Question 1

Assume the total expense for your current year in college equals $20,000.How much would your parents have needed to invest 21 years ago in an account paying 8% compounded annually to cover this amount?

Accepted Answer

PV = $20,000 / (1.08)²¹PV = $3,973.11

Question 2

An annual percentage rate (APR)is determined by annualizing the rate using compound interest.

Accepted Answer

The answer of An annual percentage rate (APR)is determined by...

Question 3

Three thousand dollars is deposited into an account paying 10% annually to provide three annual withdrawals of $1,206.34 beginning in one year.How much remains in the account after the second payment has been withdrawn?

Accepted Answer

The present value of three annual withdrawals of $1,206.34 at a 10% annual interest rate is $3,000. Therefore, the initial deposit is fully utilized to provide the withdrawals. After the first payment of $1,206.34 is withdrawn, the remaining balance in the account is $3,000 + ($3,000 x 0.10) - $1,206.34 = $2,493.66. After the second payment of $1,206.34 is withdrawn, the remaining balance in the account is $2,493.66 + ($2,493.66 x 0.10) - $1,206.34 = $1,096.69, which is option C.

Question 4

You invested $1,200 three years ago.During the three years,you earned annual rates of return of 4.8%,9.2%,and 11.6%.What is the value of this investment today?

Accepted Answer

FV = PV(1 + r)^tFV = PV(1 + r)^t (1 + r)^t (1 + r)^tFV = $1,200(1.048)¹ (1.092)¹ (1.116)¹FV = $1,532.60

Question 5

How much interest can be accumulated during one year on a $1,000 deposit paying continuously compounded interest at an APR of 10%?

Accepted Answer

The formula for continuously compounded interest is A = Pe^(rt), where A is the ending balance, P is the principal, r is the interest rate in decimal form, and t is the time in years. Plugging in the given values, we get A = 1000e^(0.1*1), which simplifies to A = 1105.17. The interest earned is the difference between the ending balance and the principal, which is 1105.17 - 1000 = 105.17. Therefore, the answer is B.

Question 6

The appropriate manner of adjusting for inflationary effects is to discount nominal cash flows with real interest rates.

Accepted Answer

The appropriate manner of adjusting for inflationary effects is to discount real cash flows with real interest rates or to discount nominal cash flows with nominal interest rates.

Question 7

An effective annual rate must be greater than an annual percentage rate.

Accepted Answer

The answer of An effective annual rate must be greater...

Question 8

Nominal dollars refer to the amount of purchasing power.

Accepted Answer

Nominal dollars refer to the actual dollar amount, without taking into account inflation or changes in purchasing power. Therefore, they do not refer to the amount of purchasing power.

Question 9

How much interest will be earned in the next year on an investment paying 12% compounded annually if $100 was just credited to the account for interest?

Accepted Answer

The amount credited to the account for interest ($100) is not the initial investment, but rather the interest earned on the investment. To calculate the initial investment, we need to subtract the interest earned from the total amount, which can be found using the formula for compound interest:

A = P(1 + r/n)^(nt)

where:
A = total amount
P = principal (initial investment)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = time in years

Plugging in the given values, we get:

100 + P = P(1 + 0.12/1)^(1*1)
100 + P = P(1.12)
100/0.12 = P
P = $833.33

So the interest earned in the next year would be:

I = P*r = $833.33*0.12 = $100

Therefore, the total amount in the account after one year would be:

A = P + I = $833.33 + $100 = $933.33

The interest earned on the investment (not including the $100 credited to the account) is:

$933.33 - $833.33 = $100

Therefore, the answer is choice C, $112, since this is the only option that matches the correct amount of interest earned on the investment.

Question 10

How much more is a perpetuity of $1,000 worth than an annuity of the same amount for 20 years? Assume an interest rate of 10% and cash flows at the end of each period.

Accepted Answer

The formula for calculating the value of a perpetuity is V = C/r, where V is the present value, C is the cash flow, and r is the interest rate. Therefore, the present value of a $1,000 perpetuity at 10% is $10,000.

The formula for calculating the value of an annuity is V = C * [(1 - (1 + r)^-n)/r], where n is the number of periods. For a 20-year annuity with a payment of $1,000 and an interest rate of 10%, the present value is $10,513.56.

The difference between the value of the perpetuity and the annuity is $513.56, which is the amount by which the perpetuity is worth more than the annuity.

Therefore, the best choice is B, $1,486.44.

Question 11

What will be the monthly payment on a home mortgage of $75,000 at 12% interest compounded monthly,to be amortized over 30 years?

Accepted Answer

To calculate the monthly payment, we need to use the formula:
M = P [ i(1 + i)^n ] / [ (1 + i)^n – 1],
where:
P = principal amount (in this case, $75,000)
i = monthly interest rate (12% per year divided by 12 months = 1%)
n = total number of payments (30 years multiplied by 12 months = 360 payments)
Plugging in these values, we get:
M = 75,000 [ 0.01(1 + 0.01)^360 ] / [ (1 + 0.01)^360 – 1]
M = $771.46
Therefore, the best choice is A, with a monthly payment of $771.46.

Question 12

A loan officer states,"Thousands of dollars can be saved by switching to a 15-year mortgage from a 30-year mortgage." Calculate the difference in payments on a 30-year mortgage at 9% interest versus a 15-year mortgage with 8.5% interest.Both mortgages are for $100,000 and have monthly payments.What is the difference in total dollars that will be paid to the lender under each loan? (Round the monthly payment amounts to 2 decimal places.Both interest rates are compounded monthly.)

Accepted Answer

For a 30-year mortgage at 9% interest on $100,000, the monthly payment would be:

P = (100000*(0.09/12))/(1-(1+(0.09/12))^(-360))
P = $805.23

For a 15-year mortgage at 8.5% interest on $100,000, the monthly payment would be:

P = (100000*(0.085/12))/(1-(1+(0.085/12))^(-180))
P = $978.28

The difference in monthly payments is:

978.28 - 805.23 = $173.05

Over the life of the 30-year mortgage, the total paid to the lender would be:

805.23 * 360 = $289,883.60

Over the life of the 15-year mortgage, the total paid to the lender would be:

978.28 * 180 = $176,473.40

The difference in total payments is:

289,883.60 - 176,473.40 = $113,410.20, which is closest to answer choice C.

Question 13

The present value of an annuity stream of $100 per year is $614 when valued at a 10% rate.By approximately how much would the value change if these were annuities due?

Accepted Answer

The value of an annuity due is simply the value of an ordinary annuity multiplied by (1 + the interest rate). Since the interest rate is 10%, the value of the annuity due would be $614 × (1 + 0.10) = $675.40. The difference between this value and the value of the ordinary annuity ($614) is approximately $61.40, which is the answer choice B.

Question 14

What is the present value of $100 to be deposited today into an account paying 8%,compounded semiannually for 2 years?

Accepted Answer

The answer of What is the present value of $100...

Question 15

What is the present value of a five-period annuity of $3,000 if the interest rate per period is 12% and the first payment is made today?

Accepted Answer

PV_AD = PV_OA × (1 + r)PV_AD = {$3,000[1 / 0.12 − 1 / 0.12(1.12)⁵]} × 1.12PV_AD = $12,112.05

Question 16

You will be receiving cash flows of: $1,000 today,$2,000 at end of year 1,$4,000 at end of year 3,and $6,000 at end of year 5.What is the present value of these cash flows at an interest rate of 7%?

Accepted Answer

To find the present value of these cash flows, we need to discount each cash flow back to its present value using the given interest rate of 7%. Using the present value formula, we get:

PV = CF / (1+r)^n

where PV is the present value, CF is the cash flow, r is the interest rate, and n is the number of years.

So, the present value of the cash flow of $1,000 today is simply $1,000 since it is already in present value terms.

The present value of the cash flow of $2,000 at the end of year 1 is:

PV = 2,000 / (1 + 0.07)^1 = $1,869.16

The present value of the cash flow of $4,000 at the end of year 3 is:

PV = 4,000 / (1 + 0.07)^3 = $3,198.64

The present value of the cash flow of $6,000 at the end of year 5 is:

PV = 6,000 / (1 + 0.07)^5 = $4,344.47

Adding up all the present values, we get:

PV = $1,000 + $1,869.16 + $3,198.64 + $4,344.47 = $10,412.27

Therefore, the best choice is B.

Question 17

The term "constant dollars" refers to equal payments for amortizing a loan.

Accepted Answer

Constant dollars refer to a measure of inflation-adjusted value, rather than a loan amortization term.

Question 18

You're ready to make the last of four equal,annual payments on a $1,000 loan with a 10% interest rate.If the amount of the payment is $315.47,how much of that payment is accrued interest?

Accepted Answer

$315.47 − ($315.47 / 1.1) = $28.68

Question 19

How much interest is earned in just the third year on a $1,000 deposit that earns 7% interest compounded annually?

Accepted Answer

To find the interest earned in just the third year, we need to first calculate the balance at the end of the second year, which would be the principal ($1,000) plus the interest earned in the first year (7% of $1,000 = $70), compounded for a second year.

Ending balance at the end of the second year: $1,000 + $70 = $1,070

Now, we can calculate the interest earned in the third year on this balance of $1,070, which would be 7% of $1,070 = $74.90.

Therefore, the correct answer is B) $80.14, as this is the total interest earned after three years (interest earned in the first year = $70, interest earned in the second year = $74.90, interest earned in the third year = $80.14).

Question 20

Approximately how much must be saved for retirement in order to withdraw $100,000 per year for the next 25 years if the balance earns 8% annually,and the first payment occurs one year from now?

Accepted Answer

This problem can be solved using the present value of an annuity formula, which is:

PV = (PMT / r) * (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 payments.

In this case, PMT = $100,000, r = 8%, and n = 25. Plugging in these values, we get:

PV = ($100,000 / 0.08) * (1 - (1 + 0.08)^-25)
PV = $1,067,477.62

Therefore, approximately $1,067,477.62 must be saved for retirement in order to withdraw $100,000 per year for the next 25 years if the balance earns 8% annually, and the first payment occurs one year from now. The correct answer is A.

Quiz 5: The Time Value of Money