pezblanco wrote: ↑
Fri Mar 08, 2019 6:50 pm
So, after posting this, I began thinking that their idea is that a young person is going to be adding to the allocation by a certain (growing amount) every year ... that should help making the optimal values of leverage higher, shouldn't it? I'll have to see if I can quantify that from their framework.
Yes that's precisely the idea. If the money was all fully fronted (like the scenarios Samuelson entertains), they would not leverage it. The Kelly criterion you keep calculating assumes you do have a fronted bankroll and recommends leverage (all the way to 1.5+). So lifecycle investing will be much, much more conservative overall.
Another way to look at it is that your simulation recommends leveraging 1.5 of your bankroll or wealth. The point of lifecycle investing is that your wealth isn't just your current savings; it's also your future contributions, your social security, etc. If you think of THAT as your wealth, you're not really leveraging at all. And I'd argue you should think of it because in your simulation, losing X% a year is losing X% of your bankroll. It will determine the next year's bet to be smaller by that amount.
With lifecycle investing, a loss of X% does not translate to a smaller bet the next year necessarily. Your salary might more than make it up so your bankroll is intact the year after. Any way you slice it, if you are constantly adding money to the Kelly bankroll, you should be able to make far more aggressive bets than otherwise.
How aggressive? Here's a quick back-of-the-envelope calculation (which might be totally wrong but seems to make some sense, at least to first-order) I made to convince myself that 2:1 is not quite the Kelly Criterion for someone still saving:
Let's say the ideal historical Kelly leverage is 1.5, like you said. From a couple of sources, I'm seeing the ideal Kelly bet is as follows:
(1) F* = (u - r)/s^2 = 1.5 Where u is the mean return, r the risk free rate and s the standard deviation.
But an unaccounted for savings contribution yearly of 10% of your bankroll (and this part is very individual) would make the new mean 0.1 higher. How much does that change the Kelly bet?
Solve for (1) in terms of u and plug it into here:
(2) F*_1 = (u_1 - r_1)/s_1^2 If u_1 = u + 0.1, r_1 = r and s_1 = s (meaning, historical results continue but you now add savings), you get that the new, more aggressive Kelly bet is
F*_1 = F* + 0.1/s^2
The new, optimal Kelly bet is the savings rate as a percent of your current portfolio, divided by the volatility, plus the previous historical Kelly bet without savings contributions. If you use 1.5 for F*, a very modest saving of 10% of the portfolio (someone with 100k saves 10k that year) and the historical volatility of ~25% st. dev, you get that the ideal Kelly bet is 3.1:1
leverage. Just a rough estimate but helps put some numbers.
I will say what you've presented is very important because it tells me that 2:1 isn't just sunshine if you don't you aggressively save. If you barely make a contribution dent but stick to 2:1 leverage that year, you might very well overbet that year. This is a perspective I hadn't considered until I read MT's recommended book and is certainly very relevant. I appreciate that you've brought up this point about historical Kelly bet sizes.