Question 1

Your friend offers to pay you an annuity of $2,500 at the end of each year for 3 years in return for cash today. 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) $5,493.71
B) $5,782.85
C) $6,087.21
D) $6,407.59
E) $6,744.83

Accepted Answer

$\begin{array}{lr}
\text {N } & 3\
\text {  I/YR  } & 5.5 \% \
\text { PMT } & \$ 2,500 \
\text { FV } & \$ 0.00 \
\text { PV } & \$6,744.83
\end{array}$

Question 2

When a loan is amortized, a relatively low percentage of the payment goes to reduce the outstanding principal in the early years, and the principal repayment's percentage increases in the loan's later years.

Accepted Answer

This is because most of the initial payments go towards paying off the interest on the loan, while the remaining amount goes towards reducing the principal. As the loan term progresses, the interest amount decreases and a larger percentage of the payment goes towards reducing the principal balance. This is known as an amortization schedule.

Question 3

Which of the following statements regarding a 20-year (240-month) $225,000, fixed-rate mortgage is CORRECT? (Ignore taxes and transactions costs.)
A) the outstanding balance declines at a slower rate in the later years of the loan's life.
B) the remaining balance after three years will be $225,000 less one third of the interest paid during the first three years.
C) because it is a fixed-rate mortgage, the monthly loan payments (which include both interest and principal payments) are constant.
D) interest payments on the mortgage will increase steadily over time, but the total amount of each payment will remain constant.
E) the proportion of the monthly payment that goes towards repayment of principal will be lower 10 years from now than it will be the first year.

Accepted Answer

c is the correct answer. Thinking through the question, the other answers can all be eliminated. One could also set up an amortization schedule to prove that only statement c is correct.

Question 4

A $150,000 loan is to be amortized over 6 years, with annual end-of-year payments. Which of these statements is CORRECT?
A) the proportion of interest versus principal repayment would be the same for each of the 7 payments.
B) the annual payments would be larger if the interest rate were lower.
C) if the loan were amortized over 10 years rather than 6 years, and if the interest rate were the same in either case, the first payment would include more dollars of interest under the 6-year amortization plan.
D) the proportion of each payment that represents interest as opposed to repayment of principal would be higher if the interest rate were lower.
E) the proportion of each payment that represents interest versus repayment of principal would be higher if the interest rate were higher.

Accepted Answer

a, b, and d are obviously incorrect. c is also incorrect because interest in the first year would be Loan amount × interest rate regardless of the life of the loan. That makes e the "most logical guess." One could also set up an amortization schedule and change the numbers to confirm that only e is correct.

Question 5

Now that your uncle has decided to retire, 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

BEGIN Mode
 $\begin{array}{lr}
\text {N } & 25 \
\text {  I/YR  } & 5.15 \% \
\text { PMT } & \$ 85,000 \
\text { FV } & \$ 0.00 \
\text { PV } & \$1,240,960
\end{array}$

Question 6

After receiving a reward for information leading to the arrest of a notorious criminal, you are considering investing it in an annuity that pays $5,000 at the end of each year for 20 years. You could earn 5% on your money in other investments with equal risk. What is the most you should pay for the annuity?
A) $50,753
B) $53,424
C) $56,236
D) $59,195
E) $62,311

Accepted Answer

$\begin{array}{lr}
\text {N } & 20 \
\text {  I/YR  } & 5.0 \% \
\text { PMT } & \$ 5,000 \
\text { FV } & \$ 0.00 \
\text { PV } & \$62.311
\end{array}$

Question 7

A new investment opportunity for you is 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

BEGIN Mode
 $\begin{array}{lr}
\text {N } & 3 \
\text {  I/YR  } & 5.5 \% \
\text { PMT } & \$ 550 \
\text { FV } & \$ 0.00 \
\text { PV } & \$1,565.48
\end{array}$

Question 8

A salt mine you inherited 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 mine 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

BEGIN Mode
 $\begin{array}{lr}
\text {N } & 25 \
\text {  I/YR  } & 7.5 \% \
\text { PMT } & \$ 25,000 \
\text { FV } & \$ 0.00 \
\text { PV } & \$299,574
\end{array}$

Question 9

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

$\begin{array}{lr}
\text {N } & 4 \
\text {  I/YR  } & 5.0 \% \
\text { PMT } & \$ 2,250 \
\text { FV } & \$ 3.000 \
\text { PV } & \$10,446
\end{array}$ 
Alternative setup:
 $\begin{array} { c c c c c } 
0 & 1 & 2 & 3 & 4 \
\hline & \$ 2,250 & \$ 2,250 & \$ 2,250 & \$ 2,250 \
& & & & \$ 3,000 \
\hline  \$PV= 10,44650  & \$ 2,250 & \$ 2,250 & \$ 2,250 & \$ 5,250
\end{array}$

Question 10

Geraldine was injured in a car accident, and the insurance company has offered her 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 her 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

BEGIN Mode
 $\begin{array}{lr}
\text {N } & 15 \
\text {  I/YR  } & 7.5 \% \
\text { PMT } & \$ 25,000 \
\text { FV } & \$ 0.00 \
\text { PV } & \$237,229
\end{array}$

Question 11

Midway through the life of an amortized loan, the percentage of the payment that represents interest must be equal to the percentage that represents repayment of principal. This is true regardless of the original life of the loan or the interest rate on the loan.

Accepted Answer

There is no reason to think that this statement would always be true. The portion of the payment representing interest declines, while the portion representing principal repayment increases. Therefore, the statement is false. We could also work out some numbers to prove this point. Here's an example for a 3-year loan at a 10% and a 41.45% annual interest rate. The interest component is not equal to the principal repayment component except at the high interest rate.
 $\begin{array}{l}
\begin{array} { l c } 
\text { Original loan } & \$ 1,000 \text { Original loan } \
\text { Rate } & 10 \% \text { Rate } \
\text { Life } & 3 \text { Life } \
\text { Payment } & \$ 402.11 \text { Paymert }
\end{array}
\begin{array} { r } 
\$ 1,000 \
41.45 \% \
3\
\$ 640.98
\end{array}
\end{array}$ 
 $\begin{array}{lllllllll}
&\text { Beg. }&&&\text { End }&\text { Beg. }&&&\text { End }\
&\text { Balance }&\text {Interest }&\text { Principal }&\text { Bal. }&\text { Balance }&\text {Interest }&\text { Principal }&\text { Bal. }\
1&\$ 1,000.00 & \$ 100.00 & \$ 302.11 & \$ 697.89 & \$ 1,000.00 & \$ 414.50 & \$ 226.48 & \$ 773.52 \
2&\$ 697.89 & \$ 69.79 & \$ 332.33 & \$ 365.562 & \$ 773.52 & \$ 320.62 & \$ 320.36 & \$ 453.15 \
3&\$ 36556 & \$ 3656 & \$ 365.56 & \$ 0003 & \$ 45315 & \$ 18783 & \$ 45315 & \$ 000
\end{array}$

Question 12

Because your mother is about to retire, she wants to buy an annuity that will provide her 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 her to buy the annuity today?
A) $825,835
B) $869,300
C) $915,052
D) $963,213
E) $1,011,374

Accepted Answer

BEGIN Mode
 $\begin{array}{lr}
\text {N } & 5 \
\text {  I/YR  } & 5.25 \% \
\text { PMT } & \$ 75,000 \
\text { FV } & \$ 0.00 \
\text { PV } & \$963,213
\end{array}$

Question 13

An uncle of yours who is about to retire wants to sell some of his stock and buy an annuity that will provide him with income of $50,000 per year for 30 years, beginning a year from today. The going rate on such annuities is 7.25%. How much would it cost him to buy such an annuity today?
A) $574,924
B) $605,183
C) $635,442
D) $667,214
E) $700,575

Accepted Answer

$\begin{array}{lr}
\text {N } & 30 \
\text {  I/YR  } & 7.25 \% \
\text { PMT } & \$ 50,000 \
\text { FV } & \$ 0.00 \
\text { PV } & \$605,183
\end{array}$

Question 14

Midway through the life of an amortized loan, the percentage of the payment that represents interest could be equal to, less than, or greater than to the percentage that represents repayment of principal. The proportions depend on the original life of the loan and the interest rate.

Accepted Answer

This statement is true. The portion of the payment representing interest declines, while the portion representing principal repayment increases. The interest portion could be equal to, greater than, or less than the principal portion. We can work out some numbers to prove this point. Here's an example for a 3-year loan at a 10% and a 41.45% annual interest rate. The interest component is less than the principal at 10%, equal at about 41.45%, and greater at rates above 41.45%.
 $\begin{array}{l}
\begin{array} { l c } 
\text { Original loan } & \$ 1,0000 \text { Original loan } \
\text { Rate } & 10 \% \text { Rate } \
\text { Life } & 3 \text { Life } \
\text { Payment } & \$ 402.11 \text { Paymert }
\end{array}
\begin{array} { r } 
\$ 1,000 \
41.45 \% \
3\
\$ 640.98
\end{array}
\end{array}$ 
 $\begin{array}{lllllllll}
&\text { Beg. }&&&\text { End }&\text { Beg. }&&&\text { End }\
&\text { Balance }&\text {Interest }&\text { Principal }&\text { Bal. }&\text { Balance }&\text {Interest }&\text { Principal }&\text { Bal. }\
1&\$ 1,000.00 & \$ 100.00 & \$ 302.11 & \$ 697.89 & \$ 1,000.00 & \$ 414.50 & \$ 226.48 & \$ 773.52 \
2&\$ 697.89 & \$ 69.79 & \$ 332.33 & \$ 365.562 & \$ 773.52 & \$ 320.62 & \$ 320.36 & \$ 453.15 \
3&\$ 36556 & \$ 3656 & \$ 365.56 & \$ 0003 & \$ 45315 & \$ 18783 & \$ 45315 & \$ 000
\end{array}$

Question 15

The payment made each period on an amortized loan is constant, and it consists of some interest and some principal. The closer we are to the end of the loan's life, the smaller the percentage of the payment that will be a repayment of principal.

Accepted Answer

As we get closer to the end of the loan's life, the percentage of the payment that is a repayment of principal increases, while the interest portion decreases.

Question 16

A $250,000 loan is to be amortized over 8 years, with annual end-of-year payments. Which of these statements is CORRECT?
A) the proportion of interest versus principal repayment would be the same for each of the 8 payments.
B) the annual payments would be larger if the interest rate were lower.
C) if the loan were amortized over 10 years rather than 8 years, and if the interest rate were the same in either case, the first payment would include more dollars of interest under the 8-year amortization plan.
D) the proportion of each payment that represents interest as opposed to repayment of principal would be lower if the interest rate were lower.
E) the last payment would have a higher proportion of interest than the first payment.

Accepted Answer

a, b, and e can be ruled out as incorrect by simple reasoning. c is also incorrect because interest in the first year would be Loan amount × interest rate regardless of the life of the loan, so the interest payment would be identical for the first payment. Think about the situation where r = 0%, statement d is the "most logical guess." One could also set up an amortization schedule and change the numbers to confirm that only d is correct.

Question 17

The payment made each period on an amortized loan is constant, and it consists of some interest and some principal. The closer we are to the end of the loan's life, the greater the percentage of the payment that will be a repayment of principal.

Accepted Answer

This is the concept of amortization, where the payment is designed to reduce the outstanding principal balance over time, resulting in a larger portion of the payment going towards the principal as the loan nears its end.

Question 18

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

Accepted Answer

BEGIN Mode
 $\begin{array}{lr}
\text {N } & 5 \
\text {  I/YR  } & 5.5 \% \
\text { PMT } & \$ 2,500 \
\text { FV } & \$ 0.00 \
\text { PV } & \$11,262.88
\end{array}$

Question 19

Your father is considering purchasing an annuity that pays $5,000 at the beginning of each year for 5 years. He could earn 4.5% on his money in other investments with equal risk. What is the most he should pay for the annuity?
A) 20,701
B) $21,791
C) $22,938
D) $24,085
E) $25,289

Accepted Answer

BEGIN Mode
 $\begin{array}{lr}
\text {N } & 5 \
\text {  I/YR  } & 4.5 \% \
\text { PMT } & \$ 5,000 \
\text { FV } & \$ 0.00 \
\text { PV } & \$22.938
\end{array}$

Question 20

When a loan is amortized, a relatively high percentage of the payment goes to reduce the outstanding principal in the early years, and the principal repayment's percentage declines in the loan's later years.

Accepted Answer

In an amortized loan, the early payments consist of a higher percentage of interest and a lower percentage of principal reduction. As the loan matures, the portion of the payment reducing the principal increases, while the interest portion decreases.

Quiz 28: Time Value of Money

Your friend offers to pay you an annuity of $2,500 at the end of each year for 3 years in return for cash today. You could earn 5.5% on your money in other investments with equal risk. What is the most you should pay for the annuity?

When a loan is amortized, a relatively low percentage of the payment goes to reduce the outstanding principal in the early years, and the principal repayment's percentage increases in the loan's later years.

Which of the following statements regarding a 20-year (240-month) $225,000, fixed-rate mortgage is CORRECT? (Ignore taxes and transactions costs.)

A $150,000 loan is to be amortized over 6 years, with annual end-of-year payments. Which of these statements is CORRECT?

Now that your uncle has decided to retire, 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 new investment opportunity for you is 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 salt mine you inherited 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 mine 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%?

Geraldine was injured in a car accident, and the insurance company has offered her 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 her as well off financially as with the annuity?

Midway through the life of an amortized loan, the percentage of the payment that represents interest must be equal to the percentage that represents repayment of principal. This is true regardless of the original life of the loan or the interest rate on the loan.

Because your mother is about to retire, she wants to buy an annuity that will provide her 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 her to buy the annuity today?

An uncle of yours who is about to retire wants to sell some of his stock and buy an annuity that will provide him with income of $50,000 per year for 30 years, beginning a year from today. The going rate on such annuities is 7.25%. How much would it cost him to buy such an annuity today?

Midway through the life of an amortized loan, the percentage of the payment that represents interest could be equal to, less than, or greater than to the percentage that represents repayment of principal. The proportions depend on the original life of the loan and the interest rate.

The payment made each period on an amortized loan is constant, and it consists of some interest and some principal. The closer we are to the end of the loan's life, the smaller the percentage of the payment that will be a repayment of principal.

A $250,000 loan is to be amortized over 8 years, with annual end-of-year payments. Which of these statements is CORRECT?

The payment made each period on an amortized loan is constant, and it consists of some interest and some principal. The closer we are to the end of the loan's life, the greater the percentage of the payment that will be a repayment of principal.

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

Your father is considering purchasing an annuity that pays $5,000 at the beginning of each year for 5 years. He could earn 4.5% on his money in other investments with equal risk. What is the most he should pay for the annuity?

When a loan is amortized, a relatively high percentage of the payment goes to reduce the outstanding principal in the early years, and the principal repayment's percentage declines in the loan's later years.