Tuesday, 22 April 2014 00:00

Changes in Percent

By  Gideon
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Changes in percent can be counter-intuitive.

For example, if the value of a stock falls by fifty percent on one day, and then rises by fifty percent on the next, it will not have recovered all of its value.

It will in fact be seventy-five percent of its original value.

It would need to increase by one hundred percent to recover all its value.

Say that the value of a certain stock is \(X\) dollars on Monday morning when the market opens. By the time the market closes it has lost half its value--it is down fifty percent--and so is \(\frac{X}{2}\) dollars.

On Tuesday morning the stock starts out at \(Y = \frac{X}{2}\) dollars. When the market closes on Tuesday the stock is up fifty precent (from where it was in the morning) and so is worth

\[ \frac{100 + 50}{100}Y = \frac{150}{100}Y = 1.5 Y \text{ dollars.} \]

But \(Y = \frac{X}{2},\) and so \(1.5 Y = \frac{1.5}{2} X = .75 X\)

The reason for the confusion is that when we say on Monday that a stock is "down fifty-percent," we mean down fifty-percent from its value that morning, that is it is down fifty-percent relative to \(X.\) When, on Tuesday afternoon, we say that the stock is "up fifty-percent," we mean up fifty-percent relative to the value on Tuesday morning, that is \(Y = \frac{X}{2}.\)

As another example, suppose a certain stock is worth \(X\) dollars on Monday morning, and suppose that its price falls by ten percent every day that week. How much will the stock be worth, as a percent of \(X\) on Friday afternoon?

On Monday afternoon, the stock is down ten percent and so is worth \( \frac{ 100 - 10}{100} X\) dollars.

That is

\[ \frac{100 -10}{100} X = \frac{ 90}{100} X = .9 X.\]

On Tuesday morning the stock is worth \(Y = .9 X\) dollars.

On Tuesday afternoon the stock is worth \(.9 Y = (.9)^{2} X.\)

By Friday afternoon, the stock has gone down ten percent for five days in a row, and so is now worth:

\[ (.9)^{5} X = .59 X = \frac{59}{100}X = \frac{100 -41}{100} X.\]

And so by Friday afternoon the stock is worth 59 percent of \(X,\) or is down by 41 percent from its value on Monday morning.

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