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( 100 .)


(1 + r)25
What is the interest rate?
There are several ways to approach this. First, you might use a table of discount fac-
tors. You need to find the interest rate for which the 25-year discount factor = .129.
Look at Table A.2 at the end of the material and run your finger along the row corre-


if you borrow $100 today at an interest rate of 12 per- should get an answer of “993.77. The answer is dis-
cent, you receive money now (a positive cash flow), but played as a negative number because you need to
you will have to pay back $112 in a year, a negative make a cash outflow (an investment) of $993.77 now in
cash flow at that time. Therefore, the calculator displays order to enjoy a cash inflow of $10,000 in 30 years.
FV as a negative number. The following time line of cash
Finding the Interest Rate
flows shows the reasoning employed. The final negative
cash flow of $112 has the same present value as the
The 25-year Coca-Cola Enterprises IOU in Example 3.3
$100 borrowed today.
sold at $129 and promised a final payment of $1,000.
PV = $100 We may obtain the market interest rate by entering n =
25, FV = 1,000, PV = “129, and PMT = 0. Compute i and
Year: 0 1 you will find that the interest rate is 8.53 percent. This is
v the value we computed directly (but with more work) in
FV = $112 the example.
If, instead of borrowing, you were to invest $100
How Long an Investment?
today to reap a future benefit, you would enter PV as a
negative number (first press 100, then press the +/“ key In Example 3.5, we consider how long it would take for
to make the value negative, and finally press PV to enter an investment to double in value. This sort of problem
the value into the PV register). In this case, FV would is easily solved using a calculator. If the investment is to
appear as a positive number, indicating that you will double, we enter FV = 2 and PV = “1. If the interest rate
reap a cash inflow when your investment comes to is 9 percent, enter i = 9 and PMT = 0. Compute n and
fruition. you will find that n = 8.04 years. If the interest rate is
9.05 percent, the doubling period falls to 8 years, as we
Present Values found in the example.
Suppose your savings goal is to accumulate $10,000 by 1 The BAII Plus requires a little extra work to initialize the calculator.
the end of 30 years. If the interest rate is 8 percent, how When you buy the calculator, it is set to automatically interpret each
period as a year but to assume that interest compounds monthly. In
much would you need to invest today to achieve your
our experience, it is best to change the compounding frequency to
goal? Again, there is no recurring payment involved, so
once per period. To do so, press 2nd {P/Y} 1 ENTER , then press “
PMT is zero. We therefore enter the following: n = 30; i 1 ENTER , and finally press 2nd {QUIT} to return to standard calcu-
= 8; FV = 1,000; PMT = 0. Now compute PV, and you lator mode. You should need to do this only once, even if the calcula-
tor is shut off.

sponding to 25 years. You can see that an interest rate of 8 percent gives too high a dis-
count factor and a rate of 9 percent gives too low a discount factor. The interest rate on
the Coca-Cola loan was about halfway between at 8.5 percent.
Second, you can rearrange the equation and use your calculator.
$129 — (1 + r)25 = $1,000
(1 + r)25 = = 7.75
(1 + r) = (7.75)1/25 = 1.0853
r = .0853, or 8.53%
In general this is more accurate. You can also use a financial calculator (see the nearby


Double Your Money
How many times have you heard of an investment adviser who promises to double your
money? Is this really an amazing feat? That depends on how long it will take for your
money to double. With enough patience, your funds eventually will double even if they
earn only a very modest interest rate. Suppose your investment adviser promises to dou-
ble your money in 8 years. What interest rate is implicitly being promised?
The adviser is promising a future value of $2 for every $1 invested today. Therefore,
we find the interest rate by solving for r as follows:
PV — (1 + r)t
Future value =
$1 — (1 + r)8
$2 =
1+r = 21/8 = 1.0905
r = .0905, or 9.05%
By the way, there is a convenient rule of thumb that one can use to approximate the an-
swer to this problem. The Rule of 72 states that the time it will take for an investment
to double in value equals approximately 72/r, where r is expressed as a percentage.
Therefore, if the doubling period is 8 years, the Rule of 72 implies an (approximate) in-
terest rate of 9 percent (since 72/9 = 8 years). This is quite close to the exact solution
of 9.05 percent.

The nearby box discusses the Rule of 72 as well as other issues of compound inter-
est. By now you easily should be able to explain why, as the box suggests, “10 + 10 =
21.” In addition, the box considers the impact of inflation on the purchasing power of
your investments.

The Rule of 72 works best with relatively low interest rates. Suppose the time it will
Self-Test 4
take for an investment to double in value is 12 years. Find the interest rate. What is the
approximate rate implied by the Rule of 72? Now suppose that the doubling period is
only 2 years. Is the approximation better or worse in this case?

Multiple Cash Flows
So far, we have considered problems involving only a single cash flow. This is obviously
limiting. Most real-world investments, after all, will involve many cash flows over time.
When there are many payments, you™ll hear businesspeople refer to a stream of cash

Recall the computer you hope to purchase in 2 years (see Example 2). Now suppose that
instead of putting aside one sum in the bank to finance the purchase, you plan to save
some amount of money each year. You might be able to put $1,200 in the bank now, and

Confused by Investing?
Maybe It™s the New Math
If there™s something about your investment portfolio What Goes Down Comes Back Slowly
that doesn™t seem to add up, maybe you should check
In the investment world, winning is nice, but losses can
your math.
really sting. Let™s say you invest $100, which loses 10%
Lots of folks are perplexed by the mathematics of in-
in the first year, but bounces back 10% the next. Back
vesting, so I thought a refresher course might help.
to even? Not at all. In fact, you™re down to $99.
Here™s a look at some key concepts:
Here™s why. The initial 10% loss turns your $100 into
$90. But the subsequent 10% gain earns you just $9,
10 Plus 10 Is 21
boosting your account™s value to $99. The bottom line:
Imagine you invest $100, which earns 10% this year To recoup any percentage loss, you need an even
and 10% next. How much have you made? If you an- greater percentage gain. For instance, if you lose 25%,
swered 21%, go to the head of the class. you need to make 33% to get back to even.
Here™s how the math works. This year™s 10% gain
turns your $100 into $110. Next year, you also earn Not All Losses Are Equal
10%, but you start the year with $110. Result? You earn
Which is less damaging, inflation of 50% or a 50% drop
$11, boosting your wealth to $121.
in your portfolio™s value? If you said inflation, join that
Thus, your portfolio has earned a cumulative 21%
other bloke at the head of the class.
return over two years, but the annualized return is just
Confused? Consider the following example. If you
10%. The fact that 21% is more than double 10% can
have $100 to spend on cappuccino and your favorite
be attributed to the effect of investment compounding,
cappuccino costs $1, you can buy 100 cups. What if
the way that you earn money each year not only on your
your $100 then drops in value to $50? You can only buy
original investment, but also on earnings from prior
50 cups. And if the cappuccino™s price instead rises
years that you™ve reinvested.
50% to $1.50? If you divide $100 by $1.50, you™ll find
The Rule of 72 you can still buy 66 cups, and even leave a tip.
To get a feel for compounding, try the rule of 72. What™s
that? If you divide a particular annual return into 72,
Source: Republished with permission of Dow Jones, from “Getting
you™ll find out how many years it will take to double your Confused by Investing: Maybe It™s the New Math,” by Jonathan
money. Thus, at 10% a year, an investment will double Clements, Wall Street Journal, February 20, 1996. Permission con-
in value in a tad over seven years. veyed through Copyright Clearance Center.

another $1,400 in 1 year. If you earn an 8 percent rate of interest, how much will you
be able to spend on a computer in 2 years?
The time line in Figure 1.7 shows how your savings grow. There are two cash inflows
into the savings plan. The first cash flow will have 2 years to earn interest and therefore
will grow to $1,200 — (1.08)2 = $1,399.68 while the second deposit, which comes a year
later, will be invested for only 1 year and will grow to $1,400 — (1.08) = $1,512. After
2 years, then, your total savings will be the sum of these two amounts, or $2,911.68.

Even More Savings
Suppose that the computer purchase can be put off for an additional year and that you
can make a third deposit of $1,000 at the end of the second year. How much will be
available to spend 3 years from now?

Future value of two cash $1,400
flows $1,200

0 1 2
Future value in Year 2

$1,512.00 $1,400 1.08
$1,399.68 $1,200


Again we organize our inputs using a time line as in Figure 1.8. The total cash avail-
able will be the sum of the future values of all three deposits. Notice that when we save
for 3 years, the first two deposits each have an extra year for interest to compound:
$1,200 — (1.08)3 = $1,511.65
$1,400 — (1.08)2 = 1,632.96
$1,000 — (1.08) = 1,080.00
Total future value = $4,224.61

We conclude that problems involving multiple cash flows are simple extensions of
single cash-flow analysis.

To find the value at some future date of a stream of cash flows, calculate what
each cash flow will be worth at that future date, and then add up these future

As we will now see, a similar adding-up principle works for present value calculations.

Future value of a stream of $1,400
cash flows $1,200

Year Future value in Year 3
0 1 2 3
$1,080.00 $1,000 1.08
$1,632.96 $1,400
$1,511.65 $1,200

The Time Value of Money 49

When we calculate the present value of a future cash flow, we are asking how much
that cash flow would be worth today. If there is more than one future cash flow, we sim-
ply need to work out what each flow would be worth today and then add these present

Cash Up Front versus an Installment Plan
Suppose that your auto dealer gives you a choice between paying $15,500 for a new car
or entering into an installment plan where you pay $8,000 down today and make pay-
ments of $4,000 in each of the next two years. Which is the better deal? Before reading
this material, you might have compared the total payments under the two plans: $15,500
versus $16,000 in the installment plan. Now, however, you know that this comparison
is wrong, because it ignores the time value of money. For example, the last installment
of $4,000 is less costly to you than paying out $4,000 now. The true cost of that last pay-
ment is the present value of $4,000.


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