Q: You win a lottery that pays $$1 million per year for 30 years, with the first payment today. If the discount rate is 8%, what is the present value of the remaining 29 payments (excluding the first payment)? Use PV = C$$ \times \frac{1 - \frac{1}{(1+r)^n}}{r} $.

$$ PV = 1,000,000 \times \frac{1 - \frac{1}{(1+0.08$$)^{29}$$}}{0.08} = 11,160,000 $$

Q: Ellen saves $$10,000 at the end of each year for 30 years in an account earning 10% per year. What is the future value of her savings at the end of 30 years? Use FV = C$$ \times \frac{(1+r)^n - 1}{r} $.

$$ FV = 10,000 \times \frac{(1.10$$)^{30}$$ - 1}{0.10} = 1,644,940 $$

Q: A company wants to buy a warehouse for $$100,000. The bank requires a 20% down payment and offers a 30-year loan at 8% interest. What is the annual loan payment? Use c = P$$ \times \frac{r}{1 - \frac{1}{(1+r)^n}} $$ where P = 80$$,000 $.

$$ c = 80,000 \times \frac{0.08}{1 - \frac{1}{(1.08$$)^{30}$$}} = 7,106.19 $$

Question 1

Which of the following best describes the 'time value of money' principle?

Accepted Answer

Money today is worth more than the same amount in the future because it can be invested to earn interest.

Question 2

You are offered a choice between receiving $$10,000 today or 12$$,000 in three years. If the annual interest rate is 5%, which option has a higher present value?

Accepted Answer

Receiving $10,000 today, because its present value is higher.

Question 3

What is the present value of a $$1,000 payment to be received in 5 years if the discount rate is 7% per year? Use the formula  PV$$ = \frac{C}{(1+r)^n} $.

Accepted Answer

$$ PV = \frac{1000}{(1+0.07)^5$} = 712.99 $$

Question 4

A perpetuity pays $$2,000 per year, with the first payment one year from now. If the interest rate is 8%, what is the present value? Use  PV$$ = \frac{C}{r} $.

Accepted Answer

$$ PV = \frac{2000}{0.08} = 25,000 $$

Question 5

Which of the following is a key difference between an annuity and a perpetuity?

Accepted Answer

A perpetuity has payments that last forever, while an annuity has a fixed number of payments.

Question 6

You win a lottery that pays $$1 million per year for 30 years, with the first payment today. If the discount rate is 8%, what is the present value of the remaining 29 payments (excluding the first payment)? Use  PV = C$$ 	imes \frac{1 - \frac{1}{(1+r)^n}}{r} $.

Accepted Answer

$$ PV = 1,000,000 	imes \frac{1 - \frac{1}{(1+0.08$$)^{29}$$}}{0.08} = 11,160,000 $$

Question 7

Ellen saves $$10,000 at the end of each year for 30 years in an account earning 10% per year. What is the future value of her savings at the end of 30 years? Use  FV = C$$ 	imes \frac{(1+r)^n - 1}{r} $.

Accepted Answer

$$ FV = 10,000 	imes \frac{(1.10$$)^{30}$$ - 1}{0.10} = 1,644,940 $$

Question 8

A growing perpetuity pays $$30,000 next year, growing at 4% per year forever. If the discount rate is 8%, what is the present value? Use  PV$$ = \frac{C}{r-g} $.

Accepted Answer

$$ PV = \frac{30,000}{0.08 - 0.04} = 750,000 $$

Question 9

A company wants to buy a warehouse for $$100,000. The bank requires a 20% down payment and offers a 30-year loan at 8% interest. What is the annual loan payment? Use  c = P$$ 	imes \frac{r}{1 - \frac{1}{(1+r)^n}} $$ where  P = 80$$,000 $.

Accepted Answer

$$ c = 80,000 	imes \frac{0.08}{1 - \frac{1}{(1.08$$)^{30}$$}} = 7,106.19 $$

Question 10

Suppose you invest $$1,000 today and will receive 2$$,000 in 6 years. What is the annual rate of return, $$ r $$? Use $$ 1000 	imes (1+r)^6$ = 2000 .$$

Accepted Answer

$$ r = (\frac{2000}{1000}$$)^{1/6}$$ - 1 = 0.1225  or 12.25$$%

Question 11

A growing annuity pays $$10,000 in the first year, growing at 5% per year for 30 years. The discount rate is 10%. What is the present value? Use  PV$$ = $$C_1$$ 	imes \frac{1 - ((1+g)/(1+r))^n}{r-g} $.

Accepted Answer

$$ PV = 10,000 	imes \frac{1 - ((1.05)/(1.10$$))^{30}$$}{0.10 - 0.05} = 150,463 $$

Question 12

If you want to accumulate $$60,000 for a house down payment, already have 10$$,050 saved, and can save $5,000 per year at 7.25% interest, which formula would you use to solve for the number of years required?

Accepted Answer

$$ FV = PV 	imes (1+r)^n + C 	imes \frac{(1+r)^n - 1}{r} $$ and solve for $$ n $$

Question 13

Which of the following statements about discounting and compounding is correct?

Accepted Answer

Discounting moves future cash flows to present value, while compounding moves present cash flows to future value.

Question 14

A company is considering two options to purchase equipment: pay $$40,000 cash today or four annual payments of 15$$,000. Which equation would you use to find the implied interest rate?

Accepted Answer

$$ 40,000 = 15,000 	imes \frac{1 - \frac{1}{(1+r)^4$}}{r} $$ and solve for $$ r $$

Question 15

Which of the following is NOT a correct application of the Law of One Price in the context of time value of money?

Accepted Answer

A future cash flow can be compared directly to a present cash flow without discounting.

Question 16

If you invest $100 at 5% interest per year forever and withdraw the interest each year, what is the present value of this perpetuity?

Accepted Answer

$$ PV = \frac{5}{0.05} = 100 $$

Question 17

Which formula would you use to calculate the present value of a stream of cash flows $$ C_0$, C_1$, ..., C_n  at $$discount rate $$ r $$?

Accepted Answer

$$ PV_0$ = C_0$ + \frac{C_1$}{(1+r)} + \frac{C_2$}{(1+r)^2$} + ... + \frac{C_n}{(1+r)^n} $$

Chapter 4: The Time Value of Money (Fundamentals of Corporate Finance)

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