(c – cy) + (y + ε)(c – cy) < c, where ε > 0

Yahoo!’s share price has plummeted 45% from its recent high in October of 34.08. And despite a modest increase to 19.40 in the ensuing months, yesterday’s 47% increase on news of Microsoft’s takeover offer was not enough for the share price to surpass the October high, and it closed the day at 28.60. So how come a 45% fall followed by a more than 47% rise leaves the stock short of its original value?

It’s a similar conundrum to the one I alluded to in my post two and a half years ago about the area of the Guardian’s Berliner compared to its broadsheet and tabloid competitors. The two percentages are of different things, so can’t be compared against one another.

Let’s take apples. If you have ten apples and I take away 40% of them, you’re left with six. If I then add 50% more apples to your fruit bowl, you’re left with nine. The 40% was applied to the ten apples you started with, but the 50% was applied to the six you were left with after the snatch.

If Yahoo!’s stock was measured against an index, with its October peak being pegged at 100, then it dropped to 55 and then rallied Friday to 84: a 45 point drop followed by a 29 point rise.

Taking the above inequality:

(c – cy) + (y + ε)(c – cy) < c
c – cy + cy – cy2 + εc – cyε < c
y2 – ε + yε > 0

In the apples example, y is your 40% reduction, ε is 10%, the absolute difference between the 50% drop and the 40% rise.

0.16 – 0.1 + 0.04 is 0.1, which is greater than zero, so you end up with a lower value than that with which you started. To bridge the gap exactly, you’d need to add 67% of the remaining apples, which would make the inequality into an equation: 0.16 – 0.27 + 0.11 = 0.

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