=127.63

Output

Also, some calculators require the user to press a Compute key before

pressing the FV key. Finally, ¬nancial calculators permit specifying the number

of decimal places that are displayed, even though 12 (or more) signi¬cant

digits are actually used in the calculations. Two places are generally used for

answers in dollars or percentages, and four places for decimal answers. The

¬nal answer, however, should be rounded to re¬‚ect the accuracy of the input

values; it makes no sense to say that the return on a particular investment is

14.63827 percent when the cash ¬‚ows are highly uncertain. The nature of

the analysis dictates how many decimal places should be displayed.

Spreadsheet Spreadsheet programs, such as Excel, are frequently used in time value anal-

Solution ysis. Many common time value solutions are preprogrammed, and users can

create their own formulas to perform tasks that have not been preprogrammed.

The time value formulas that are preprogrammed in spreadsheets are called

functions (or in some software, @ functions”pronounced “at functions”).

Like any formula, a time value function consists of a number of arithmetic

calculations combined into one statement. By using functions, spreadsheet

users can save the time and tedium of building formulas from scratch.

Each function begins with a unique function name that identi¬es the

calculation to be performed, along with one or more arguments (the input

values for the calculation) enclosed in parentheses. There is no spreadsheet

function for ¬nding the future value of a lump sum because it can be quickly

calculated by formula. For example, the Excel formula for solving the Merid-

ian Clinics example over ¬ve years is:

= 100 — (1.05)§ 5

where = tells the spreadsheet that a formula is being entered into the cell, — is

the spreadsheet multiplication sign, and § is the spreadsheet exponential (or

power) sign. When this formula is entered into a spreadsheet cell, the value

127.63 appears in the cell.3

259

Chapter 9: Time Value Analysis

The most ef¬cient way to solve most problems that involve time value

is to use a ¬nancial calculator or spreadsheet.4 However, the basic mathematics

behind the calculations must be understood to set up complex problems

before solving them. In addition, the underlying logic must be understood

to comprehend stock and bond valuation, lease analysis, capital budgeting

analysis, and other important healthcare ¬nancial management topics.

The Power of Compounding

The “power of compounding” is a phrase that emphasizes the fact that a

relatively small starting value can grow to a large amount, even when the

rate of growth (interest rate) is modest, when invested over a long period.

For example, assume that a new parent places $1,000 in a mutual fund to

help pay the child™s college expenses, which are expected to begin in 18 years.

The investment is assumed to earn a return of 10 percent per year, which is

a reasonable estimate by historical standards. After 18 years, the value of the

mutual fund account would be $5,560, which is not an inconsequential sum.

Now, assume that the money was meant to help fund the child™s retire-

ment, which is assumed to occur 65 years into the future. The value of the

mutual fund account at that time would be $490,371, or nearly a half-million

dollars. Imagine that: $1,000 grows to nearly half a million all because of the

power of compounding. The moral of this story is clear: When saving for

retirement, or for any other purpose, start early.

Self-Test

1. What is a lump sum?

Questions

2. What is compounding? What is interest on interest?

3. What are three solution techniques for solving lump sum compounding

problems?

4. How does the future value of a lump sum change as the time is

extended and as the interest rate increases?

5. What is meant by the power of compounding?

Present Value of a Lump Sum (Discounting)

Suppose that GroupWest Health Plans, which has premium income reserves

to invest, has been offered the chance to purchase a low-risk security from

a local broker that will pay $127.63 at the end of ¬ve years. A local bank is

currently offering 5 percent interest on a ¬ve-year certi¬cate of deposit (CD),

and GroupWest™s managers regard the security offered by the broker as having

the same risk as the bank CD. The 5 percent interest rate available on the bank

CD is GroupWest™s opportunity cost rate. (Opportunity costs are discussed in

detail in the next section.) How much would GroupWest be willing to pay for

the security that promises to pay $127.63 in ¬ve years?

260 Healthcare Finance

The future value example presented in the previous section showed

that an initial amount of $100 invested at 5 percent per year would be worth

$127.63 at the end of ¬ve years. Thus, GroupWest should be indifferent to

the choice between $100 today and $127.63 at the end of ¬ve years. Today™s

$100 is de¬ned as the present value, or PV, of $127.63 due in ¬ve years when

the opportunity cost rate is 5 percent. If the price of the security being offered

is exactly $100, GroupWest could buy it or turn it down because that is the

security™s “fair value.” If the price is less than $100, GroupWest should buy it,

while if the price is greater than $100, GroupWest should turn the offer down.

Conceptually, the present value of a cash ¬‚ow due N years in the future

is the amount which, if it were on hand today, would grow to equal the future

amount when compounded at the opportunity cost rate. Because $100 would

grow to $127.63 in ¬ve years at a 5 percent interest rate, $100 is the present

value of $127.63 due ¬ve years in the future when the opportunity cost rate is

5 percent. In effect, the present value tells us what amount would have to be

invested to earn the opportunity cost rate. If the investment can be obtained

for a lesser amount, a higher rate will be earned. If the investment costs

more than the present value, the rate earned will be less than the opportunity

cost rate.

Finding present values is called discounting, and it is simply the reverse

of compounding: if the PV is known, compound to ¬nd the FV; if the FV

is known, discount to ¬nd the PV. Here are the solution techniques used to

solve this discounting problem.

0 1 2 3 4 5

5%

? $127.63

To develop the discounting equation, solve the compounding equation for

PV:

FVN = PV — (1 + I)N

Compounding:

FVN

PV =

Discounting:

(1 + I)N

The equations show us that compounding problems are solved by multiplica-

tion, while discounting problems are solved by division.

Regular Enter $127.63 and divide it ¬ve times by 1.05:

Calculator

Solution 0 1 2 3 4 5

5%

$100 = 1.05 · 1.05 · 1.05 · 1.05 · 1.05 · $127.63

As shown by the arrows, discounting is moving left along a time line.

261

Chapter 9: Time Value Analysis

Financial

Inputs 5 5 127.63

Calculator

Solution

= ’100

Output

=127.63/(1.05)§ 5 Spreadsheet

Cell formula

Solution

Cell display 100.00

Discounting at Work

At relatively high interest rates, funds due in the future are worth very little

today, and even at moderate discount rates, the present value of a sum due

in the distant future is quite small. To illustrate discounting at work, consider

100-year bonds. A bond is a type of debt security in which an investor loans

some amount of principal”say, $1,000”to a company (borrower), which in

turn promises to pay interest over the life of the bond and to return the princi-

pal amount at maturity. Typically, the longest maturities for bonds are 30“40

years, but, in the early 1990s, several companies, including Columbia/HCA

Healthcare (now HCA), issued 100-year bonds.

At ¬rst blush, it might appear that anyone who would buy a 100-year

bond must be irrational because there is little assurance that the borrower

will even be around in 100 years to repay the amount borrowed. However,

consider the present value of $1,000 to be received in 100 years. If the

discount rate is 7.5 percent, which is roughly the interest rate that was set on

the bond, the present value is a mere $0.72. Thus, the time value of money

eroded the value of the bond™s principal repayment to the point that it was

worth less than $1 at the time the bond was issued. This tells us that the

value of the bond when it was sold was based primarily on the interest stream

received in the early years of ownership, and that the payments expected

during the later years contributed little to the bond™s initial $1,000 value.

Self-Test

1. What is discounting? How is it related to compounding?

Questions

2. What are the three techniques for solving lump sum discounting

problems?

3. How does the present value of a lump sum to be received in the future

change as the time is extended and as the interest rate increases?

4. What is meant by discounting at work?

Opportunity Costs

In the last section, the opportunity cost concept was used to set the discount

rate on GroupWest™s investment offer. This concept plays a very important

262 Healthcare Finance

role in time value analysis. To illustrate the concept, suppose an individual

found the winning ticket for the Florida lottery and now has $1 million to

invest. Should the individual assign a cost to these funds? At ¬rst blush, it

might appear that this money has zero cost because its acquisition was purely

a matter of luck. However, as soon as the lucky individual thinks about what

to do with the $1 million, he or she has to think in terms of the opportunity

costs involved. By using the funds to invest in one alternative, for example,

in the stock of HCA, the individual forgoes the opportunity to make some

other investment, for example, buying U.S. Treasury bonds. Thus, there is an

opportunity cost associated with any investment planned for the $1 million

even though the lottery winnings were “free.”

Because one investment decision automatically negates all other pos-

sible investments with the same funds, the cash ¬‚ows expected to be earned

from any investment must be discounted at a rate that re¬‚ects the return that

could be earned on forgone investment opportunities. The problem is that

the number of forgone investment opportunities is virtually in¬nite, so which

one should be chosen to establish the opportunity cost rate? The opportunity

cost rate to be applied in time value analysis is the rate that could be earned

on alternative investments of similar risk. It would not be logical to assign a

very low opportunity cost rate to a series of very risky cash ¬‚ows, or vice versa.

This concept is one of the cornerstones of healthcare ¬nance, so it is worth

repeating. The opportunity cost rate (i.e., the discount rate) applied to

investment cash ¬‚ows is the rate that could be earned on alternative in-

vestments of similar risk.

It is very important to recognize that the discounting process itself

accounts for the opportunity cost of capital (i.e., the loss of use of the capital

for other purposes). In effect, discounting a potential investment at, say, 10

percent, produces a present value that provides a 10 percent return. Thus,

if the investment can be obtained for less than its present value, it will earn

more than its opportunity cost of capital and hence is a good investment.

Alternatively, if the cost of the investment is greater than its present value, it

will earn less than its opportunity cost of capital and hence is a bad investment.

It is also important to note that the opportunity cost rate does not depend on

the source of the funds to be invested. Rather, the primary determinant of

this rate is the riskiness of the cash ¬‚ows being discounted. Thus, the same

10 percent opportunity cost rate would be applied to this potential investment

regardless of whether the funds to be used for the investment were won in a

lottery, taken out of petty cash, or obtained by selling some securities.

Generally, opportunity cost rates are obtained by looking at rates that

could be earned, or more precisely, rates that are expected to be earned,

on securities such as stocks or bonds. Securities are usually chosen to set

opportunity cost rates because their expected returns are more easily estimated

than rates of return on real assets such as hospital beds, MRI machines, and

the like. Furthermore, as discussed in Chapter 12, securities generally provide

263

Chapter 9: Time Value Analysis

the minimum return appropriate for the amount of risk assumed, so securities

returns provide a good benchmark for other investments.

To illustrate the opportunity cost concept, assume that Oakdale Com-

munity Hospital is considering building a nursing home. The ¬rst step in the

¬nancial analysis is to forecast the cash ¬‚ows that the nursing home is expected

to produce. These cash ¬‚ows, then, must be discounted at some opportunity

cost rate to determine their present value. Would the hospital™s opportunity

cost rate be (1) the expected rate of return on a bank CD; (2) the expected rate

of return on the stock of Beverly Enterprises, which operates a large number

of nursing homes and assisted living centers; or (3) the expected rate of return

on pork belly futures? (Pork belly futures are investments that involve com-

modity contracts for delivery at some future time.) The answer is the expected

rate of return on Beverly Enterprises™s stock because that is the rate of return

available to the hospital on alternative investments of similar risk. Bank

CDs are very low-risk investments, so they would understate the opportunity

cost rate in owning a nursing home. Conversely, pork belly futures are very

high-risk investments, so that rate of return is probably too high to apply to

Oakdale™s nursing home investment.5

The source of the funds used for the nursing home investment is not

relevant to the analysis. Oakdale may obtain the needed funds by borrowing,

by soliciting contributions, or by using excess cash accumulated from pro¬t

retention. The discount rate applied to the nursing home cash ¬‚ows depends

only on the riskiness of those cash ¬‚ows and the returns available on alternative

investments of similar risk, not on the source of the investment funds.

At this point, you may question the ability of real-world analysts to

assess the riskiness of a cash ¬‚ow stream or to choose an opportunity cost rate

with any con¬dence. Fortunately, the process is not as dif¬cult as it may appear

here because businesses have benchmarks that can be used as starting points.

(Chapter 13 contains a discussion of how benchmark opportunity cost rates

are established for capital investments, while Chapter 15 presents a detailed

discussion on how the riskiness of a cash ¬‚ow stream can be assessed.)

Self-Test

1. Why does an investment have an opportunity cost rate even when the

Questions

funds employed have no explicit cost?

2. How are opportunity cost rates established?

3. Does the opportunity cost rate depend on the source of the investment

funds?

Solving for Interest Rate and Time

At this point, it should be obvious that compounding and discounting are

reciprocal processes. Furthermore, four time value analysis variables have been

presented: PV, FV, I, and N. If the values of three of the variables are known,

264 Healthcare Finance

the value of the fourth can be found with the help of a ¬nancial calculator

or spreadsheet. Thus far, the interest rate, I, and the number of years, N,

plus either PV or FV have been given in the illustrations. In some situations,

however, the analysis may require solving for either I or N.6

Solving for Interest Rate (I)

Suppose that Family Practice Associates (FPA), a primary care group practice,

can buy a bank CD for $78.35 that will return $100 after ¬ve years. In this

case PV, FV, and N are known, but I, the interest rate that the bank is paying,

is not known.

0 1 2 3 4 5

?

’$78.35 $100

FVN = PV — (1 + I)N

$100 = $78.35 — (1 + I)5

Financial ’78.35

Inputs 5 100

Calculator

Solution

= 5.0

Output

Spreadsheet = RATE(N, 0, PV, FV)

Function

Solution

= RATE(5, 0, ’78.35, 100)

Cell formula

Cell display 5%

In this case, a spreadsheet function named RATE is used to solve for I. Note

that some spreadsheet programs display the answer in decimal form, unless

the cell is formatted to display in percent.

Solving for Time (N)

Suppose that the bank told FPA that a certi¬cate of deposit pays 5 percent

interest each year, that it costs $78.35, and that at maturity the group would

receive $100. How long must the funds be invested in the CD? In this case,

PV, FV, and I are known, but N, the number of periods, is not known.

0 1 2 N’1 N

5%

.. .

’$78.35 $100

265

Chapter 9: Time Value Analysis

FVN = PV — (1 + I)N

$100 = $78.35 — (1.05)N

’78.35

Inputs 5 100 Financial

Calculator

Solution

= 5.0

Output

Spreadsheet

= NPER (I, 0, PV, FV)

Function

Solution

= NPER (0.05, 0, ’78.35, 100)

Cell formula

Cell display 5.00

Note in this example that the interest rate is entered as a decimal in function

arguments.

Self-Test

1. What are a few real-world situations that may require you to solve for

Questions

interest rate or time?

2. Can ¬nancial calculators and spreadsheets easily solve for interest rate or

time?

Annuities

Whereas lump sums are single values, an annuity is a series of equal payments

at ¬xed intervals for a speci¬ed number of periods. Annuity payments, which

are given the symbol PMT, can occur at the beginning or end of each period. If

the payments occur at the end of each period as they typically do, the annuity

is an ordinary, or deferred, annuity. If payments are made at the beginning of

each period, the annuity is an annuity due. Because ordinary annuities are far

more common in time value problems, when the term annuity is used in this

book (or in general), payments are assumed to occur at the end of each period.

Furthermore, we begin our discussion of annuities by focusing on ordinary

annuities.

Ordinary Annuities

If Meridian Clinics were to deposit $100 at the end of each year for three

years in an account that paid 5 percent interest per year, how much would

Meridian accumulate at the end of three years? The answer to this question

is the future value of the annuity, which for ordinary annuities coincides with

the ¬nal payment.

266 Healthcare Finance

Regular One approach to the problem is to compound each individual cash ¬‚ow to

Calculator Year 3.

0 1 2 3

Solution 5%

$100 $100 $100

105

110.25

$315.25

Financial

’100

Inputs 3 5

Calculator

Solution

= 315.25

Output

In annuity problems, the PMT key is used in conjunction with either the PV

or FV key.

Spreadsheet = FV(I, N, PMT)

Function

Solution

= FV(0.05, 3, ’100)

Cell formula

Cell display $315.25

Suppose that Meridian Clinics was offered the following alternatives: a three-

year annuity with payments of $100 at the end of each year or a lump sum

payment today. Meridian has no need for the money during the next three

years. If it accepts the annuity, it would deposit the payments in an account

that pays 5 percent interest per year. Similarly, the lump sum payment would

be deposited into the same account. How large must the lump sum payment

be today to make it equivalent to the annuity? The answer to this question

is the present value of the annuity, which for ordinary annuities occurs one

period prior to the ¬rst payment.

0 1 2 3

Regular 5%

Calculator $100 $100 $100

Solution $ 95.24

90.70

86.38

$ 272.32

Financial

’100

Inputs 3 5

Calculator

Solution

= 272.32

Output

267

Chapter 9: Time Value Analysis

Spreadsheet

= PV(I, N, PMT)

Function

Solution

= PV(0.05, 3, ’100)

Cell formula

Cell display $272.32

One especially important application of the annuity concept relates to

loans with constant payments, such as mortgages, auto loans, and many bank

loans to businesses. Such loans are examined in more depth in a later section

on amortization.

Annuities Due

If the three $100 payments in the previous example had been made at the

beginning of each year, the annuity would have been an annuity due. The

future value of an annuity due occurs one period after the ¬nal payment, while

the future value of a regular annuity coincides with the ¬nal payment.

0 1 2 3

Regular

5%

Calculator

$100 $100 $100

Solution

$ 105

110.25

115.76

$ 331.01

In the case of an annuity due, as compared with an ordinary annuity, all the

cash ¬‚ows are compounded for one additional period, and hence the future

value of an annuity due is greater than the future value of a similar ordinary

annuity by (1 + I). Thus, the future value of an annuity due also can be found

as follows:

FV (Annuity due) = FV of a regular annuity — (1 + I)

= $315.25 — 1.05 = $331.01.

Most ¬nancial calculators have a switch or key marked DUE or BEGIN that Financial

permits the switching of the mode from end-of-period payments (ordinary an- Calculator

nuity) to beginning-of-period payments (annuity due). When the beginning- Solution

of-period mode is activated, the calculator will normally indicate the changed

mode by displaying the word BEGIN or some other symbol. To deal with an-

nuities due, change the mode to beginning of period and proceed as before.

Because most problems will deal with end-of-period cash ¬‚ows, do not forget

to switch the calculator back to the END mode.

Spreadsheet

= FV(I, N, PMT) — (1 + I)

Function

Solution

= FV(0.05, 3, ’100) — (1.05)

Cell formula

Cell display $331.01

268 Healthcare Finance

The present value of an annuity due is found in a similar manner.

0 1 2 3

Regular 5%

Calculator

$100 $100 $100

Solution

95.24

90.70

$ 285.94

Because the payments are shifted to the left, each one is discounted for one

less year. Thus, the present value of an annuity due is larger than that of a

similar regular annuity.

Note that the present value of an annuity due can be thought of as the

present value of an ordinary annuity that is compounded for one additional

period, so it also can be found as follows:

PV(Annuity due) = PV of a regular annuity — (1 + I)

= $272.32 — 1.05 = $285.94

Financial Activate the beginning of period mode (i.e., the BEGIN mode), and then

Calculator proceed as before. Again, because most problems will deal with end-of-period

Solution cash ¬‚ows, do not forget to switch the calculator back to the END mode.

Spreadsheet = PV(I, N, PMT) — (1 + I)

Function

Solution

= PV(0.05, 3, ’100) — (1.05)

Cell formula

Cell display $285.94

Self-Test 1. What is an annuity?

Questions 2. What is the difference between an ordinary annuity and an annuity due?

3. Which annuity has the greater future value: an ordinary annuity or an

annuity due? Why?

4. Which annuity has the greater present value: an ordinary annuity or an

annuity due? Why?

Perpetuities

Most annuities call for payments to be made over some ¬nite period of time”

for example, $100 per year for three years. However, some annuities go on

inde¬nitely, or perpetually, and hence are called perpetuities. The present value

of a perpetuity is found as follows:

Payment PMT

PV (Perpetuity) = = .

Interest rate I

269

Chapter 9: Time Value Analysis

Perpetuities can be illustrated by some securities issued by the Canadian

Healthcare Board. Each security promises to pay $100 annually in perpetuity

(forever). What would each security be worth if the opportunity cost rate, or

discount rate, is 10 percent? The answer is $1,000:

$100

PV (Perpetuity) = = $1,000.

0.10

Suppose interest rates, and hence the opportunity cost rate, rose to 15

percent. What would happen to the security™s value? The interest rate increase

would lower its value to $666.67:

$100

PV (Perpetuity) = = $666.67.

0.15

Assume that interest rates fell to 5 percent. The rate decrease would increase

the perpetuity™s value to $2,000:

$100

PV (Perpetuity) = = $2,000.

0.05

The value of a perpetuity changes dramatically when opportunity costs

(interest rates) change. All securities™ values are affected by interest rate

changes, but some, like perpetuities, are more sensitive to interest rate changes

than others, such as short-term government bonds. The risks associated with

interest rate changes are discussed in more detail in Chapter 11.

Self-Test

1. What is a perpetuity?

Questions

2. What happens to the value of a perpetuity when interest rates increase

or decrease?

Uneven Cash Flow Streams

The de¬nition of an annuity (or perpetuity) includes the words “constant

amount,” so annuities involve payments that are the same in every period. Al-

though some ¬nancial decisions, such as bond valuation, do involve constant

payments, most important healthcare time value analyses involve uneven, or

nonconstant, cash ¬‚ows. For example, the ¬nancial evaluation of a proposed

outpatient clinic or MRI facility rarely involves constant cash ¬‚ows.

In general, the term lump sum is used with a single cash ¬‚ow; the term

payment (PMT) is reserved for annuity situations in which there are multiple

constant lump sums; and the term cash ¬‚ow (CF) is used when there is a

series of uneven lump sums. Financial calculators are set up to follow this

convention. When dealing with uneven cash ¬‚ows, the CF function, rather

than the PMT key, is used.

270 Healthcare Finance

Present Value

The present value of an uneven cash ¬‚ow stream is found as the sum of

the present values of the individual cash ¬‚ows of the stream. For example,

suppose that Wilson Memorial Hospital is considering the purchase of a new

x-ray machine. The hospital™s managers forecast that the operation of the new

machine would produce the following stream of cash in¬‚ows (in thousands of

dollars):

0 1 2 3 4 5

$100 $120 $150 $180 $250

What is the present value of the new x-ray machine investment if the appro-

priate discount rate (i.e., the opportunity cost rate) is 10 percent?

Regular The PV of each lump sum cash ¬‚ow can be found using a regular calculator,

Calculator and then these values are summed to ¬nd the present value of the stream,

Solution $580,950:

Financial The present value of an uneven cash ¬‚ow stream can be solved with most

Calculator ¬nancial calculators by using the following steps:

Solution

• Input the individual cash ¬‚ows, in chronological order, into the cash

¬‚ow register, where they usually are designated as CF0 and CFj (CF1,

CF2, CF3, and so on) or just CFj (CF0, CF1, CF2, CF3, and so on).

• Enter the discount rate.

• Push the NPV key.

For this problem, enter 0, 100, 120, 150, 180, and 250 in that order into

the calculator™s cash ¬‚ow register; enter I = 10; then push NPV to obtain the

answer, 580.95. Note that an implied cash ¬‚ow of zero is entered for CF0.

Three points should be noted about the calculator solution. First, when

dealing with the cash ¬‚ow register, the term NPV, rather than PV, is used to

represent present value. The letter N in NPV stands for the word net, so NPV

271

Chapter 9: Time Value Analysis

is the abbreviation for net present value. Net present value means the sum or

net of the present values of a cash ¬‚ow stream. Often, the stream will consist

of both in¬‚ows and out¬‚ows, but the stream here contains all in¬‚ows.

Second, annuity cash ¬‚ows within any uneven cash ¬‚ow stream can be

entered into the cash ¬‚ow register most ef¬ciently on most calculators by using

the Nj key. This key allows the user to specify the number of times a constant

payment occurs within the stream. (Some calculators prompt the user to enter

the number of times each cash ¬‚ow occurs.)

Finally, amounts entered into the cash ¬‚ow register remain there until

the register is cleared. Thus, if a problem had been previously worked with

eight cash ¬‚ows, and a problem is worked with only four cash ¬‚ows, the

calculator assumes that the ¬nal four cash ¬‚ows from the ¬rst calculation

belong to the second calculation. Be sure to clear the register before starting

a new time value analysis.

The NPV function calculates the present value of a stream, called a spread- Spreadsheet

sheet range, of cash ¬‚ows. First, the cash ¬‚ow values must be entered into Solution

consecutive cells in the spreadsheet. For example:

Cell Address: A10 B10 C10 D10 E10

Value: 100 120 150 180 250

The NPV function then is placed in an empty cell, for example, A5:

= NPV(I, range)

Function

= NPV(0.10, A10 : E10)

Cell formula

Cell display $580.95

The NPV function assumes that cash ¬‚ows occur at the end of each

period, so NPV is calculated as of the beginning of the period of the ¬rst cash

¬‚ow speci¬ed in the range, which is one period before that cash ¬‚ow occurs.

Because the cash ¬‚ow speci¬ed as the ¬rst ¬‚ow in the range is a Year 1 value,

the calculated NPV occurs at the beginning of Year 1, or the end of Year 0,

which is correct for this illustration. However, if a Year 0 cash ¬‚ow is included

in the range, the NPV would be calculated at the beginning of Year 0, or the

end of Year -1, which typically is incorrect. This problem will be addressed in

the next major section.

Future Value

The future value of an uneven cash ¬‚ow stream is found by compounding

each payment to the end of the stream and then summing the future values.

The future value of each lump sum cash ¬‚ow can be found, using a regular Regular

calculator, by summing these values to ¬nd the future value of the stream, Calculator

Solution

$935,630:

272 Healthcare Finance

Financial Some ¬nancial calculators have a net future value key (NFV) that, after the

Calculator cash ¬‚ows have been entered into the cash ¬‚ow register, can be used to obtain

Solution the future value of an uneven cash ¬‚ow stream. However, analysts generally

are more concerned with the present value of a cash ¬‚ow stream than with

its future value. The reason, of course, is that the present value represents the

value of the investment today, which then can be compared to the cost of the

investment”whether a stock, bond, x-ray machine, or new clinic”to make

the investment decision.

Spreadsheet Most spreadsheet programs do not have a function that computes the future

Solution value of an uneven cash ¬‚ow stream. However, future values can be found by

building a formula in a cell that replicates the regular calculator solution.

Self-Test 1. Give two examples of ¬nancial decisions that typically involve uneven

Questions cash ¬‚ows.

2. Describe how present values of uneven cash ¬‚ow streams are calculated

using a regular calculator, using a ¬nancial calculator, and using a

spreadsheet.

3. What is meant by net present value?

Using Time Value Analysis to Measure Financial Returns

In most investments, an individual or a business spends cash today with the

expectation of receiving cash in the future. The ¬nancial attractiveness of such

investments is measured by ¬nancial return, or just return. There are two

basic ways of expressing ¬nancial return: in dollar terms and in percentage

terms.

To illustrate the concept, let™s reexamine the cash ¬‚ows expected to

be received if Wilson Memorial Hospital buys its new x-ray machine (shown

on the time line in thousands of dollars). In the last section, we determined

that the present value of these ¬‚ows, when discounted at a 10 percent rate, is

$580,950:

273

Chapter 9: Time Value Analysis

Dollar Return

The $580,950 calculated above represents the present value of the cash ¬‚ows

that the x-ray machine is expected to provide to Wilson Memorial Hospital,

assuming a 10 percent discount rate (opportunity cost of capital). This result

tells us that a 10 percent return on a $580,950 investment would produce a

cash ¬‚ow stream that is identical to one being discounted.

To measure the dollar return on the investment, the cost of the x-ray

machine must be compared to the present value of the expected bene¬ts (the

cash in¬‚ows). If the machine will cost $500,000, and the present value of

the in¬‚ows is $580,950, then the expected dollar return on the machine is

$580,950 ’ $500,000 = $80,950. Note that this measure of dollar return

incorporates time value, and hence opportunity costs, through the discount-

ing process. The opportunity cost inherent in the use of the $500,000 is ac-

counted for because the 10 percent discount rate re¬‚ects the return that could

be earned on alternative investments of similar risk. By virtue of the $80,950

excess, the x-ray machine has an expected present value that is $80,950 more

than would occur if it had only a 10 percent return, which is the opportunity

cost rate. Thus, the x-ray machine makes sense ¬nancially because it creates

an excess dollar return for the hospital.

The dollar return process can be combined into a single calculation by

adding the cost of the x-ray machine to the time line:

274 Healthcare Finance

Financial Now, with the investment outlay (cost) added to the time line, the following

Calculator cash ¬‚ows would be entered into the cash ¬‚ow register: ’500, 100, 120, 150,

Solution 180, and 250 in that order. Then, enter I = 10 and push NPV to obtain the

answer, 80.95.

Spreadsheet As in the ¬nancial calculator solution, the cost of the machine must be added

Solution to the cash ¬‚ow data. Here, it is added to the spreadsheet range:

Cell Address: A10 B10 C10 D10 E10 F10

’500 100 120 150 180 250.

Value:

The NPV function then is placed in an empty cell, for example, A5:

= NPV(I, range)

Function

= NPV(0.10, A10 : F10)

Cell formula

Cell display $73.59

Oops! We have a problem. As discussed previously, the NPV function

assumes that cash ¬‚ows occur at the end of each period. Thus, NPV is calcu-

lated as of the beginning of the period of the ¬rst cash ¬‚ow speci¬ed in the

range, so the NPV incorrectly occurs at the beginning of Year 0, or the end

of Year -1. One solution to the problem is to compound the calculated NPV

one period at 10 percent. The effect is to move the NPV one year to the right

along the time line. The spreadsheet cell would look like this:

= NPV(I, range including CF0 ) — (1 + I)

Function

= NPV(0.10, A10 : F10) — 1.10

Cell formula

Cell display $80.95

A second solution is to change the range in the argument to force the

¬rst payment in the range to occur at Year 1, so the present value will be

calculated at Year 0. However, because there is a Year 0 cash ¬‚ow that must

be included in the calculation, the Year 0 cash ¬‚ow must be added to the

spreadsheet-calculated NPV. This approach would look like this:

= NPV(I, range without CF0 ) + Year 0 Cell

Function

= NPV(0.10, B10 : F10) + A10

Cell formula

Cell display $80.95

Rate of Return

The second way to measure the ¬nancial return on an investment is by rate

of return, or percentage return. This measures the interest rate that must be

earned on the investment outlay to generate the expected cash in¬‚ows. In

other words, this measure provides the expected periodic rate of return on

275

Chapter 9: Time Value Analysis

the investment. If the cash ¬‚ows are annual, as in this example, the rate of

return is an annual rate. In effect, we are solving for I”the interest rate that

equates the sum of the present values of the cash in¬‚ows to the dollar amount

of the cash outlay.

Mathematically, if the sum of the present values of the cash in¬‚ows

equals the investment outlay, then the NPV of the investment is forced to $0.

This relationship is shown here:

Note that the rate of return on an investment, particularly an invest-

ment in plant or equipment, typically is called the internal rate of return

(IRR). Although a trial-and-error procedure could be used on a regular cal-

culator to determine the rate of return, it is better to use a ¬nancial calculator

or spreadsheet.

Use the same cash ¬‚ows that were entered to solve for NPV: ’500, 100, Financial

120, 150, 180, and 250. However, now push the IRR button to obtain the Calculator

Solution

answer”15.3 percent.

Spreadsheet

Use the same spreadsheet format as earlier:

Solution

Cell Address: A10 B10 C10 D10 E10 F10

Value: 500 100 120 150 180 250.

But now, place the IRR function in an empty cell”for example, A6:

= IRR (range, starting guess)

Function

= IRR (A10 : F10, 0.10)

Cell formula

Cell display 15.3

A starting guess is required to calculate the IRR because the methodology

used by the spreadsheet IRR function is actually a trial-and-error process that

requires a starting point.

The IRR of 15.3 percent tells the hospital™s managers that the expected

rate of return on the x-ray machine exceeds the opportunity cost rate by 15.3

’ 10.0 = 5.3 percentage points. Thus, the expected rate of return is higher

276 Healthcare Finance

than that available on alternative investments of similar risk (the required rate

of return), and hence the x-ray machine makes ¬nancial sense. Note that both

the dollar (NPV) return and the percentage (IRR) return indicate that the

x-ray machine should be acquired. In general, the two methods lead to the

same conclusion regarding the ¬nancial attractiveness of an investment.

We will have much more to say about ¬nancial returns in Chapters 11,

12, and 14. For now, an understanding of the basic concept is suf¬cient.

Self-Test 1. Differentiate between dollar return and rate of return.

Questions 2. Is the calculation of ¬nancial return an application of time value analysis?

Explain your answer.

3. What role does the opportunity cost rate play in calculating ¬nancial

returns?

Semiannual and Other Compounding Periods

In all the examples thus far, we assumed that interest is earned (compounded)

once a year, or annually. This is called annual compounding. Suppose, how-

ever, that Meridian Clinics puts $100 into a bank account that pays 6 percent

annual interest, but it is compounded semiannually. How much would the

clinic accumulate at the end of one year, two years, or some other period?

Semiannual compounding means that interest is paid each six months, so in-

terest is earned more often than under annual compounding.

The Effect of Semiannual Compounding

To illustrate semiannual compounding, assume that the $100 is placed into

the account for three years. The following situation occurs under annual

compounding:

0 1 2 3

6%

’$100 ?

FVN = PV — (1 + I)N = $100 — (1.06)3

Regular 0 1 2 3

6%

Calculator

Solution $100 1.06 1.06 1.06 = $119.10

277

Chapter 9: Time Value Analysis

’100

Inputs 3 6 Financial

Calculator

Solution

= 119.10

Output

= 100 — (1.06)§ 3 Spreadsheet

Cell formula

Solution

Cell display 119.10

Now, consider what happens under semiannual compounding. Be-

cause interest rates usually are stated as annual rates, this situation would be

described as 6 percent interest, compounded semiannually. With semiannual

compounding, N = 2 — 3 = 6 semiannual periods, and I = 6 / 2 = 3% per

semiannual period. Here is the solution.

Years 0 1 2 3

Semiannual periods 0 1 2 3 4 5 6

3%

’$100 ?

FVN = PV — (1 + I)N = $100 — (1.03)6

Regular

Calculator

Solution

’100

Inputs 6 3 Financial

Calculator

Solution

= 119.41

Output

= 100 — (1.03)§ 6 Spreadsheet

Cell formula

Solution

Cell display 119.41

The $100 deposit grows to $119.41 under semiannual compounding, but

grows only to $119.10 under annual compounding. This result occurs be-

cause interest on interest is being earned more frequently under semiannual

compounding.

278 Healthcare Finance

Stated Versus Effective Interest Rates

Throughout the economy, different compounding periods are used for differ-

ent types of investments. For example, bank accounts often compound interest

monthly or daily, most bonds pay interest semiannually, and stocks generally

pay quarterly dividends.7 Furthermore, the cash ¬‚ows that stem from capital

investments such as hospital wings or diagnostic equipment can be analyzed in

monthly, quarterly, or annual periods or even some other interval. To properly

compare time value analyses with different compounding periods, they need

to be put on a common basis, which leads to a discussion of stated interest

rates versus effective annual rates.

The stated interest rate in the Meridian Clinics™s semiannual com-

pounding example is 6 percent. The effective annual rate is the rate that pro-

duces the same ending (i.e., future) value under annual compounding. In the

example, the effective annual rate is the rate that would produce a future value

of $119.41 at the end of Year 3 under annual compounding. The solution

is 6.09 percent:

’100

Inputs 3 119.41

= 6.09

Output

Thus, if one bank offered to pay 6 percent interest with semiannual com-

pounding on a savings account, while another offered 6.09 percent with an-

nual compounding, they both would be paying the same effective annual rate

because the ending value is the same under both sets of terms:

In general, the effective annual rate (EAR) can be determined, given

the stated rate and number of compounding periods per year, by using this

equation:8

Effective annual rate (EAR) = (1 + IStated /M)M ’ 1.0,

where IStated is the stated (i.e., the annual) interest rate and M is the number

279

Chapter 9: Time Value Analysis

of compounding periods per year. The term IStated / M is the periodic interest

rate, so the EAR equation can be recast as:

Effective annual rate (EAR) = (1 + Periodic rate)M ’ 1.0.

To illustrate use of the EAR equation, the effective annual rate when the

stated rate is 6 percent and semiannual compounding occurs is 6.09 percent:

EAR = (1 + 0.06/2)2 ’ 1.0

= (1.03)2 ’ 1.0

= 1.0609 ’ 1.0 = 0.0609 = 6.09%,

which con¬rms the answer that we obtained previously.

As shown in the preceding calculations, semiannual compounding, or

for that matter any compounding that occurs more than once a year, can be

handled two ways. First, the input variables can be expressed as periodic vari-

ables rather than annual variables. In the Meridian Clinics example, use N = 6

periods rather than N = 3 years, and I = 3% per period rather than I = 6% per

year. Second, ¬nd the effective annual rate and then use this rate as an annual

rate over the number of years. In the example, use I = 6.09% and N = 3 years.

For another illustration, consider the interest rate charged on credit

cards. Many banks charge 1.5 percent per month and, in their advertising,

state that the annual percentage rate (APR) is 18.0 percent.9 However, the

true cost rate to credit card users is the effective annual rate of 19.6 percent:

EAR = (1 + Periodic rate)M ’ 1.0

= (1.015)12 ’ 1.0 = 0.196 = 19.6%.

Self-Test

1. What changes must be made in the calculations to determine the future

Questions

value of an amount being compounded at 8 percent semiannually

versus one being compounded annually at 8 percent?

2. Why is semiannual compounding better than annual compounding

from an investor™s standpoint?

3. How does the effective annual rate differ from the stated rate?

4. How does the periodic rate differ from the stated rate?

Amortized Loans

One important application of time value analysis involves loans that are to be

paid off in equal installments over time, including automobile loans, home

mortgage loans, and most business debt other than very short-term loans and

bonds. If a loan is to be repaid in equal periodic amounts”monthly, quarterly,

280 Healthcare Finance

or annually” it is said to be an amortized loan. The word amortize comes

from the Latin mors, meaning death, so an amortized loan is one that is killed

off over time.

To illustrate, suppose Santa Fe Healthcare System borrows $1 million

from the Bank of New Mexico, to be repaid in three equal installments at the

end of each of the next three years. The bank is to receive 6 percent interest

on the loan balance that is outstanding at the beginning of each year. The ¬rst

task in analyzing the loan is to determine the amount Santa Fe must repay

each year, or the annual payment. To ¬nd this value, recognize that the loan

amount represents the present value of an annuity of PMT dollars per year for

three years, discounted at 6 percent.

0 1 2 3

6%

$1,000,000 PMT PMT PMT

Inputs 3 6 1000000

Financial

Calculator

Solution

= ’374,100

Output

Spreadsheet = PMT(I, N, PV)

Function

Solution

= PMT(0.06, 3, 1000000)

Cell formula

’ $374,110

Cell display

Therefore, if Santa Fe pays the bank $374,110 at the end of each of the next

three years, the percentage cost to the borrower, and the rate of return to the

lender, will be 6 percent.

Each payment made by Santa Fe consists partly of interest and partly of

repayment of principal. This breakdown is given in the amortization schedule

TABLE 9.1

Beginning Repayment Remaining

Loan a

of Principalb

Amount Payment Interest Balance

Amortization Year (1) (2) (3) (4) (5)

Schedule

1 $1,000,000 $ 374,110 $ 60,000 $ 314,110 $685,890

2 685,890 374,110 41,153 332,957 352,933

3 352,933 374,110 21,177 352,933 0

$1,122,330 $122,330 $1,000,000

a

Interest is calculated by multiplying the loan balance at the beginning of each year by the interest rate. Therefore,

interest in Year 1 is $1,000,000 — 0.06 = $60,000; in Year 2 it is $685,890 — 0.06 = $41,153; and in Year 3 it is

$352,933 — 0.06 = $21,177.

b

Repayment of principal is equal to the payment of $374,110 minus the interest charge for each year.

281

Chapter 9: Time Value Analysis

shown in Table 9.1. The interest component is largest in the ¬rst year, and

it declines as the outstanding balance of the loan is reduced over time. For

tax purposes, a taxable business borrower reports the interest payments in

Column 3 as a deductible cost each year, while the lender reports these same

amounts as taxable income.

Financial calculators are often programmed to calculate amortization

schedules; simply key in the inputs, and then press one button to get each

entry in Table 9.1.

Self-Test

1. When constructing an amortization schedule, how is the periodic

Questions

payment amount calculated?

2. Does the periodic payment remain constant over time?

3. Do the principal and interest components remain constant over time?

Explain your answer.

Key Concepts

Financial decisions often involve situations in which future cash ¬‚ows must

be valued. The process of valuing future cash ¬‚ows is called time value

analysis. The key concepts of this chapter are:

• Compounding is the process of determining the future value (FV) of a

lump sum or a series of payments.

• Discounting is the process of ¬nding the present value (PV) of a future

lump sum or series of payments.

• An annuity is a series of equal, periodic payments (PMT) for a speci¬ed

number of periods.

• An annuity that has payments that occur at the end of each period is called

an ordinary annuity.

• If each annuity payment occurs at the beginning of the period rather than

at the end, the annuity is an annuity due.

• A perpetuity is an annuity that lasts forever.

• If an analysis that involves more than one lump sum does not meet the

de¬nition of an annuity, it is called an uneven cash ¬‚ow stream.

• The ¬nancial consequence of an investment is measured by return, which

can be expressed either in dollar terms or in percentage (rate of return)

terms.

• An amortized loan is one that is paid off in equal amounts over some

speci¬ed number of periods. An amortization schedule shows how much

of each payment represents interest, how much is used to reduce the

principal, and how much of the principal balance remains on each

payment date.

• The stated rate is the annual rate normally quoted in ¬nancial contracts.

282 Healthcare Finance

• The periodic rate equals the stated rate divided by the number of

compounding periods per year.

• If compounding occurs more frequently than once a year, it is often

necessary to calculate the effective annual rate, which is the rate that

produces the same results under annual compounding as obtained with

more frequent compounding.

Time value analysis will be applied in subsequent chapters, so the contents

of this chapter are very important. Readers should feel comfortable with this

material before moving ahead.

Questions

9.1 a. What is an opportunity cost rate?

b. How is this rate used in time value analysis?

c. Is this rate a single number that is used in all situations?

9.2 What is the difference between a lump sum, an annuity, and an unequal

cash ¬‚ow stream?

9.3 Great Lakes Health Network™s net income increased from $3.2 million

in 1994 to $6.4 million in 2004. The total growth rate over the ten years

is 100 percent, while the annual growth rate is only about 7.2 percent,

which is much less than 100 percent divided by ten years.

a. Why does this relationship hold?

b. Which growth rate has more meaning”the total rate over ten years

or the annualized rate?

9.4 Would you rather have a savings account that pays 5 percent compounded

semiannually or one that pays 5 percent compounded daily? Explain

your answer.

9.5 The present value of a perpetuity is equal to the payment divided by the

opportunity cost (interest) rate: PV = PMT/I. What is the future value

of a perpetuity?

9.6 When a loan is amortized, what happens over time to the size of the

total payment, interest payment, and principal payment?

9.7 Explain the difference between the stated rate, periodic rate, and

effective annual rate.

9.8 What are three techniques for solving time value problems?

9.9 Explain the concept of investment return and the two different

approaches to measuring return.

Problems

9.1 Find the following values for a lump sum assuming annual

compounding:

a. The future value of $500 invested at 8 percent for one year

b. The future value of $500 invested at 8 percent for ¬ve years

283

Chapter 9: Time Value Analysis

c. The present value of $500 to be received in one year when the

opportunity cost rate is 8 percent

d. The present value of $500 to be received in ¬ve years when the

opportunity cost rate is 8 percent

9.2 Repeat Problem 9.1 above, but assume the following compounding

conditions:

a. Semiannual

b. Quarterly

9.3 What is the effective annual rate (EAR) if the stated rate is 8 percent

and compounding occurs semiannually? Quarterly?

9.4 Find the following values assuming a regular, or ordinary, annuity:

a. The present value of $400 per year for ten years at 10 percent

b. The future value of $400 per year for ten years at 10 percent

c. The present value of $200 per year for ¬ve years at 5 percent

d. The future value of $200 per year for ¬ve years at 5 percent

9.5 Repeat Problem 9.4, but assume the annuities are annuities due.

9.6 Consider the following uneven cash ¬‚ow stream:

Year Cash Flow

0 $0

1 250

2 400

3 500

4 600

5 600

a. What is the present (Year 0) value if the opportunity cost (discount)

rate is 10 percent?

b. Add an out¬‚ow (or cost) of $1,000 at Year 0. What is the present

value (or net present value) of the stream?

9.7 Consider another uneven cash ¬‚ow stream:

Year Cash Flow

0 $2,000

1 2,000

2 0

3 1,500

4 2,500

5 4,000

a. What is the present (Year 0) value of the cash ¬‚ow stream if the

opportunity cost rate is 10 percent?

b. What is the value of the cash ¬‚ow stream at the end of Year 5 if the

cash ¬‚ows are invested in an account that pays 10 percent annually?

c. What cash ¬‚ow today (Year 0), in lieu of the $2,000 cash ¬‚ow, would

be needed to accumulate $20,000 at the end of Year 5? (Assume

that the cash ¬‚ows for Years 1 through 5 remain the same.)

284 Healthcare Finance

d. Time value analysis involves either discounting or compounding

cash ¬‚ows. Many healthcare ¬nancial management decisions,

such as bond refunding, capital investment, and lease versus buy,

involve discounting projected future cash ¬‚ows. What factors must

executives consider when choosing a discount rate to apply to

forecasted cash ¬‚ows?

9.8 What is the present value of a perpetuity of $100 per year if the

appropriate discount rate is 7 percent? Suppose that interest rates

doubled in the economy and the appropriate discount rate is now 14

percent. What would happen to the present value of the perpetuity?

9.9 Assume that you just won $35 million in the Florida lottery, and

hence the state will pay you 20 annual payments of $1.75 million each

beginning immediately. If the rate of return on securities of similar risk

to the lottery earnings (e.g., the rate on 20-year U.S. Treasury bonds)

is 6 percent, what is the present value of your winnings?

9.10 An investment that you are considering promises to pay $2,000

semiannually for the next two years, beginning six months from now.

You have determined that the appropriate opportunity cost (discount)

rate is 8 percent, compounded quarterly. What is the value of this

investment?

9.11 Consider the following investment cash ¬‚ows:

Year Cash Flow

0 ($1,000)

1 250

2 400

3 500

4 600

5 600

a. What is the return expected on this investment measured in dollar

terms if the opportunity cost rate is 10 percent?

b. Provide an explanation, in economic terms, of your answer.

c. What is the return on this investment measured in percentage terms?

d. Should this investment be made? Explain your answer.

9.12 Epitome Healthcare has just borrowed $1,000,000 on a ¬ve-year,

annual payment term loan at a 15 percent rate. The ¬rst payment is due

one year from now. Construct the amortization schedule for this loan.

9.13 Assume that $10,000 was invested in the stock of General Medical

Corporation with the intention of selling after one year. The stock pays

no dividends, so the entire return will be based on the price of the stock

when sold. The opportunity cost of capital on the stock is 10 percent.

a. To begin, assume that the stock sale nets $11,500. What is the dollar

return on the stock investment? What is the rate of return?

285

Chapter 9: Time Value Analysis

b. Assume that the stock price falls and the net is only $9,500 when

the stock is sold. What is the dollar return and rate of return?

c. Assume that the stock is held for two years. Now, what is the dollar

return and rate of return?

Notes

1. Even if no investment opportunities existed, a dollar in hand would still be

worth more than a dollar to be received in the future because a dollar today can

be used for immediate consumption, whereas a future dollar cannot.

2. On some ¬nancial calculators, the keys are buttons on the face of the calculator;

on others, the time value variables are shown on the display after accessing the

time value menu. Also, some calculators use different symbols to represent the

number of periods and interest rate. Finally, ¬nancial calculators today are quite

powerful in that they can easily solve relatively complex time value of money

problems. To focus on concepts rather than mechanics, all the illustrations in

this chapter and the remainder of the book assume that cash ¬‚ows occur at the

end or beginning of a period and that there is only one cash ¬‚ow per period.

Thus, to follow the illustrations, ¬nancial calculators must be set to one period

per year, and it is not necessary to use the calendar function.

3. In constructing spreadsheets, it is most useful to construct a formula that can

accommodate changing input values. For this example, the formula might be

entered as

= A1 — (1 + B1)§ C1

where the present value ($100) would be contained in Cell A1, the interest rate

(0.05) in Cell B1, and the number of periods (5) in Cell C1. With this formula,

the future value over one, two, three, or more years can be calculated, as shown

in the example. Finally, different spreadsheet programs use slightly different

syntax in their time value functions. The examples presented in this book use

Excel syntax.

4. Time value problems also can be solved using mathematical multipliers obtained

from tables. At one time, tables were the most ef¬cient way to solve time value

problems, but calculators and spreadsheets have made tables obsolete.

5. Actually, owning a single nursing home is riskier than owning the stock of a ¬rm

that has a large number of nursing homes with geographical diversi¬cation. Also,

an owner of Beverly Enterprises™s stock can easily sell the stock if things go sour,

whereas it would be much more dif¬cult for Oakdale to sell its nursing home.

These differences in risk and liquidity suggest that the true opportunity cost

rate is probably higher than the return that is expected from owning the stock

of Beverly Enterprises. However, direct ownership of a nursing home implies

control, while ownership of the stock of a large ¬rm usually does not. Such

control rights would tend to reduce the opportunity cost rate. The main point

here is that in practice it may not be possible to obtain a “perfect” opportunity

cost rate. Nevertheless, an imprecise one is better than none at all.

286 Healthcare Finance

6. The Rule of 72 gives a simple and quick method for judging the effect of

different interest rates on the growth of a lump sum deposit. To ¬nd the number

of years required to double the value of a lump sum, merely divide the number

72 by the interest rate paid. For example, if the interest rate is 10 percent, it

would take 72 / 10 = 7.2 years for the money in an account to double in value.

The calculator solution is 7.27 years, so the Rule of 72 is relatively accurate, at

least when reasonable interest rates are applied. In a similar manner, the Rule of