Managerial Finance Chapter 4

If an annuity pays $50 every 3 years over a 30-year period at 8 percent per year, then the present value equals ____.

$173.39

Suppose you have a car loan that lasts 6 years, a discount rate of 7 percent, and a loan balance of $15,000 requiring annual payments. What is the annual payment?

$3,146.94

The standard amortization time for a Stafford loan is ___ years.

10

In the formula for continuous compounding, a constant equal to _____ is included.

2.72

How long will it take to double your money at 10 percent per year?

7.2 years 72 ÷ 10 = 7.2 years.

Which of the following spreadsheet (Excel) functions will calculate the $614.46 present value of an ordinary annuity of $100 per year for 10 years at 10 percent per year?

= PV(0.10, 10, −100, 0, 0)

The ______ is the annual interest rate without consideration of compounding.

annual percentage rate

An annuity due is a series of payments that are made ____.

at the beginning of each period

Assume interest is compounded monthly. The ______ annual rate will express this rate as though it were compounded annually.

effective

When using an annuity table to find the present value of an annuity, you multiply the annuity cash flow by the present value interest _____________for annuities.

factor

A growing annuity has a(n) ____.

finite number of growing cash flows

Which type of amortization is most commonly used in the real world for mortgages and car loans?

fixed payment

Which of the following payment methods amortizes a loan?

fixed payments that result in a zero loan balance interest plus fixed amount

The present value of a growing perpetuity requires the interest rate to be ______ the growth rate.

greater than

The present value of a growing perpetuity will be ______ the present value of a zero-growth perpetuity, all else equal.

greater than

In an infrequent annuity, the payments occur ____.

less than once a year

The concept of future value implies that a dollar today is worth ______ a dollar in the future, assuming positive interest rates.

more than

Annuity time value of money tables are based on _____.

ordinary annuities

Payments in a partial amortization loan are based on the amortization period, not the loan period. The remaining balance is then ____.

paid off in a lump sum bullet payment

The loan balance on partial amortization loans declines so slowly because the ___.

payments are mostly interest

The value of a future cash flow stated in today's dollars is referred to as the _____.

present value

The value of a firm can be found by taking the _____ value of all _____ cash flows.

present; future

Amortization is the process of paying off loans by regularly reducing the _________.

principal

If you invest $1,000 and your NPV is $200, then the present value of your future cash flows is ______.

$1,200 NPV = −Cost + PV → PV = NPV + Cost PV = $200 + 1,000 = $1,200.

Find the future value of an annuity of $75 per year for 15 years at 6% per year.

$1,745.70

What is the future value of an annuity due of $100 per year for 10 years at 10 percent per year?

$1,753.12

At an annual interest rate of 10 percent, what is the present value of a perpetuity growing at 4 percent per year if next year's cash flow is $6?

$100 $6/(0.10 − 0.04)= $100.

Which one of the following constitutes an infrequent annuity?

$100 once every 2 years

What is the future value of $100 compounded for 50 years at 10 percent annual interest?

$11,739.09 FV = $100 × 1.1050 = $11,739.09.

What is the present value of a perpetuity of $100 per year if the annual interest rate is 10 percent and the growth rate is 6 percent per year?

$2,500

If you invest $100 at 10 percent simple interest, how much will you have in 10 years?

$200 FV = $100 + 10($100 × 0.10) = $200.

What is the present value of an annuity of $100 per year that begins at the end of year 4 and lasts for 5 years if the interest rate is 10 percent per year?

$284.81 PV3 = $100[(1 − 1/1.10^5)/0.1] = $379.08. PV0= $379.08/1.10^3 = $284.81.

Ralph has $1,000 in an account that pays 10 percent per year. Ralph wants to give this money to his favorite charity by making three equal donations at the end of the next 3 years. How much will Ralph give to the charity each year?

$402.11 Calculate the payment using the PV of an annuity at 10 percent for 3 years. $1,000/[(1 − 1/1.10^3)/0.10] = $402.11.

Suppose you paid off a $1,200 loan by paying $400 in principal each year plus 10 percent annual interest over a 3-year period. What is the total payment (interest plus principal) in Year 3?

$440 $400 + ($1,200 − 800) × 0.10 = $440.

The minimum monthly payment on a Stafford loan is ____.

$50

What is the present value of an annuity that makes payments of $100 per year for 10 years if the first payment is made immediately and the discount rate is 10 percent per year?

$675.90

To find the future value annuity factor using the time value of money table, read down the rows to find T = 2, then across the columns for an interest rate of 10 percent. The intersection of that row and column will show the factor ____.

2.100

Dependent undergraduate students can borrow a maximum of $_____________ for a subsidized Stafford loan.

23,000

If the cash flows of an annuity start at the end of year 4, the present value of an annuity formula will discount all of the annuity cash flows back to the end of year _______________

3

At the end of 5 days, you repay your $1,000 loan plus $50 in interest. What is the EAR?

3,422.24% [($1,050/$1,000)^365/5 − 1] = 34.2224, or 3,422.24%.

If the cash flows of an annuity start at the end of year 6 (date 6), the present value of an annuity formula will discount all of the annuity cash flows back to the end of year ___.

5 If the cash flows of an annuity start at the end of year 6 (date 6), the present value of an annuity formula will discount all of the annuity cash flows back to the end of year 5. The annuity value is as of the period before the payments begin.

To find the present value of an annuity of $100 per year for 10 years at 10 percent per year using the tables, look up the present value interest factor which is ______ and multiply that by ______.

6.1446; $100

When finding the present value of an annuity using a spreadsheet (Excel), the interest rate should be entered as a whole number.

False When finding the present value of an annuity using a spreadsheet (Excel), the interest rate should be entered as a decimal or percentage.

What are two ways to calculate a balloon payment?

Find the present value of the payments remaining after the loan term. Amortize the loan over the loan life to find the ending balance.

For a subsidized Stafford loan,

Interest does not accrue until repayment begins

Which of the following is the formula for the present value of a growing perpetuity?

PV = C/(r − g).

Which one of the following is the correct formula for the one-period present value?

PV = FV/(1 + r)

Which of the following are ways to amortize a loan?

Pay principal and interest every period in a fixed payment. Pay the interest each period plus some fixed amount of the principal.

Which of the following is true about a growing annuity?

The cash flows grow for a finite period.

Which of the following are true about the growing perpetuity model assumptions?

The interest rate must exceed the growth rate. The cash flow used is that for next year. The cash flows occur at regular intervals.

Which of the following are true about the amortization of a fixed payment loan?

The principal amount paid increases each period. The amount of interest paid decreases each period.

Which of the following are ways in which a $5,000 fixed principal repayment loan differs from a comparable $5,000 fixed payment loan?

The total payment each period for each type of loan is different. The amount of interest paid each period is different. The amount of principal paid each period is different.

True or false: The formula for the present value of an annuity factor is {1-1/(1+r) / tr}.

True

True or false: More money can be earned with compound interest than with simple interest.

True With compound interest, you earn interest on interest as well as interest on the principal.

Which of the following is a perpetuity?

a constant stream of cash flows forever

Which of the following would lower the present value of a future amount?

a higher interest rate a longer period of time a higher level of risk

Which of the following represents an infinite and constant stream of cash flows?

a perpetuity

Which of the following will increase wealth?

a positive NPV

Which of the following can be used to calculate present value?

a time value of money table a financial calculator an algebraic formula

Another term for an annuity due is _____.

an annuity in advance

Another term for a partial amortization loan is a(n) ____ loan.

balloon bullet

Which party benefits from delaying interest accrual for a subsidized Stafford loan?

borrower

If the discount rate increases, the net present value will ______.

decrease

The principal balance ______ over time of a fixed payment loan.

decreases

A ________ annuity begins in the future

delayed

Cf*((1+r)t−1 / r) is the formula for the ________ value of an annuity.

future

In the formula for the future value of an annuity, the expression in brackets is equal to the ______.

future value interest factor for an annuity

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

greater than

As the compounding frequency increases, the future value will ______.

increase

A loan might be repaid in equal ________ over a specific period of time.

installments, payments, or instalments

Compared to a comparable fixed payment loan, the total interest on a fixed principal loan is ___.

less because the principal is reduced faster.

A delayed annuity (or perpetuity) is one that begins ______.

many periods in the future

Fixed payment loans are typically used for which of the following ______.

student loans mortgages car loans

Balloon payments on partial amortization loans are typically quite large because ____.

the loan balance declines slowly

The real world has moved away from using ______ for calculating future and present values.

time value of money tables

Present value represents what an amount of money promised or expected in the future is worth ______.

today

One of the most basic principles of finance is that rational individuals prefer to receive a dollar ____ than a dollar ______.

today; tomorrow

True or false: The first step in calculating the present value of a delayed annuity is to find the present value of the annuity one period prior to the first payment using the present value of an annuity formula.

true

A firm has cash flows of $100 at the end of years 1-4. How much will the present value of the firm change if the discount rate rises to 12 percent from 10 percent?

−$13.25

A 5-year $10,000 loan with a 15-year amortization period paid monthly at 10 percent, compounded monthly, will have monthly payments of ____.

$107.46

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

$1593.74

A firm has cash flows of $100 at the end of years 1-4. What is the net present value of an investment in this firm if we pay $300 to purchase the firm and the discount rate is 10 percent?

$16.99 $100 × [(1 − (1/1.10)^4))/0.10] − $300 = $16.99. NPV = PV - Cost

Suppose you paid a $1,200 loan off by paying $400 in principal each year plus 10 percent annual interest. How much is the interest payment in the second year of the loan?

$80 You are repaying $400 each year. Interest is computed on the principal outstanding for the year, which is ($1,200 − 400) = 800. $800 × 0.1 = $80.

What is the present value of $100 each year for 20 years at 10 percent per year?

$851.36

If you invest $100 at a stated annual rate of 10 percent compounded quarterly, how much more money will you have in 10 years than if the rate was compounded annually?

$9.13

When evaluating NPV, the future is typically ______.

uncertain

What is the difference in the future value of $100 at 7 percent interest for 5 years if the interest is compounded semiannually rather than annually?

$0.80 ($100 × 1.035^10) − ($100 × 1.07^5) = $0.80.

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

$1,000 $100/0.10 = $1,000.

Assume $100 earns a stated 10 percent rate compounded quarterly. What will the value of the $100 be after 1 year?

$110.38

Assume a $100 investment earns a stated interest rate of 10 percent, compounded monthly. What will be the investment value after 1 year?

$110.47 First, find the monthly interest rate: 0.1/12 = 0.008333. Next, you can find the future value: $100 x (1 + 0.008333)12 = $110.47.

A perpetuity will pay $5 next year with payments increasing at a rate of 2 percent annually. What is the value of the 2 percent growth if the interest rate is 10 percent?

$12.50 $5/(0.10 − 0.02) − $5/0.10 = $12.50.

What is the future value of $100 at 10 percent simple interest for 2 years?

$120 $100 + 2(0.10 × $100) = $120.

If you invest $100 at 10 percent per year for 2 years, your future value with annual compounding will be ______.

$121

If $100 earns compound interest for 2 years at 10 percent per year, the future value will be ____.

$121.00

If you invest $100 at 10 percent compounded annually, how much money will you have at the end of 3 years?

$133.10

If you invest $100 at 10 percent per year for 3 years, your future value with annual compounding will be ___.

$133.10

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

$248.69 $100{[1 − (1/(1.10)^3)]/0.10}

If the future value is $500 in 1 year and the interest rate is 12 percent per year, what is the present value?

$446.43 $500/1.12 = $446.43.

Suppose you are promised $100 at the end of year 1, $200 at the end of year 2, and $300 at the end of year 3. If the interest rate is 10 percent per year, what is the present value of these promises?

$481.59 $100/1.1 + $200/1.1^2 + $300/1.1^3 = $481.59.

Assume 12 percent annual interest is compounded semiannually on a $500 investment. What will that investment be worth after 1 year?

$561.80 $500 × (1.06)2 = $561.80

You invest $500 at 10 percent interest per annum. At the end of 2 years with simple interest, you will have ____ and with compound interest, you will have ____.

$600; $605

If the future value is $750 in 1 year and the interest rate is 15 percent, what is the present value?

$652.17 $750/1.15 = $652.17

How much is $50 at 7 percent interest at the end of each year forever worth today?

$714.29

If the interest rate is 10 percent per year, then what is today's value of $100 received one year from today?

$90.91 $100/1.10 = $90.91.

If you invest $1,000 and the present value of the expected cash inflows is $1,300, then the NPV is ______.

+$300 NPV = −$1,000 + 1,300 = $300.

The formula for finding the net present value of a cash outflow now, a positive cash flow in 1 year, a positive cash flow in 2 years, and a positive cash flow in 3 years is ______.

-C0+ C1/(1 + r)^1 + C2/(1 + r)^2+ C3/(1 + r)^3

From highest to lowest, rank the following compounding periods effective annual rates:

1. continuous 2.. weekly 3. semiannual 4. annual

You agree to repay $1,200 in 2 weeks for a $1,000 payday loan. What is your EAR assuming that there are 52 weeks in a year?

11,347.55% ($1,200/$1,000)^26 − 1 = 11,347.55%.

According to the Rule of 72, at 18 percent per year, it will take _________ years to double your money. Hint: round your answer to the nearest whole number of years.

4

Your bank quotes a 9% APR on your car loan (.75 percent interest each month). What is the EAR?

9.38% 1.0075^12 - 1 = 9.38%

The present value interest factor for a 30-year annuity with an interest rate of 10 percent per year is ______.

9.4269

What are the implications of the time value of money concept?

A dollar tomorrow is worth less than a dollar today. A dollar today is worth more than a dollar tomorrow.

What is the formula for computing future value with continuous compounding?

C0 × e^rT

What is the general compounding formula for calculating the annual return on investment when there is more than one compounding period in a year?

C0(1 + r/m)^m

The future value formula is ______.

FV = C0 × (1 + r)^τ

How frequently does continuous compounding occur?

every infinitesimal instant

Discounting is the process of converting ______ dollars into a ______ value.

future; present

The present value of a perpetuity can be found as the limit of a(n) ______ series.

geometric

The future value of $100 at 10 percent compounded semiannually is ______ the future value of $100 at 10 percent compounded annually.

greater than

You invest $100 today. With positive interest rates, the concept of future value implies that the future value of your $100 will be ____ $100.

greater than If interest rates are positive, $100 given to you today will be worth more than $100 in the future (for example, if the interest rate is 5%, your $100 will be worth $100 x (1 + 0.05)1 = $105 in 1 year).

A stream of cash flows that grow at a constant rate for a finite period is called a(n) _____.

growing annuity

PV = C/(r − g) is the formula for the present value of a ______.

growing perpetuity

When there are multiple time periods, which of the following would lower the present value of a single future cash flow, all else constant?

higher interest rates more risk more time periods

A positive NPV will ______ wealth.

increase

Compound interest (increases or decreases) ___________ with time.

increases

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

infinite

Time value of money tables is not as common as they once were because ______.

it is easier to use inexpensive financial calculators instead they are available for only a relatively small number of interest rates

A dollar tomorrow is worth ______ a dollar today.

less than A dollar tomorrow is worth less than a dollar today, because if you invest the dollar you have today, you'll have more than a dollar tomorrow.

True or false: A one-period formula for present value is PV = FV/(1 + r).

True

A geometric series has a(n) ______ sum.

finite

True or false: Receiving $10 today has the same value as receiving $1 today and $9 one year from now.

False

True or false: The multi-period formula for compounding is FV = (1 + r)^t.

False FV = PV * (1+r)^t

True or false: Discounting is the opposite of compounding.

True

True or false: The formula for finding the net present value of a cash outflow now, a positive cash flow in 1 year, and a positive cash flow in 2 years is -C0+ C1/(1 + r)^1 + C2/(1 + r)^2.

True Present value is the sum of the discounted cash flows over time.

Which compounding interval will result in the lowest future value assuming everything else is held constant?

annual

The EAR is meaningful by itself, but the ______ is only meaningful when the number of compounding periods per year is given.

annual percentage rate

Discounting brings money ______ in time and compounding brings money ______ in time.

back; forward

When investing in large U.S. stocks, the reinvestment of dividends and capital gains generates ______.

compound interest

The annual percentage rate is the annual interest rate without consideration of _____.

compounding

The effective annual rate (EAR) takes into account the ______ of interest that occurs within a year.

compounding

The idea behind ______ is that interest is earned on interest.

compounding

The APR is meaningful for comparisons only when the number of ______ per year is given.

compounding periods

The limiting case of compounding periods is ______________ compounding.

continuous

If you are promised $100 in 1 year, $200 in 2 years, and $300 in 3 years, then if you can earn a positive interest rate on your investments, those promises combined equal ______ $600 today.

less than The sum of those cash flows is $600, so if you deposited $600 today at a positive interest rate and withdrew the series of payments described, there would be money left in the account at the end because of accumulated interest. So you would need to deposit less than $600, which means the present value is less than $600.

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

level

If reinvestment of interest or dividends does not occur, then the future value of an investment will be _____ and the realized yield will be ____ than if reinvestment had occurred.

lower; lower If there is no reinvestment, there will be less money in the investment at the end of the time period, so the realized yield will be less.

Which of the following are annuities?

monthly rent payments in a lease installment loan payments

Which of the following will result in a lower present value for a given future cash flow?

more risk a higher interest rate more time

Which of the following are real-world examples of annuities?

mortgages pensions

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 interest rate (r) used in the general compounding formula is the ______ interest rate.

quoted

In reality, perfect certainty of future cash flows occurs ____.

rarely

The difference between _______ interest and compound interest is that compound interest (increases or decreases) ___________ with time.

simple; increases

Which of the following can be used to calculate present value?

spreadsheet software a financial calculator a time value of money table

The first cash flow at the end of week 1 is $100, the second cash flow at the end of month 2 is $100, and the third cash flow at the end of year 3 is $100. This cash flow pattern is a(n) ______ type of cash flow.

uneven The cash flows for an annuity must happen at regular intervals. These do not (week, month, year).