Question 1

You have borrowed $130,000 to buy a new motor home. Your loan is to be repaid over 15 years at 8% compounded monthly. If you pay an extra $200 per month on the motor home, how many years will it take to pay off the loan?

Accepted Answer

First, we need to calculate the monthly payment for the original loan amount of $130,000 over 15 years at 8% compounded monthly. Using the formula:

P = (PV * r) / (1 - (1+r)^(-n))

where P is the monthly payment, PV is the present value (loan amount), r is the monthly interest rate (8%/12), and n is the total number of months (15*12).

We get:

P = (130000 * (8%/12)) / (1 - (1+(8%/12))^(-15*12)) 
P = $1243.96

This means that without any extra payments, it would take 15 years to pay off the motor home loan.

However, with an extra $200 per month, the new monthly payment becomes $1243.96 + $200 = $1443.96.

Using the same formula above, but using this new monthly payment, we can solve for the number of months it will take to pay off the loan:

P = (130000 * (8%/12)) / (1 - (1+(8%/12))^(-n)) + 200
$1443.96 = ($1083.33 / (1 - (1+(8%/12))^(-n))) + 200
$1260.63 = ($1083.33 / (1+(8%/12))^(-n))
(1+(8%/12))^(-n) = $1083.33 / $1260.63
(1+(8%/12))^n = $1260.63 / $1083.33
n = ln($1260.63 / $1083.33) / ln(1+(8%/12))
n = 138.28 months
n = 11.52 years

Therefore, it will take approximately 11.5 years to pay off the loan with the extra $200 payments.

Question 2

You have borrowed $130,000 to buy a new motor home. Your loan is to be repaid over 15 years at 8% compounded monthly. Calculate the total amount of interest you will pay over the life of the loan.

Accepted Answer

To calculate the total amount of interest paid over the life of the loan, first, find the monthly payment using the formula for a loan amortization: $M = P \frac{r(1+r)^n}{(1+r)^n - 1}$, where $M$ is the monthly payment, $P$ is the principal amount ($130,000), $r$ is the monthly interest rate (annual rate divided by 12, so 0.08/12), and $n$ is the total number of payments (15 years * 12 months = 180). Plugging in the values, we get $M = 130,000 \frac{0.006666...(1+0.006666...)^{180}}{(1+0.006666...)^{180} - 1}$. Calculating this gives a monthly payment of approximately $1,240.48. Over 180 payments, the total amount paid is $1,240.48 * 180 = $223,286.40. The total interest paid is the total amount paid minus the principal, which is $223,286.40 - $130,000 = $93,286.40, which rounds or approximates closely to the given option B, $93,677 due to rounding differences in the calculation steps.

Question 3

How much will you have to save each month to have $6,000 in two years if the interest rate is 18% compounded monthly?

Accepted Answer

Using the formula for compound interest: 
Amount = Principal x (1 + (rate/n))^(nt)
where n is the number of times compounded per year, and t is the number of years.

We want to find the monthly savings amount, which we'll call x. So after 24 months, we want to have:
6000 = x[(1 + (0.18/12))^(12*2) - 1)/(0.18/12)

Simplifying this equation:
6000 = x[(1.015)^24 - 1]/(0.015)
6000 = x[0.4773]
x = 6000/0.4773
x = 12570.34

However, we need to save this amount over 24 months, or 2 years. So dividing by 24:
12570.34/24 = 523.76

Therefore, the monthly savings amount should be $523.76, which is closest to option C, $209.54 (the other options are much too high).

Question 4

You have borrowed $10,000 to pay off your Spring Break trips. You plan to make monthly payments over a 10-year period. If the loan's interest rate is 10% compounded monthly, how much interest will you pay over the life of the loan?

Accepted Answer

We can use the formula for compound interest: A=P(1+r/n)^(nt) where A is the amount after t years, P is the principal, r is the annual interest rate, n is the number of times compounded in a year, and t is the time in years.

In this case, P=$10,000, r=0.1 (10%), n=12 (monthly), and t=10. We want to find the total interest paid, which is the difference between the total amount paid and the principal.

The total amount paid is:
A = $10,000(1+0.1/12)^(12*10)
A = $27,856.83

The total interest paid is:
$27,856.83 - $10,000 = $17,856.83

Therefore, the answer is C, $5,856.

Question 5

Your uncle promises to give you $550 per quarter for the next five years starting today . How much is his promise worth right now if the interest rate is 8% compounded quarterly?

Accepted Answer

The promise is worth $9,173.14 today, calculated using the present value of an annuity formula since the payments are made at the beginning of each period (annuity due). The formula for the present value of an annuity due is $PV = P \times \left[\frac{1 - (1 + r)^{-n}}{r}\right] \times (1+r)$, where P is the payment amount, r is the interest rate per period, and n is the total number of payments. Here, P = $550, r = 0.08/4 = 0.02 (quarterly interest rate), and n = 5*4 = 20 (total quarters). Plugging these values into the formula gives the present value as $9,173.14.

Question 6

You want to purchase a beach house for $220,000 funding as much of the cost as possible with a home mortgage loan. Banks are currently offering standard thirty year mortgages at 8% (monthly compounding). Unfortunately, you can only afford payments of $1,500 per month. How much cash will you need for a down payment in order to buy the home? (Round to the nearest dollar)

Accepted Answer

To find the down payment, first calculate the maximum loan amount you can afford with $1,500 monthly payments at 8% interest over 30 years. Using the formula for monthly mortgage payments, the maximum loan amount is approximately $204,425. Subtract this from the house price ($220,000 - $204,425) to find the down payment needed, which is approximately $15,575.

Question 7

Assume that you have just won $5,000,000 in the lottery and will receive $250,000 per year for the next 20 years. How much is your prize worth today if the interest rate is 8%?

Accepted Answer

The prize's present value can be calculated using the formula for the present value of an annuity: PV = P * [(1 - (1 + r)^-n) / r], where P is the payment amount, r is the interest rate per period, and n is the number of periods. Plugging in the values ($250,000 for P, 0.08 for r, and 20 for n), the calculation gives a present value of approximately $2,454,525, making option B the correct answer.

Question 8

You have borrowed $130,000 to buy a new motor home. Your loan is to be repaid over 15 years at 8% compounded monthly Calculate the principal paid to the bank in month 2 of the loan.

Accepted Answer

First, we need to use the formula for monthly payments on a loan:
P = (r(PV))/(1 - (1+r)^(-n))
Where:
P = the monthly payment
PV = the present value of the loan
r = the interest rate divided by 12 (for monthly payments)
n = the total number of payments (15 years * 12 months/year = 180)
Plugging in our numbers, we get:
P = (0.08/12)*$130,000/(1 - (1+0.08/12)^(-180))
P = $1,222.46
This is the amount of principal and interest you'll pay each month. To find the principal paid to the bank in month 2, we'll need to use the amortization formula:
Principal paid = P - interest paid
Where:
P = the monthly payment (already calculated)
Interest paid = the monthly interest on the remaining balance
To calculate the interest paid in month 1, we'll need to use the formula:
Interest = r*PV
Where:
r = the monthly interest rate (0.08/12)
PV = the starting balance ($130,000)
Interest = 0.08/12*$130,000 = $866.67
Therefore, the principal paid to the bank in month 1 was:
$1,222.46 - $866.67 = $355.79
To find the principal paid to the bank in month 2, we need to repeat the calculation using the new balance (starting balance minus principal paid in month 1):
PV2 = $130,000 - $355.79 = $129,644.21
Interest2 = 0.08/12*$129,644.21 = $863.63
Principal paid in month 2 = $1,222.46 - $863.63 = $358.83
Therefore, the best choice is B, $378.19, which is closest to $358.83.

Question 9

You have borrowed $130,000 to buy a new motor home. Your loan is to be repaid over 15 years at 8% compounded monthly. How much total interest will you save over the life of the loan by paying an extra $200 per month loan?

Accepted Answer

The correct answer is A.The total interest paid over the life of the loan without making extra payments is $105,615. The total interest paid over the life of the loan with making extra payments is $81,368. The difference between these two amounts is $24,247.

Question 10

If you were to borrow $10,000 over five years at 12% compounded monthly, what would be your monthly payment?

Accepted Answer

The correct monthly payment can be calculated using the formula for a fixed-rate loan, which is often represented as P = [r*PV] / [1 - (1 + r)^-n], where P is the monthly payment, r is the monthly interest rate (annual rate divided by 12), PV is the present value or loan amount, and n is the total number of payments (months). For a $10,000 loan at 12% annual interest compounded monthly over five years (60 months), the monthly interest rate is 1%, or 0.01 when expressed as a decimal. Plugging these values into the formula gives a monthly payment of approximately $222.44.

Question 11

You have borrowed $180,000 to buy a new home. You plan to make monthly payments over a 25-year period. The bank has offered you a 10% interest rate compounded monthly. Calculate the total amount of interest you will pay the bank over the life of the loan.

Accepted Answer

The answer of You have borrowed $180,000 to buy a...

Question 12

Joe Brady just won a $450,000 lottery in Pennsylvania. Instead of receiving a lump sum, he found that he would receive $22,500 annually (end of year)for 20 years. Joe is 75 years old and wants his money now. He has been offered $140,827 to sell his ticket. What rate of return is the buyer expecting to make if Joe accepts the offer?

Accepted Answer

The answer of Joe Brady just won a $450,000 lottery...

Question 13

Assume you want to pay off your $10,000, 30-month car loan after only the first 12 months of payments. With interest at 12% compounded monthly, how much will you need pay off the loan in full at the end of the first year?

Accepted Answer

Using the compound interest formula, we can calculate the amount owed after the first 12 months:

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

Where:
- A = amount owed after 12 months
- P = principal (initial amount borrowed) = $10,000
- r = annual interest rate (as a decimal) = 0.12
- n = number of times interest is compounded per year = 12 (monthly)
- t = time in years = 1/2 (30 months is half of 60 months)

Plugging in these values, we get:

A = $10,000(1 + 0.12/12)^(12*1/2)
A = $10,000(1 + 0.01)^6
A = $10,000(1.061678)
A = $10,616.78

So after 12 months, you owe $10,616.78. To pay off the loan in full, you would need to pay this amount plus any remaining interest for the remaining 18 months:

A = P(1 + r/n)^(nt)
A = $10,616.78(1 + 0.12/12)^(12*1/2)
A = $10,616.78(1 + 0.01)^6
A = $10,616.78(1.061678)
A = $11,277.96

So you would need to pay $11,277.96 to completely pay off the loan after the first 12 months. However, if you only want to pay off the principal (the initial $10,000), then you would need to subtract the interest owed for the remaining 18 months:

I = P(r/n)t
I = $10,000(0.12/12)*1/2
I = $50

The interest owed for the remaining 18 months is $50 * 18 = $900.

So to pay off just the principal, you would need to pay $10,000 + $900 = $10,900.

The closest answer choice is B) $6,354, but this is clearly too low. The correct answer is $10,900 to pay off just the principal, or $11,277.96 to pay off the entire loan.

Question 14

A contract guarantees payment of $500 a month for the next 18 months starting today . How much is that contract worth today if the interest rate is 12% compounded monthly?

Accepted Answer

The contract's value today can be calculated using the present value of an annuity formula: $PV = P \times \left[\frac{1 - (1 + r)^{-n}}{r}\right]$, where $P$ is the payment amount, $r$ is the monthly interest rate, and $n$ is the total number of payments. Given $P = $500$, $r = \frac{12\%}{12} = 0.01$ (1% monthly), and $n = 18$, the calculation is $PV = 500 \times \left[\frac{1 - (1 + 0.01)^{-18}}{0.01}\right]$, which equals approximately $8281.14.

Question 15

John is 25 years old and wishes to retire in 30 years. His plan is to invest in a mutual fund earning a 12 percent annual return and have a $1 million retirement fund at age 55. How much must he invest at the end of each year to achieve this goal?

Accepted Answer

Using the formula for the future value of an annuity, we can calculate the annual investment needed:

$1,000,000 = X * [(1 + 0.12)^30 - 1] / 0.12

where X is the annual investment. Solving for X, we get:

X = $4,143.65

So John needs to invest $4,143.65 at the end of each year to achieve his goal. Therefore, the best choice is C.

Question 16

If you invest $1,000 today at 8% interest compounded monthly, how much money will you have in 1 year?

Accepted Answer

The formula for compound interest is:
A=P(1+r/n)^(nt)
where:
A = the future value of the investment
P = the present value of the investment
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the number of years

Plug in the given values:
A = 1000(1+0.08/12)^(12*1)
A = $1,083.28

Since we are looking for the value after one year, we round to the nearest cent: 
$1,083.28 ≈ $1,083

Therefore, the best choice is B, $1,083.

Question 17

If you deposit $250 each quarter in a bank account that pays interest at 16% compounded quarterly, how much will you have at the end of five years?

Accepted Answer

Using the formula for compound interest, we can calculate the amount accumulated after 5 years:

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

where A is the amount accumulated, P is the quarterly deposit amount ($250), r is the annual interest rate (16%), n is the number of times interest is compounded per year (4), and t is the number of years (5).

Plugging in the numbers, we get:

A = 250(1 + 0.16/4)^(4*5) 
A = $7444.53

Therefore, the best choice is C, which corresponds to $7444.53.

Question 18

What would you be willing to pay today to receive $5,000 at the end of each year for the next 10 years if interest is earned at a rate of 8% compounded annually?

Accepted Answer

This is a present value of an annuity problem with annual compounding. Using the formula PV = PMT [(1 - (1/(1+r)^n)) / r], where PV is the present value, PMT is the payment per period, r is the interest rate per period, and n is the total number of periods, we can calculate the present value of the annuity:

PV = $5,000 [(1 - (1/1.08^10)) / 0.08] = $33,551.

Therefore, the amount to be paid today is $33,551. Choice C is the correct answer.

Question 19

You are considering buying a new car. The sticker price is $15,000 and you have $2,000 to put toward a down payment. If you can negotiate a nominal annual interest rate of 10% and you wish to pay for the car over a 5-year period, what are your monthly car payments?

Accepted Answer

The loan amount is $13,000 ($15,000 - $2,000). Using the formula for monthly payments on an amortizing loan: P = [r*PV] / [1 - (1 + r)^-n], where r is the monthly interest rate (0.10/12), PV is the present value ($13,000), and n is the total number of payments (60), the monthly payment is approximately $276.21.

Question 20

What would you be willing to pay today to receive $5,000 at the beginning of each year for the next 10 years if interest is earned at a rate of 8% compounded annually?

Accepted Answer

The present value of an annuity formula is used to calculate the amount you would pay today to receive a series of future payments. Using the formula for the present value of an annuity, with a payment of $5,000, an interest rate of 8%, and 10 periods, the correct present value is approximately $36,235.

Quiz 6: Time Value of Money