So the majority is wrong the correct answer is answer 2. https://twitter.com/bankofvol/status/1388105086770024448

1/If the non levered investment Max drawdown is noted MD then cumulative return of levered investment is
2*R if MD>-50% else 0.
But what’s more interesting is how wrong the statement is. Indeed for the 2 levered investment to generate 2*R is has to be levered potentially more.

2/Indeed whenever there is drawdown the leverage increases:
If the investment drops initially by -25% the leverage becomes 3 to 1 & you have to accept to stay at 3 to 1 if you want to get back your (prayer needed) 2*R return.

3/if you adjust the leverage back to 2 to 1 after a DD then you’ll never get 2*R and the return you get is independent of R. So if you trully are levered at 2 to 1 all the time you never get 2*R and the return is totally independent of R, it only depends on the random return path

4/So when someone says I am levered 2 to 1 and I will get 2*R what that person is saying without really realizing it, is I am startingst at 2 to 1 & will accept thereafter any type of leverage that the random walk of return path will impose to me.

5/So what happens if I try to keep the leverage at 2 by adjustijg from time to time? Well it might be ugly,
If I have -25% inital DD, I adjust back to 2, then I have a recovery +33.33%, non levered investement return R is zero and levered is -16.66%.

6/ So now I understand I can’t adjust, but then what’s the risk?
Well if I am levered at 3 a 25% DD brings me to 9 to 1, when I am at 9 to 1 a mere -10% DD brings me to 81 to 1

7/LTCM was running a 27 to 1 leverage (reminder they had to Nobel prize winners Merton and Scholes on their board). Well do the math a -3.32% DD brings their leverag to 250 to 1.

8/When they were bailed out by the Fed and the banks who took over their portfolio opened their books that’s exactly what they found a 250 to 1 leverage.

9/ Now do Archegos.

10/Let’s now see where the beauty of leverage <1 comes from, if you have a DD your leverage automatically decreases!
Suppose I start at 0.7 leverage and non levered drops -25% then leverage becomes 0.63 it’s convex. Bingo.

11/ Now let’s see how controling for DD is important in the context of model assumptions. So let suppose I invest in the S&P500 buy and hold, and I assume my next 20 years of returns to be more or less the same as my last 20 years.

12/I can certainly try to get the best returns and l find out very quickly that 1.5 leverage I get what on the surface seems that be the “optimal” risk reward assuming future will look like the past. (This is -75% Max DD during the Financial Crisis hmmm).

13/But what if future doesn’t look like the past? What If I apply a shock -20% to my dataset and that’s the future?

14/At 1.5 leverage my Max DD is now -82% instead of -75% and I only get 23% of the returns I was expecting.

15/Then comes the magic, if I control the expected DD at around -10%, then in case of a 20% shock to expecations, I retain 60% of my expected returns and my DD is -13.4% instead of -10.6%

16/If I apply different shocks, 30% or 40%, things do not change at -10% DD I obtain the maximum actual returns/expected returns Ratio (even thoug a lowest ratio as shocks ), decreaseing DD further does not improve actual returns/expected returns ratio. Double Bingo