Going DOWN
... then going up 
the PORTFOLIO BLUES 
Think about this:
"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 (1r). To regain that loss, your portfolio must be divided by
(1r) and that means a return of 1/(1r)  1 = r/(1r)
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?"
>And your answer is ...?
Wait.
>Wait? That's an answer?
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(1r) / 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(10.7)/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 $(1R) in N months, so, in one month,
it's reduced to
(1R)^{1/N}, so, in one year (that's 12 months!) it's reduced to
(1R)^{12/N} which is 1r, so
r = 1  (1R)^{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:
>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 19571958 chart, the calculator suggests 8 months when,
in fact, it took about 9 months.
Also, for the 19611963 chart, the calculator says 10 months, but it took 15.
For the 19901991 chart ...
>So it always takes longer, eh?
As I was about to say, for the 19901991 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!
