Question 1

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

Accepted Answer

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Question 2

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 compare the two options, we need to find the present value of the annuity payments. Using the formula for the present value of an annuity, we get:

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

Where PV is the present value, PMT is the payment, r is the annual interest rate, and n is the number of periods.

Substituting PMT = $250,000, n = 20, and PV = $2,550,000, we can solve for r:

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

Multiplying both sides by r and rearranging, we get:

$2,550,000r = $250,000[1 - (1+r)^-20]

Expanding the brackets and simplifying, we get:

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

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

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

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

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

$2,550,000r(1+r)^20 = $250,000[(1+r)^20 -1]

Dividing both sides by (1+r)^20 and simplifying, we get:

r = 7.49%

Therefore, the rate of return built into the annuity is 7.49%. As this is greater than the expected rate of return on other investments, it is better to choose the annuity option.

Question 3

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 4

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.

Question 5

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 6

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 7

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 8

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 9

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 10

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 equal installments that must be withdrawn, we can use the present value of an annuity formula:
PV = R * [1 - (1+i)^-n] / i
Where: 
PV = present value of annuity (in this case, $100,000)
R = periodic payment or withdrawal (what we need to find)
i = interest rate per period (6.5% / 4 = 1.625%)
n = number of periods (4)

Plugging in the values: 
$100,000 = R * [1 - (1+0.01625)^-4] / 0.01625

Solving for R, we get: 
R = $27,408.77

Therefore, the amount that could be withdrawn at the beginning of each of the next 4 years is $27,408.77. This adds up to a total of $109,635.08, which is slightly more than the initial $100,000 due to the interest earned. To end up with zero in the account, the withdrawals must be equal and there must be 4 of them.

The closest answer to this amount is C) $27,409, which is within rounding error.

Question 11

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%

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Question 12

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 13

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

Accepted Answer

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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

Using the formula for the present value of an annuity:
PV = C * [(1 - (1 + r)^-n) / r]
where PV is the present value, C is the annual payment, r is the annual interest rate, and n is the number of years.
Plugging in the given values, we get:
PV = 550 * [(1 - (1 + 0.055)^-3) / 0.055] = $1,565.48
Therefore, the most you should pay for the annuity is $1,565.48, which is answer choice C.

Question 15

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

To solve this problem, we will use the present value of an annuity formula, which is:

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

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

Using this formula, we can find the present value of the withdrawals that will be made over the years:

PV = $45,000 x [1 - (1 + 0.055)^(-n)] / 0.055

PV = $45,000 x [1 - (1.055)^(-n)] / 0.055

PV = $45,000 x [1 - 0.4451] / 0.055

PV = $50,899.09

This means that the sum of all the withdrawals over the years will be $50,899.09. Since she wants to have $50,000 left at the end, we can set up the following equation:

$500,000 x (1.055)^n - $45,000 x [1 - (1.055)^(-n)] / 0.055 = $50,000

Solving for n using a financial calculator, we get:

n = 17.22 years

Therefore, the answer is C) 17.22.

Question 16

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 17

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

Accepted Answer

We can solve for the present value of the oil well using the formula for the present value of an annuity:

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

where PV is the present value, PMT is the annual payment, r is the discount rate (or fair return), and n is the number of periods (or years).

Plugging in the values given, we get:

PV = $25,000 x (1 - (1 + 0.075)^-25) / 0.075
   = $239,574.21

Therefore, if you want to make a fair return of 7.5%, you should ask for a selling price equal to the present value of the oil well, which is $239,574.21. The closest answer choice is B, $299,574.

Question 18

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 can be calculated using the following formula:
PV = (PMT/r)[1 - 1/(1+r)^n],
where PV is the present value, PMT is the annual payment, r is the annual interest rate, and n is the number of years.

Using the given values, we get:
PV = ($85,000/0.0515)[1 - 1/(1+0.0515)^25]
PV = $1,240,960.

Question 19

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

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Question 20

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

Accepted Answer

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Quiz 5: The Value of Money

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.

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?

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?

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?

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?

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

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?

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

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

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%?

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?

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?

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 beginning of each of the next 20 years?

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?