The Mathematics of Money

Copyright © 2008, The McGraw-Hill Companies, Inc.

Nonetheless, mathematical formulas can be useful tools for prediction, for estimating the

dollar value of something even if we don’t actually want to offer it up for sale, and in other

cases where we need to place a reasonable dollar value on something. In this chapter we will

examine some of these situations and the mathematical tools that can be useful in dealing

with them.

Calculating Price Appreciation

Let’s take on the question of appreciation fi rst. The following example will illustrate how

this works mathematically, and will also illustrate that we already know how to do it!

Example 8.4.1 According to a wine expert, the market price for a very rare bottle

of Chateau la Plonque wine is $3,650. In an interview in a wine trade publication, he

states that he expects this particular bottle to appreciate at a 7% annual rate for the

next 10 years. If his prediction turns out to be correct, what will the price be 10 years

from now?

This problem is yet another example of a case where even though the 7% appreciation rate

is not compound interest, it is a percentage rate of growth and is mathematically equivalent

to compound interest. Therefore, we can use the compound interest formula:

FV 
 PV(1  i)n

FV 
 $3,650(1.07)^10

FV 
 $7,180.10

It’s worth also noting that we could have estimated that the bottle would a little less than

double in value, since Rule of 72 says that the doubling time at 7% is approximately 72/7 


10.28 years.

Price appreciation is very often stated as a percent growth rate, and so using the compound

interest formula as we did in Example 8.3.1 will handle price appreciation questions quite

effectively.

Depreciation as a Percent

Just as price appreciation is often expressed in terms of a percent rate, price depreciation

is also often expressed as a percent rate. The difference, of course, is that whereas the

appreciation rate is a rate of growth, a depreciation rate is a rate of decline (sometimes also

called decay). However, since in both cases the dollar value is assumed to be changing at a

percent rate, the same mathematical formulas can be used. The difference, though, is that

since depreciation means a price is going down, we treat this as a negative growth rate.

The following example will illustrate how this works.

Example 8.4.2 To dd just bought a new car for $23,407. According to an online used

car pricing service, the value of this car will decline at a 15% annual rate. Assuming

this is correct, what will the car’s value be in 5 years?

We take on this problem in just the same way as the previous example, except that here the

rate is understood to be 15%.

FV 
 PV(1  i)n

FV 
 $23,407[1  (0.15)]^5

Adding a negative number is the same as subtracting a positive one, so this becomes:

FV 
 $23,407(1  0.15)^5

FV 
 $23,407(0.85)^5

FV 
 $10,386

So depreciation works in basically the same way as appreciation, which itself is essentially

the same way as compound interest, even using the same formula as compound interest

does. There is, however, one very significant difference. With appreciation, over time we

8.4 Depreciation 363