The Mathematics of Money

364 Chapter 8 Mathematics of Pricing

are increasing the price by a percent of a growing value; so with each passing year, the

price not only rises, but does so at an accelerating rate. If we look at the price of the col-

lectible bottle of wine from Example 8.4.1 year by year, we can see this:

Year Starting Price Increase Ending Price

1 $3,650.00 $255.50 $3,905.50

2 $3,905.50 $273.39 $4,178.89

3 $4,178.89 $292.52 $4,471.41

4 $4,471.41 $313.00 $4,784.41

5 $4,784.41 $334.91 $5,119.32

6 $5,119.32 $358.35 $5,477.67

7 $5,477.67 $383.44 $5,861.11

8 $5,861.11 $410.28 $6,271.39

9 $6,271.39 $439.00 $6,710.39

10 $6,710.39 $469.73 $7,180.12

This is exactly the same thing we saw with compound interest.^3

With depreciation, though, matters are a little different. The price is not increasing, it is

decreasing, and so, as we calculate the percent of a decreasing price, we get a decreasing

amount. So the price is declining at a decelerating (slowing down) rate. If we look at the

price of the car from Example 8.4.2 year by year, we get:

Year Starting Price Decrease Ending Price

1 $23,407.00 $3,511.05 $19,895.95

2 $19,895.95 $2,984.39 $16,911.56

3 $16,911.56 $2,536.73 $14,374.83

4 $14,374.83 $2,156.22 $12,218.61

5 $12,218.61 $1,832.79 $10,385.82

Even though this is a big change from what we are used to seeing from the compound inter-

est formula, this slowing down of price decreases is pretty much what we would expect for

the value of a used car. As a car ages through its fi rst year, it goes from being a new car to

a used one. There is a big difference in what people are willing to pay for a new car versus

what they are willing to pay for a used one. (There is truth in the old saying that the biggest

drop in the car’s value occurs “when you drive it off the lot.”) On the other hand, there is

usually less difference in the value people will assign to a 4-year-old car than to a 5-year-

old one. This is refl ected in the pricing of the table.

If we carry this further, we would expect to see even smaller differences in the car’s

value as years go by; we would not expect people to see much of a price difference between

a 9-year-old car versus a 10-year-old one. Either way, it’s a pretty old car. If we extend our

car price projection out to year 10, we see this is exactly what happens:

6 $10,385.82 $1,557.87 $8,827.95

7 $8,827.95 $1,324.19 $7,503.76

8 $7,503.76 $1,125.56 $6,378.20

9 $6,378.20 $956.73 $5,421.47

10 $5,421.47 $813.22 $4,608.25

The car is predicted to lose just $813.22 in value between years 9 and 10, versus $3,511.05

in the fi rst year. This agrees with common sense.

Just how low will the value of Todd’s car go? The car’s value each year is 85% of its

value in the prior year. For this reason, even though the value will continue to decline with

(^3) The 2 cent difference between the ending value in this table and the ending value we calculated in Example 8.4.1
is due to rounding the price change each year.