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f. What is the nominal value of the amount you need to save during the first year? (Assume

the savings are put aside at the end of each year.) The thirtieth year?

74. Retirement and Inflation. Redo part (a) of problem 63, but now assume that the inflation

rate over the next 50 years will average 4 percent.

The Time Value of Money 77

a. What is the real annual savings the couple must set aside?

b. How much do they need to save in nominal terms in the first year?

c. How much do they need to save in nominal terms in the last year?

d. What will be their nominal expenditures in the first year of retirement? The last?

75. Annuity Value. What is the value of a perpetuity that pays $100 every 3 months forever?

The discount rate quoted on an APR basis is 12 percent.

76. Changing Interest Rates. If the interest rate this year is 8 percent and the interest rate next

year will be 10 percent, what is the future value of $1 after 2 years? What is the present value

of a payment of $1 to be received in 2 years?

77. Changing Interest Rates. Your wealthy uncle established a $1,000 bank account for you

when you were born. For the first 8 years of your life, the interest rate earned on the account

was 8 percent. Since then, rates have been only 6 percent. Now you are 21 years old and

ready to cash in. How much is in your account?

1 Value after 5 years would have been 24 Ã— (1.05)5 = $30.63; after 50 years, 24 Ã— (1.05)50 =

Solutions to $275.22.

2 Sales double each year. After 4 years, sales will increase by a factor of 2 Ã— 2 Ã— 2 Ã— 2 = 24

Self-Test

= 16 to a value of $.5 Ã— 16 = $8 million.

Questions 3 Multiply the $1,000 payment by the 10-year discount factor:

1

PV = $1,000 Ã— = $441.06

(1.0853)10

4 If the doubling time is 12 years, then (1 + r)12 = 2, which implies that 1 + r = 21/12 = 1.0595,

or r = 5.95 percent. The Rule of 72 would imply that a doubling time of 12 years is con-

sistent with an interest rate of 6 percent: 72/6 = 12. Thus the Rule of 72 works quite well

in this case. If the doubling period is only 2 years, then the interest rate is determined by (1

+ r)2 = 2, which implies that 1 + r = 21/2 = 1.414, or r = 41.4 percent. The Rule of 72 would

imply that a doubling time of 2 years is consistent with an interest rate of 36 percent: 72/36

= 2. Thus the Rule of 72 is quite inaccurate when the interest rate is high.

5 Gift at Year Present Value

1 10,000/(1.07) = $ 9,345.79

2 10,000/(1.07)2 = 8,734.39

3 10,000/(1.07)3 = 8,162.98

4 10,000/(1.07)4 = 7,628.95

$33,872.11

Gift at Year Future Value at Year 4

1 10,000/(1.07)3 = $12,250.43

2 10,000/(1.07)2 = 11,449

3 10,000/(1.07) = 10,700

4 10,000 = 10,000

$44,399.43

6 The rate is 4/48 = .0833, about 8.3 percent.

7 The 4-year discount factor is 1/(1.08)4 = .735. The 4-year annuity factor is [1/.08 â€“ 1/(.08

Ã— 1.084)] = 3.312. This is the difference between the present value of a $1 perpetuity start-

ing next year and the present value of a $1 perpetuity starting in Year 5:

78 SECTION ONE

1

PV (perpetuity starting next year) = = 12.50

.08

1

1

= 12.50 Ã— .735 = 9.188

Ã—

â€“ PV (perpetuity starting in Year 5) =

.08 (1.08)4

= PV (4-year annuity) = 12.50 â€“ 9.188 = 3.312

8 Calculate the value of a 19-year annuity, then add the immediate $465,000 payment:

1 1

19-year annuity factor = â€“

r r(1 + r)19

1 1

= â€“

.08 .08(1.08)19

= 9.604

PV = $465,000 Ã— 9.604 = $4,466,000

Total value = $4,466,000 + $465,000

= $4,931,000

Starting the 20-year cash-flow stream immediately, rather than waiting 1 year, increases

value by nearly $400,000.

9 Fifteen years means 180 months. Then

100,000

Mortgage payment =

180-month annuity factor

100,000

=

83.32

= $1,200.17 per month

$1,000 of the payment is interest. The remainder, $200.17, is amortization.

10 You will need the present value at 7 percent of a 20-year annuity of $55,000:

Present value = annual spending Ã— annuity factor

The annuity factor is [1/.07 â€“ 1/(.07 Ã— 1.0720)] = 10.594. Thus you need 55,000 Ã— 10.594

= $582,670.

11 If the interest rate is 5 percent, the future value of a 50-year, $1 annuity will be

(1.05)50 â€“ 1

= 209.348

.05

Therefore, we need to choose the cash flow, C, so that C Ã— 209.348 = $500,000. This re-

quires that C = $500,000/209.348 = $2,388.37. This required savings level is much higher

than we found in Example 3.12. At a 5 percent interest rate, current savings do not grow as

rapidly as when the interest rate was 10 percent; with less of a boost from compound in-

terest, we need to set aside greater amounts in order to reach the target of $500,000.

12 The cost in dollars will increase by 5 percent each year, to a value of $5 Ã— (1.05)50 = $57.34.

If the inflation rate is 10 percent, the cost will be $5 Ã— (1.10)50 = $586.95.

13 The weekly cost in 1980 is $250 Ã— (370/100) = $925. The real value of a 1980 salary of

$30,000, expressed in real 1947 dollars, is $30,000 Ã— (100/370) = $8,108.

14 a. If thereâ€™s no inflation, real and nominal rates are equal at 8 percent. With 5 percent in-

flation, the real rate is (1.08/1.05) â€“ 1 = .02857, a bit less than 3 percent.

b. If you want a 3 percent real interest rate, you need a 3 percent nominal rate if inflation

is zero and an 8.15 percent rate if inflation is 5 percent. Note 1.03 Ã— 1.05 = 1.0815.

The Time Value of Money 79

15 The present value is

$5,000

PV = = $4,629.63

1.08

The real interest rate is 2.857 percent (see Self-Test 3.14a). The real cash payment is

$5,000/(1.05) = $4,761.90. Thus

$4,761.90

PV = = $4,629.63

1.02857

16 Calculate the real annuity. The real interest rate is 1.10/1.05 â€“ 1 = .0476. Weâ€™ll round to 4.8

percent. The real annuity is

$3,000,000

Annual payment =

30-year annuity factor

= $3,000,000

1 1

â€“

.048 .048(1.048)30

$3,000,000

= = $190,728

15.73

You can spend this much each year in dollars of constant purchasing power. The purchas-

ing power of each dollar will decline at 5 percent per year so youâ€™ll need to spend more in

nominal dollars: $190,728 Ã— 1.05 = $200,264 in the second year, $190,728 Ã— 1.052 =

$210,278 in the third year, and so on.

17 The quarterly rate is 8/4 = 2 percent. The effective annual rate is (1.02)4 â€“ 1 = .0824, or 8.24

percent.

MINICASE inflation. That is, they will be automatically increased in propor-

Old Alfred Road, who is well-known to drivers on the Maine

tion to changes in the consumer price index.

Turnpike, has reached his seventieth birthday and is ready to re-

Mr. Roadâ€™s main concern is with inflation. The inflation rate

tire. Mr. Road has no formal training in finance but has saved his

has been below 3 percent recently, but a 3 percent rate is unusu-

money and invested carefully.

ally low by historical standards. His social security payments will

Mr. Road owns his homeâ€”the mortgage is paid offâ€”and

increase with inflation, but the interest on his investment portfo-

does not want to move. He is a widower, and he wants to bequeath

lio will not.

the house and any remaining assets to his daughter.

What advice do you have for Mr. Road? Can he safely spend

He has accumulated savings of $180,000, conservatively in-

all the interest from his investment portfolio? How much could he

vested. The investments are yielding 9 percent interest. Mr. Road

withdraw at year-end from that portfolio if he wants to keep its

also has $12,000 in a savings account at 5 percent interest. He

real value intact?

wants to keep the savings account intact for unexpected expenses

Suppose Mr. Road will live for 20 more years and is willing

or emergencies.

to use up all of his investment portfolio over that period. He also

Mr. Roadâ€™s basic living expenses now average about $1,500

wants his monthly spending to increase along with inflation over

per month, and he plans to spend $500 per month on travel and

that period. In other words, he wants his monthly spending to stay

hobbies. To maintain this planned standard of living, he will have

the same in real terms. How much can he afford to spend per

to rely on his investment portfolio. The interest from the portfolio

month?

is $16,200 per year (9 percent of $180,000), or $1,350 per month.

Assume that the investment portfolio continues to yield a 9

Mr. Road will also receive $750 per month in social security

percent rate of return and that the inflation rate is 4 percent.

payments for the rest of his life. These payments are indexed for

FINANCIAL PLANNING

What Is Financial Planning?

Financial Planning Focuses on the Big Picture

Financial Planning Is Not Just Forecasting

Three Requirements for Effective Planning

Financial Planning Models

Components of a Financial Planning Model

An Example of a Planning Model

An Improved Model

Planners Beware

Pitfalls in Model Design

The Assumption in Percentage of Sales Models

The Role of Financial Planning Models

External Financing and Growth

Summary

Financial planning?

Financial planners donâ€™t guess the future, they prepare for it.

SuperStock

81

tâ€™s been said that a camel looks like a horse designed by committee. If a

I firm made all its financial decisions piecemeal, it would end up with a

financial camel. Therefore, smart financial managers consider the overall

effect of future investment and financing decisions. This process is called fi-

nancial planning, and the end result is called a financial plan.

New investments need to be paid for. So investment and financing decisions cannot

be made independently. Financial planning forces managers to think systematically

about their goals for growth, investment, and financing. Planning should reveal any in-

consistencies in these goals.

Planning also helps managers avoid some surprises and think about how they should

react to those surprises that cannot be avoided. We stress that good financial managers

insist on understanding what makes projects work and what could go wrong with them.

The same approach should be taken when investment and financing decisions are con-

sidered as a whole.

Finally, financial planning helps establish goals to motivate managers and provide

standards for measuring performance.

We start by summarizing what financial planning involves and we describe the con-

tents of a typical financial plan. We then discuss the use of financial models in the plan-

ning process. Finally, we examine the relationship between a firmâ€™s growth and its need

for new financing.

After studying this material you should be able to

Describe the contents and uses of a financial plan.

Construct a simple financial planning model.

Estimate the effect of growth on the need for external financing.

What Is Financial Planning?

Financial planning is a process consisting of:

1. Analyzing the investment and financing choices open to the firm.

2. Projecting the future consequences of current decisions.

3. Deciding which alternatives to undertake.

4. Measuring subsequent performance against the goals set forth in the financial plan.

Notice that financial planning is not designed to minimize risk. Instead it is a process

of deciding which risks to take and which are unnecessary or not worth taking.

Firms must plan for both the short-term and the long-term. Short-term planning

rarely looks ahead further than the next 12 months. It is largely the process of making

sure the firm has enough cash to pay its bills and that short-term borrowing and lend-

ing are arranged to the best advantage.

82

Financial Planning 83

Here we are more concerned with long-term planning, where a typical planning

PLANNING HORIZON

Time horizon for a financial horizon is 5 years (although some firms look out 10 years or more). For example, it can

plan. take at least 10 years for an electric utility to design, obtain approval for, build, and test

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