Question 1

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 2

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

To calculate the effective interest rate in the first two years, we need to use the formula for APY (Annual Percentage Yield):

APY = (1 + periodic interest rate)n - 1

Where n is the number of periods in a year (12 for monthly payments).

For the first two years, the periodic interest rate is 2.5% / 12 = 0.20833%.

APY = (1 + 0.0020833)12 - 1 = 0.0252848 or 2.53%

To calculate the effective interest rate after the second year, we can simply use the given APR of 4.25%.

Therefore, the effective interest rate in the first two years is 2.53% and the effective interest rate after the second year is 4.33%. The answer is C.

Question 3

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

The future value of an annuity due is simply the future value of an ordinary annuity multiplied by (1 + i), where i is the interest rate per period. Therefore, the future value of the same annuity due is $100,000 x 1.05 = $105,000.00. Choice C is the correct answer.

Question 4

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 5

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 three factors listed (number of periods, amount of annuity, and interest rate) contribute to the future value of an annuity. Increasing any of these factors will lead to a higher future value. Therefore, choice D is the correct answer.

Question 6

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 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 calculate how much Paige needs to contribute each month, we need to use the present value formula:

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

Where PV is the present value, PMT is the monthly contribution, r is the interest rate per period (5 percent per year divided by 12 months), and n is the total number of periods (40 years times 12 months per year).

Using this formula and solving for PMT, we get:

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

We know that Paige wants her retirement savings to provide her with $2,500 per month for 20 years, which means she needs a total of:

$2,500 x 12 months x 20 years = $600,000

This is the future value that Paige is aiming for. To find the present value, we need to discount it back 40 years using an annual interest rate of 5 percent:

PV = $600,000 / (1 + 0.05)^40 = $63,360.19

Now we can plug in all the values into the PMT formula and solve for the monthly contribution:

PMT = $63,360.19 / ((1 - (1 + 0.05 / 12)^-(40 x 12)) / (0.05 / 12)) = $248.24

Therefore, Paige needs to contribute $248.24 per month to reach her retirement savings goal.

Question 8

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 can be calculated using the following formula:
FV = PMT * [(1 + r/n)^(n*t) - 1] / (r/n)
where PMT = $90, r = 0.06 (6% APR), n = 12 (compounded monthly), and t = 10 (years).

Plugging in the values, we get:
FV = $90 * [(1 + 0.06/12)^(12*10) - 1] / (0.06/12) = $14,794.14

The present value of the annuity is the value today of all the future payments, discounted at the same interest rate. It can be calculated using the following formula:
PV = PMT * [1 - (1 + r/n)^(-n*t)] / (r/n)

Plugging in the values, we get:
PV = $90 * [1 - (1 + 0.06/12)^(-12*10)] / (0.06/12) = $8,106.61

Therefore, the best choice is A, with a future value of $14,794.14 and a present value of $8,106.61.

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

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.Car Company: PV = 40000, I = 0, FV = 0, N = 4 × 12 = 48, PMT = −833.33

Accepted Answer

To compare the two deals, we need to calculate the total cost of each option.

Option A: With the car company's 0% APR, 4-year loan, the buyer will pay a total of $40,000 (the price of the car) over the course of 4 years, without any interest. Therefore, the total cost of this option is $40,000.

Option B: With the $4,000 cash back on the purchase, the buyer will need to finance $36,000 ($40,000 - $4,000) through their credit union loan. Using the formula for a loan, we can calculate that the total cost of this loan is $41,957.04.

Therefore, option A is cheaper than option B.

It is not necessary to compare the car company's offer to the credit union's loan at 6% APR, since option B already includes the credit union's loan.

Question 11

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 12

A furniture company is offering a choice of deals. You can receive $500 cash back on the purchase, or a 3 percent APR, 2-year loan. The price of the dining room set is $5,000 and you could obtain a 2-year loan from your credit union at 4 percent APR. What is the cost per month of each deal?
A)cash back: $193.42, 4 percent APR: $217.12
B)cash back: $193.42, 2 percent APR: $214.91
C)cash back: $195.41, 2 percent APR: $217.12
D)cash back: $195.41, 2 percent APR: $214.91

Accepted Answer

To calculate the cost per month for the cashback option, we need to subtract the $500 from the price of the dining room set, which gives us $4,500. Then, we can divide this by 24 months to get a cost per month of $187.50. 
For the 3 percent APR, 2-year loan option, we can use a loan calculator to determine the monthly payment, which comes out to $217.12. 
Comparing the two options, the cashback option has a lower cost per month ($187.50 vs. $217.12). Therefore, the best choice is to take the cashback option.

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

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?

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?

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?

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

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?

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 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?

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

A furniture company is offering a choice of deals. You can receive $500 cash back on the purchase, or a 3 percent APR, 2-year loan. The price of the dining room set is $5,000 and you could obtain a 2-year loan from your credit union at 4 percent APR. What is the cost per month of each deal?