(1 + r)^t In Regency Bank, you will have: FV = $9,100(1.0075)^228 FV = $49,993.54 And in King Bank, you will have: FV = $9,100(1.09)^19 FV = $46,789.12
Your coin collection contains 85 1952 silver dollars. If your grandparents purchased them for their face value when they were new, how much will your collection be worth when you retire in 2062, assuming they appreciate at an annual rate of 5.1 percent?
To find the FV of a lump sum, we use: FV = PV(1 + r)^t FV = $85(1.051)^110 FV = $20,217.05
Solve for the unknown interest rate in each of the following: Present Value Years Interest Rate Future Value 770 5 ? 1,491 960 6 ? 1,828 20,500 17 ? 147,332 75,800 20 ? 323,815
We will use the FV formula, that is: FV = PV(1 + r)t Solving for r, we get: r = (FV / PV)^1/t - 1 FV = $1,491 = $770(1 + r)^5 r = ($1,491 / $770)^1/5 - 1 r = .1413, or 14.13% FV = $1,828 = $960(1 + r)^6 r = ($1,828 / $960)^1/6 - 1 r = .1133, or 11.33% FV = $147,332 = $20,500(1 + r)^17 r = ($147,332 / $20,500)^1/17 - 1 r = .1230, or 12.30% FV = $323,815 = $75,800(1 + r)^20 r = ($323,815 / $75,800)^1/20 - 1 r = .0753, or 7.53%
In March 2015, Daniela Motor Financing (DMF), offered some securities for sale to the public. Under the terms of the deal, DMF promised to repay the owner of one of these securities $2,000 in March 2045, but investors would receive nothing until then. Investors paid DMF $790 for each of these securities; so they gave up $790 in March 2015, for the promise of a $2,000 payment 30 years later. a. Assuming you purchased the bond for $790, what rate of return would you earn if you held the bond for 30 years until it matured with a value $2,000? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.) b. Suppose under the terms of the bond you could redeem the bond in 2026. DMF agreed to pay an annual interest rate of .9 percent until that date. How much would the bond be worth at that time? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) c. In 2026, instead of cashing in the bond for its then current value, you decide to hold the bond until it matures in 2045. What annual rate of return will you earn over the last 19 years?
a. To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)^t Solving for r, we get: r = (FV / PV)^1/t - 1 r = (FV / PV)^1/t - 1 r = ($2,000 / $790)^1/30 - 1 r = .0314, or 3.14% b. Using the FV formula, we get: FV = PV(1 +r)^t FV = $790(1 + .009)^11 FV = $871.83 c. Using the FV formula and solving for the interest rate, we get: r = (FV / PV)^1/t - 1 r = ($2,000 / $871.83)^1/19 - 1 r = .0447, or 4.47%
Lester had $6,270 in his savings account at the beginning of this year. This amount includes both the $6,000 he originally invested at the beginning of last year plus the $270 he earned in interest last year. This year, Lester earned a total of $282.15 in interest even though the interest rate on the account remained constant. This $282.15 is best described as:
compound interest
Computing the present value of a future cash flow to determine what that cash flow is worth today is called:
discounted cash flow valuation
Given an interest rate of zero percent, the future value of a lump sum invested today will always:
remain constant, regardless of the investment time period
