<<

. 10
( 21)



>>




=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

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