Tag Archives: volatility

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”.

Portfolio Optimization With Leveraged Bond Funds

Summary Bond funds are great because they generate alpha and usually have negative correlation with stocks. Using the leveraged version of a bond fund can drastically improve portfolio optimization (i.e. produce greater expected returns for a given level of volatility). I use SPY/TLT and SPY/TMF to illustrate. SPY/TLT Portfolio Optimization Consider a two-fund portfolio optimizaton problem based on the SPDR S&P 500 ETF Trust (NYSEARCA: SPY ) and the iShares 20+ Year Treasury Bond ETF (NYSEARCA: TLT ). Often the goal is to maximize the ratio of expected returns to volatility (Sharpe ratio). I don’t like that approach, because when you maximize Sharpe ratio, you tend to get a portfolio with great risk-adjusted returns but relatively small raw returns. Instead, let’s say the goal is to choose an asset allocation that maximizes expected returns for some level of volatility that you can tolerate. A good way to do that is to look at a plot of mean vs. standard deviation of daily returns for various asset allocations. Here is that plot using SPY and TLT data going back to 2002. (click to enlarge) The red curve shows mean and standard deviation of daily portfolio gains for various asset allocations. The points represent 10% asset allocation increments. The top-right point is 100% SPY, 0% TLT; the next point is 90% SPY, 0% TLT; and so on until the bottom-most point on the other end of the curve, which is 0% SPY, 100% TLT. Suppose you want no more than three-fourths the volatility of SPY, or a standard deviation no greater than 0.93%. Looking at the graph, we want to be right around the third data point from the upper-right end of the curve. That data point represents 80% SPY, 20% TLT. This is the optimal allocation for an investor who wants to maximize returns at three-fourths the volatility of SPY. SPY/3x TLT Portfolio Optimization Let’s see how replacing TLT with a perfect 3x daily TLT fund (no expense ratio, no tracking error) affects the portfolio optimization problem. (click to enlarge) The red curve shows the same data as in the first figure, it just looks different because I had to zoom out to accommodate the SPY/3x TLT curve. Here I show asset allocations in 5% increments for the blue curve. The lowest point on the blue curve is 100% SPY, 0% 3x TLT; the next point is 95% SPY, 5% 3x TLT; and so on until the rightmost point, which is 0% SPY, 100% 3x TLT. Interestingly, increasing 3x TLT exposure from 0% reduces volatility and increases mean returns up until about 25% 3x TLT. Over the volatility range 0.884%-1.235%, you can do substantially better in terms of maximizing mean returns for a given level of volatility with SPY/3x TLT compared to SPY/TLT. Going back to the first example, at a volatility of 0.93%, or three-fourths the volatility of SPY, the best mean return you can achieve with SPY/TLT is 0.039%, with 80.1% SPY and 19.9% TLT. The best you can do with SPY/3x TLT is 0.059%, with 65.5% SPY and 34.5% 3x TLT. Daily returns of 0.059% and 0.039% correspond to CAGRs of 16.0% and 10.3%, respectively. For another interesting special case, suppose you can tolerate the volatility of SPY. With SPY/TLT, the optimal portfolio is 100% SPY and 0% TLT, with a mean daily return of 0.040%. With SPY/3x TLT, the optimal portfolio is 48.4% SPY and 51.6% 3x TLT, with a mean daily return of 0.069%. Also noteworthy is the fact that SPY/3x TLT portfolios are capable of achieving volatility greater than SPY, while SPY/TLT portfolios are not. This could be appealing to aggressive investors. A Real 3x Bond Fund: TMF So far, I’ve shown that a perfect 3x daily TLT fund would be extremely useful for portfolio optimization. The next question is whether such a fund exists, and how “perfect” it is in regard to expense ratio and tracking error. There are a few options, but I think the most relevant is the Direxion Daily 20+ Year Treasury Bull 3x Shares (NYSEARCA: TMF ). TMF was introduced on April 16, 2009, and has a net expense ratio of 0.95%. The next figure shows that indeed TMF effectively multiplies daily TLT gains by a factor of 3. The correlation between actual TMF gains and 3x TLT gains over TMF’s 6.5-year lifetime is 0.996. (click to enlarge) I realize that TMF does not attempt to track 3x TLT, but rather 3x the NYSE 20 Year Plus Treasury Bond Index (AXTWEN). But practically speaking TMF operates very much like a 3x TLT ETF. Let’s go ahead and re-examine the mean vs. standard deviation plot for SPY/TLT, SPY/3x TLT, and SPY/TMF over TMF’s lifetime. (click to enlarge) This is interesting, and slightly disappointing. As in the previous plot, we see that SPY/3x TLT achieves drastically better mean returns for particular levels of volatility compared to SPY/TLT. The orange curve for SPY/TMF is also higher than SPY/TLT, but not as much so as SPY/3x TLT. It seems that TMF’s reasonable expense ratio and tiny tracking error do detract somewhat from the optimization problem. But we still see that increasing exposure to TMF from 0% to about 20% reduces volatility and increases expected returns, and SPY/TMF does much better than SPY/TLT for those who can tolerate volatility between 0.722% and 1.022%. Leveraged Bond Funds Multiply Alpha and Beta As I’ve argued in other articles (e.g. SPY/TLT and SPXL/TMF Strategies Work Because of Positive Alpha, not Negative Correlation ), the reason bond funds compliment stocks so well is that they generate positive alpha. A bond fund with zero or negative alpha has no place in any portfolio; you would be better off using cash to adjust volatility and expected returns. Anyway, bond funds are special because they generate alpha. Ignoring tracking error and expense ratio, a leveraged version of a bond fund multiples both the alpha and beta of the underlying bond index. We can see this with TLT and TMF. Over TMF’s lifetime, their alphas are 0.061 and 0.173, and their betas are -0.492 and -1.493, respectively. TMF’s alpha is 2.84 times that of TLT’s, and its beta is 3.03 times that of TLT’s. 3x greater alpha does not immediately render 3x TLT the better choice for portfolio optimization. You have to look at the effect on both expected returns and volatility, which are both functions of alpha and beta. Suppose you can achieve the same portfolio volatility with c allocated to SPY and (1-c) to TLT, or with d allocated to SPY and (1-d) to 3x TLT. If you subtract the expected return of the SPY/TLT portfolio from the expected return of the SPY/3x TLT portfolio, you get: (d-c) E[X] + [3(1-d) – (1-c)] E[Y] where X represents the daily return of SPY, and Y the daily return of TLT. Whether this expression is positive or negative depends on d, c, E[X], and E[Y] (which can also be expressed as alpha + beta E[X]). For SPY and TLT, the expression is always positive, which means that SPY/3x TLT provides better expected returns than SPY/TLT for any level of volatility that both can achieve. Conclusions Leveraged bond funds appear to be extremely useful for portfolio optimization. In the case of SPY and TLT, we saw that using a 3x version of TLT, like TMF, allows us to: Improve expected returns for a particular level of volatility. Achieve the same volatility as SPY, but with drastically better expected returns. Take on extra volatility beyond SPY’s in pursuit of greater raw returns. In practice, TMF’s expense ratio and tracking error detract somewhat from the performance of an ideal SPY/3x TLT portfolio. But SPY/TMF still allows for substantial improvements over SPY/TLT in terms of maximizing returns for a given level of volatility.

What Should Investors Know About Volatility, As We Approach The Year End?

Summary Volatility is a powerful tool that allows to profit from market sentiment. Seasonality data suggests volatility is set for more downside. U.S. economic data will take a center stage for the remainder of the year, but unexpected developments overseas may bring about more volatility. The oldest and strongest emotion of mankind is fear, and the oldest and strongest kind of fear is fear of the unknown. H. P. Lovecraft The wounds are still fresh. The last couple of months were tragic for global capital markets. It was all about the turmoil in equities, caused by global economic uncertainty, slowdown in China, and the Fed’s indecisiveness to raise interest rates due to inflationary and employment concerns. This has pushed the U.S. and global capital markets into correction territory, the second worldwide decline in less than a year. The markets bottomed on August 23rd, and reversed in ironic fashion, climbing back up to the pre-correction levels of mid-August. As of November 3rd, the U.S. major indices were mostly flat for the year. Following the action in the equity markets, CBOE’s Volatility index , or VIX, advanced rapidly, as investors rushed out of their positions. The Volatility Index (VIX), often referred to as the gauge of fear, has gained 140% over the course of one week, following the dovish announcement from the Federal Reserve to keep interest rates unchanged at the FOMC meeting on August 19. VIX, of course, has a negative correlation with the S&P 500 index, which lost close to 10% during the same period of time. (click to enlarge) As markets reversed their trend, so did the VIX. It declined steeply to the lows of early August, to indicate the rising confidence of market participants. Investors who have not watched the markets closely during that period of time might have spared themselves some sleepless nights. For some people however, the global turmoil and rising uncertainty in the stock market had equally presented an incredible opportunity to profit, by investing around wild swings in volatility. With the correction behind us, I’d like to give my projection for the VIX, as we now entered the last two months of the trading season. Short-term volatility outlook. Following the recovery in equity markets, volatility has declined more than 70% from it’s peak on August 23rd. While that is undoubtedly a significant move to the downside, I believe that the VIX has yet to hit its bottom. Below are several factors that are set to weight heavily on VIX performance going forward. Seasonality It is no secret that the behavior of certain investment instruments differ throughout the year or from one stage to another within the economic cycle. Constantly recurring seasonal events fundamentally affect the performance of individual stocks, indices, and overall markets. Many traders look out for these patterns on the chart to determine the momentum of the instrument in question. Volatility is one of those affected by the seasonality. (click to enlarge) If history holds any clue, both November and December comprise the period of declining volatility, that generally peaks in October, as per chart above, which indicates the average annual performance of the VIX over the last 20 year period, ending 12/31/2014. Interestingly enough, the Volatility Index behaved astonishingly similar in 2015. And although certain skepticism still prevails in the market, it is unlikely we see another correction any time soon, that might propel volatility higher. In fact, investors should now feel more confident, after tackling a pullback in equities, believed by many to be overdue. Specificities . December is certainly the month to watch, as both year-end portfolio rebalances and more surprises from the Fed may cause investors to become uncertain. The U.S. economic data will take center stage during the last two months of the year, as non-farm payrolls for October and November will be dissected for clues in regards to the timing of the initial rate hike. As futures indicated a 50% chance of December rate hike as of November 3rd, I predict any negative number to add volatility to the market. The first GDP estimate for the third quarter was driving U.S. markets this past Thursday. Nevertheless, this figure will be subjected to further revisions in November and December, and will not cause much volatility, as market participants have already priced in the increase of only 1.5% for the quarter. More stimulus from China and Japan, as well as positive remarks from ECB’s Mario Draghi on situation in Europe, will definitely propel markets higher. China, on the other hand, still poises the biggest threat for global capital markets stability. An additional slew of worse-than-expected data from Beijing will likely trigger GDP downgrades, that will re-ignite the fear of global slowdown. Lastly, expect the end of the year trading activity to be light. The holiday season is likely to keep investors on the sidelines until the new year sets in. This does not necessarily mean a quiet period for the markets. Investors should follow the news closely, as illiquid exchanges are less likely to absorb selling pressure caused by global developments. The ways to use volatility to your advantage Every investor who would like to trade around market sentiment might want to consider gaining an exposure to the CBOE’s Volatility Index (VIX). Although it’s impossible to purchase VIX directly, there are ways to gain such exposure either through options, or by taking a position in one of the designated ETFs. Volatility as an investment vehicle has unquestionably gained in popularity among investors. Almost every ETF provider now issues products that are linked to the volatility of a broader market. These products vary substantially in terms of leverage, liquidity, and underlying assets, but they all try to replicate either a long or short position in the Volatility Index. VIX currently trades at $14, which implies roughly 15% downside, if traded against its long term support. One of the ways to speculate on that is by shorting the Barclays S&P 500 VIX Short Term Futures ETF (NYSEARCA: VXX ), that re-invests in volatility futures on a rolling basis. From my observations, VXX incorporates 30% to 70% movement of the VIX during short periods, suggesting 5% to 15% move to the downside, excluding contango drag down. For investors who might want to gain a leveraged exposure to the VIX, the VelocityShares Daily 2x VIX Short Term ETN (NASDAQ: TVIX ) would be the second best instrument to short. It currently trades for $5.85, or more than 15% above it’s pre-correction levels. In addition to that, the current state of contango is expected to weigh heavily on volatility ETFs if shorted for an extended period of time, suggesting even more downside. Risk-reward, however, is not as attractive as it was a month ago. After falling significantly from its peak, volatility still preserves enough momentum to justify an ongoing decline. However, investors now face far greater risk associated with the strategy, and should be aware of any potential loss related to it. Despite my bearish volatility outlook, I generally seek better risk-reward potential to initiate a position. Preserving the money in this case is far greater concern. Nevertheless, any small speculative bet at this point is definitely warranted. I urge market participants to be more cautious when putting money to work at this time of the year. Best of luck to all investors!