Question

An investment of \(P\) dollars that gains \(r\) percent of its value in one year is worth \(P(1+r)\) at the end of that year. An investment that loses \(r\) percent of its value in one year is worth \(P(1-r)\) at the end of that year. If the investment gains \(r\) percent the first year and loses \(r\) percent the second year, what is the increase or decrease in the value of the investment?

Short answer

The change in value of the investment over the two years is \(P\)[\((1+r)(1-r) - 1\)]. This means that the investment will typically decrease in value due to asymmetric percentage change, unless there is no change (\(r = 0\)).

Step-by-step solution

Calculate the value of investment after the first year

Given an initial investment value \(P\) and a percentage gain \(r\), the value of the investment after the first year is given as \(P(1+r)\).

Calculate the value of investment after the second year

The value of the investment at the end of the first year is \(P(1+r)\) and it loses \(r\) percent of its value in the second year. Therefore, the value at the end of the second year would be \(P(1+r)(1-r)\).

Find the difference in value

To find the increase or decrease in the value of the investment, subtract the original investment from the value at the end of the second year. The change is thus \(P(1+r)(1-r) - P\).