Question 1

The present value of an ordinary annuity of $350 each year for five years, assuming an opportunity cost of 4 percent, is

Accepted Answer

The formula for present value of an ordinary annuity is:

PV = [(1 - (1/(1+r)^n)) / r] x PMT

Where PV is the present value, r is the discount rate (which is 4% in this case), n is the number of periods (which is 5 in this case), and PMT is the annual payment (which is $350 in this case).

Plugging in the numbers, we get:

PV = [(1 - (1/(1+0.04)^5)) / 0.04] x 350
PV = [(1 - (1/1.2167)) / 0.04] x 350
PV = [(1 - 0.8227) / 0.04] x 350
PV = (0.1773 / 0.04) x 350
PV = 4.4325 x 350
PV = $1,558.88

Therefore, the answer is D.

Question 2

Calculate the present value of a $10,000 perpetuity at a 6 percent discount rate.

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Question 3

Jia has just won a $20 million lottery, which will pay her $1 million at the end of each year for 20 years. An investor has offered her $10 million for this annuity. She estimates that she can earn 10 percent interest, compounded annually, on any amounts she invests. She asks your advice on whether to accept or reject the offer. What will you tell her? (Ignore Taxes)

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Question 4

To pay for her college education, Gina is saving $2,000 at the beginning of each year for the next eight years in a bank account paying 12 percent interest. How much will Gina have in that account at the end of 8th year?

Accepted Answer

We can use the formula for future value of an annuity to calculate the amount in the account at the end of 8 years:
FV = PMT * ((1 + r)^n - 1)/r
where PMT is the annual payment ($2,000), r is the interest rate (12% or 0.12), and n is the number of periods (8). Plugging in the values, we get:
FV = 2000 * ((1 + 0.12)^8 - 1)/0.12
FV = $27,552
Therefore, Gina will have $27,552 in the account at the end of the 8th year. Option D is the correct answer.

Question 5

Nico establishes a seven-year, 8 percent loan with a bank requiring annual end-of-year payments of $960.43. Calculate the original principal amount.

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Question 6

The present value of an ordinary annuity of $2,350 each year for eight years, assuming an opportunity cost of 11 percent, is

Accepted Answer

The present value of an ordinary annuity formula is:

PV = C * [(1 - (1 + r)^(-n)) / r]

Where:
PV = Present Value
C = Cash Flow per period
r = Opportunity cost or discount rate
n = Number of periods

Using the given values:

PV = $2,350 * [(1 - (1 + 0.11)^(-8)) / 0.11]
PV = $2,350 * [(1 - 0.31799) / 0.11]
PV = $2,350 * 6.2372
PV = $14,668.82

Therefore, the present value of the annuity is $14,668.82, which corresponds to option D.

Question 7

A generous philanthropist plans to make a one-time endowment to a renowned heart research center which would provide the facility with $250,000 per year into perpetuity. The rate of interest is expected to be 8 percent for all future time periods. How large must the endowment be?

Accepted Answer

Using the perpetuity formula, we can determine that the endowment must be:

Endowment = Annual Payment / Interest Rate

Endowment = $250,000 / 0.08

Endowment = $3,125,000

Therefore, the best choice is C.

Question 8

You have been offered a project paying $300 at the beginning of each year for the next 20 years. What is the maximum amount of money you would invest in this project if you expect 9 percent rate of return to your investment?

Accepted Answer

The formula for calculating the present value of an annuity is:
PV = C x ((1 - (1 + r)^-n) / r)
where PV is the present value, C is the annual cash flow, r is the discount rate, and n is the number of periods.
In this case, C = $300, r = 9%, and n = 20.
Plugging these values into the formula, we get:
PV = $300 x ((1 - (1 + 0.09)^-20) / 0.09) = $2,985.18
Therefore, the maximum amount of money that should be invested in this project is $2,985.18.

Question 9

Mary will receive $12,000 per year for the next 10 years as royalty for her work on a finance book. What is the present value of her royalty income if the opportunity cost is 12 percent?

Accepted Answer

We can use the present value formula to calculate the present value of Mary's royalty income:
PV = PMT x [1 - (1+r)^(-n)] / r
where PV is the present value, PMT is the annual payment, r is the opportunity cost, and n is the number of years.

Plugging in the given values, we get:
PV = $12,000 x [1 - (1+0.12)^(-10)] / 0.12
PV = $67,800

Therefore, the present value of Mary's royalty income is $67,800. Choice B is the correct answer.

Question 10

Dottie has decided to set up an account that will pay her granddaughter (Lexi) $5,000 a year indefinitely. How much should Dottie deposit in an account paying 8 percent annual interest?

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Question 11

Calculate the future value of an annuity of $5,000 each year for eight years, deposited at 6 percent.

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Question 12

In their meeting with their advisor, Mr. and Mrs. O'Rourke concluded that they would need $40,000 per year during their retirement years in order to live comfortably. They will retire 10 years from now and expect a 20-year retirement period. How much should Mr. and Mrs. O'Rourke deposit now in a bank account paying 9 percent to reach financial happiness during retirement?

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Question 13

Mr. Knowitall has been awarded a bonus for his outstanding work. His employer offers him a choice of a lump-sum of $5,000 today, or an annuity of $1,250 a year for the next five years. Which option should Mr. Knowitall choose if his opportunity cost is 9 percent?

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Question 14

A wealthy industrialist wishes to establish a $2,000,000 trust fund which will provide income for his grandchild into perpetuity. He stipulates in the trust agreement that the principal may not be distributed. The grandchild may only receive the interest earned. If the interest rate earned on the trust is expected to be at least 7 percent in all future periods, how much income will the grandchild receive each year?

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Question 15

James plans to fund his individual retirement account, beginning today, with 20 annual deposits of $2,000, which he will continue for the next 20 years. If he can earn an annual compound rate of 8 percent on his deposits, the amount in the account upon retirement will be

Accepted Answer

Using the formula for future value of an annuity:

FV = PMT x (((1 + r)^n - 1) / r)

Where:
PMT = Annual deposit = $2,000
r = Annual interest rate = 8% or 0.08
n = Number of deposits = 20

FV = $2,000 x (((1 + 0.08)^20 - 1) / 0.08)
FV = $2,000 x (6.352) 
FV = $12,703.77 (amount at the end of 20 years)

Then, using the formula for future value of a lump sum:

FV = PV x (1 + r)^n

Where:
PV = Present value = $12,703.77
r = Annual interest rate = 8% or 0.08
n = Number of years = 20

FV = $12,703.77 x (1 + 0.08)^20
FV = $98,845.88

Therefore, the amount in the account upon retirement will be approximately $98,846. Answer choice C is correct.

Question 16

A lottery administrator has just completed the state's most recent $50 million lottery. Receipts from lottery sales were $50 million and the payout will be $5 million at the end of each year for 10 years. The expenses of running the lottery were $800,000. The state can earn an annual compound rate of 8 percent on any funds invested.
(a) Calculate the gross profit to the state from this lottery.
(b) Calculate the net profit to the state from this lottery (no taxes).

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Question 17

A generous benefactor to the local ballet plans to make a one-time endowment which would provide the ballet with $150,000 per year into perpetuity. The rate of interest is expected to be 5 percent for all future time periods. How large must the endowment be?

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Question 18

A college received a contribution to its endowment fund of $2 million. They can never touch the principal, but they can use the earnings. At an assumed interest rate of 9.5 percent, how much can the college earn to help its operations each year?

Accepted Answer

To calculate the earnings, we multiply the principal by the interest rate as a decimal: $2,000,000 x 0.095 = $190,000. Therefore, the college can earn $190,000 per year to help its operations.

Question 19

If the present value of a perpetual income stream is increasing, the discount rate must be

Accepted Answer

The present value of a perpetual income stream increases when the discount rate decreases, as the present value is inversely related to the discount rate.

Question 20

Calculate the present value of an annuity of $3,900 each year for four years, assuming an opportunity cost of 10 percent.

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Quiz 5: Time Value of Money