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Or, if we enter the expression all at once:Operation Result
20000*1.0625^(1000/365) 23613.70The reason the ( ) are required is order of operations. Without them, the calculator will take
the exponent prior to division, and so 1.0625 will be raised to the 1,000 power, and then the
result will be divided by 365. That is not the same as using 1,000/365 as the exponent. The
good news is that it is so different that if you accidentally forget to put in the parentheses
in most cases the result will be so absurdly far from a reasonable answer that it should be
immediately obvious that it is not correct.
Alternatively, we could divide “1000/365” and store the result in the calculator’s
memory, reclaiming that value to use in the exponent in the next step.When “Interest” Isn’t Really Interest
In all of our discussions so far, we have considered only situations where the growth from
present value to future value was due to the action of interest. There are, though, many
similar situations which don’t involve actual interest—“rent”paid for the temporary use of
money—but do look quite similar. Consider the following examples:According to an article in an industry publication, the suggested retail price of a pair of
Treadworthy Sneaker Company’s TWS-42 shoes was $75.49 seven years ago, and has
increased at an annual rate of 4.8% since then. How much would the suggested retail
price of the shoes be today?
Amazingly Awesome Internet Stuff Corp.’s stock has increased in value over the last
5 years at a rate of 28.4% annually. If you had invested $2,000 in the company’s stock
5 years ago, how much would it be worth today?
A union contract sets a worker’s hourly wage at $16.35 an hour this year, and also
states that this hourly rate will be increased by 3.5% in each of the next 4 years. What
will the hourly rate be at the end of the 4 years?
The population of the Republic of Freedonia was 1,534,670 according to the 2005
national census. If the population grows at an annual rate of 2.4%, what will the popu-
lation be in 2015?Let’s look at the first example. Clearly, this is not a situation where interest is involved. When
you buy a pair of sneakers, you are not lending the manufacturer or retailer any money. You
are buying their product. This is a purchase, not a loan.^7 However, the situation seems entirely
analogous to the one where the increases were due to interest. The price grows by a percent-
age each year, just as an account value grows by earning interest. Each year’s increase is a
percentage on top of the prior year’s price, just as each year’s interest is earned on top of the
previous year’s. Even though the price increases really have nothing to do with any actual
interest, it seems as though the mathematics could be worked out in much the same way.
The other examples are similar. If you buy stock in a company, you are putting your
money to use, but you are not lending it to the company or to anyone else for that matter.
Stock represents ownership of a company, and so when you buy stock, you are buying a
piece of the company. Likewise, hourly wage rates are not in any sense a loan, and their
contractual growth most certainly is not interest. It would be even more absurd to think of
a country’s population as a loan, or its growth rate as interest on one.
However, each of these cases does bear a strong similarity to compound interest. In
each case something is growing, the rate of growth is a set percent, and each year’s growth
builds upon the previous years. These were the essential attributes of compound interest(^7) Even if you borrow the money to buy the shoes, that is a separate issue, and has nothing to do with the
suggested retail price that the company sets.
3.3 Effective Interest Rates 121