When Warren Buffett was asked how he would ask his wife to manage their money after he was gone,
his advice was simple.
My advice to the trustee could not be more simple: Put 10% of the cash in short-term government bonds and 90% in a very low-cost S&P 500 index fund. (I suggest Vanguard’s.) I believe the trust’s long-term results from this policy will be superior to those attained by most investors…
Can we try and understand this choice through the lens of the Kelly criterion, which we have
explored in some depth in
previous posts?
Setup
Suppose that returns \(r\) from the stock market obey a Gaussian distribution, with the mean \(\mu = 1.1\) and scale factor \(\sigma = 0.3\). Note that this \[r = 1 + \text{annual rate of return}.\] This choice essentially assumes average annual returns of 10% from the market, and anticipates typical swings (\(2\sigma\)) to lie between -50% and +70%.
We will play with some of these parameters later, but right now, they seem reasonable enough; so lets not fixate on them.
I will also assume that the worst possible return is \(r = 0\), which implies a return of -100%, which means everything is completely wiped out. This has never happened in the past, and it seems like a good bet that this will "never" happen in the future. Similarly, I will assume that the best possible returns are 300%. It seems like a reasonable cap.
So my annual returns are expected to be random variables drawn from a truncated normal distribution. Let's see what this would look like.
Kelly Criterion
In the Kelly criterion, the crucial expression to develop is the
mean of the logarithm of the expected return.
In this case, the probability of obtaining a return of \(r\) is \[\mathcal{N}_{T}(r|\mu, \sigma, r_{min}, r_{max}) dr,\] where the subscript "T" reinforces the "truncated" aspect of the normal distribution.
If I wager a fraction \(f\) of my investment pot, then I should expect to end up with, \(1 - f + f r\), where I assume that the uninvested fraction is kept safe in cash, and earns 0%.
Thus I can write the logarithm of the pay off as:
\[\log S = \langle \log R \rangle = \int_{r_{min}}^{r_{max}} \mathcal{N}_{T}(r) \log(1 + f(r-1)) dr\] Let's visualize log S for this choice of \(\mathcal{N}_{T}(r)\).
Python codes used to generate these (and subsequent results) are at the end of this post.
The optimal allocation to the S&P portfolio using the Kelly criterion (by maximizing log S) is 82.8%, which is reasonably close to Buffett's suggested portfolio.
Scenarios
Let us consider three scenarios: (i) conservative (\(\mu = 1.05\), \(\sigma = 0.5\)), (ii) moderate (\(\mu = 1.08\), \(\sigma = 0.4\)), (iii) aggressive (\(\mu = 1.12\), \(\sigma = 0.3\)).
The optimal allocations for the conservative, moderate, and aggressive assumptions are approximately, 55%, 83%, and 100%, respectively.
If we assume a \(\sigma = 0.3\), then a 90% stock allocation corresponds to a \(\mu \approx 1.09\), or an expected 9% rate of return from the S&P.
Historically, the standard deviation of S&P returns has been closer to 0.2. I like to use a higher number because the histogram of the actual returns is not perfectly normal, and the tails have been fatter than one would expect from a \(\sigma = 0.2\).
Python Code
