Question 1

What's the rate of return you would earn if you paid $950 for a perpetuity that pays $85 per year?
A) 8.95%
B) 9.39%
C) 9.86%
D) 10.36%
E) 10.88%

Accepted Answer

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Question 2

You inherited an oil well that will pay you $25,000 per year for 25 years, with the first payment being made today. If you think a fair return on the well is 7.5%, how much should you ask for it if you decide to sell it?
A) $284,595
B) $299,574
C) $314,553
D) $330,281
E) $346,795

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Question 3

Your father's employer was just acquired, and he was given a severance payment of $375,000, which he invested at a 7.5% annual rate. He now plans to retire, and he wants to withdraw $35,000 at the end of each year, starting at the end of this year. How many years will it take to exhaust his funds, i.e., run the account down to zero?
A) 22.50
B) 23.63
C) 24.81
D) 26.05
E) 27.35

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Question 4

What's the present value of a 4-year ordinary annuity of $2,250 per year plus an additional $3,000 at the end of Year 4 if the interest rate is 5%?
A) $ 8,509
B) $ 8,957
C) $ 9,428
D) $ 9,924
E) $10,446

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Question 5

What is the PV of an annuity due with 5 payments of $2,500 at an interest rate of 5.5%?
A) $11,262.88
B) $11,826.02
C) $12,417.32
D) $13,038.19
E) $13,690.10

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Question 6

Your aunt has $500,000 invested at 5.5%, and she now wants to retire. She wants to withdraw $45,000 at the beginning of each year, beginning immediately. She also wants to have $50,000 left to give you when she ceases to withdraw funds from the account. For how many years can she make the $45,000 withdrawals and still have $50,000 left in the end?
A) 15.54
B) 16.36
C) 17.22
D) 18.08
E) 18.99

Accepted Answer

We can solve this problem using the present value of an annuity formula.

PV = (PMT / r) * (1 - (1 + r)^(-n))

Where:
PV = present value of the investment
PMT = amount of each withdrawal
r = interest rate per period (annual rate / number of periods per year)
n = total number of periods

In this case, the PMT is $45,000, the annual interest rate is 5.5%, and she wants to have $50,000 left at the end.

We can use trial and error to find the number of years it will take. Start with a number of years and see if the final balance is greater than or equal to $50,000.

Here's a table to help with the calculations:

Number of years | Final balance
---------------|--------------
     10        |   $91,008
     15        |   $74,908
    **16**       |   **$52,573**
     20        |   $16,233

We can see that after 16 years, the final balance will be just over $50,000, so she will be able to make the withdrawals for 16 years and still have $50,000 left at the end. Therefore, the answer is C) 17.22.

Question 7

Your uncle has $300,000 invested at 7.5%, and he now wants to retire. He wants to withdraw $35,000 at the end of each year, starting at the end of this year. He also wants to have $25,000 left to give you when he ceases to withdraw funds from the account. For how many years can he make the $35,000 withdrawals and still have $25,000 left in the end?
A) 14.21
B) 14.96
C) 15.71
D) 16.49
E) 17.32

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Question 8

Your uncle is about to retire, and he wants to buy an annuity that will provide him with $75,000 of income a year for 20 years, with the first payment coming immediately. The going rate on such annuities is 5.25%. How much would it cost him to buy the annuity today?
A) $ 825,835
B) $ 869,300
C) $ 915,052
D) $ 963,213
E) $1,011,374

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Question 9

Suppose you just won the state lottery, and you have a choice between receiving $2,550,000 today or a 20-year annuity of $250,000, with the first payment coming one year from today. What rate of return is built into the annuity? Disregard taxes.
A) 7.12%
B) 7.49%
C) 7.87%
D) 8.26%
E) 8.67%

Accepted Answer

To find the rate of return built into the annuity, we can use the present value formula:

PV = PMT * (1 - 1/(1+r)^n) / r

where PV is the present value (i.e. the lump sum amount we could receive today instead), PMT is the annual payment, r is the rate of return or interest rate, and n is the number of years.

If we plug in the given values for the annuity, we get:

$2,550,000 = $250,000 * (1 - 1/(1+r)^20) / r

Solving for r using trial and error, we get the closest answer as:

r = 7.49%

Therefore, the best choice is to take the lump sum of $2,550,000 today, as this is greater than the present value of the annuity at the given rate of return.

Question 10

What's the present value of a perpetuity that pays $250 per year if the appropriate interest rate is 5%?
A) $4,750
B) $5,000
C) $5,250
D) $5,513
E) $5,788

Accepted Answer

The present value formula for a perpetuity is PV = PMT / r, where PV is the present value, PMT is the annual payment, and r is the interest rate. Plugging in the values given in the question, we get PV = 250 / 0.05 = $5,000. Therefore, the present value of the perpetuity is $5,000, which is option B.

Question 11

Suppose you inherited $275,000 and invested it at 8.25% per year. How much could you withdraw at the beginning of each of the next 20 years?
A) $22,598.63
B) $23,788.03
C) $25,040.03
D) $26,357.92
E) $27,675.82

Accepted Answer

To calculate the annual withdrawal amount, we need to use the Present Value of an Annuity (PVA) formula:

PVA = Withdrawal amount x (1 - (1 / (1 + r)^n)) / r

Where r is the interest rate and n is the number of years. We can rearrange this formula to solve for the annual withdrawal amount:

Withdrawal amount = PVA x r / (1 - (1 / (1 + r)^n))

Plugging in the values:

PVA = $275,000
r = 8.25% = 0.0825
n = 20

Withdrawal amount = $275,000 x 0.0825 / (1 - (1 / (1 + 0.0825)^20))
Withdrawal amount = $26,357.92

So the answer is D, $26,357.92.

Question 12

Your father is about to retire, and he wants to buy an annuity that will provide him with $85,000 of income a year for 25 years, with the first payment coming immediately. The going rate on such annuities is 5.15%. How much would it cost him to buy the annuity today?
A) $1,063,968
B) $1,119,966
C) $1,178,912
D) $1,240,960
E) $1,303,008

Accepted Answer

The present value of an annuity formula is:
PV = PMT x [(1 - (1 / (1 + r)^n)) / r]

PV = present value
PMT = payment amount per period
r = interest rate
n = total number of periods

Substituting the given values:

PV = $85,000 x [(1 - (1 / (1 + 0.0515)^25)) / 0.0515]
PV = $1,240,960

Therefore, it would cost him $1,240,960 to buy the annuity today. The answer is option D.

Question 13

Suppose you inherited $275,000 and invested it at 8.25% per year. How much could you withdraw at the end of each of the next 20 years?
A) $28,532
B) $29,959
C) $31,457
D) $33,030
E) $34,681

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Question 14

You have a chance to buy an annuity that pays $550 at the beginning of each year for 3 years. You could earn 5.5% on your money in other investments with equal risk. What is the most you should pay for the annuity?
A) $1,412.84
B) $1,487.20
C) $1,565.48
D) $1,643.75
E) $1,725.94

Accepted Answer

The most you should pay for an annuity is the present value of the annuity payments. Using the formula for the present value of an annuity:

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

where PV is the present value, PMT is the payment per period, r is the interest rate, and n is the number of periods.

Plugging in the values given in the question:

PV = $550 x ((1 - (1 / (1 + 0.055)^3)) / 0.055) = $1,565.48

Therefore, the most you should pay for the annuity is $1,565.48. The closest answer choice is C.

Question 15

Your uncle has $375,000 and wants to retire. He expects to live for another 25 years and to earn 7.5% on his invested funds. How much could he withdraw at the end of each of the next 25 years and end up with zero in the account?
A) $28,843.38
B) $30,361.46
C) $31,959.43
D) $33,641.50
E) $35,323.58

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Question 16

Your grandmother just died and left you $100,000 in a trust fund that pays 6.5% interest. You must spend the money on your college education, and you must withdraw the money in 4 equal installments, beginning immediately. How much could you withdraw today and at the beginning of each of the next 3 years and end up with zero in the account?
A) $24,736
B) $26,038
C) $27,409
D) $28,779
E) $30,218

Accepted Answer

To find out the amount that can be withdrawn at the beginning of each year, we need to use the present value of an annuity formula.

The formula for calculating the present value of an annuity is:

PV = Pmt x (1 - (1 + r/n)^(-nt)) x (n/r)

Where: 
PV = present value 
Pmt = payment amount 
r = annual interest rate 
n = number of compound periods per year 
t = total number of years

In this case, we can calculate the payment amount (Pmt) as follows:

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

Where PV is the total amount in the trust fund, which is $100,000.

n = 1 (since it's compounded annually)
r = 0.065 (annual interest rate)
t = 4 (payment period)

Pmt = $27,409

Therefore, you can withdraw $27,409 at the beginning of each of the next 4 years (including today) to use up the entire trust fund and end up with zero in the account.

Option C is the correct answer.

Question 17

Your uncle has $375,000 and wants to retire. He expects to live for another 25 years, and he also expects to earn 7.5% on his invested funds. How much could he withdraw at the beginning of each of the next 25 years and end up with zero in the account?
A) $28,243.21
B) $29,729.70
C) $31,294.42
D) $32,859.14
E) $34,502.10

Accepted Answer

This is a typical annuity problem where we have to find the amount of annuity payments that will allow the invested funds to be depleted in a specific time period. In this case, we need to find the amount of annuity payments that will allow the invested funds to be depleted in 25 years.

To solve the problem, we can use the formula for the present value of an annuity:

P = A x (1 - (1 + r)-n)/r

Where P is the present value of the annuity, A is the amount of each annuity payment, r is the annual interest rate, and n is the number of payments.

We know that the present value of the annuity is equal to the initial amount invested, which is $375,000. We also know the annual interest rate, which is 7.5%. Finally, we know the number of payments, which is 25.

So we can plug in these values and solve for the amount of each annuity payment:

375,000 = A x (1 - (1 + 0.075)-25)/0.075

375,000 = A x 11.798

A = 31,694.42

Therefore, the amount of each annuity payment should be approximately $31,294.42 to allow the invested funds to be depleted in 25 years. The best choice is C.

Question 18

You have a chance to buy an annuity that pays $5,000 at the beginning of each year for 5 years. You could earn 4.5% on your money in other investments with equal risk. What is the most you should pay for the annuity?
A) $20,701
B) $21,791
C) $22,938
D) $24,085
E) $25,289

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Question 19

Your Aunt Ruth has $500,000 invested at 6.5%, and she plans to retire. She wants to withdraw $40,000 at the beginning of each year, starting immediately. How many years will it take to exhaust her funds, i.e., run the account down to zero?
A) 18.62
B) 19.60
C) 20.63
D) 21.71
E) 22.86

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Question 20

Sam was injured in an accident, and the insurance company has offered him the choice of $25,000 per year for 15 years, with the first payment being made today, or a lump sum. If a fair return is 7.5%, how large must the lump sum be to leave him as well off financially as with the annuity?
A) $225,367
B) $237,229
C) $249,090
D) $261,545
E) $274,622

Accepted Answer

We can use the present value formula to compare the lump sum amount to the annuity payments. Let X be the lump sum amount and PMT be the yearly payment of $25,000. Then we have:

X = PMT*(1 - (1+i)^(-n))/i + (1+i)^(-n)*PMT

where i is the interest rate (7.5% or 0.075), and n is the number of years (15).

Simplifying this formula, we get:

X = 25,000*(1 - 1.075^(-15))/0.075 + 1.075^(-15)*25,000
X = 237,229.09

So the lump sum amount would need to be $237,229.09 to leave Sam as well off financially as the annuity payments. Therefore, option B is the best choice.

Quiz 5: Time Value of Money

What's the rate of return you would earn if you paid $950 for a perpetuity that pays $85 per year?

You inherited an oil well that will pay you $25,000 per year for 25 years, with the first payment being made today. If you think a fair return on the well is 7.5%, how much should you ask for it if you decide to sell it?

What's the present value of a 4-year ordinary annuity of $2,250 per year plus an additional $3,000 at the end of Year 4 if the interest rate is 5%?

What is the PV of an annuity due with 5 payments of $2,500 at an interest rate of 5.5%?

Your uncle is about to retire, and he wants to buy an annuity that will provide him with $75,000 of income a year for 20 years, with the first payment coming immediately. The going rate on such annuities is 5.25%. How much would it cost him to buy the annuity today?

Suppose you just won the state lottery, and you have a choice between receiving $2,550,000 today or a 20-year annuity of $250,000, with the first payment coming one year from today. What rate of return is built into the annuity? Disregard taxes.

What's the present value of a perpetuity that pays $250 per year if the appropriate interest rate is 5%?

Suppose you inherited $275,000 and invested it at 8.25% per year. How much could you withdraw at the beginning of each of the next 20 years?

Your father is about to retire, and he wants to buy an annuity that will provide him with $85,000 of income a year for 25 years, with the first payment coming immediately. The going rate on such annuities is 5.15%. How much would it cost him to buy the annuity today?

Suppose you inherited $275,000 and invested it at 8.25% per year. How much could you withdraw at the end of each of the next 20 years?

You have a chance to buy an annuity that pays $550 at the beginning of each year for 3 years. You could earn 5.5% on your money in other investments with equal risk. What is the most you should pay for the annuity?

Your uncle has $375,000 and wants to retire. He expects to live for another 25 years and to earn 7.5% on his invested funds. How much could he withdraw at the end of each of the next 25 years and end up with zero in the account?

Your uncle has $375,000 and wants to retire. He expects to live for another 25 years, and he also expects to earn 7.5% on his invested funds. How much could he withdraw at the beginning of each of the next 25 years and end up with zero in the account?

You have a chance to buy an annuity that pays $5,000 at the beginning of each year for 5 years. You could earn 4.5% on your money in other investments with equal risk. What is the most you should pay for the annuity?

Your Aunt Ruth has $500,000 invested at 6.5%, and she plans to retire. She wants to withdraw $40,000 at the beginning of each year, starting immediately. How many years will it take to exhaust her funds, i.e., run the account down to zero?

Sam was injured in an accident, and the insurance company has offered him the choice of $25,000 per year for 15 years, with the first payment being made today, or a lump sum. If a fair return is 7.5%, how large must the lump sum be to leave him as well off financially as with the annuity?