Going DOWN ... then going up

 the PORTFOLIO BLUES
"A 50% loss in your portfolio means that you must get a 100% gain in order to recover to your initial portfolio"
In general, if your loss is r (r = 0.08 meaning 8% loss), then it means that your portfolio has been multiplied by (1-r). To regain that loss, your portfolio must be divided by (1-r) and that means a return of 1/(1-r) - 1 = r/(1-r)

In general, it looks like this:

>Scary, eh?
Yeah. If you lose 40% you'd have to gain 67% to get back to where you started

>I suppose you have a solution for this?
No, it's just that friends of mine, knowing the sad facts of Figure 1, have asked,
"How can I ever recoup my losses?"

Wait.

If the gains occurred at the same rate as the losses, you'd just just have to wait longer. For example, if you lost 40%, then (see Figure 2) you'd have to wait 1.52 times longer.
For a 40% loss in one year, then, using 1.52 ...

>Then I'd get my portfolio back in 1.52 years. Right?
Assuming your gains occur at the same rate as the losses.

>And do they?
Here are some examples. You decide

Figure 1

Figure 2

>I assume there's a formula, eh? You always have a ...
Yes, assuming losses occur at the same rate as subsequent gains:

Years to recover from a loss of r (in a year) = - { log(1-r) / log(1+r) } years
where a loss of 12.3% means r = 0.123

>Okay, I lost 70% in the past year so r = 0.7 so I get: -log(1-0.7)/log(1+0.7) = -(-.523)/(0.230) ... uh, that's more than 2 years!

Wait.

>Suppose I lost that 70% in, say, 15 months?
Okay. If the loss in N months is R, then every portfolio \$1.00 is reduced to \$(1-R) in N months, so, in one month, it's reduced to
(1-R)1/N, so, in one year (that's 12 months!) it's reduced to (1-R)12/N which is 1-r, so r = 1 - (1-R)12/N and that's what we'd use in the formula above ... which would then give the answer in years.

>You gotta be kidding!
Okay, use this calculator:

 Total Percentage Loss = % Months over which you suffered the above loss = months Time to Recover = months     assuming gains are at the same rate as losses :^)

>So, is this a good estimate for, say, the period after the '87 crash?
No. The calculator says that, after a 33% decline in 3 months (after Aug/87) it should recover in about 8 months, but ...

>Ha! It looks more like 20 months!
Well ... yes, but for the 1957-1958 chart, the calculator suggests 8 months when, in fact, it took about 9 months.
Also, for the 1961-1963 chart, the calculator says 10 months, but it took 15.
For the 1990-1991 chart ...

>So it always takes longer, eh?
As I was about to say, for the 1990-1991 chart the calculator says 6 months but it only took about 5 months.

>Okay. The Nasdaq dropped 72% (from 5000 in Mar/00) to Jul/02. That's 28 months, so ...
So, it'll take about 44 months to recover.

>That makes it ... uh, from July, 2002 ... uh ...
It'll recover to 5000 on Feb 10, 2006 ... at 10.37 in the morning

>And what if I don't believe any of this stuff?
Wait ... and, to feel better while you're waiting, Click!