Apply Kelly Formula To Investing: Is Volatility Just Risk?
Summary Kelly Formula is one of the most important formulas in the investment theories. It is also very interesting and useful since it is against our intuition. Volatility is commonly seen as just “risks”, but it is much more than that, since volatility can affect performance too. Theoretically, Kelly Bet is also the “optimal bet”, but that is often not true in practice. Since reducing volatility can help performance too, I will also talk about many methods to reduce the volatility of a portfolio. Kelly Formula Kelly formula or Kelly bet was found by John Kelly in 1956. This formula gives the optimal bet, given fixed odds in a gambling game. Although it has some fairly simple math behind, it didn’t get much attention from the financial world until much later. On the surface, its application is limited. Financial activities such as investments don’t always have fixed odds, and may not have a fixed period for its return either. However, what is profound in that formula is that it gives a “maximum bet” that is optimal even if one is completely non-risk-averse. If we really think about it, it is actually against regular people’s intuition. Normally we would think the returns are always related to the risk we are willing to take. The more risk we can take, the more returns we will get. So the investment return is a function of our risk tolerance. The Kelly Formula, however, says returns don’t always go up when we take more risk, even if we can ignore the risk completely, and have a very good risk appetite: there is a maximum risk we should take, more risk taking will only bring worse returns. In other words, “volatility” is more than just the risk we have to experience during the process, or more than just the wider dispersion of possible returns; higher volatility may also reduce the eventual expected return. Why is that? That is because the returns of sequential investments multiply on each other, instead of “adding” onto each other. For example, if you lose 50% in the first year, you have to make 100% gain on the next year to get back even. It is (100% – 50%) * (100% + 100%) = 100%, rather than (50% + 100%) = 150%. In math terms, the returns are multiplicative, instead of additive. (The log-returns or log-assets are additive instead). This concept is very interesting and very useful, when we start to apply it to many financial decisions. For example, many years ago, I used to think that I should invest 100% on stocks since historically stocks had higher returns than bonds or bank CDs, and since I was very young, I shouldn’t be too concerned about the risks of stocks. That seems very logical, right? Well, not after you get familiar with the Kelly Formula. The fact is, if stocks are very volatile, 100% invested in stocks may not give your best returns even if the stocks do turn out to have better returns than bonds/CD’s, and even when we assume you don’t care about the risks at all. Let’s work through an example to better understand this: Suppose you had $100 at the beginning of 2008, and stock market dropped 50% and it became $50 at the beginning of 2009. Two years later, the stock market fully recovered, rose 140% and your asset got back to $120, therefore you had a 20% gain in 3 years. Very bumpy and scary roller-coaster ride indeed, but assuming you have very good risk tolerance, that didn’t matter much to you. What if you had 70% on stocks and 30% on bonds? At the beginning of 2009, because of the government’s monetary policy, the bond interest rate dropped significantly, and your bonds had a 20% return in 1 year, but your stocks had a 50% loss. Together, it is $70 * 0.5 + $30 * 1.2 = $71, or 29% total loss. At this point, you should do a rebalancing (assuming you rebalance every year), and get back to 70% stocks and 30% bonds, so you would sell some bonds and buy more stocks. After that, you would have $49.7 in stocks, and $21.3 in bonds. Then assuming 2 years later, stocks went up 140%, and bonds had a return of 0% during these 2 years, your total asset became $49.7 * 2.4 + $21.3 = $140.58. This is a total return of 40.58% in 3 years, which is much more than the 20% return in the 100% stock case. What is more interesting here is that both underlying assets (stock and bond) only had 20% return in 3 years, yet the portfolio had 40% return, much more than the return of any of the underlying assets. (See the “magic” of financial engineering can sometimes turn toads into princes!) Now it seems to be a no-brainer for you to always invest some of your capital in bonds? After all, if it gives you more return and less risk, why not? Not too fast. In the example above, I used 2008 – 2010 as an example, and my figures are hypothetical. After all, you won’t see a lot of 2008s happening down the road. That said, the basic reasoning here still applies: Reducing the volatility of a portfolio can also help to improve returns, not just reduce the risks. The ultimate decision of portfolio allocation depends on how risky the underlying asset is. Maybe 100% stocks is optimal, maybe not, but it really requires you to have a fairly accurate estimate of the future volatility of your stock assets. Is Kelly Bet The Optimal Bet? Another interesting property of Kelly Bet is that it is the “optimal bet”. Well, I just said it is the “maximum bet”, but why am I also calling it the “optimal bet”? The reality is that the “maximum bet” part of the theory is actually agreed upon by almost everyone: normally you never want to bet more than Kelly Bet, unless you were too conservative on estimating your winning odds. In other words, if you bet more than Kelly Bet, you are not just aggressive, you are “insane”, since it will bring higher risk AND worse performance too. But calling it “optimal bet” becomes much more controversial among investors and traders. The theory does show that it is indeed the optimal bet, but that has a lot assumptions attached to it, such as: the bet can be made frequently (not exactly true for long term investments), the bets have the same odds, the odds could be estimated accurately, and you could never lose 100% of your asset. As you can see, the first 3 conditions are not true for long term investments, and the last one is probably true if you have fairly good diversification and don’t use any leverage. This is where the theory diverts from reality, and why we have to be careful when assuming Kelly Bet is the optimal bet. If you want to learn more about Kelly Bet, you can check out my blogs here and here . Common Methods on Reducing Volatilities As I mentioned above, reducing volatilities is so important that it not only helps to reduce your risk and overcome your emotions, but can also help to improve your performance. Therefore, managing volatilities of a portfolio becomes a central topic of risk management and money management. A following question is: how can we reduce the volatility? As you will find out below, this is much more than just diversification. Diversification Despite all the caveats and potential drawbacks, diversification is still the most powerful concept in financial engineering on reducing the volatility. However, people sometimes either over-extend this concept or didn’t apply it in full scale. Diversification should not be just among stocks Normally, money managers often talk about how many stocks they should hold, or how big each position should be, such as whether each position should be 1%, 5% or 20% of the portfolio. However, diversification should not just be among the stocks you hold. First, stocks usually have high correlations, or we can call it “systematic risk”. If they are in the same sector and the same country, they will have even higher correlations. So when you have more than 7 – 20 stocks in your portfolio, additional stocks may not do much good to your portfolio at all. On the contrary, it may harm you more than help you, especially when you are a small investor. This is because over-diversification will spread you too thin, make you have less information edge over the stocks you own. It may also make your performance suffer because you have to put money into your less favorable ideas. One thing we should all realize is that investment is hard and highly competitive. For this reason, the chance that you can find a good idea is slim. You may get 1 or 2 really great ideas in a year, but expecting to get many great ideas is not realistic, even for those superinvestors. As Charlie Munger said, if you remove the top 20 best ideas Buffett had in the last 40 years, the rest of the ideas’ performance is not much better than the average index. In this sense, while having low volatility is important to improve performance, having higher expected returns is just as important. (I’d like to think higher expected return as “offense”, and lower risk as “defense”.) Also, diversification is not limited to just stocks, since it can be done in many other ways: Diversify over different asset classes Bonds, cash, commodities, gold and real estate are all asset classes that could be used for diversifying risks. Historically, bonds are one of the most favorable choices since they usually have negative correlation with stocks, which helps to reduce the volatility even more. But since bonds are not attractive right now due to all the QEs, cash and gold are probably good choices, too. Cash is stable, but has inflation risk. Gold has no inflation risk, and is especially helpful in doom scenarios, but its value is more dependent on supply-demand since it has no clear “fundamental value”. Diversify over different strategies This method may not be very practical for small investors, but money managers can often utilize different investment strategies to diversify the risks, such as allocating capital among trading strategies and investment strategies, so that they have less correlations. Or they can maintain short positions in addition to long positions. Downside Protection Is As Important Instead of diversification, value investors often make very restrict requirement on the downside protection on their stock picks. In Buffett’s words, the secret of investing is his: Rule No.1: Never lose money. Rule No.2: Never forget rule No.1. Here, “don’t lose money” doesn’t mean that there should be no loss at all, because that is certainly impossible. It only means “no significant loss” and you have to be really careful about protecting yourself from any significant downside on each individual stock you select. Usually this protection requires many of the following traits in the company you are investing in: Sufficient Margin Of Safety Low P/B ratio, and good tangible book value or liquidation value Durable competitive strength and low/reasonable P/E ratio Long product cycles Not overly dependent on one product, or one customer Low financial leverage and low operating leverage Good pricing power Recurring Revenue Good management Conservative accounting Non-cyclical industry Non-commodity product or has durable low-cost advantage “Certainty” is the basis of all investment theses As mentioned above, diversification has limited effectiveness and has significant drawbacks too. For this reason, successful value investors often use the downside protection of the business itself to reduce volatility. However, any investment thesis requires “certainty” or “information edge” as its basis. Without “certainty”, any conclusion could be built on imagination instead of facts. Therefore, to reduce volatility, investors have to devote their efforts to achieve an information edge and achieve high “certainty”, instead of just focusing on diversification. In other words, “certainty” and downside protection of each stock pick reduces the volatility of each position, and diversification reduces the overall volatility of the entire portfolio. Both of these methods are needed, but it is more of an art than science to find the balance between these two. In some sense, that also depends on personal style and personal strategies. Conclusion I remember Charlie Munger once said many smart people should better devote their talents to real engineering projects instead of financial engineering, since he thinks financial engineering doesn’t really generate much value for our society. While I am a fan of Munger, I don’t really agree with this particular comment. I believe understanding the math behind financial engineering can not only achieve better returns for our investments, but also help us to make better capital allocation decisions in general. There are many financial engineered products that are very useful to us, such index, index fund, ETFs, options, interest rate swaps, ABS or even the infamous CDOs. Many of these products used the powerful concept of “diversification”. It is undeniable that there are new problems coming up along with these new things, such as the loss of insight with over-diversification, or lack of incentives to ensure the quality. However, I would compare that with stock exchanges. While so many small investors unconsciously used the stock exchange as a casino (except that they wouldn’t bet all their savings in a casino like they did in the stock market), and sometimes lost all their lifetime savings (especially in immature markets, like the Chinese stock market), overall, stock exchanges still provided tremendous value for both businesses and investors. It does take time to get regulations in place to make it more mature though, as what we have seen in the US since 1930s. All in all, it is my opinion that financial engineering and the math behind it do provide good value to us, although it is also important to recognize its limitations and don’t lose our “common sense”.