Question 1

Jane has been saving $500 in her retirement account each month for the last 20 years and plans to continue contributing $500 each month for the next 20 years. Her account has been earning an 8 percent annual interest rate and she expects to earn the same rate for the next 20 years. Her twin brother, Hal, has not saved anything for the last 20 years. Due to sibling rivalry, he wants to have as much as Jane is expected to have at the end of 20 years. If Hal expects to earn the same annual interest rate as Jane, how much must Hal save each month to achieve his goal?
A) $1,043.71
B) $1,517.92
C) $2,007.53
D) $2,963.40

Accepted Answer

The answer of Jane has been saving $500 in her...

Question 2

Jane has been saving $450 in her retirement account each month for the last 20 years and plans to continue contributing $450 each month for the next 20 years. Her account has been earning a 9 percent annual interest rate and she expects to earn the same rate for the next 20 years. Her twin brother, Hal, has not saved anything for the last 20 years. Due to sibling rivalry, he wants to have as much as Jane is expected to have at the end of 20 years. If Hal expects to earn the same annual interest rate as Jane, how much must Hal save each month to achieve his goal?
A) $1,791.34
B) $2,109.28
C) $2,872.91
D) $3,154.12

Accepted Answer

The answer of Jane has been saving $450 in her...

Question 3

Which of the following will increase the present value of an annuity?
A) The number of periods decreases.
B) The interest rate decreases.
C) The amortization schedule decreases.
D) The effective rate is calculated over fewer years.

Accepted Answer

The present value of an annuity increases when the interest rate decreases because the discount factor used in calculating present value is inversely related to the interest rate. Lower interest rates result in higher present values for future cash flows.

Question 4

Your current $155,000 mortgage calls for monthly payments over 25 years at an annual interest rate of 6 percent. If you pay an additional $50 each month beginning with the first payment, how much interest expense do you save by pre-paying?
A) $15,981.28
B) $16,009.62
C) $17,152.22
D) $19,001.69

Accepted Answer

By paying an additional $50 each month, you reduce the principal faster, which decreases the total interest paid over the life of the loan. The correct calculation shows a savings of $17,152.22 in interest.

Question 5

Sally saves $500 per month in her retirement plan. She plans on making monthly contributions for 35 years. If her account earns a 12 percent annual interest rate, how much will she have at the end of 35 years and what percent of the total are her out-of-pocket contributions?
A) $1,113,879.14; 43.72 percent
B) $2,452,905.33; 12.07 percent
C) $3,215,479.74; 6.53 percent
D) $3,691,003.27; 8.28 percent

Accepted Answer

The answer of Sally saves $500 per month in her...

Question 6

You started your first job after graduating from college. Your company offers a retirement plan for which the company contributes 50 percent of what you contribute each year. You expect to contribute $2,000 per year from your salary. You decide to invest the contributions in assets that you expect to earn 10 percent per year. If you plan to retire in 40 years, how big will you expect that retirement account to be?
A) $442,592.56
B) $885,185.11
C) $1,327,777.67
D) $1,527,787.70

Accepted Answer

The future value of an annuity formula is used to calculate the expected retirement account value: FV = P * [((1 + r)^n - 1) / r]. Here, P = $3,000 (your $2,000 contribution plus $1,000 from the company), r = 0.10, and n = 40. Plugging in these values gives approximately $1,327,777.67.

Question 7

Paige has decided that she wants to build enough retirement wealth that, if invested at 5 percent per year, will provide her with $2,500 monthly income for 20 years. To date, she has saved nothing, but she still has 40 years until she retires. How much money does she need to contribute per month to reach her goal?
A) $180.02
B) $248.24
C) $460.81
D) $921.61

Accepted Answer

To solve this, we first need to calculate the present value (PV) of the retirement wealth needed to provide Paige with $2,500 monthly for 20 years at a 5% annual interest rate, converted to a monthly rate (5%/12 months = 0.4167%). Using the present value of an annuity formula, PV = Pmt * [(1 - (1 + r)^-n) / r], where Pmt is the monthly payment ($2,500), r is the monthly interest rate (0.4167%), and n is the total number of payments (20 years * 12 months = 240). This calculation gives us the lump sum Paige needs at the start of her retirement.Next, we calculate how much Paige needs to save monthly for the next 40 years (40 years * 12 months = 480 months) to reach this lump sum, using the future value of a series formula in reverse. We use the same interest rate (0.4167% monthly) to find the monthly contribution.The correct answer involves calculating the present value of the retirement goal and then determining the monthly contribution needed to reach this goal over 40 years at a 5% annual interest rate. The calculations show that Paige needs to contribute $248.24 per month to reach her goal, making option B the correct choice.

Question 8

Which of the following will increase the future value of an annuity?
A) The number of periods increases.
B) The amount of the annuity increases.
C) The interest rate increases.
D) All of these choices are correct.

Accepted Answer

All of these factors—increasing the number of periods, increasing the amount of the annuity, and increasing the interest rate—will increase the future value of an annuity. Each one contributes to a higher accumulation of value over time.

Question 9

After saving diligently your entire career, you and your spouse are ready to retire with a nest egg of $500,000. You need to invest this money in a mix of stocks and bonds that will allow you to earn $4,000 per month for 30 years. What annual interest rate (APR) do you need to earn?
A) 6.92 percent
B) 7.45 percent
C) 8.94 percent
D) 9.17 percent

Accepted Answer

To find the required annual interest rate to earn $4,000 per month for 30 years from a $500,000 investment, we can use the formula for the present value of an annuity. The monthly withdrawal of $4,000 translates to $48,000 annually. Using financial calculators or present value of annuity formulas, we find that an annual interest rate of approximately 8.94% is needed to sustain these withdrawals for 30 years from a $500,000 nest egg.

Question 10

After saving diligently your entire career, you and your spouse are ready to retire with a nest egg of $600,000. You need to invest this money in a mix of stocks and bonds that will allow you to earn $5,000 per month for 30 years. What annual interest rate (APR) do you need to earn?
A) 9.40 percent
B) 10.13 percent
C) 8.37 percent
D) 9.61 percent

Accepted Answer

To find the required annual interest rate, you can use the formula for the present value of an annuity. Solving for the interest rate that allows $600,000 to provide $5,000 monthly for 30 years results in an APR of approximately 9.40 percent.

Question 11

Bethany purchased a $35,000 car three years ago using a 6 percent, 5-year loan. She has decided that she would sell the car now, if she could get a price that would pay off the balance of her loan. What is the minimum price Bethany would need to receive for her car?
A) $9,680,67
B) $15,267.12
C) $22,242.11
D) $23,429.19

Accepted Answer

To find the minimum price Bethany would need to receive for her car to pay off the balance of her loan, we can calculate the remaining balance of a 5-year loan at 6 percent interest after 3 years. Using the formula for the remaining balance of an amortizing loan or an amortization calculator, we find that the correct answer is $15,267.12. This accounts for the principal and interest payments made over the first three years, reducing the initial loan amount.

Question 12

Jane has been saving $200 in her retirement account each month for the last 20 years and plans to continue contributing $200 each month for the next 20 years. Her account has been earning an 8 percent annual interest rate and she expects to earn the same rate for the next 20 years. Her twin brother, Hal, has not saved anything for the last 20 years. Due to sibling rivalry, he wants to have as much as Jane is expected to have at the end of 20 years. If Hal expects to earn the same annual interest rate as Jane, how much must Hal save each month to achieve his goal?
A) $400.00
B) $1,185.36
C) $1,569.85
D) $2,909.17

Accepted Answer

To solve this, we use the future value of an annuity formula for both Jane and Hal. For Jane, who has been saving for 20 years and will save for another 20 years, the future value of her savings can be calculated in two parts: the future value of her current savings and the future value of her future savings. For Hal, who wants to match Jane's amount in 20 years, we calculate how much he needs to save each month using the same future value of an annuity formula, setting it equal to the total future value of Jane's savings. Given the same interest rate and time frame for Hal's savings, the calculation will show that Hal needs to save a significantly higher amount each month to match Jane's savings due to the power of compound interest and the head start Jane has. The correct calculation based on the given interest rate and time frames will show that $1,185.36 is the monthly amount Hal needs to save to match Jane's expected savings in 20 years.

Question 13

If you start making $90 monthly contributions today and continue them for ten years, what is their future value if the compounding rate is 6 percent APR? What is the present value of this annuity?
A) $14,794.14; $8,106.61
B) $14,794.14; $8,147.14
C) $14,822.89; $8,106.61
D) $14,822.89; $8,147.14

Accepted Answer

The future value of the annuity is $14,794.14 and the present value is $8,106.61. The future value is calculated using the formula FV = PMT * (((1 + r)^n - 1) / r), where PMT is the monthly contribution, r is the annual interest rate, and n is the number of years. The present value is calculated using the formula PV = PMT * ((1 - (1 + r)^-n) / r), where PMT is the monthly contribution, r is the annual interest rate, and n is the number of years.

Question 14

Payday loans are very short-term loans that charge very high interest rates. You can borrow $500 today and repay $550 in ten weeks. What is the compound annual rate implied by this 10 percent rate charged for only ten weeks?
A) 5.20 percent
B) 10.41 percent
C) 59.94 percent
D) 64.15 percent

Accepted Answer

The compound annual rate (CAR) can be calculated using the formula for compound interest, which is A = P(1 + r/n)^(nt), where A is the amount on maturity, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time the money is borrowed for in years. In this case, we're looking for r, the annual interest rate, given that $500 grows to $550 in 10 weeks (which is 10/52 years since there are 52 weeks in a year). The interest rate for 10 weeks is 10%, but we need to find the equivalent annual rate if that rate were compounded weekly for a year. The correct calculation involves finding the rate that would lead to the same growth over a year as the 10% does over 10 weeks, compounded weekly. The correct answer, 64.15%, reflects the annualized interest rate, taking into account the effect of compounding over the course of a year.

Question 15

Your current $95,000 mortgage calls for monthly payments over 30 years at an annual interest rate of 6 percent. If you pay an additional $50 each month beginning with the first payment, how soon do you pay off your mortgage?
A) 329.67 months
B) 311.56 months
C) 291.78 months
D) 288.45 months

Accepted Answer

Paying an additional $50 each month on a $95,000 mortgage with a 6% annual interest rate over 30 years reduces the total loan term to approximately 291.78 months. This acceleration in payment reduces the interest paid over the life of the loan and shortens the payoff period.

Question 16

If the present value of an ordinary, 10-year annuity is $25,000 and interest rates are 7 percent, what is the present value of the same annuity due?
A) $23,644.49
B) $24,997.51
C) $25,000.00
D) $26,750.00

Accepted Answer

An annuity due is an annuity where the payments are made at the beginning of each period instead of the end. To find the present value of an annuity due, you can calculate the present value of an ordinary annuity and then multiply it by (1 + r), where r is the interest rate. Given that the present value of the ordinary annuity is $25,000 and the interest rate is 7% (or 0.07 as a decimal), the present value of the annuity due is $25,000 * (1 + 0.07) = $25,000 * 1.07 = $26,750.

Question 17

Your current $115,000 mortgage calls for monthly payments over 30 years at an annual interest rate of 7 percent. If you pay an additional $50 each month beginning with the first payment, how much interest expense do you save by prepaying?
A) $32,764.43
B) $30,718.29
C) $29,503.14
D) $22,008.73

Accepted Answer

Paying an additional $50 each month on a mortgage reduces the principal balance faster, which in turn reduces the total amount of interest paid over the life of the loan. The correct calculation involves comparing the total interest paid over the life of the loan with and without the additional $50 payments. The savings of $32,764.43 indicates the difference in interest paid due to the accelerated repayment schedule.

Question 18

If the future value of an ordinary, 5-year annuity is $100,000 and interest rates are 5 percent, what is the future value of the same annuity due?
A) $95,238.10
B) $100,000.00
C) $105,000.00
D) $107,000.00

Accepted Answer

An annuity due generates higher future value than an ordinary annuity because each payment is invested for an additional period. To find the future value of an annuity due, you can calculate the future value of an ordinary annuity and then multiply by (1 + interest rate). Here, $100,000 * 1.05 = $105,000.

Question 19

A car company is offering a choice of deals. You can receive $4,000 cash back on the purchase, or a 0 percent APR, 4-year loan. The price of the car is $40,000 and you could obtain a 4-year loan from your credit union, at 6 percent APR. Which deal is cheaper?
A) The car company's 0 percent 4-year loan.
B) The rebate with the credit union's 6 percent 4-year loan.
C) There is not enough information given to determine which deal is cheaper.

Accepted Answer

The 0 percent APR deal is cheaper because the interest saved over a 4-year loan at 6 percent APR on $40,000 exceeds the $4,000 cash back offer.

Question 20

A mortgage broker is offering a 30-year mortgage with a teaser rate. In the first two years of the mortgage, the borrower makes monthly payments on only a 2.5 percent APR interest rate. After the second year, the mortgage interest charged increases to 4.25 percent APR. What is the effective interest rate in the first two years? What is the effective interest rate after the second year?
A) 2.08 percent, 3.54 percent respectively
B) 2.50 percent, 4.25 percent respectively
C) 2.53 percent, 4.33 percent respectively
D) 5.00 percent, 8.50 percent respectively

Accepted Answer

The effective interest rate is calculated using the formula for converting APR to the effective annual rate. For the first two years, the effective rate is approximately 2.53%, and after the second year, it is approximately 4.33%.

Quiz 5: Time Value of Money 2: Analyzing Annuity Cash Flows

Which of the following will increase the present value of an annuity?

Your current $155,000 mortgage calls for monthly payments over 25 years at an annual interest rate of 6 percent. If you pay an additional $50 each month beginning with the first payment, how much interest expense do you save by pre-paying?

Sally saves $500 per month in her retirement plan. She plans on making monthly contributions for 35 years. If her account earns a 12 percent annual interest rate, how much will she have at the end of 35 years and what percent of the total are her out-of-pocket contributions?

Which of the following will increase the future value of an annuity?

After saving diligently your entire career, you and your spouse are ready to retire with a nest egg of $500,000. You need to invest this money in a mix of stocks and bonds that will allow you to earn $4,000 per month for 30 years. What annual interest rate (APR) do you need to earn?

After saving diligently your entire career, you and your spouse are ready to retire with a nest egg of $600,000. You need to invest this money in a mix of stocks and bonds that will allow you to earn $5,000 per month for 30 years. What annual interest rate (APR) do you need to earn?

Bethany purchased a $35,000 car three years ago using a 6 percent, 5-year loan. She has decided that she would sell the car now, if she could get a price that would pay off the balance of her loan. What is the minimum price Bethany would need to receive for her car?

If you start making $90 monthly contributions today and continue them for ten years, what is their future value if the compounding rate is 6 percent APR? What is the present value of this annuity?

Payday loans are very short-term loans that charge very high interest rates. You can borrow $500 today and repay $550 in ten weeks. What is the compound annual rate implied by this 10 percent rate charged for only ten weeks?

Your current $95,000 mortgage calls for monthly payments over 30 years at an annual interest rate of 6 percent. If you pay an additional $50 each month beginning with the first payment, how soon do you pay off your mortgage?

If the present value of an ordinary, 10-year annuity is $25,000 and interest rates are 7 percent, what is the present value of the same annuity due?

Your current $115,000 mortgage calls for monthly payments over 30 years at an annual interest rate of 7 percent. If you pay an additional $50 each month beginning with the first payment, how much interest expense do you save by prepaying?

If the future value of an ordinary, 5-year annuity is $100,000 and interest rates are 5 percent, what is the future value of the same annuity due?

A car company is offering a choice of deals. You can receive $4,000 cash back on the purchase, or a 0 percent APR, 4-year loan. The price of the car is $40,000 and you could obtain a 4-year loan from your credit union, at 6 percent APR. Which deal is cheaper?