Tell us why this document is inappropriate

Offensive Content
Adult Content
Copyright Violation
Other

More information

We will review and remove this document in timely manner.

View the PDF version

Share on Facebook

Chapter 4  The Time Value of Money 39

11.When a loan is amortized, the largest portion of the periodic payment goes to reduce principal in the early years of the loan such that the accumulated interest can be spread out over the life of the loan.

ANS: F

DIF: Medium

TOP: Amortization

12.The effective annual rate is always greater than the simple rate as a result of compounding effects.

ANS: F

DIF: Medium

TOP: Effective and simple rates

13.Because we usually assume positive interest rates in time value analyses, the present value of a three-year annuity will always be less than the future value of a single lump sum, if the annuity payment equals the original lump sum investment.

ANS: F

DIF: Medium

TOP: Lump sum and annuity

14.All else equal, a dollar received sooner is worth more than a dollar received at some later date, because the sooner the dollar is received the more quickly it can be invested to earn a positive return.

ANS: T

DIF: Medium

TOP: Time value concepts

15.An annuity is a series of equal payments made at fixed equal-length intervals for a specified number of periods.

ANS: T

DIF: Medium

TOP: Annuities

16.The difference between an ordinary annuity and an annuity due is that each of the payments of the annuity due earns interest for one additional year (period).

ANS: T

DIF: Medium

TOP: Annuities

17.The difference between the PV of an annuity due and the PV of an ordinary annuity is that each of the payments of the annuity due is discounted by one more year.

ANS: T DIF: Medium TOP: Annuities

18. The effective annual rate is less than the simple rate when we have monthly compounding. ANS: F DIF: Medium TOP: Effective annual rate

MULTIPLE CHOICE

1.Given some amount to be received several years in the future, if the interest rate increases, the present value of the future amount will

a.Be higher.

b.Be lower.

c.Stay the same.

d.Cannot tell.

e.Be variable.

ANS: B

DIF: Easy

OBJ: TYPE: Conceptual TOP: PV of a sum

2.You have determined the profitability of a planned project by finding the present value of all the cash flows form that project. Which of the following would cause the project to look more appealing in terms of the present value of those cash flows?

40Chapter 4  The Time Value of Money

a.The discount rate decreases.

b.The cash flows are extended over a longer period of time, but the total amount of the cash flows remains the same.

c.The discount rate increases.

d.Answers b and c above.

e.Answers a and b above.

ANS:	A	DIF: Easy	OBJ: TYPE: Conceptual
TOP:	PV and discount rate

3.As the discount rate increases without limit, the present value of the future cash inflows

a.Gets larger without limit.

b.Stays unchanged.

c.Approaches zero.

d.Gets smaller without limit, i.e., approaches minus infinity.

e.Goes to ern.

ANS:	C	DIF: Easy	OBJ: TYPE: Conceptual
TOP:	PV and discount rate

4.Which of the following statements is most correct?

a.If annual compounding is used, the effective annual rate equals the simple rate.

b.If annual compounding is used, the effective annual rate equals the periodic rate.

c.If a loan has a 12 percent simple rate with semiannual compounding, its effective annual rate is equal to 11.66 percent.

d.Both answers a and b are correct.

e.Both answers a and c are correct.

ANS: D

Statement d is correct. The equation for EAR is as follows:

If annual compounding is used, m = 1 and the equation above reduces to EAR = rSIMPLE. The equation for the periodic rate is:

If annual compounding is used then m = 1 and rPER = rSIMPLE and since EAR = rSIMPLE then rPER = EAR.

DIF: Easy OBJ: TYPE: Conceptual TOP: Effective annual rate

5.Why is the present value of an amount to be received (paid) in the future less than the future amount?

a.Deflation causes investors to lose purchasing power when their dollars are invested for greater than one year.

b.Investors have the opportunity to earn positive rates of return, so any amount invested today should grow to a larger amount in the future.

c.Investments generally are not as good as those who sell them suggest, so investors usually are not willing to pay full face value for such investments, thus the price is discounted.

d.Because investors are taxed on the income received from investments they never will buy an investment for the amount expected to be received in the future.

e.None of the above is a correct answer.

ANS:	B	DIF: Easy	OBJ: TYPE: Conceptual
TOP:	Time value concepts

42Chapter 4  The Time Value of Money

11.Which of the following statements is correct?

a.For all positive values of k and n, FVIFr, n 1.0 and PVIFAr, n n.

b.You may use the PVIF tables to find the present value of an uneven series of payments. However, the PVIFA tables can never be of use, even if some of the payments constitute an annuity (for example, $100 each year for Years 3, 4, and 5), because the entire series does not constitute an annuity.

c.If a bank uses quarterly compounding for saving accounts, the simple rate will be greater than the effective annual rate.

d.The present value of a future sum decreases as either the simple interest rate or the number of discount periods per year increases.

e.All of the above statements are false.

ANS:	D	DIF: Medium	OBJ: TYPE: Conceptual
TOP:	Time value concepts

12.Which of the following statements is correct?

a.Other things held constant, an increase in the number of discounting periods per year increases the present value of a given annual annuity.

b.Other things held constant, an increase in the number of discounting periods per year increases the present value of a lump sum to be received in the future.

c.The payment made each period under an amortized loan is constant, and it consists of some interest and some principal. The later we are is the loan's life, the smaller the interest portion of the payment.

d.There is an inverse relationship between the present value interest factor of an annuity and the future value interest factor of an annuity, (i.e., one is the reciprocal of the other).

e.Each of the above statements is true.

ANS:	C	DIF: Medium	OBJ: TYPE: Conceptual
TOP:	Time value concepts

13.A $10,000 loan is to be amortized over 5 years, with annual end-of-year payments. Given the following facts, which of these statements is correct?

a.The annual payments would be larger if the interest rate were lower.

b.If the loan were amortized over 10 years rather than 5 years, and if the interest rate were the same in either case, the first payment would include more dollars of interest under the 5-year amortization plan.

c.The last payment would have a higher proportion of interest than the first payment.

d.The proportion of interest versus principal repayment would be the same for each of the 5 payments.

e.The proportion of each payment that represents interest as opposed to repayment of principal would be higher if the interest rate were higher.

ANS: E

If the interest rate were higher, the payments would all be higher, and all of the increase would be attributable to interest. So, the proportion of each payment that represents interest would be higher.

Note that statement b is false because interest during Year 1 would be the interest rate times the beginning balance, which is $10,000. With the same interest rate and the same beginning balance, the Year 1 interest charge will be the same, regardless of whether the loan is amortized over 5 or 10 years.

DIF: Medium

OBJ: TYPE: Conceptual

TOP: Time value concepts

44Chapter 4  The Time Value of Money

19.Suppose that the present value of receiving a guaranteed $450 in two years is $385.80. The opportunity rate of return on similar risk investments is 8 percent. According to this information, all else equal, which of the following statements is correct?

a.It always would be preferable to wait two years to receive the $450 because this value is greater than the present value.

b.Risk averse investors always would prefer to take the $385.80 today because it is a guaranteed amount whereas there is uncertainty as to whether the future amount will be paid.

c.No investor should be willing to pay more than $385.80 for such an investment.

d.It is apparent the present value was computed incorrectly because the present value of a future amount always should be greater than the future value.

e.None of the above is a correct answer.

ANS:	C	DIF: Medium	OBJ: TYPE: Conceptual
TOP:	Time value concepts

20.You plan to invest an amount of money in five-year certificate of deposit (CD) at your bank. The stated interest rate applied to the CD is 12 percent, compounded monthly. How much must you invest if you want the balance in the CD account to be $8,500 in five years?

a.$4,678.82

b.$4,823.13

c.$13,600.00

d.$14,979.90

e.$7,589.29

ANS:	A	DIF: Medium	OBJ: TYPE: Conceptual
TOP:	PV of a lump sum

21.Vegit Corporation needs to borrow funds to support operations during the summer. Vegit's CFO is trying to decide whether to borrow from the Bank of Florida or the Bank of Georgia. The loan offered by Bank of Florida has a 12.5 percent simple interest rate with annual interest payments, whereas the loan offered by the Bank of Georgia has a 12 percent simple interest rate with monthly payments. Which bank should Vegit use for the loan?

a.Bank of Georgia, because the 12 percent simple interest is cheaper than the 12.5 percent simple interest at Bank of Florida.

b.Bank of Georgia, because the effective interest rate on the loan is less than 12 percent, whereas the effective interest rate on the loan at the Bank of Florida is greater than 12.5 percent.

c.Bank of Florida, because the simple interest rate is higher, which means that Vegit will be able to invest the proceeds from the loan at a higher rate of return.

d.Bank of Florida, because the effective interest rate on the loan is 12.5 percent, which is less than the 12.7 percent effective interest rate on the loan offered by the Bank of Georgia.

e.There is not enough information to answer this question.

ANS:	D	DIF: Medium	OBJ: TYPE: Conceptual
TOP:	Effective annual rate

Chapter 4  The Time Value of Money 45

22.Alice's investment advisor is trying to convince her to purchase an investment that pays $250 per year. The investment has no maturity; therefore the $250 payment will continue every year forever. Alice has determined that her required rate of return for such an investment should be 14 percent and that she would hold the investment for 10 years and then sell it. If Alice decides to buy the investment, she would receive the first $250 payment one year from today. How much should Alice be willing to pay for this investment?

a.$1,304.03, because this is the present value of an ordinary annuity that pays $250 a year for 10 years at 14 percent.

b.$1,486.59, because this is the present value of an annuity due that pays $250 a year for 10 years at 14 percent.

c.$1,785.71, because this is the present value of a $250 perpetuity at 14 percent.

d.There is not enough information to answer this question, because the selling price of the investment in 10 years is not known today.

e.None of the above is correct.

ANS: C

DIF: Medium

OBJ: TYPE: Conceptual TOP: Perpetuity

23.At approximately what rate would you have to invest a lump-sum amount today if you need the amount to triple in six years? Assume interest is compounded annually.

a.20%

b.12%

c.24%

d.Not enough information is provided to answer the question.

e.None of the above is a correct answer.

ANS:	A	DIF: Medium	OBJ: TYPE: Conceptual
TOP:	Time value concepts

24.Sarah is thinking about purchasing an investment from HiBond Investing. If she buys the investment, Sarah will receive $100 every three months for five years. The first $100 payment will be made as soon as she purchases the investment. If Sarah's required rate of return is 16 percent, to the nearest dollar, how much should she be willing to pay for this investment?

a.$1,359

b.$1,413

c.$1,112

d.$1,519

e.$1,310

ANS:	B	DIF: Medium	OBJ: TYPE: Conceptual
TOP:	PV of an annuity

25.Which of the following statements is most correct?

a.The first payment under a 3-year, annual payment, amortized loan for $1,000 will include a smaller percentage (or fraction) of interest if the interest rate is 5 percent than if it is 10 percent.

b.If you are lending money, then, based on effective interest rates, you should prefer to lend at a 10 percent simple, or quoted, rate but with semiannual payments, rather than at a 10.1 percent simple rate with annual payments. However, as a borrower you should prefer the annual payment loan.

c.The value of a perpetuity (say for $100 per year) will approach infinity as the interest rate used to evaluate the perpetuity approaches zero.

d.Statements a, b, and c are all true.

e.Only statements b and c are true.

ANS:	D	DIF: Tough	OBJ: TYPE: Conceptual
TOP:	Time value concepts

46Chapter 4  The Time Value of Money

26.A recent advertisement in the financial section of a magazine carried the following claim: "Invest your money with us at 14 percent, compounded annually, and we guarantee to double your money sooner than you imagine." Ignoring taxes, how long would it take to double your money at

asimple rate of 14 percent, compounded annually?

a.Approximately 3.5 years

b.Approximately 5 years

c.Exactly 7 years

d.Approximately 10 years

e.Exactly 14 years

ANS: B

Tabular solution:

$1 (FVIF14%, n) = $2 FVIF14%, n = 2.000 n = 5+ years.

Financial calculator solution: Inputs: I = 14; PV = -1; FV =2. Output: N = 5.29 years.

DIF: Easy

OBJ: TYPE: Problem

TOP: Time for a sum to double

27.At an effective annual interest rate of 20 percent, how many years will it take a given amount to triple in value? (Round to the closest year.)

a.5

b.8

c.6

d.10

e.9

ANS: C

Cash flow time line:

Tabular solution:

$1 = $3 (PVIF20%, n) PVIF20%, n = 0.3333 n = 6 periods (years).

Financial calculator solution: Inputs: I = 20; PV = -1; FV =3. Output: N = 6.026 = 6 years.

DIF: Easy

OBJ: TYPE: Problem

TOP: Time for a sum to triple

Chapter 4  The Time Value of Money 47

28.You deposited $1,000 in a savings account that pays 8 percent interest, compounded quarterly, planning to use it to finish your last year in college. Eighteen months later, you decide to go to the Rocky Mountains to become a ski instructor rather than continue in school, so you close out your account. How much money will you receive?

a.$1,171

b.$1,126

c.$1,082

d.$1,163

e.$1,008

ANS: B

Tabular solution:

FV = $1,000 (FVIF2%, 6) = $1,000 (1.1262) = $1,126.20 $1,126.

Financial calculator solution:

Inputs: N = 6; I = 2; PV = -1,000.

Output: FV = $1,126.16 $1,126.

DIF: Easy

OBJ: TYPE: Problem

TOP: FV of a sum

29.What is the future value of a 5-year ordinary annuity with annual payments of $200, evaluated at

a15 percent interest rate?

a.$670.44

b.$842.91

c.$1,169.56

d.$1,522.64

e.$1,348.48

ANS: E

Tabular solution:

FB = $200 (FVIFA15%, 5) = $200 6.7424 = $1,348.48.

Financial calculator solution:

Inputs: N = 5; I = 15; PMT = -200.

Output: FV = $1,348.48.

DIF: Easy

OBJ: TYPE: Problem

TOP: FV of an annuity

Chapter 4  The Time Value of Money 49

Tabular solution:

PV = $2,000 (PVIFA14%, 5) + $3,000 (PVIFA14%, 5) (PVIF14%, 9)

=$2,000 (3.4331) + $3,000 (2.3216) (0.5194) + $4,000 (0.3075)

=$6,866.20 + $3,617.52 + $1,230.00 = $11,713.72 $11,714.

Financial calculator solution:

Using cash flows

Inputs: C0 = 0; C1 = 2,000; Nj = 5; C2= 3,000; Nj = 3; C3 = 4,000; I = 14. Output: NPV = $11,713.54 $11,714.

DIF: Easy

OBJ: TYPE: Problem

TOP: PV of an uneven CF stream

33.If $100 is placed in an account that earns a simple 4 percent, compounded quarterly, what will it be worth in 5 years?

a.$122.02

b.$105.10

c.$135.41

d.$120.90

e.$117.48

ANS: A

Tabular solution:

$100 (FVIF1%, 20) = $100 (1.2202) = $122.02.

Financial calculator solution: Inputs: N = 20; I = 1; PV = -100. Output: FV = $122.02.

DIF: Easy

OBJ: TYPE: Problem

TOP: Quarterly compounding

34.In 1958 the average tuition for one year at an Ivy League school was $1,800. Thirty years later, in 1988, the average cost was $13,700. What was the growth rate in tuition over the 30-year period?

a.12%

b.9%

c.6%

d.7%

e.8%

ANS: D

Cash flow time line:

Tabular solution:

$13,700 = $1,800 (FVIFi, 30) FVIFi, 30 = 7.6111

I 7%

50 Chapter 4  The Time Value of Money

Financial calculator solution:

Inputs: N = 30; PV = -1,800; FV = 13,700.

Output: I = 7.0%

DIF: Easy

OBJ: TYPE: Problem

TOP: Growth rate

35.At an inflation rate of 9 percent, the purchasing power of $1 would be cut in half in 8.04 years. How long to the nearest year would it take the purchasing power of $1 to be cut in half if the inflation rate were only 4%?

a.12 years

b.15 years

c.18 years

d.20 years

e.23 years

ANS: C

Cash flow time line:

Tabular solution:

0.5 = $1 (PVIF4%, n) PVIF4%, n = 0.5

PVIF4%, 18 = 0.4936; PVIF4%, 17 = 0.5134 n 18 years.

Although a financial calculator or interpolation might be used to solve precisely, Response c is clearly the closest and best answer of those given.

Financial calculator solution: Inputs: I = 4; PV = 1; PV = -0.50. Output: N = -17.67 = 18 years.

DIF: Easy

OBJ: TYPE: Problem

TOP: Effect of inflation

36.Gomez Electronics needs to arrange financing for its expansion program. Bank A offers to lend Gomez the required funds on a loan where interest must be paid monthly, and the quoted rate is 8 percent. Bank B will charge 9 percent, with interest due at the end of the year. What is the difference in the effective annual rates charged by the two banks?

a.0.25%

b.0.50%

c.0.70%

d.1.00%

e.1.25%

ANS: C

Bank A: 8%, monthly

EARA =

Bank B: 9%, interest due at end of year EARB = 9%.

9.00% - 8.30% = 0.70%.

52 Chapter 4  The Time Value of Money

Tabular solution:

FVYear 20	= $50 (FVIFA10$, 20) = $50 (57.275) = $2,863.75.
FVYear 30	= $2,863 (FVIFA10$, 10) = $2,863.75 (2.5937) = $7,427.71.

Financial calculator solution:

Calculate FV at Year 20, then take that lump sum forward 10 years to Year 30 at 10%.

Inputs: N = 20; I = 10; PMT = -50. OutputYear 20: FV = $2,863.75.

At Year 30

Inputs: N = 10; I = 10; PV = -2,863.75. OutputYear 30: FV = $7,427.83.

DIF: Medium

OBJ: TYPE: Problem

TOP: FV of an annuity

39.You expect to receive $1,000 at the end of each of the next 3 years. You will deposit these payments into an account which pays 10 percent compounded semiannually. What is the future value of these payments, that is, the value at the end of the third year?

a.$3,000

b.$3,310

c.$3,318

d.$3,401

e.$3,438

ANS: C

Tabular solution:

FV = $1,000(FVIF5%,4) + $1,000(FVIF5%,2) + $1,000

=$1,000(1.2155) + $1,000(1.1025) + $1,000

=$1,215.50 + $1,102.50 + $1,000 = $3,318.00.

Financial calculator solution:

Convert rSIMPLE to EAR using interest rate conversion

Inputs: P/YR = 2; NOM% =10.

Output: EFF% = EAR = 10.25%.

Solve for FV on annual basis using EAR

Inputs: N = 3; I = 10.25; PMT = -1,000.

Output: FV = $3,318.006 $3,318.00.

DIF: Medium

OBJ: TYPE: Problem

TOP: FV of an annuity

Chapter 4  The Time Value of Money 53

40.You just graduated, and you plan to work for 10 years and then to leave for the Australian "Outback" bush country. You figure you can save $1,000 a year for the first 5 years and $2,000 a year for the next 5 years. These savings cash flows will start one year from now. In addition, your family has just given you a $5,000 graduation gift. If you put the gift now, and your future savings when they start, into an account which pays 8 percent compounded annually, what will your financial "stake" be when you leave for Australia 10 years from now?

a.$21,432

b.$28,393

c.$16,651

d.$31,148

e.$20,000

ANS: D

Cash flow time line:

Tabular solution:

FV = (FVIFA8%, 10) + $1,000 (FVIFA8%, 5) + $5,000 (FVIF8%, 10)

=$1,000 (14.487) + $1,000 (5.866) + $5,000 (.1589)

=$14,487 + $5,866 + $10,794.50 = $31,147.50 $31,148

Financial calculator solution:

Solution using NFV (Note: Some calculators do not have net future value function. Cash flows can be grouped and carried forward or PV can be used; see alternative solution below.)

Inputs:	= 5,000;	= 1,000; Nj = 5;	= 2,000; Nj = 5; I = 8
Output: NFV = $31,147.79 $31,148

Alternative solution: calculate PV of the cash flows, then bring them forward to FV using the interest rate.

Inputs: = 5,000; = 1,000; Nj = 5; = 2,000; Nj = 5; I = 8 Output: PV = $14,427.45

Inputs: N = 10; I = 8; PV = -14,427.45 Output: FV = $31,147.79 $31,148

DIF: Medium

OBJ: TYPE: Problem

TOP: FV of an uneven CF stream

41.As the winning contestant in a television game show, you are considering the prizes to be awarded. You must indicate to the sponsor which of the following two choices you prefer, assuming you want to maximize your wealth. Assume it is now January 1, and there is no danger whatever that the sponsor won't pay off.

(1)$1,000 now and another $1,000 at the beginning of each of the 11 subsequent months during the remainder of the year, to be deposited in an account paying 12 percent simple annual rate, but compounded monthly (to be left on deposit for the year).

(2)$12,750 at the end of the year.

54 Chapter 4  The Time Value of Money

Which one would you choose?

a.Choice 1

b.Choice 2

c.Choice 1, if the payments were made at the end of the year.

d.The choice would depend on how soon you need the money.

e.Either one, since they have the same present value.

ANS: A

Tabular solution:

PVChoice 1	= $1,000 (PVIFA 1%, 11 + 1.0) = $1,000 (11.3676) = $11,367.60
PVChoice 2	= $12,750 (PVIF 1%, 12) = $12,750 (0.8874) = $11,314.35

Financial calculator solution:

Choice 1

BEGIN mode, Inputs N = 12; I = 1; PMT = 1,000.

Output: PV = -$11,367.63

Choice 2

END mode, Inputs: N = 12; I = 1; FV - 12,750.

Output: PV = -$11,314.98.

DIF: Medium

OBJ: TYPE: Problem

TOP: PV of an annuity

42.You want to buy a Nissan 350Z on your 27th birthday. You have priced these cars and found that they currently sell for $30,000. You believe that the price will increase by 5 percent per year until you are ready to buy. You can presently invest to earn 14 percent. If you just turned 20 years old, how much must you invest at the end of each of the next 7 years to be able to purchase the Nissan in 7 years?

a.$4,945.57

b.$3,933.93

c.$7,714.72

d.$3,450.82

e.$6,030.43

ANS: B

Cash flow time lines:

Chapter 4  The Time Value of Money 55

Tabular solution:

Price of car on 27th birthday

FV = $30,000 (FVIF 5%, 7) = $30,000 (1.4071) = $42,213.

Annual investment required

FV of annuity = FVAn = PMT (FVIFA i, n)

$42,213 = PMT (FVIFA 14%, 7)

PMT = $42,213/10.7305 = $3,933.93.

Financial calculator solution:

Price of car on 27th birthday

Inputs: N = 7; I = 5; PV = -30,000.

Output: FV = $42,213.01 $42,213.

Annual investment required Inputs: N = 7; I = 14; FV = 42,213. Output: PMT = -$3,933.93.

DIF: Medium

OBJ: TYPE: Problem

TOP: Annuity payment

43.Assume that your required rate of return is 12 percent and you are given the following stream of cash flows:

Year

Cash Flow

0$10,000

115,000

215,000

315,000

415,000

520,000

If payments are made at the end of each period, what is the present value of the cash flow stream?

a.$66,909

b.$57,323

c.$61,815

d.$52,345

e.$62,029

ANS: A

Tabular solution:

PV = $10,000 + $15,000 (PVIFA12%, 4) + $20,000 (PVIF12%, 5)

=$10,000 + $15,000 (3.0373) + $20,000 (0.5674)

=$10,000 + $45,559.50 + $11,348 = $66,907.50.

Financial calculator solution:

Using cash flows

Inputs:	= 10,000;	= 15,000; Nj = 4 times;	= 20,000; I = 12.
Output: NPV = $66,908.77 $66,909.

Note: Tabular solution differs from calculator solution due to interest factor rounding.

DIF: Medium

OBJ: TYPE: Problem

TOP: PV of an uneven CF stream

Chapter 4  The Time Value of Money 57

Years 1-5

PV of annuity Years 1-5 = $$20,000 (PVIFA12%, 5) = $20,000 (3.6048) = $72,096.

Years 6-10 Value of annuity Years 6-10 on Day 1 of Year 6

PV5 = $25,000 (PVIFA10%, 5) = $25,000 (3.7908) = $94,770

PV of annuity Years 6-10 at time = 0

PV0 = $94,770 (PVIF 12%, 5) = $94,770 (0.5674) = $53,772.50

PV0 of both annuities

$72,096 + $53, 772.50 = $125.868.50 $125.870

Financial calculator solution:

Solve for PV at time = 0 of $20,000 annuity

Inputs:

Solve for PV at time = 5 pf $25,000 annuity using its value at t = 5

Inputs:

Solve for PV at time = 0 0f $25,000 annuity

Inputs: N = 5; I = 12; FV = -94,769.669. Output: PV = $53,774.855

Add the two PVs together

PVBoth annuities = $72,095.524 + $53,774.855 = $125,870.38 $125.870

Note: Tabular solution differs from calculator solution due to interest factor rounding.

DIF: Medium

OBJ: TYPE: Problem

TOP: PV of an uneven CF stream

46.A project with a 3-year life has the following probability distributions for possible end of year cash flows in each of the next three years:

	Year 1		Year 2		Year 3
Prob	Cash Flow	Prob	Cash Flow	Prob	Cash Flow
0.30	$300	0.15	$100	0.25	$200
0.40	500	0.35	200	0.75	800
0.30	700	0.35	600
		0.15	900

Using an interest rate of 8 percent, find the expected present value of these uncertain cash flows. (Hint: Find the expected cash flow in each year, then evaluate those cash flows.)

a.$1,204.95

b.$835.42

c.$1,519.21

d.$1,580.00

e.$1,347.61

ANS: E

Calculate expected cash flows
E(CF1)	= (0.30) ($300) + (0.40)	($500)	+ (0.30)	($700)	= $500
E(CF2)	= (0.15) ($100) + (0.35)	($200)	+ (0.35)	($600)	+ (0.15) ($900) = $430
E(CF3)	= (0.25) ($200) + (0.75)	($800)	= $650

Chapter 4  The Time Value of Money 59

Tabular solution:

PV = $0.26 = $0.38 (PVIFi, 4) PVIF i, 4 = $0.26/$0.38 = 0.6842

i1 10%

PV = $0.38 = $0.535 (PVIF i, 7) PFIV i,7 = $0.38/$0.535 = 0.7103

i2 5%

Financial calculator solution:

Inputs: N = 4; PV = -0.26; FV = 0.38. Output: I = 9.95% 10% Inputs: N = 7; PV = -0.38; FV = 0.535. Output: I = 5.01% 5%

DIF: Medium

OBJ: TYPE: Problem

TOP: Growth rate

49.Steaks Galore needs to arrange financing for its expansion program. One bank offers to lend the required $1,000,000 on a loan which requires interest to be paid at the end of each quarter. The quoted rate is 10 percent, and the principal must be repaid at the end of the year. A second lender offers 9 percent, daily compounding (365-day year), with interest and principal due at the end of the year. What is the difference in the effective annual rates (EFF%) charged by the two banks?

a.0.31%

b.0.53%

c.0.75%

d.0.96%

e.1.25%

ANS: D

EARQuarterly =

EARDaily =

Difference = 10.38% - 9.42% = 0.96%

Alternatively, with a financial calculator, for the quarterly loan enter P/YR = 4, NOM% = 10, and press EFF% to get EAR = 10.38%.

For the daily loan, enter P/YR = 365, NOM = 9%, and press EFF% to get EAR = 9.42%.

DIF: Medium

OBJ: TYPE: Problem

TOP: Effective annual rate

50.You are currently at time period 0, and you will receive the first payment on an annual payment annuity of $100 in perpetuity at the end of this year. Six full years from now you will receive the first payment on an additional $150 in perpetuity, and at the end of time period 10 you will

62 Chapter 4  The Time Value of Money

PVCosts = $17,000, I = 8%

PVAcct = $25,000, I = 9.31% Financial calculator solution:

Step 1: Use the current law school costs and inflation rate to calculate the withdrawals to cover law school costs at T = 4, 5, 6:

At T = 4, Inputs: N = 4, PV = -17,000; I = 8. Output: FV4 = $23,128.31 At T = 5, Inputs: N = 5; PV = -17,000; I = 8. Output: FV 5 = $24,978.58 At T = 6, Inputs: N = 6; PV = -14,000; I = 8. Output: FV6 = $26,976.86

Step 2: Use interest rate conversion feature to calculate the effective annual rate of the 9% account, compounded quarterly.

Inputs: NOM% = 9; P/YR = 4. Output: EFF% = 9.31%

Step 3: Use the EAR from Step 2 to determine the PV of law school payments at T = 4, 5, 6 as of T = 4.

Inputs: Output: NPV = $68,556.73 which equals PVT=4, costs

Step 4: Determine the VF at T = 4 of the $25,000 in the account as of T = 0:

Inputs: N = 4; I = 9.31; PV = -25,000. Output: FV = $35,692.72

Step 5: Calculate shortfall between what the account needs to have and will actually have at T = 4:

$68,556.73 - $35,692.72 = $32,864.01

Step 6: Calculate the annuity payments, which will earn 9.31% EAR and accumulate to an FV of $32,864.01 at T = 4:

Inputs: N = 4; I = 9.31%; FV = 32,864.01. Output: PMT = $7,153.56

DIF: Tough

OBJ: TYPE: Problem

TOP: Annuity payments

66 Chapter 4  The Time Value of Money

Tabular solution:

$20,000 = $444.89 (PVIFA r, 60) PVIFA r, 60 = 44.9549

r = 1%

EAR

Financial calculator solution:

Calculate periodic rate and simple rate

Inputs: N = 60; PV = -20,000; PMT = 444.89

Output: I = 1.0. NOM% = 1.0% 12 = 12.00%

Use interest rate conversion feature

Inputs: P/YR = 12; NOM% = 12.0.

Output: EFF% = EAR = 12.68%

DIF: Medium

OBJ: TYPE: Financial Calculator TOP: Effective annual rate

59.Bank A offers a 2-year certificate of deposit (CD) that pays 10 percent compounded annually. Bank B offers a 2-year CD that is compounded semi-annually. The CDs have identical risk. What is the stated, or simple, rate that Bank B would have to offer to make you indifferent between the two investments?

a.9.67%

b.9.76%

c.9.83%

d.9.87%

e.9.93%

ANS: B

Numerical solution:

1.10= (1 + r/2)2

1.0488	= 1 + r/2
r/2	= 0.0488
r	= 0.0976 = 9.76%

Financial calculator solution:

Use interest rate conversion feature

Inputs: P/YR = 2; EFF% = 10.0%.

Output: NOM% = 9.462% 9.76%

	Chapter 4  The Time Value of Money 67
DIF: Medium	OBJ: TYPE: Financial Calculator TOP: Effective interest rates

60.Assume that you inherited some money. A friend of yours is working as an unpaid intern at a local brokerage firm, and her boss is selling some securities which call for four payments, $50 at the end of each of the next 3 years, plus a payment of $1,050 at the end of Year 4. Your friend says she can get you some of these securities at a cost of $900 each. Your money is now invested in a bank that pays an 8 percent simple (quoted) interest rate, but with quarterly compounding. You regard the securities as being just as safe, and as liquid, as your bank deposit, so your required effective annual rate of return on the securities is the same as that on your bank deposit. You must calculate the value of the securities to decide whether they are a good investment. What is their present value to you?

a.$1,000

b.$866

c.$1,050

d.$901

e.$893

ANS: E

Financial calculator solution:

Calculate the EAR on the bank deposit

Inputs: P/YR = 4, NOM% = 8; Output: EFF% = EAR = 8.24%

Calculate the PV of the investment

Inputs: N = 4; I = 8.24; PMT = 50; FV = 1,000

Output: PV = -$893.26 $893

Alternate method: Using cash flows

Inputs:

Output: NPV = $893.26 $893

DIF: Medium

OBJ: TYPE: Financial Calculator TOP: PV and effective annual rate

61.Your company is planning to borrow $1,000,000 on a 5-year, 15 percent, annual payment, fully amortized term loan. What fraction of the payment made at the end of the second year will represent repayment of principal?

a.29.83%

b.57.18%

c.35.02%

d.64.45%

e.72.36%

ANS: B Tabular solution:

$1,000,000 = PMT(PVIFA15%,5)

PMT = $1,000,000/3.3522 = $298,311.56.

Construct amortization table

Year

Beg Balance

Payment

Interest

Principal

End Balance

Chapter 4  The Time Value of Money 69

63.You have a 30-year mortgage with a simple annual interest rate of 8.5 percent. The monthly payment is $1,000. What percentage of your total payments over the first three years goes toward the repayment of principal?

a.1.50%

b.3.42%

c.5.23%

d.6.75%

e.8.94%

ANS: E

Enter the information into the calculator to use its amortization feature:

N	= 360
I/YR	= 8.5/12 = 0.7063
PMT	= 1,000
FV	= 0

Solve for PV = -$130,053.64 = Original value of mortgage.

Enter: 1 INPUT 36 AMORT

Int 1 - 36 = $32,782.14

Prin 1 - 36 = $3,217.86

Total payments 1 - 36 = $36,000.

Percentage of total payments which is principal =

DIF: Medium

OBJ: TYPE: Financial Calculator TOP: Amortization

64.Your company must make payments of $100,000 each year for 10 years, with the first payment to be made 10 years from today. To prepare for these payments, your company must make 10 equal annual deposits into an account which pays a simple interest rate of 7 percent, daily compounding (360-day year). Funds will remain in the account during both the accumulation period (the first 10 years) and the distribution period (the last 10 years), and the same interest rate will be earned throughout the entire 20 years. The first deposit will be made immediately. How large must each deposit be?

a.$47,821.11

b.$49,661.86

c.$51,234.67

d.$52,497.33

e.$53,262.39

ANS: B

The FV of the DEP annuity at T = 10 must be sufficient to make the 10 payments of $100,000 each.

Step 1: Find the PV of the $100,00 payments at the end of Year 10. This is a 10-year annuity due. What rate do we use? 7% is not correct, and if we use the periodic that won't work either in an annuity setup. We want a rate that's consistent with an annual annuity. That means we must use the EAR.

70 Chapter 4  The Time Value of Money

Use the interest conversion feature on your financial calculator to find EAR = 7.2501%

P/YR = 360

NOM% = 7

Solve for EFF% = 7.2501%

Now find the PV of the annuity:

Step 2: Determine the amount of the annuity due by using the present value of the $100,000 payments at Year 10 as the future value of the annuity due.

Deposits of $49.661.86 will provide the needed funds.

DIF:	Medium	OBJ: TYPE: Financial Calculator
TOP:	Annuities and daily compounding

65.Your lease calls for payments of $500 at the end of each month for the next 12 months. Now your landlord offers you a new 1-year lease which calls for zero rent for 3 months, then rental payments of $700 at the end of each month for the next 9 months. You keep your money in a bank time deposit that pays a simple annual rate of 5 percent. By what amount would your net worth change if you accept the new lease? (Hint: Your return per month is 5%/12 = 0.4166667%.)

a.-$509.81

b.-$253.62

c.+$125.30

d.+$253.62

e.+$509.81

ANS: B

Chapter 4  The Time Value of Money 71

Solve for NPV = -$6,094.23

Therefore, the PV of payments under the proposed lease would be greater than the PV of payments under the old lease by $6,094.23 - $5,840.61 = $253.62. Thus, your net worth would decrease by $253.62.

DIF:	Medium	OBJ: TYPE: Financial Calculator
TOP:	NPV and non-annual discounting

66.You plan to invest $2,500 in a money market account which will pay an annual stated (simple) interest rate of 8.75 percent, but which compounds interest on a weekly basis. If you leave this money on deposit for one year (52 weeks), what will be your ending balance when you close the account?

a.$2,583.28

b.$2,611.72

c.$2,681.00

d.$2,703.46

e.$2,728.50

ANS: E

Numerical solution:

FV =

Financial calculator solution:

Convert simple rate to EAR

Inputs: P/YR = 52; NOM% = 8.75.

Output: EFF% = EAR + 9.1362% 9.14%

Calculate FV

Inputs: N = 1; I = 9.14; PV = -2,500.

Output: FV = $2,728.50

DIF: Medium

OBJ: TYPE: Financial Calculator TOP: Non-annual compounding

67.You have just purchased a life insurance policy that requires you to make 40 semiannual payments of $350 each, where the first payment is due in 6 months. The insurance company has guaranteed that these payments will be invested to earn you an effective annual rate of 8.16 percent, although interest is to be compounded semiannually. At the end of 20 years (40 payments), the policy will mature. The insurance company will pay out the proceeds of this policy to you in 10 equal annual payments, with the first payment to be made one year after the policy matures. If the effective interest rate remains at 8.16 percent, how much will you receive during each of the 10 years?

a.$6,113.20

b.$5,244.62

c.$5,792.21

d.$4,992.39

e.$4,723.81

72 Chapter 4  The Time Value of Money

ANS: D

Tabular solution: (Part a)

Value of the policy at the end of 20 years

FV = $350 (FVIFA 4%, 40) = $350 (95.026) = $33,259.10 This amount is then to be paid out over a 10-year period.

Note: There is no tabular solution presented for part b due to fractional interest rate (EAR = 8.16%)

Financial calculator solution: (Part a)

Inputs: N = 40; I = 4; PMT = -350. Output: FV = $33,258.93

Financial calculator solution (Part b)

Inputs: N = 10; I = 8.16; PV = 33,258.93.

Output: PMT = $4,992.39.

DIF: Tough

OBJ: TYPE: Financial Calculator TOP: Annuity payments

68.Assume that you just had a child, and you are now planning for her college education. You would like to make 43 equal payments over the next 21 years (the first payment to be made immediately, all other payments to be made at 6-month intervals, with the final payment to be made at her 21st birthday) so that you will be able to cover her expected expenses while in school. You expect to pay expenses on her 18th, 19th, 20th, and 21st birthdays. Assume that the current (time period 0) annual cost of college is $6,000, that you expect annual inflation to be 8 percent for the next 5 years, and then 5 percent thereafter. If you expect to be able to earn a return of 4 percent every 6 months on your investments (a simple rate of 8 percent with semiannual compounding), what will be the amount of each of the 43 payments?

a.$705.86

b.$731.93

c.$692.15

d.$650.46

e.$785.72

ANS: B

Chapter 4  The Time Value of Money 75

Step 1: The retirement payments, which begin at t = 2, must be: t = 2: $40,000 (1.05)2 = $44,100

t = 3: $44,100 (1.05) = $46,305 t = 4: $46, 305 (1.05) = $48.620

Step 2: Now we need enough at t = 2 to make the 3 retirement payments as calculated in Step 1. We cannot use the annuity method, but we can enter, in the cash flow register, the following:

Then enter I = 8, and press NPV to find NPV = PV = $128,659

Step 3: The $100,000 now on hand will compound at 8% for 2 years: $100,000 (1.08)2 = $116,640.

Step 4: The net funds needed are:

Need at t = 2	$128,659
Will have	(116,640)
Net needed	$ 12,019

Step 5: Find the payments needed to accumulate $12,019. Set the calculator to "Begin" and then enter:

N = 2; I = 8; PV = 0; FV = 12,019. Solve for PMT = $5,350.

DIF: Tough

OBJ: TYPE: Financial Calculator TOP: Annuity payments

71.Your client just turned 75 years old and plans on retiring in 10 years on her 85th birthday. She is saving money today for her retirement and is establishing a retirement account with your office. She would like to withdraw money from her retirement account on her birthday each year until she dies. She would ideally like to withdraw $50,000 on her 85th birthday, and increase her withdrawals 10 percent a year through her 89th birthday (i.e., she would like to withdraw $73,205 on her 89th birthday). She plans to die on her 90th birthday, at which time she would like to leave $200,000 to her descendants. Your client currently has $100,000. You estimate that the money in the retirement account will earn 8 percent a year over the next 15 years. Your client plans to contribute an equal amount of money each year until her retirement. Her first contribution will come in one year; her tenth and final contribution will come in ten years (on her 85th birthday). How much should she contribute each year in order to meet her objectives?

78 Chapter 4  The Time Value of Money

Step 1: Calculate college costs at t = 10, 11, 12, 13: t = 10; (10,000) (1.05)10 = $16,288.95

t = 11: (10,000) (1.05)11 = $17,103.39 t = 12: (10,000) (1.05)12 = $17,958.56 t = 13: (10,000) (1.05)13 = $18,856.49

Step 2: Find the NPV of the cash flows:

	=	25,000
	=	0
	=	16,288.95
	=	-17,103.39
	=	-17,958.56
	=	-18,856.49
I	=	6

Solve for NPV = -$10,871.03

Step 3: Find the payment stream which equates to the NPV.

BEGIN

N = 10

I = 6

PV = -10,871.03

FV = 0

Solve for PMT = $1,393.42

DIF: Tough

OBJ: TYPE: Financial Calculator TOP: Annuity payments

74.You will receive a $100 annual perpetuity, the first payment to be received now, at Year 0, a $300 annual perpetuity payable starting at the end of Year 5, and a $200 semiannual (2 payments per year) perpetuity payable starting midway through Year 10. If you require an effective annual interest rate of 14.49 percent, what is the present value of all three perpetuities together at Year 0? (Hint: The semiannual annuity can be thought of as two annual annuities.)

a.$2,091.86

b.$2,785.14

c.$4,213.51

d.Infinite; the present value of any perpetuity is infinite.

e.Cannot determine the value since some payments are annually and some semiannually.

ANS: B

Numerical solution:

PV = $100 + ($100/0.1449)+ ($300/0.1449) [1/(1.1449)4] + ($200/0.1449) [1/(1.449)9]

+($200/0.1449) [1/(1.1449)9.5]

=$100 + $690.13 + $1,204.99 + $408.37 + $318.65 = $2785.14

Financial calculator solution:

Step 1: Calculate the values of the respective perpetuities as their starting points; t =

semiannual periods.
PVp1 = 100 + 100/0.1449 = 790.13	n = 0

		Chapter 4  The Time Value of Money 79
FVp2, t = 8 = 300/0.1449 = 2,070.3		n = 8 semiannual periods
1/2FVp3, t = 18 = 200/0.1449 = 1,380.26		n = 18 semiannual periods
1/2FVp3, t = 19 = 200/0.1449 = 1,380.26		n = 19 semiannual periods
Step 2: Use interest at conversion to convert EAR to NOM%
Inputs: P/YR = 2; EFF% = EAR = 14.49. Output: NOM% = 14.0%
Periodic rate = 14/2 = 7.0%
Inputs:	N = 8; I = 7; FV = -2,070.39	Output	PV	=	$1,204.99
	N = 18; I = 7; FV = -1,380.26		PV	=	408.37
	N = 19; I = 7; FV = -1,380.26		PV	=	381.65
	Plus PV of perpetuity one		PV	=	790.13
					$2,785.14
PV of all perpetuities = $2,785.14
DIF: Tough	OBJ: TYPE: Financial Calculator TOP:		PV of an annuity

75.Hillary is trying to determine the cost of health care to college students, and parents' ability to cover those costs. She assumes that the cost of one year of health care for a college student is $1,000 today, that the average student is 18 when he or she enters college, that inflation in health care cost is rising at the rate of 10 percent per year, and that parents can save $100 per year to help cover their children's costs. All payments occur at the end of the relevant period, and the $100/year savings will stop the day the child enters college (hence 18 payments will be made). Savings can be invested at a simple rate of 6 percent, annual compounding. Hillary wants a health care plan which covers the fully inflated cost of health care for a student for 4 years, during years 19 through 22 (with payments made at the end of years 19 through 22). How much would the government have to set aside now (when a child is born), to supplement the average parent's share of a child's college health care cost? The lump sum the government sets aside will also be invested at 6 percent, annual compounding.

a.$1,082.76

b.$3,997.81

c.$5,674.23

d.$7,472.08

e.$8,554.84

ANS: D
Parent's savings:	Health Care Costs, Years 19-22
N = 18	-$1,000 (1.1)19	= -$6,115.91
I = 8	$1,000 (1.1)20	= -$6,727.50
PMT = 100	$1,000 (1.1)21	= -$7,400.25
FV = 0	-$1,000 (1.1)22	= -$8,140.27
Solve for PV = $1,082.76

-$8,554.84 PV of Health care costs 1,082.76 PV of parents' savings

-$7,472.08 Lump sum government must set aside

= 0 = 0

80 Chapter 4  The Time Value of Money

	=	-6,115.91
	=	-6,727.50
	=	-7,400.25
	=	-8,140.27
I	=	6

Solve for NPV = -8,554.84 = PV of Health care costs.

Consequently, the government must set aside $8,554.84 - $1,082.76 = $7,472.08 Alternatively,

= 0

= 100

= -6,115.91

= -6,727.50

= -7,400.25

= -8,140.27 I = 6

Solve for NPV = -$7,472.08 = Lump sum government must set aside.

DIF: Tough

OBJ: TYPE: Financial Calculator TOP: PV of an uneven CF stream

76.You have some money on deposit in a bank account which pays a simple (or quoted) rate of 8.0944 percent, but with interest compounded daily (using a 365-day year). Your friend owns a security which calls for the payment of $10,000 after 27 months. The security is just as safe as your bank deposit, and your friend offers to sell it to you for $8,000. If you buy the security, by how much will the effective annual rate of return on your investment change?

a.1.87%

b.1.53%

c.2.00%

d.0.96%

e.0.44%

ANS: C

Numerical solution:

Step 1: Find the effective annual rate (EAR) of interest on the bank deposit

EARDaily = (1 + 0.080944/365)365 -1 = 8.43% Step 2: Find the EAR of the investment

$8,000 = $10,000/(1 + r)2.25 (1 + r)2.25 = 1.25

1 + r = 1.25(1/2.25)

1 + r = 1.10426

r = 0.10426 10.43%

Step 3: Difference = 10.43% - 8.43% = 2.0%

Financial calculator solution:

Chapter 4  The Time Value of Money 81

Calculate EARDaily using interest rate conversion feature

Inputs: P/YR = 365; NOM% = 8.0944; Output: EFF% = EAR = 8.43%

Calculate EAR of the equal risk investment

Inputs: N = 2.25; PV = -8,000; FV = 10,000; Output: I = 10.4259 10.43%

Difference: 10.43% - 8.43% = 2.0%

DIF: Tough

OBJ: TYPE: Financial Calculator TOP: Effective annual rate

77.Your employer has agreed to make 80 quarterly payments of $400 each into a trust account to fund your early retirement. The first payment will be made 3 months from now. At the end of 20 years (80 payments), you will be paid 10 equal annual payments, with the first payment to be made at the beginning of Year 21 (or the end of Year 20). The funds will be invested at a simple rate of 8.0 percent, quarterly compounding, during both the accumulation and the distribution periods. How large will each of your 10 receipts be? (Hint: You must find the EAR and use it in one of your calculations.)

a.$7,561

b.$10,789

c.$11,678

d.$12,342

e.$13,119

ANS: B

PMT = ?

Find the FV at t = 80 of $400 quarterly payments: N = 80; I = 2; PV = 0; and PMT = 400.

Solve for FV = $77,508.78

Find the EAR of 8%, compounded quarterly, so you can determine the value of each of the receipts.

EAR =

Now, determine the value of the receipts, remembering that this is an annuity due. With a financial calculator, input the following:

N = 10; I = 8.2432; PV = -77,508.78; and FV = 0. Solve for PMT = $10,788.78 $10,789

DIF:	Tough	OBJ: TYPE: Financial Calculator
TOP:	PMT and quarterly compounding