Fundamentals of corporate finance brealey chapter 04 the time value of money

You should compare the present values of the two annuities.Discount Present Value of Present Value of Rate 10-year, $1000 annuity 15-year, $800 annuity When the interest rate is low, as

Solutions to Chapter 4 The Time Value of Money

Note: Unless otherwise stated, assume that cash flows occur at the end of each year

6 You should compare the present values of the two annuities.

Discount Present Value of Present Value of

Rate 10-year, $1000 annuity 15-year, $800 annuity

When the interest rate is low, as in part (a), the longer (i.e., 15-year) but smaller annuity is more valuable because the impact of discounting on the present value of future payments is less severe When the interest rate is high, as in part (b), the shorter but higher annuity is more valuable In this case, with the 20 percent interest rate, the present value of more distant payments is substantially reduced, making it better to take the shorter but higher annuity

7 PV = 200/1.05 + 400/1.052 + 300/1.053

= 190.48 + 362.81 + 259.15 = $812.44

8 In these problems, you can either solve the equation provided directly, or you can use your financial calculator setting PV = ()400, FV = 1000, PMT = 0, i as specified by the problem Then compute n on the calculator

Period

Per PeriodRate, APR/

m Effective annual rate

Number ofPeriods peryear, m Per period rate,(1+EAR)1/m -1 APR, m × perperiod rate

= 02

4×.02 = 08 = 8%

13 We need to find the value of n for which 1.08n = 2 You can solve to find that

n = 9.01 years On a financial calculator you would enter PV = ()1, FV = 2,

PMT = 0, i = 8 and then compute n

14 Semiannual compounding means that the 8.5 percent loan really carries interest of

4.25 percent per half year Similarly, the 8.4 percent loan has a monthly rate of

.7 percent

= (1 + per period rate)m – 1

Choose the 8.5 percent loan for its slightly lower effective rate

15 APR = 1%  52 = 52%

EAR = (1 + 01)52  1 = 6777 = 67.77%

16 Our answer assumes that the investment was made at the beginning of 1900 and

now it is the end of 2008 Thus the investment was for 106 years (2008 – 1900 + 1)

a 1000  (1.05)109 = $204,001.61

b PV  (1.05)109 = 1,000,000 implies that PV = $4,901.92

17 $1000  1.05 = $1050.00 First-year interest = $50

$1050  1.05 = $1102.50 Second-year interest = $1102.50  $1050 = $52.50After 9 years, your account has grown to 1000  (1.05)9 = $1551.33

After 10 years, your account has grown to 1000  (1.05)10 = $1628.89

Interest earned in tenth year = $1628.89  $1551.33 = $77.56

18 Method 1: If you earned simple interest (without compounding) then the total

growth in your account after 25 years would be 4% per year  25 years = 100%, and your money would double With compound interest, your money would growfaster, and therefore would require less than 25 years to double

Method 2: Another quick way to answer the question is with the Rule of 72 Dividing 72 by 4 gives 18 years, which is less than 25

The exact answer is 17.673 years, found by solving 2000 = 1000  (1.04)n [On your calculator, input PV = (-) 1000, FV = 2000, i = 4, PMT = 0, and compute the number of periods.]

19 We solve 422.21  (1 + r)10 = 1000 This implies that r = 9% [On your

calculator, input PV = (-)422.21, FV = 1000, n = 10, PMT = 0, and compute the interest rate.]

20 The number of payment periods: n = 12 × 4 = 48

If the payment is denoted PMT, then

PMT  annuity factor( %, 48 periods) = 8,000

PMT = $202.90

The monthly interest rate is 10/12 = 8333 percent Therefore, the effective annual interest rate on the loan is (1.008333)12  1 = 1047 = 10.47 percent.

21 a PV = 100  annuity factor(6%, 3 periods)

= 100  = $267.30

b If the payment stream is deferred by an extra year, each payment will be

discounted by an additional factor of 1.06 Therefore, the present value is

reduced by a factor of 1.06 to 267.30/1.06 = $252.17

22 a This is an annuity problem with PV = (-)80,000, PMT = 600, FV = 0,

n = 20  12 = 240 months Use a financial calculator to solve for i, the

monthly rate on this annuity: i = 5479%

b Your present value is 10,000 (1  d), and the future value you pay back is 10,000 Therefore, the annual interest rate is determined by:

c With a discount interest loan, the discount is calculated as a fraction of the future value of the loan In fact, the proper way to compute the interest rate is as a fraction of the funds borrowed Since PV is less than FV, the interest payment is a smaller fraction of the future value of the loan than it is of the present value Thus,the true interest rate exceeds the stated discount factor of the loan.

24 If we assume cash flows come at the end of each period (ordinary annuity) when infact they actually come at the beginning (annuity due), we discount each cash flow

by one period too many Therefore we can obtain the PV of an annuity due by

multiplying the PV of an ordinary annuity by (1 + r) Similarly, the FV of an

annuity due also equals the FV of an ordinary annuity times (1 + r) Because each cash flow comes at the beginning of the period, it has an extra period to earn

interest compared to an ordinary annuity

25 a Solve for i in the following equation:

Repeat the steps in (a) to find the EAR of this car loan to see which loan is

charging the lower interest rate:

Solve for i in the following equation:

Using the calculator, set PV = -10,000, FV = 0, n = 4, i = 15.3271 and solve for

PMT = $3,525.86 By comparison, 12 times $275 per month is $3,300 The annual payment on a 4-year loan equivalent to $275 per month for 48 months is greater than 12 times the monthly payment of $275 because of the benefit of delaying payment to the end of each year The borrower gets to delay payment and therefore is better off If Little Bank doesn't charge at least $3,525.86

annually, it earns less on its loan than Big Bank earns on its loan

26 a Compare the present value of the lease to cost of buying the truck

PV lease = 8,000 × PVIFA(7%, 6) = -$38,132.32

It is cheaper to lease than buy because by leasing the truck will cost only

$38,132.32, rather than $40,000 Of course, the crucial assumption here is that the

truck is worthless after 6 years If you buy the truck, you can still operate it after 6years If you lease it, you must return the truck and replace it.

b If the lease payments are payable at the start of each year, then the present value ofthe lease payments are:

PV annuity due lease = 8,000 + 8,000 × PVIFA(7%, 5) = 8,000 + 32,801.58 =

$40,801.58 Note too that PV of an annuity due = PV of ordinary annuity  (1 + r) Therefore, with immediate payment, the value of the lease payments increases from its value in the previous problem to $38,132  1.07 = $40,801 which is greater than $40,000 (the cost of buying a truck) Therefore, if the first payment onthe lease is due immediately, it is cheaper to buy the truck than to lease it

27 a Compare the PV of the payments Assume the product sells for $100

Installment plan:

Down payment = 25 × 100 = 25

Three installments of 25 × 100 = 25

PV = 25 + 25  annuity factor(6%, 3 years) = $91.83

Pay in full: Payment net of discount = $90 Choose this payment plan for its lower present value of payments

Note: The pay-in-full payment plan will have the lowest present value of

payments, regardless of the chosen product price

b Installment plan: PV = 25  annuity factor(6%, 4 years) = $86.63 Now the installment plan offers the lower present value of payments

28 a PMT  annuity factor(12%, 5 years) = 1000

PMT  3.6048 = 1000

PMT = $277.41

b If the first payment is made immediately instead of in a year, the annuity

factor will be greater by a factor of 1.12 Therefore

PMT  (3.6048  1.12) = 1000

PMT = $247.69

29 This problem can be approached in two steps First, find the PV of the $10,000,

10-year annuity as of year 3, when the first payment is exactly one year away (and

is therefore an ordinary annuity) Then discount the value back to today

Using a financial calculator,

1) PMT = 10,000; FV = 0; n = 10; i = 6%

Compute PV3 = $73,600.87

2) PV0 = = = $61,796.71

A second way to solve the problem is the take the difference between a 13-year

annuity and a 3-year annuity, valued as of the end of year 0:

PV of delayed annuity = 10,000 × PVIFA(6%,13) – 10,000 × PVIFA(6%,3)

= 10,000 × (8.852683 – 2.673012) = 10,000 × 6.179671 = $61,796.71

30 Note: Assume that this is a Canadian mortgage

The monthly payment is based on a $175,000 loan with a 300-month (12 × 25

years) amortization The posted interest rate of 6 percent has a 6-month

compounding period Its EAR is (1 + 06/2)2 – 1 = 0609, or 6.09% The monthly interest rate equivalent to 6.09% annual is (1.0609)1/12 – 1 = 004939, or 0.4939%

PMT  annuity factor(.4939%, 300) = 175,000

PMT = $1,119.71

When the mortgage expires in 5 years, there will be 20 years remaining in the

amortization period, or 240 months The loan balance in five years will be the

present value of the 240 payments:

Loan balance in 5 years = $1,119.71  Annuity factor (.4939%, 240 periods)

= $157,208

31 The EAR of the posted 7% rate is (1 + 07/2)2 – 1 = 071225 The monthly interest rate equivalent is (1.071225)1/12 – 1 = 00575, or 0.575% The payment on the

mortgage is computed as follows:

PMT  annuity factor (.575%, 300 periods) = 350,000

PMT = $2,451.44 per month

If you pay the monthly mortgage payment in two equal installments, you will pay

$2,451.44/2, or $1,225.72 every two weeks Thus each year you make 26 payments The bi-weekly equivalent of the 7% posted interest rate is (1.071225)1/26 – 1

= 002649, or 2649% every two weeks Now calculate the number of periods it will take to pay off the mortgage:

$1,225.72  Annuity factor (.2649%, n periods) = $350,000

Using the calculator: PMT = 1,225.49, PV = (-)350,000, i = 2649 and compute n = 533.84 This is the number of bi-weekly periods Divide by 26 to get the number of years: 533.84/26 = 20.5 years If you pay bi-weekly, the mortgage is paid off 5.5 years sooner than if you pay monthly

32 a Input PV = (-)1,000, FV = 0, i = 8%, n = 4, compute PMT which equals

$301.92

The effective annual rate on the loan is (1.01)12  1 = 1268 = 12.68%

34 The present value of the $2 million, 20-year annuity, discounted at 8%, is

$19,636,295

If the payment comes one year earlier, the PV increases by a factor of 1.08 to

$21,207,198

35 The real rate is zero With a zero real rate, we simply divide her savings by the

years of retirement: $450,000/30 = $15,000 per year

36 Per month interest = 6%/12 = 5% per month

FV in 1 year (12 months) = 1000  (1.005)12 = $1,061.68

FV in 1.5 years (18 months) = 1000  (1.005)18 = $1,093.93

37 You are repaying the loan with an annuity of payments The PV of those paymentsmust equal $100,000 Therefore,

804.62  annuity factor(r, 360 months) = 100,000

which implies that the interest rate is 750% per month

[On your calculator, input PV = ()100,000, FV = 0, n = 360, PMT = 804.62, and compute the interest rate.]

The effective annual rate is (1.00750)12  1 = 0938 = 9.38%

If the lender is a Canadian financial institution, the quoted rate will be the APR for

a 6-month compounding period:

$250  annuity factor(1%, 48 months) = $9,493.49

Option (b) is the better deal

42 The present value of your payments to the bank equals:

$100  annuity factor(8%, 10 years) = $671.01

The present value of your receipts is the value of a $100 perpetuity deferred for 10 years:

 = $578.99

This is a bad deal if you can earn 8% on your other investments.

43 If you live forever, you will receive a $100 perpetuity which has present value 100/r Therefore, 100/r = 2500, which implies that r = 4 percent

44 r = 10,000/125,000 = 08 = 8 percent

45 Suppose the purchase price is $1 If you pay today, you get the discount and pay only $.97 If you wait a month, you must pay $1 Thus, you can view the deferred payment as saving a cash flow of $.97 today, but paying $1 in a month The

monthly rate is therefore 03/.97 = 0309, or 3.09% The effective annual rate is (1.0309)12  1 = 4408 = 44.08%

46 You borrow $1000 and repay the loan by making 12 monthly payments of $100

We find that r = 2.923% by solving:

100  annuity factor(r, 12 months) = 1000[On your calculator, input PV = ()1,000, FV = 0, n = 12, PMT = 100, and

compute the interest rate.]

The APR is therefore 2.923%  12 = 35.08%

and the effective annual rate is (1.02923)12  1 = 4130 = 41.30%

How do we know that the true rates must be greater than 20%? If you borrow

$1000 and repay $1200 in one year, the rate of interest on the loan is 20% Here, with add-on interest, you make the $1200 repayment sooner Because of the time value of money, the effective interest rate must be higher than 20%

47 You will have to pay back the original $1000 plus 3  20% = 60% of the loan amount, or $1600 over the three years This implies monthly payments of

$1600/36 = $44.44

The monthly interest rate is obtained by solving:

44.44  annuity factor(r, 36) = 1000

On your calculator, input PV = ()1,000, FV = 0, n = 36, PMT = 44.44, and

compute the interest rate as 2.799% per month

The APR is therefore 2.799%  12 = 33.59%,

and the effective annual rate is (1.02799)12  1 = 3927 = 39.27%

48 For every $1000 you borrow, your present value is 1000 (1  d), and the future value you pay back is 1000 Therefore, the annual interest rate is determined by:

PV  (1 + r) = FV

[10,000 (1 – d)]  (1 + r) = 10,000

1 + r =  r =  1 = > d

If d = 20%, then the effective annual interest rate is 2/.8 = 25 = 25%

49 The semi-annual interest rate paid at First National is 0.062/2 = 031, or 3.1% every six months After one year, each dollar invested will grow to:

$1  (1.031)2 = $1.06296 and the EAR is 6.296%

The monthly interest rate paid at Second National is 06/12 = 005, or 0.5% every month After one year, each dollar invested will grow to:

$1  (1.005)12 = $1.06168 and the EAR is 6.168%

First National pays the higher effective annual ratẹ

50 Since the $20 origination fee is taken out of the initial proceeds of the loan, the amount actually borrowed is $1000  $20 = $980 The monthly rate is found by solving the following equation for r:

90  annuity factor(r, 12) = 980

r = 1.527% per month

The effective rate is (1.01527)12 -1 = 1994 = 19.94%

51 Monthly interest rate = (1.08)1/12 – 1 = 006434 or 6434%

Football Quarterback:

Total salary paid over contract = 5 years × 3 million/year = $15 million

Monthly salary = 3 million/12 months per year = $250,000 at the end of each month for 12 ×5, or 60 months

PV = 250,000 PVIFẶ6434%,60 months) = $12,411,236.45

Hockey Player

Total salary paid over contract = $4 million + 5 × $2.1 million = $14.5 millionMonthly salary = $2.1 million/12 months per year = $175,000 at the end of each

PV = 4 million + 175,000 PVIFẶ6434%,60 months) = $12,687,865.52

The quarterback is wrong The hockey player’s contract has a higher present valuẹ

52 ạ Per month interest rate = 7%/12 = 005833333

48-month loan: PV = 400 × PVIFĂ7%/12,48) = $16,704.08

60-month loan: PV = 400 × PVIFĂ7%/12,60) = $20,200.80

Bill will buy a car for $16,704.08 if he arranges a 48-month loan and will buy acar for $20,200.80 if he arranges a 60-month loan

b To fairly compare the two loans, both time periods must be the samẹ We assume that Bill will keep the car for 5 years, regardless of which loan he takes

Bill’s wealth at the end of 5 years will depend on the value of the car and the balance in his bank account

Wealth in five years if take the 48-month loan and buy the $16,704.08 car:(1) Value of car at the end of 5 years: starting value × (1 – depreciation rate)5 = $16,704.08 × (1 - 18)5 = $6,192.87

(2) Savings = $400 invested each month for one year:

FV = 400 × FVIFĂ5%/12,12) = $4,911.54

(3) Total wealth = $6,192.87 + $4,911.54 = $11,104.41

Wealth in five years if take the 60-month loan and buy the $20,200.80 car:(1) Value of car at the end of 5 years: starting value × (1 – depreciation rate)5 = $20,200.80 × (1 - 18)5 = $7,489.24

(2) Savings = none, spent all spare cash on car payments

(3) Total wealth = $7,489.24

We haven’t compared Bill’s happiness from owning the more expensive car to his happiness from owning the less expensive car On the other hand, we’ve notcompare the cost of fuel or maintenance either Whether the more expensive car is worth it has not been established

53 ạ The present value of the ultimate sales price is 4 million/(1.08)5 = $2.722

million

b The present value of the sales price is less than the cost of the property, so this would not be an attractive opportunitỵ

c The value of the total cash flows from the property is now

PV = 2  annuity factor(8%, 5 years) + 4/(1.08)5

= 80 + 2.72 = $3.52 million

To solve with a calculator, enter: PMT = 2, FV = 4, i = 8%, n = 5 and compute PV.

Therefore, the property is an attractive investment if you can buy it for $3 million

54 PV of cash inflows

= [120,000/1.12 + 180,000/1.122 + 300,000/1.123]

= $464,172

This exceeds the cost of the factory, so the investment is attractive

55 a The present value of the future payoff is 2000/(1.05)10 = $1227.83 This is a

good deal: PV exceeds the initial investment

You can solve this also by looking at the future value of investing $1,000 at the opportunity cost of 5% for 10 years: FV = 1.0510 × 1000 = $1628.9

This is less than the $2000 payoff expected from the investment The

investment is a good deal

Another way to answer this question is to figure out the interest rate that theinvestment is offering: Find r such that (1 + r)10 × 1000 = 2000 Either

using the calculator or solving directly, gives r = 07177, or 7.177% Clearly

it is better to invest at 7.177% than at 5%

b The PV is now only 2000/(1.10)10 = $771.09, which is less than the initial investment Therefore, this is a bad deal

Another way to look at the investment: Since we know that the investment offers a 7.177% return, it makes no sense to undertake the investment if we can invest elsewhere and earn 10%

56 The future value of the payments into your savings fund must accumulate to

$500,000 We assume that payments are made at the end of the year We choose the payment so that PMT  future value of an annuity = $500,000 On your

calculator, enter: n = 40; i = 5; PV = 0; FV = 500,000 Compute PMT to be

$4,139.08

57 If you invest the $100,000 received in year 10 until your retirement in year 40, it will grow to $100,000  (1.05)30 = $432,194 Therefore, your savings plan would need to generate a future value of only $500,000 – $432,194 = $67,806 This

would require a savings stream of only $561.31

58 By the time you retire you will need

$40,000  future value annuity factor(5%, 20 periods) = $498,488.41

The future value of the payments into your savings fund must accumulate to

$498,488.41 We choose the payment so that PMT  future value of an annuity =

$498,488.41 On your calculator, enter: n = 40; i = 5; PV = 0; FV = 498,488.41 Compute PMT to be $4,126.57

59 After 30 years the couple will have accumulated the future value of a $3,000 annuity, plus the present value of the $10,000 gift The sum of the savings from these sources is:

$3,000  future value annuity factor (30,8%) = $339,849.63

$408,334.38

If they wish to accumulate $800,000 by retirement, they need to save an

additional amount per year to provide additional accumulations of $391,665.62

This requires additional annual savings of $3,457.40 [On your calculator, input i

= 8; n = 30;

PV = 0; FV = 391,665.62 and compute PMT.]

60 a The present value of the planned consumption stream as of the retirement

date will be $30,000  annuity factor(25,8%) = $320,243.29 Therefore,

they need to have accumulated this level of savings by the time they retire

So their savings plan must provide a future value of $320,243.29 With 50

years to save at 8%, the savings annuity must be $558.14

Another way to think about this is to recognize that the present value of the savings stream must equal the present value of the consumption stream The

PV of consumption as of today is = $320,243.29/(1.08)50 = $6,827.98

Therefore, we set the present value of savings equal to this value, and solve

for the required savings stream Using the calculator: enter PV = (-)6,827.98

n = 50, i = 8, PV = 0 and solve for PMT

b The couple needs to accumulate additional savings with a present value of

$60,000/(1.08)20 = $12,872.89 The total PV of savings is now $12,872.89 +

$6,827.98 = $19,700.77 Now we solve for the required savings stream as follows: n = 50; i = 8; PV = ()19,700.77; FV = 0; and solve for PMT as

$1,610.40 They need to save $1,610.40 each year for the next 50 years

61 Note: Ignore taxes

Monthly interest rate = 08/12 = 00666666

Borrow and buy the copier

Monthly loan payments : 20,000 = PMT × PVIFĂ8%/12, 60)

PMT = $405.53

Cash flows if borrow/buy:

MonthCash

PV(Lease cash flows) = +20,000 – X – X × PVIFẶ08/12, 59)

Find X such that PV(borrow/buy cash flows) = PV(lease cash flows):

3,356.05 = +20,000 – X – X × PVIFẶ08/12, 59)

16,643.95 = X + X × PVIFẶ08/12, 59)

= X × annuity paid at the beginning of the period (.08/12, 60)

This can be solved using a financial calculator Set the calculator for payments at the beginning of the period (With a BAII Plus: enter 2nd PMT (which is BEG), followed by

2nd Enter (which is SET) You should see the word END on the screen change to the word BEG.)

b Years to retirement = 2008 – 1950 = 58 years

Salary inflation rate, s: (1 + s)58 × $6,000 = $60,000

s = (60,000/6,000)1/58 – 1 = 0405, or 4.05%

Cost of goods inflation rate, c: (1 + c)58 × 1 = 8.2

66 Standard & Poor’s

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Based on the annual income statement, the compound annual growth rates (CAGR) over the period of Dec 03 to Dec 07 will be calculated as follow: