Ch 5 Managerial Finance

How much is $100 received at the end of each year forever, at 10% interest, worth today?

$1,000

Today you deposit $1000 in an account paying 6% interest. At the end of years 1, 2, and 3, you will deposit $100 in that account. What is the present value of that stream of cash flows?

$1,267.30 (Reason: $1000 + $100/(1.06)^1 + $100/(1.06)^2 + $100/(1.06)^3 = $1,267.30.)

Find the future value of an annuity of $100 per year for 10 years at 10% per year.

$1,593.75 (First, find the PV by using the 10 year annuity factor: PV = $100 × 10 year annuity factor = $100 × [1/0.1 − 1/0.1x(1.1)^10]= $614.46. To find the future value, multiply $614.46 x (1.1)^10= $1,593.75.)

Today you deposit $1000 in an account paying 6% interest. At the end of years 1, 2, and 3, you will deposit $100 in that account. How much will you have at the end of year 4?

$1,599.94 (Reason: $1000(1.06)^4 + $100(1.06)^3 + $100(1.06)^2 + $100(1.06)^1 = $1,599.94.)

A dollar invested today at 7.5% simple annual interest will be worth _______ one year from now.

$1.075 FV = $1.00 + $0.075

A dollar invested today at 7.5% interest compounded annually will be worth _______ one year from now.

$1.075 (FV = $1.00(1+0.075))

A dollar invested today at 8.0% simple annual interest will be worth _________ three years from now.

$1.24 (With simple interest, the bank calculates interest only on the principal investment: $1.00 + $0.08 + $0.08 + $0.08 = $1.24. Do not confuse this with compound interest, which computes interest earned on interest.)

A dollar invested today at 8.0% interest compounded annually will be worth _______ three years from now.

$1.2597 FV = $1.00(1 + 0.08)^3

Your insurance agent wants to sell you an annuity consisting of 20 equal end of year payments of $10,000 each, starting at the end of this year. Your desired rate of return for investments of this type is 7 percent. What is the most you would pay for this annuity today?

$105,940.14 (PV = 10,000[1/0.07 − 1/0.07(1.07)^20] = 105,940.14.

In 2013, the CPI was about 2.5 times its level in 1981. If the cost of one semester of college was $5,000 in 1981, what should the nominal cost of a semester of college be in 2013, assuming the real price is constant?

$12,500 (2.5 × $5000 = $12,500.)

What is the future value of $100 invested for 4 years at 10% interest?

$146.41 FV = $100 × (1 + r)^t = $100 × (1 + 0.1)^4 = $146.41.

Assume you have $100 to invest today. Investing it at 5% interest compounded annually will yield _______ in 10 years, while investing it at 6% interest compounded annually will yield _______ in 10 years.

$162.89; $179.08

$200 at the end of each year forever at 10% per year is worth how much today?

$2,000.00 ($200/0.10 = $2,000.)

What is the present value of an ordinary annuity that pays $100 per year for 3 years if the interest rate is 10% per year?

$248.69 (100[(1/0.10) − (1/(0.10(1.10)^3))] = 248.69.

You will receive $100 in 1 year, $200 in 2 years, and $300 in 3 years. If you can earn a 7.5% rate of interest, what is the present value of this stream of cash flows? (Please note that you receive nothing immediately—there is no initial payment)

$507.58 (Reason: $100/(1.075)^1 + $200/(1.075)^2 + $300/(1.075)^3 = $507.58.)

You put $100 in the bank now, $200 in the bank a year from now, and $300 in the bank in 2 years. How much money will you have available 3 years from now if you earn a 7.5% rate of interest? (Calculate the future value of this stream of cash flows. Refer to Example 5.6)

$677.85 (Reason: $100 × (1.075)^3 + $200 × (1.075)^2 + $300 × (1.075) = $677.85.)

The present value of $100 paid annually at year-end for 20 years at 10% per year is ______.

$851.36 (100[(1/0.10) − (1/(0.10(1.10)^20))] = 851.36.)

If the interest rate is 10% per year, then what is the present value (PV) of $100 received 1 year from today?

$90.91 PV = $100/1.10 = $90.91

The effective annual interest rate is equal to ______.

(1 + APR/m)^m − 1

Compound growth means that value increases after t periods by ______.

(1 + growth rate)^t

______ dollars refer to the actual number of dollars of the day, whereas ______ dollars refer to the amount of purchasing power.

- Current or Real - Nominal, real - Nominal or Constant - Current, constant

Which of the following statements are true regarding the present value of a stream of cash payments?

- Real cash payments should be discounted using a real interest rate. - Nominal cash payments should be discounted using a nominal interest rate.

Which of the following are annuities?

- annual installment loan payments - yearly lease payments

If a bank quotes a loan with an APR of 15%, compounded monthly, what is the periodic rate on this loan?

1.25% (15/12 = 1.25%.)

Which of the following is the correct formula for the discount factor?

1/(1 + r)^t

Your neighborhood bank is offering investors a money market account that pays 3.5% interest on deposits. If the current annual rate of inflation is 1.2%, how much is the exact real rate for this account?

2.27% ([1.035/1.012] − 1 = 2.27%.)

A mortgage company is advertising a 30 year fixed rate mortgage with monthly payments and an APR of 3.0%. What is the effective annual interest rate of this loan?

3.04% (First, find the monthly interest rate: 0.03/12= 0.0025%. Next, convert this to an annually compounded interest rate: 1 + effective annual rate = (1 + monthly rate)12 = (1 + 0.0025)12 = 1.0304. The effective annual interest rate is 3.04%.)

If a bank account pays a monthly interest rate on deposits of 0.5%, what is the APR the bank will quote for this account?

6% (12 × 0.5 = 6%.)

The rate quoted by Big Bank on a car loan is 8%. The annual rate of inflation is currently 1.5%. What is the approximate real interest rate paid by the consumer on this loan?

6.5% (8 − 1.5 = 6.5%.)

A bank offers a savings account with an APR of 9% with monthly compounding. What is the effective annual interest rate of this investment?

9.38% (EAR = (1 + 0.09/12)12 − 1 = 9.38%.)

Inflation can be defined as ______.

An overall general rise in prices.

Which of the following is the correct equation for the present value of an annuity with regular payment C for t periods at interest rate r?

PV = C[1/r − 1/r(1 + r)^t]

If the interest rate (r) increases, what will happen to the present value (PV) over time?

PV will decline.

The real interest rate can be defined as ______.

The real change in value of an investment (or a real cost of a loan) after adjustment for inflation

True or false: The nominal interest rate can be defined as an interest rate quoted today by a financial institution on a loan or investment, such as an APR or a periodic rate.

True

True or false: The discount factor refers to the present value of a $1 future payment.

True Reason: The discount factor measures the present value of $1 received in year t.

Which of the following is a perpetuity?

a constant stream of cash flows forever

The effective annual interest rate is also known as the ___________.

annually compounded rate

A fixed stream of cash flows that ends after a specified number of years is called a(n) ______.

annuity

The present value of an annuity of $1 per period is called the __________.

annuity factor

Joseph signs a contract with a company that will pay him $25,000. Following the principles of the time value of money, Joseph would be best off if he received payment ______.

at the beginning of the project

An ordinary annuity is a series of level payments that begin ____.

at the end of one payment period

If interest rates go up, the present value of a perpetuity will ______.

decrease

At a rate of interest of 10% (r), the present value (PV) of $100 will _________ as the time period (t) ____________.

decrease; increases (Reason: Present value will decrease as the time period increases. This follows the time value of money concept that a dollar today is worth more than a dollar tomorrow.)

Interest income is _________ to interest rate.

directly proportional

Another name for the interest rate used to calculate PV is called the ______ rate.

discount

Discounting a future value (FV) at interest rate r over time t is termed a __________ calculation.

discounted cash-flow

Present value is calculated using a __________ calculation.

discounted cash-flow

The ________ is the rate at which invested funds will grow over the course of a year.

effective annual interest rate

A traditional (non-growing) annuity consists of a(n) ________ stream of cash flows for a fixed period of time.

fixed

The value in t years of an investment made today at interest rate r is called the ___________ of your investment.

future value

If the interest rate is greater than zero, the present value of an annuity due is always ______ an ordinary annuity.

greater than (REASON: Cash flows for annuities due always come one period earlier than the corresponding cash flows for ordinary annuities. Therefore, each is discounted for one less period, and the present value for the annuity due increases by a factor of (1 + r) over that of the ordinary annuity.)

An annuity due is a series of level payments that begin ____.

immediately

If interest rates go down, the present value of a perpetuity will _________.

increase

In 2013, the CPI was about 2.5 times its level in 1981. If the price of a pack of cigarettes was $1.00 in 1981 and $5.00 in 2013, then the real price has ______ since 1981. The inflation-adjusted price today should be _______ if there had been no real growth in the price of a pack.

increased; $2.50

A perpetuity is a constant stream of cash flows for a(n) ______ period of time.

infinite (Reason: A perpetuity is a constant stream of cash flows for an infinite time period.)

When money is invested at compound interest, the growth rate is equal to the __________.

interest rate

The time value of money concept states that a dollar today is worth _______ a dollar tomorrow.

more than

Which type of interest rate is generally quoted for loans and by banks and other financial institutions?

nominal

A stream of cash flows means that ________.

payments are made over time

C/r is the formula for the present value of a(n) ____.

perpetuity

The present value formula for a(n) ______ is PV = C/r, where C is the constant and regularly timed cash flow to infinity, and r is the interest rate.

perpetuity

The present value of an annuity due is equal to the ______.

present value of an ordinary annuity × (1 + r)

Which type of price refers to the purchasing power of money?

real

Real cash flow must be discounted by the ______.

real interest rate

The equation used to determine the approximate real interest rate is ______.

real interest rate = nominal interest rate − inflation rate

Real-world investments often involve many payments received or paid over time. Managers refer to this as a ___________.

stream of cash flows (Reason: A stream of cash flows means that payments are made over time.)

The best-known price index used by economists who measure inflation is ________.

the consumer price index (CPI)

Which of the following is a proper definition for the effective annual interest rate?

the interest rate that is annualized using compound interest

The future value of an annuity that lasts n years is equal to ______.

the present value allowed to grow n years