Q: Suppose you invest $$100 in a bank account, and five years later it has grown to 134.39. $$What APR did you receive if the interest was compounded monthly?

$$ APR = 12 \times [(1.3439$$)^{\frac{1}$${60}} - 1] = 6\% $$

Q: Suppose the interest rate is 8% APR with monthly compounding. What is the present value of an annuity that pays $100 every six months for five years?

$$ PV = 100 \times \frac{1 - (1 + 0.06467$$)^{-10}$$}{0.06467} = 803.90 $$

Q: You have just purchased a home and took out a $500,000 mortgage. The mortgage has a 30-year term with monthly payments and an APR of 6%. What is your total interest paid in the first year?

$$ Total\ Interest = (2997.75 \times 12) - (500,000 \times (1 - (1 + \frac{0.06}{12}$$)^{-12}$$)) = 29,973 - 5,614 = 24,359 $$

Q: You have an outstanding student loan with required payments of $$500 per month for the next four years. The starting rate of the loan is 9% APR (monthly compounding). If you pay an extra 100 $$today, what is the effective rate of return (expressed as an APR with monthly compounding) you have earned on the $100?

$$ APR = 12 \times [(1 + \frac{141.14}{100}$$)^{\frac{1}$${48}} - 1] = 9\% $$

Q: You need to borrow $8000 to purchase a motorcycle. The loan is a 48-month, 5.99% APR motorcycle loan. What will your monthly payment be?

$$ C = \frac{8000 \times 0.0049925}{1 - (1 + 0.0049925$$)^{-48}$$} = 184.77 $$

Question 1

A bank offers you an account that will pay 20% interest in total for a two-year deposit. What is the equivalent discount rate for a period length of six months?

Accepted Answer

$$1 - (1 + 0.20$$)^{-\frac{1}$${4}} = 0.0466$$

Question 2

Which account would you prefer: a bank account that pays 5% per year (EAR) for three years or an account that pays 2.5% every six months for three years?

Accepted Answer

The account that pays 2.5% every six months, since $$ (1.025)^6$ = 1.1597 $$ which is greater than $$ (1.05)^3$ = 1.1576 $$

Question 3

You have found three investment choices for a one-year deposit: 10% APR compounded monthly, 10% APR compounded annually, and 9.5% APR compounded daily. Which has the highest effective annual rate (EAR)?

Accepted Answer

10% APR compounded monthly, since $$ EAR = (1 + \frac{0.10}{12}$$)^{12}$$ - 1 = 0.1047 $$

Question 4

Suppose you invest $$100 in a bank account, and five years later it has grown to 134.39. $$What APR did you receive if the interest was compounded monthly?

Accepted Answer

$$ APR = 12 	imes [(1.3439$$)^{\frac{1}$${60}} - 1] = 6\% $$

Question 5

You make monthly payments on your mortgage. It has a quoted APR of 5% (monthly compounding). What percentage of the outstanding principal do you pay in interest each month?

Accepted Answer

$$ Discount\ Rate = 1 - (1 + \frac{0.05}{12}$$)^{-1}$$ = 0.0041667 $$

Question 6

You are considering moving your money to a new bank offering a one-year CD that pays an 8% APR with monthly compounding. Your current bank’s shorter-term account pays 8% APR with semiannual compounding. What is the EAR for each account?

Accepted Answer

Monthly compounding: $$ EAR = (1 + \frac{0.08}{12}$$)^{12}$$ - 1 = 0.083 $$; Semiannual compounding: $$ EAR = (1 + \frac{0.08}{2}$$)^{2}$$ - 1 = 0.0816 $$

Question 7

Suppose the interest rate is 8% APR with monthly compounding. What is the present value of an annuity that pays $100 every six months for five years?

Accepted Answer

$$ PV = 100 	imes \frac{1 - (1 + 0.06467$$)^{-10}$$}{0.06467} = 803.90 $$

Question 8

You have just purchased a home and took out a $500,000 mortgage. The mortgage has a 30-year term with monthly payments and an APR of 6%. What is your total interest paid in the first year?

Accepted Answer

$$ Total\ Interest = (2997.75 	imes 12) - (500,000 	imes (1 - (1 + \frac{0.06}{12}$$)^{-12}$$)) = 29,973 - 5,614 = 24,359 $$

Question 9

You have an outstanding student loan with required payments of $$500 per month for the next four years. The starting rate of the loan is 9% APR (monthly compounding). If you pay an extra 100 $$today, what is the effective rate of return (expressed as an APR with monthly compounding) you have earned on the $100?

Accepted Answer

$$ APR = 12 	imes [(1 + \frac{141.14}{100}$$)^{\frac{1}$${48}} - 1] = 9\% $$

Question 10

You need to borrow $8000 to purchase a motorcycle. The loan is a 48-month, 5.99% APR motorcycle loan. What will your monthly payment be?

Accepted Answer

$$ C = \frac{8000 	imes 0.0049925}{1 - (1 + 0.0049925$$)^{-48}$$} = 184.77 $$

Question 11

You have just sold your house for $$1,000,000 in cash. Your mortgage was originally a 30-year term with monthly payments and an initial balance of 800$$,000. The mortgage is currently at 5.25% APR, and you have just made a payment. If the interest rate on the mortgage is 5.25% (APR), how much cash will you have from the sale once you pay off the mortgage?

Accepted Answer

$$ PV = 4,417.63 	imes \frac{1 - (1 + 0.004375$$)^{-138}$$}{0.004375} = 534,069 $$; Cash = 1,000,000 - 534,069 = 465,931 $

Question 12

Your mortgage has 25 years left, and has an APR of 7.425% with monthly payments of $1,449. What is the outstanding balance?

Accepted Answer

$$ PV = 1,449 	imes \frac{1 - (1 + 0.0061875$$)^{-300}$$}{0.0061875} = 194,024.13 $$

Question 13

You are offered a loan with an APR of 4% (semiannual compounding). What is the semiannual discount rate?

Accepted Answer

$$ 1 - (1 + \frac{0.04}{2}$$)^{-1}$$ = 0.0198 $$

Question 14

A professor earns $70,000 per year with a 6% EAR. What is the present value of the amount she will earn while on sabbatical for 7 years?

Accepted Answer

$$ PV = \frac{70,000}{(1.06)^7$} = 46,703 $$

Question 15

You are considering a 30-year mortgage at 4.75% APR, with monthly payments of $$1,250. If you make an extra payment of 1$$,250 every year, how many years will it take to pay off the mortgage?

Accepted Answer

$$ N = \frac{\log(1 - \frac{0.004375 	imes 200,000}{1,250})}{\log(1 + 0.004375)} = 23.2 $$ years

Question 16

If you deposit $1,000 in a savings account with 8% APR compounded monthly, what will be the balance after 5 years?

Accepted Answer

$$ 1,000 	imes (1 + \frac{0.08}{12}$$)^{60}$$ = 1,489.85 $$

Question 17

You are offered a loan with an EAR of 5%. What is the equivalent APR if the interest is compounded monthly?

Accepted Answer

$$ APR = 12 	imes [(1.05$$)^{\frac{1}$${12}} - 1] = 4.89\% $$

Chapter 5: Interest Rates: Concepts, Calculations, and Applications in Financial Accounting

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