Fortunes's Formula And Asset Allocation

by: Alexander Steinberg
Summary

Asset allocation is very similar to gamble betting.

The Kelly formula for betting is introduced and applied for stock allocation.

Half-Kelly allocation for stocks is proposed as a less risky approach which is equal to 80% stocks in low interest rate world.

In my previous article, using an example of retiree’s portfolio, I proposed a model to assess and manage risks associated with a certain investment portfolio but stopped short of constructing this portfolio.

Building an investment portfolio consists of asset allocation and securities selection. Of these two tasks, asset allocation is often considered more important. Since beating the stock market is known to be very difficult, a careful stock selection can provide only unlikely success, if any. On the other hand, asset allocation can directly influence your risk and return as supported by historical evidence.

Following this logic, we will consider a model where an investor has only two assets to choose from:

  1. stock market index with the average annual return µ and the standard deviation σ;
  2. the risk-free asset with the annual return r.

Let us further assume that at the beginning of each year our investor needs to allocate a fraction of his wealth f to index and the remaining (1-f) fraction to the risk-free asset. What would be his optimal allocation?

Before we move further, I would like to briefly review some well-known solutions to this problem. In practical portfolios, these solutions typically narrow down to splitting funds between cash, bonds, and stocks. And certainly, it is most important to deal with the percentage allocated to stocks, with the split between cash and bonds being the secondary and easier issue.

One popular approach is to allocate some fixed percentage to stocks, typically within the range of 50-65%. However, it is not uncommon to consider a more conservative allocation to stocks. We can illustrate several practical solutions using Vanguard Funds as an example. One of the most popular Vanguard funds – Wellesley Income, belonging to the elite group of Vanguard Select Funds – allocates only about 40% to stocks. Another popular Vanguard Fund – Wellington – allocates about 65% to stocks. I have always wondered how investment managers determine their target allocations and have never been completely satisfied with available explanations.

Still another allocation solution is the so-called Target Retirement funds which change allocation based on an investor’s age. The closer to retirement you are, the less funds you allocate to stocks. Vanguard offers a whole collection of these funds: when you are very young, they allocate 90% to stocks and, with age, this allocation gradually goes down. These funds seem quite popular and are based on an intuitively appealing idea that your risk, measured by fraction of stocks in your portfolio, should go down as your age increases. Still again, certain questions remain unanswered: for example, why does Vanguard start with 90% in stocks instead of 100% when you are young? And more generally: how does Vanguard figure out the optimal allocation for each age?

Now I would like to mention two solutions for asset allocation that do have clear explanations. First, in his well-known article about 4% withdrawal rule, William P. Bengen formulates the following based on careful analysis of historic data: “I think it is appropriate to advise the client to accept a stock allocation as close to 75 percent as possible, and in no cases less than 50 percent”.

And secondly, the famous investor Peter Lynch in his book “Beating the Street” advocates allocating 100% to stocks if your investment horizon is long enough. With time stocks appreciate and increase dividends and this is sufficient to overcome shorter-term market downturns. Peter Lynch provides evidence for his solution by analyzing several model portfolios. However, since it is emotionally challenging for an individual investor to tolerate stock market gyrations, Peter Lynch suggests to have an investment portfolio as close to 100% as one can stomach. A similar approach is also supported by Warren Buffett in many of his annual letters, Q&A sessions during shareholders’ meetings, and TV interviews.

Please note, that those who provide clear and logical arguments instead of appealing to intuitive risk adversity, advocate higher stock allocation than the one used by well-run and established balanced funds (for example, such as Vanguard Wellesley, Vanguard Wellington or Dodge & Cox Balanced). We can explain this discrepancy by marketing reasons. If mutual fund companies allocated more to stocks in balanced funds, they would differ little from pure stock funds and, from sales standpoint, both offerings would overlap counterproductively.

Now, let us get back to our simple allocation model, in which all the investment universe consists of 2 investable assets. The only scientific approach that I am aware of for solving this model is the Kelly formula (also known as “the Kelly criterion”, “the Fortune’s formula”, “the Kelly” and so on) proposed initially for placing optimal bets in gambling. The Kelly’s approach is well described in many publications and for entertaining reading I suggest a book by William Poundstone "Fortune's Formula". However, to understand our further line of thinking, I will review the Kelly’s approach in a very simple game:

You have $100 and are offered to place a bet in a coin toss. Tails – you win, heads – you lose. The coin is not your ordinary one: the probability of tails is 2/3 and the probability of heads is 1/3. You can play this game as long as you want. How much would you bet?

The game is profitable for you, so it makes sense to play. But you cannot bet all $100 since, if you lose, you will exhaust your funds and will not be able to recover. To minimize the risks, you can alternatively bet some small amount, say, $1. But this approach seems excessively cautious: your expected gain per toss is too small and you will not be able to fully exploit your favorable chances. There should be some optimal amount -- and the Kelly’s formula determines it. For our simple game the optimal fraction is

Or, for a more general case with probability of tails p and probability of heads q (p>q, p+q =1)

f=p-q.

So, your optimal strategy will be to bet $33.33 initially and then continue to further bet 1/3 of the funds you possess no matter the outcome. This approach can be quite counterintuitive: the Wikipedia article on the Kelly formula mentions a study, in which each participant was given $25 and asked to bet on a coin that would land tails 60% of the time. 28% of participants lost all their money!

Before moving further, I would like to emphasize what exactly the Kelly does: it maximizes growth of your wealth! The math proves rigorously that betting less or betting more than the Kelly makes your compounding slower on average! The Kelly strategy never explicitly addresses risks associated with this fast growth.

Certainly, there are similarities between coin tossing and the two-asset model that I proposed initially. But there are also significant differences: in investing you think in terms of annual returns which for stock index generally fluctuate between -50% and 50% and it is unrealistic to lose your stock index bet in full. Applying calculus, it is possible to receive an estimate for the Kelly fraction. Please allow me to skip the mathematical details here and present the final estimate I derived for our two-asset model:

Differently from our coin example, this is not theexact expression but rather just an estimate. Since for stocks we can discard the second term in the denominator and come up with a simplified formula:

This is exactly how the Wikipedia article on the Kelly criterion lists it, and the one we are going to use from now on.

For the S&P 500 index, µ ~ 0.1 (10%), σ ~ 0.2 (20%), today’s r ~ 0.025 (2.5%), and we can easily calculate that f ~ 1.6.

How to interpret f > 1? The Kelly formula indicates that for r being that low, we have to invest more than we own by borrowing. This result is purely theoretical because the Kelly formula “assumes” that we can borrow funds at the same risk-free rate r, which is clearly not the case. Unless we are prepared to borrow on margin, we should assume f = 1 and invest all we have in stocks!

Let us calculate the risk-free rate r that would justify allocating at least some funds outside of stocks. For this we need to solve a simple equation

Solving it for r produces

Inserting our normal values for average return and standard deviation, we calculate

r = 0.1 – 0.04 = 0.06 (6%).

Thus, the Kelly tells us that only for risk-free returns of more than 6%, it makes sense to allocate some money outside of stocks! Quite stunning! And so, the math of the Kelly directly corroborates William Bengen, Peter Lynch and Warren Buffett…

Let us now consider a riskier portfolio of securities consisting, perhaps, of some small-cap stocks, with higher both µ = 0.13 and σ = 0.4. For this portfolio, the Kelly fraction is equal to (0.13-0.025)/0.16 = 0.66. This is a dramatic reduction of optimal stock allocation!

So far, we treated the Kelly’s approach as optimal at least in purely mathematical sense. However, computer-based Monte-Carlo simulations demonstrated it to be quite risky in practical investing. In particular, the author of these simulations distinguished between two sources of its risk.

First of all, he compared the Kelly allocation with the half-Kelly. So, for our case of S&P 500 index, f = 1.6 or 1 in practical terms and the half Kelly is 1.6/2=0.8. Even though the Kelly allocation produces, on average, higher ending wealth, the chances to win (i.e. to have higher ending wealth than beginning wealth) are slightly higher in the half-Kelly allocation. However, this source of risk is quite trivial and expected.

Secondly, and much more dramatically, the Kelly allocation results turned out being very sensitive to imprecise nature of inputs. When we analyze some real investment portfolio, we can only estimate values of its µ and σ, knowing that these estimates obviously contain some errors. Moreover, the formula for f is an estimate as well. This use of estimates instead of precise values can render the Kelly allocation really risky. With these particular risks introduced, Monte-Carlo simulations show the half-Kelly being a superior solution.

In today’s world, the half-Kelly means 80% allocation to stocks, very much in line with recommendations of experts quoted above. Perhaps Peter Lynch put it best in his suggestion to keep your investment as close to 100% in stocks as you can possibly tolerate. In one of my next articles, I hope to suggest a way to convert “tolerance” into something more measurable.

Disclosure: I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.