The Asymmetry of Losses: The Gain You Need to Recover

0:00Tuesday evening: down by half, so half back? ▶

It's Tuesday evening, and Max is at your kitchen counter, turning his phone face down and face up again. He put money into something last spring, and tonight the app is showing him red. Say the account held ten thousand dollars, and say it has dropped by half. Five thousand remains, and Max has already done the sum out loud: down by half, so it needs half back. Then we're even, he says, and he puts the phone down like the matter is closed. So give him what he asked for: say the account climbs by half over the next year. Half of five thousand is twenty-five hundred, and the app now shows seven thousand five hundred. Max looks at it for a long time, because that is not ten thousand. He lost half, he gained half back, and he is still a quarter short. Twenty-five hundred dollars is missing, nobody took it, so where is it?

0:59One idea: a percent is a slice of the pile in front of you ▶

A percent is a slice of the pile in front of you, and Max's pile just got smaller. This is how a loss counts, not how it feels. Put a seesaw on the counter, with the ten thousand as the level line.

1:11The seesaw: dollars, not percents ▶

On the loss side, the half that went: five thousand dollars, and the plank tips down. To bring it level, the gain side needs the same weight in dollars, and the outline shows how much. Guess first: what does half of the money still on the counter put on the gain side? Half of five thousand is twenty-five hundred, and the plank barely lifts. The twenty-five hundred was never there: half of five thousand is half as much as half of ten, and the percent hides that switch. Only five thousand on the gain side levels the plank, and five on five is the whole thing: a hundred percent. Lose half, and the account has to double; so is there a rule for any loss, not just this one? Look at the plank once more, in fractions of the whole instead of dollars.

2:04The rule for any loss, read off the plank: g = L / (1 − L) ▶

Call the loss L: the pile on the loss side is L of the whole. Beside the plank sits what remains, one minus L; the gain side gets a slice of it, call it g: g times one minus L. Level means both piles weigh the same, so L equals g times one minus L. Divide everything by what remains, and the gain you need is L over one minus L. Add what remains to both piles, and what remains, grown by g, is the whole again: the account agrees with the plank. Fill in a half: a half over a half is one, a hundred percent, exactly what the plank said. And the order does not matter: half up and then half down lands on the same seven thousand five hundred, one multiplication either way. Now the surprise: fill in a tenth, and a tenth over nine tenths is eleven point one percent. A tenth down needs barely more than a tenth up, so for small losses Max was almost right. The rule bites harder the more the bottom of the fraction shrinks, and it bites fast.

3:19The check: a shortcut against the exact rule ▶

Time for a shortcut you can do in your head, and a shortcut gets checked against the exact rule. You are short by L of the whole, so add L. But L of the smaller pile is itself short by L of L, so add L times L. Stop there: the next piece is L of that, too small to matter. For a tenth: a tenth plus a hundredth is eleven percent, and the exact rule lands on eleven point one. The shortcut missed by a tenth of a point. Try it on a half: a half plus a quarter is seventy-five percent, and the plank wants a hundred. The shortcut breaks where losses get big, because the pieces shrink too slowly; so what happens past the half? Nora, back from the hallway,

4:09Ninety percent down, and the calendar ▶

asks what Max has been avoiding: what if it had dropped more? Say it had dropped by ninety percent: a thousand dollars remains of the ten. Nine tenths over one tenth is nine: the thousand needs nine hundred percent, ten times over. And a percent hides one more thing, which is time. Say the account grows seven percent a year from here, which is an assumption, nothing more. Max's half has to double, so you flip the calendar forward at seven percent. The calendar stops at about ten years, the rule of seventy-two again. Nine tenths down, ten times over: at seven percent, that is thirty-four years. The plank does not care how the loss felt. It counts what remains, and what remains has to do all the climbing. Max picks the phone back up and asks what he actually wants to know:

5:03What the math says, and what it doesn't ▶

sell, wait, or add more? This one number gets used to sell you very different things, and the math sells none of them. The plank gives him no answer, only a number, and he works it out himself: five thousand on five thousand, a hundred percent. Half down is double back, and every percent of that climb is a percent of the smaller pile. Next time an app shows you a red number, you can work out the green one it needs, and that's the math behind the money. Next time: how a one percent fee compounds,

5:38What's next: How a 1% Fee Compounds ▶

because over thirty years it quietly takes about a quarter of what it would have grown to. And for you: what is the biggest drop you have ever had to climb back from, in percent? I'm the MoneyMathician, and if you subscribe, the next one will find you. Until then: do the math!