Q: You invest $$2,000 for 5 years at an annual interest rate of 5%. What is the future value (FV) of your investment at the end of 5 years? Use the formula FV = PV$$ \times (1 + r)^n $.

$$ FV = 2,000 \times (1.05)^5$ = 2,552.56 $$

Q: You are offered an investment opportunity: invest $$10,000 today, receive 500 $$per year for 10 years, and $$10,000 in 10 years. If the interest rate is 6% per year, what is the Net Present Value (NPV) of this opportunity? Use $$ NPV = \sum \frac{CF_t}{(1 + r)^t} - Initial\ Investment $.

$$ NPV = -10,000 + 500 \times \frac{1 - (1.06$$)^{-10}$$}{0.06} + \frac{10,000}{(1.06$$)^{10}$$} = -10,000 + 3,690.59 + 5,583.95 = -725.46 $$

Q: You have a loan outstanding that requires making three annual payments of $$1,000 each. If the interest rate is 8%, what is the present value of the loan? Use the annuity formula PV = C$$ \times \frac{1 - (1 + $$r)^{-n}$$}{r} $.

$$ PV = 1,000 \times \frac{1 - (1.08$$)^{-3}$$}{0.08} = 2,577.10 $$

Question 1

Which of the following best describes the concept of the time value of money?

Accepted Answer

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

Question 2

You invest $$2,000 for 5 years at an annual interest rate of 5%. What is the future value (FV) of your investment at the end of 5 years? Use the formula  FV = PV$$ 	imes (1 + r)^n $.

Accepted Answer

$$ FV = 2,000 	imes (1.05)^5$ = 2,552.56 $$

Question 3

A bank offers you a choice between receiving $$10,000 now or 10$$,000 in 10 years. If the interest rate is 8% per year, which option is preferable?

Accepted Answer

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

Question 4

You want to have $$100,000 in 10 years for your daughter's college education. If the account pays a fixed interest rate of 5% per year, how much do you need to deposit today? Use the formula  PV$$ = \frac{FV}{(1 + r)^n} $.

Accepted Answer

$$ PV = \frac{100,000}{(1.05$$)^{10}$$} = 61,386.09 $$

Question 5

Suppose you deposit $1,000 into an account paying 8% interest per year. What is the balance in the account after 3 years? How much of this balance corresponds to interest earned?

Accepted Answer

Balance: $$1,259.71; Interest earned: 259.71$$

Question 6

You are offered an investment opportunity: invest $$10,000 today, receive 500 $$per year for 10 years, and $$10,000 in 10 years. If the interest rate is 6% per year, what is the Net Present Value (NPV) of this opportunity? Use $$ NPV = \sum \frac{CF_t}{(1 + r)^t} - Initial\ Investment $.

Accepted Answer

$$ NPV = -10,000 + 500 	imes \frac{1 - (1.06$$)^{-10}$$}{0.06} + \frac{10,000}{(1.06$$)^{10}$$} = -10,000 + 3,690.59 + 5,583.95 = -725.46 $$

Question 7

A perpetuity pays $$100 per year forever. If the current interest rate is 5% per year, what is the present value of the perpetuity? Use  PV$$ = \frac{C}{r} $.

Accepted Answer

$$ PV = \frac{100}{0.05} = 2,000 $$

Question 8

You are considering an investment that will pay $$2 million in its first year and that will grow at a rate of 5% per year for the next 16 years. If the interest rate is 10% per year, what is the present value of the investment? Use the growing annuity formula  PV$$ = \frac{C}{r - g} \left[1 - \left(\frac{1 + g}{1 + r}\right)^n\right] $.

Accepted Answer

$$ PV = \frac{2,000,000}{0.10 - 0.05} \left[1 - \left(\frac{1.05}{1.10}\$$right)^{17}$$\right] = 21,045,859.86 $$

Question 9

You have a loan outstanding that requires making three annual payments of $$1,000 each. If the interest rate is 8%, what is the present value of the loan? Use the annuity formula  PV = C$$ 	imes \frac{1 - (1 + $$r)^{-n}$$}{r} $.

Accepted Answer

$$ PV = 1,000 	imes \frac{1 - (1.08$$)^{-3}$$}{0.08} = 2,577.10 $$

Question 10

A British government bond pays £100 per year forever. If the current interest rate is 4% per year, what is the value of the bond immediately after a payment is made?

Accepted Answer

$$ PV = \frac{100}{0.04} = 2,500 $$

Question 11

You are planning to save $$5,000 per year for retirement, starting at age 25 and making the last deposit when you retire at age 65. If the account pays 8% interest per year, what is the future value of your savings at retirement? Use the annuity formula  FV = C$$ 	imes \frac{(1 + r)^n - 1}{r} $.

Accepted Answer

$$ FV = 5,000 	imes \frac{(1.08$$)^{41}$$ - 1}{0.08} = 1,295,316.06 $$

Question 12

A rich relative has bequeathed you a growing perpetuity. The first payment will occur in a year and will be $$1,000. Each payment after that will grow at 8% per year. If the interest rate is 12% per year, what is the present value of the bequest? Use  PV$$ = \frac{C}{r - g} $.

Accepted Answer

$$ PV = \frac{1,000}{0.12 - 0.08} = 25,000 $$

Question 13

You are considering two retirement plans: Plan A pays $$350,000 immediately on retirement, Plan B pays 350$$,000 five years after retirement. If the interest rate is 8% per year, which plan has the higher present value?

Accepted Answer

Plan A has the higher present value because the payment is received sooner.

Question 14

You are offered an investment that pays $$1,000 at the end of each of the next 100 years. If the interest rate is 7% per year, what is the present value of the investment? Use the annuity formula  PV = C$$ 	imes \frac{1 - (1 + $$r)^{-n}$$}{r} $.

Accepted Answer

$$ PV = 1,000 	imes \frac{1 - (1.07$$)^{-100}$$}{0.07} = 14,296.25 $$

Question 15

You are considering an investment that requires an initial outlay of $$5,000 and will pay 4$$,000 at the end of each of the next three years. If the interest rate is 2% per year, what is the Net Present Value (NPV) of the investment?

Accepted Answer

$$ NPV = -5,000 + \frac{4,000}{1.02} + \frac{4,000}{(1.02)^2$} + \frac{4,000}{(1.02)^3$} = 6,000.35 $$

Question 16

You are considering a machine that costs $$1,000 to build and will generate 100 $$per year forever. If the interest rate is 9.5% per year, should you build the machine? Use the perpetuity formula $$ PV = \frac{C}{r} .$$

Accepted Answer

Yes, because $$ PV = \frac{100}{0.095} = 1,052.63 $$ which is greater than the cost.

Question 17

You are offered a growing annuity that pays $$1,000 per year for 17 years, with payments growing at 5% per year. If the interest rate is 10% per year, what is the present value? Use  PV$$ = \frac{C}{r - g} \left[1 - \left(\frac{1 + g}{1 + r}\right)^n\right] $.

Accepted Answer

$$ PV = \frac{1,000}{0.10 - 0.05} \left[1 - \left(\frac{1.05}{1.10}\$$right)^{17}$$\right] = 10,241.65 $$

Question 18

You are planning to pay $10,000 per year for your child's private school tuition for 13 years, starting in one year. If tuition is expected to increase at a rate of 5% per year and the interest rate is 5% per year, how much would you need to deposit in the bank now to fund all 13 years of tuition?

Accepted Answer

$$ PV = 10,000 	imes \frac{1 - (1.05$$)^{-13}$$}{0.05} = 97,000 $$

Chapter 4: The Time Value of Money: Financial Accounting Study Notes

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