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FINDING THE INTEREST RATE

When we looked at Coca-Cola`s IOUs in the previous section, we used the interest rate to compute a fair market price for each IOU. Sometimes you are given the price and have to calculate the interest rate that is being offered.

For example, when Coca-Cola borrowed money, it did not announce an interest rate. It simply offered to sell each IOU for $129. Thus we know that

PV = $1,000 1 = $129 (1 + r)25

What is the interest rate?

There are several ways to approach this. First, you might use a table of discount factors. 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 corresponding to 25 years. You can see that an interest rate of 8 percent gives too high a discount 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 = $1,000 = 7.75 $129 (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 box).

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 double 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:

Future value = PV (1 + r)t $2 = $1 (1 + r)8 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 answer 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) interest rate of 9 percent (since 72/9 = 8 years). This is quite close to the exact solution of 9.05 percent.

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 flows.

FUTURE VALUE OF MULTIPLE CASH FLOWS

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 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?

Again we organize our inputs using a time line as in Figure 1.8. The total cash available 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 values.

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