If you haven't yet, I recommend reading How I Can Explain 96% Of Your Portfolio's Returns (Part 1), since it's crucial to most of the concepts discussed here.

## The Medieval Version Of Risk

Risk is a nebulous concept to most investors, when it should be 99% of every investing decision. The problem is that most investors want a single, convenient number to tell them how risky an investment would be. Let's get real... measuring risk with a single number is both lazy, and too frequently inaccurate. In the 21st century, there is so much information available that there's no excuse for failing to understand risk more intimately than a single number like beta. Let's start simple...

Pretend you're considering buying your friend's ice cream stand. How would you measure risk in the returns of an ice cream stand? If you're like me, the first risk factor that comes to mind is weather. Therefore, it would be a good idea to establish a concrete relationship between changes in the weather and changes in the ice cream stand's returns. This is quite straightforward. Get your hands on historical daily temperatures and historical daily returns from your friend's ice cream stand. You have the data, but now you have to turn that into a meaningful measure.

What comes next is a process called regression. Intuitively, you should suspect that as temperature rises above average, returns also rise, and that when temperature falls below average, returns fall as well. Regression, in theory, will tell you how much returns will change for a given deviation from the average temperature. Let's say that you run the regression and it gives you the following model:

R = 25x + 500

The model reads, "The ice cream stand's returns will be $500 plus another $25 for each degree above average that the temperature rises (or minus $25 for each degree below average that the temperature falls)." Thus, if the temperature on a given day is 5 degrees above average, we would predict that returns will be $625. This model is symmetric, which means that the magnitude of risk is equal for both changes in temperature above average and changes in temperature below average.

All of this seems quite logical and useful but this is where most people go wrong. The problem is that risk is rarely symmetric. In our example, there's a good chance that returns are more sensitive to degrees above average than degrees below average. It could be the case that returns actually rise by $50 per degree when temperature is above average, and only fall by $10 per degree when temperature is below average. However, since the regression model, by definition, must give a single measure of sensitivity to change in temperature, it blurs both the $50 and the $10 measures, and returns something in between ($25 in our example). Thus, you might pass on buying your friend's ice cream stand because you believe that its returns may fall too low if temperature is below average for awhile, when in fact, there is very little loss of returns for temperature changes below average. The moral of the story is that this single, convenient measure of risk that the model presented was highly misleading and caused an incorrect decision.

Now apply this lesson to the stock market. When you see beta, understand that you are looking at the result of a regression that is assuming symmetric risk. Risk may or may not be symmetric for a given stock, but you really have no idea unless you run the numbers yourself. I will save you the trouble by simply saying that more often than not, beta is a grossly inaccurate measure.

Believe it or not, that's what you should expect. There's no reason that returns in the market should have a direct relationship to the return of a given stock. After all, returns are simply changes in price that result from buying and selling. There is no giant panel of buyers out there that unanimously decides how much buying and selling will be done on all stocks that day. Price changes are the result of individual buying and selling decisions that sometimes happen to be influenced by what other buyers and sellers are doing in the market. In other words, if all investors were suddenly unable to see the buying and selling activity of the rest of the market, it's likely that the returns of individual stocks would become much less correlated to the returns of the market. Since this is obviously not the reality, we have to deal with the fact that most stocks will have some level of dependence on the returns of the overall market and measure it accordingly.

## The 21st Century Version Of Risk

This brings us to how we should measure risk if we can concede the fact that a single measure is going to be grossly inaccurate. As it turns out, only about 60% of returns since the 1920s can be explained through the use of beta, and this shows no signs of changing. About 20 years ago, two finance pioneers, Eugene Fama and Kenneth French, discovered 2 more risk factors which, when used alongside beta, explain about 96% of returns. In short, the first factor measures the absolute difference in returns between the largest 50% of stocks and the smallest 50% of stocks (by market cap). Thus, if small stocks return 10% for the month and large stocks return 5% for the month, we would say that the SMB or size factor is 5. Similarly, the second factor measures the difference between the 50% of stocks with the lowest P/B and the 50% of stocks with the highest P/B, known as the HML or value factor.

At first glance, there is again no reason that either of these factors should have a direct, causal relationship with the returns of a given stock, but nonetheless the calculated relationships are highly effective at predicting returns. Thus, we have 3 factors for which we cannot pinpoint a direct, causal relationship. At the end of the day, it doesn't really matter. I have my own theories and evidence as to why these factors have a strong correlation to returns, but it makes no difference whether my theories are valid. The reality is that these factors are going to continue to be correlated with returns, and investors need to understand their effects.

In the same way beta is a measure of a stock's exposure to the market's returns, every stock has a measure of exposure to the SMB and HML factors. The only difference is that these measures are not reported anywhere, and happen to be very difficult to calculate. I'll get to calculating them later, but for now, let's talk about how we can use these measures once we have them.

Since most investors believe beta alone is an accurate measure of risk, I have always wondered why high beta stocks would not be the greatest investments in the long run. If we assume the market will rise in the long run (hence, a positive average value for market returns), and beta tells us a stock's exposure to the market's returns, why would anyone invest in low beta stocks?

The reason is that in the long run, the inaccuracy of beta is magnified, but we refuse to recognize that. We scratch our heads and wonder why our high beta investments did not significantly outperform the market over the long run, but we don't take the time to understand why, and we continue to accept beta as an accurate measure "because it's the best thing we have." I beg to differ.

Most beta regressions have such little explanatory power that they are useless for practical purposes. For example, the average explanatory power of beta alone for utility stocks in the S&P 500 is 20.56%, yet it is still published and compared to other companies' betas on a daily basis. It should be quite obvious that one should NEVER use a beta whose R square value is that low to make any sort of investing decision. To illustrate, let's say we want to test for the relationship between the number of daily visitors to Elvis's Graceland home, and the daily returns of the S&P 500. With the appropriate data, when we run a regression, it will always calculate a relationship between the two, regardless of whether one exists. If we look at the factor alone, it will tell us how much the S&P 500's returns influenced Elvis's visitors on average, but without the R Squared number, we have no way of knowing whether we have likely identified a real relationship, and not one that came about from sheer coincidence. In this case, I would be extremely surprised to see an R square value above 0.00.

Look at Ruby Tuesday (NYSE:RT), whose traditionally calculated beta is 3.18. Most investors' beliefs dictate that they expect Ruby Tuesday to return 3.18% when the market returns 1.00%. What they do not know is that Ruby Tuesday's beta regression is quite statistically insignificant by R square. However, when we measure exposure to the market, SMB, and HML in the same regression model, we see that exposure to the market is actually 1.33, exposure to SMB is 3.13, and exposure to HML is 2.86, and each of these factors is reasonably significant. Therefore, investors in Ruby Tuesday should be much more concerned with the SMB and HML factors than the market return factor. Indeed, it is possible for Ruby Tuesday to return 3.18% when the market returns 1.00%, but our model shows that only 1.33% of that 3.18% is attributable to the market. The other excess returns were likely due to the SMB and/or HML factor also being positive on the same day, and coincidentally arriving at the same value.

So much explanatory power is added through the inclusion of the SMB and HML factors (now called multiple regression) that we can begin to make more practical conclusions about price movement. More importantly, both SMB and HML have been positive on average, for any lengthy period of time in history, in much the same way the market's returns have been positive for any lengthy period of time in history. It follows that stocks, which are highly exposed to these three factors, will outperform in the long run and capture the average positive value that we have seen historically. Therein lies the biggest assumption: the factors will continue to be positive. For that to be the case, there certainly must be explanations for their continued success. In Part 3, I will discuss these in detail.

## Conquer The Complexity

What has just been explained can be hard to grasp at first. Regression is the combination of high-level statistics and calculus concepts that do not come easily to most people. It's the same strategy that the world's top marketing consultants use to predict complex consumer behavior. To put in perspective, a regression involves 3 hypotheses tests for which the raw outputs look like this...

Yikes. Though most of the above measures are "sanity checks," this ugly output illustrates the point that there are 100 ways that a beta calculation can become irrelevant, and none of these sanity checks are performed when Yahoo Finance or Google Finance spit out a beta value for you. It's up to you to understand the process and prevent an irrational investing decision, or, in my case, to understand where other investors are going wrong.

## From Theory To Strategy

The hardest part of any theory is applying it. I love theory because it paints a nice rosy picture of what things should look like under a microscope; it provides a starting point. Inevitably, there are always problems that prevent that rosy picture from becoming reality. However, these problems can't be understood without first having a theory. In applying the theory discussed here, we have to make a few considerations.

First, we need to decide what level of exposure we would like for each of these factors. As I have shown, theoretically the highest possible exposure to each factor is ideal for an aggressive investor, but we need to keep in mind that we should not simply search for the most highly exposed stock and place all of our money in that company. Unsystematic risk would crush the portfolio more often than not, especially when most of the stocks that are highly exposed to these factors have extremely high standard deviations. The cost of being highly diverse is well worth the standard deviation that is eliminated, even for a highly aggressive strategy like this.

The next consideration that we must make is how we want to weight each factor. The average positive value of the three factors has not been the same: market returns have been about 9% per year on average, while the SMB and HML factors have both been roughly 1/3 of that on an average annual basis. Since the 1920s, small stocks have returned 2-4% more than large stocks on average, and value stocks have returned 2-4% more than growth stocks on average. Therefore, we would need our portfolio to have about 3 times as much statistically significant exposure to the SMB and HML factors in order for all three factors to have the same average effect on long-term returns. In this sense, Ruby Tuesday is an ideal investment for the portfolio, since its exposures to SMB and HML match this profile reasonably well. By weighting the entire portfolio in this way, the market's returns account for only 1/3 of long-term price movement.

You can begin to see the advantage of weighting our portfolio in this way. Consider again how diversifying a portfolio reduces unsystematic risk. There is no escaping the fact that unpredictable events will pervade any given stock, but it is highly improbable that they will occur simultaneously to multiple stocks. Let's say you have the choice between rolling two dice and one die, and your goal is to achieve the highest average value of your rolls. If you chose to roll the single die, you have a 1/6 chance of a one (the worst-case scenario). However, if you roll two dice, there's still a 1/6 chance that the first dice will roll a one, but there is also a 5/6 chance that the other dice will roll something other than one, making the average value of your rolls higher, and reducing the chances you will see the worst case scenario. As you add more dice rolls to average, there will be less and less chance of seeing the worst case scenario. Therefore, by including multiple stocks in a portfolio, the average effects of unpredictable events from single stocks are mitigated. One stock may have earth shattering bad news, but the weighted average effect of this event is reduced because it affects only one part of the portfolio.

The same logic applies to the strategy of including multiple independent systematic risk factors in the portfolio. While we may not be able to predict when one of the factors will be negative, it is highly improbable that all of them will be negative at the same time, since they are highly uncorrelated. Therefore, when market returns are negative, such as in a recession, there is theoretically very little probability that the SMB and HML factors will also both be negative, since they are independent. In theory, any recession should account for only 1/3 of the returns in the portfolio. If SMB and HML happen to be positive on average for the duration of a recession, there is a good chance that the portfolio's overall returns will be positive, despite the market's returns being negative. Essentially, this strategy allows us to diversify both systematic risk and unsystematic risk, whereas most investors are concerned exclusively with diversifying unsystematic risk.

## A Visual Approach

For those who consider things more visually than mathematically, think of these independent risk factors as indexes like the S&P 500. The only difference is that these indexes don't track real assets. You may be familiar with the S&P 500 Volatility Index (VIX), which attempts to measure volatility (not a real asset) by looking at the changes in put and call options on the S&P 500's constituent stocks. This is very similar. Supposing we start the SMB index at 100, each day it will gain or lose a percent amount equal to the daily calculation of SMB (the absolute difference in average returns between the smallest 50% of stocks and the largest 50% of stocks). In other words, if an asset existed whose returns were completely dictated by the daily value of SMB (exposure level of 1.00), its price chart would be identical to this index. In this way, you could compare the S&P 500 (inclusive of the risk-free rate) to the HML and SMB indexes. The last 10 years would look like this (on a monthly basis)...

RM-RFx represents the S&P 500 and risk free rate (calculated from monthly changes net of the risk free rate), SMBx represents the SMB index, and HMLx represents the HML index. Bear in mind that while SMBx and HMLx recorded modest returns over 10 years, there is a plethora of stocks, which have statistically significant exposure to these factors by as much as 7, while I have yet to find a stock with factor adjusted exposure to RM-RF of more than 2.5. In a diversified portfolio of over 30 stocks, it is possible to have an exposure profile of 1.5 for RM-RF, 2.5 for HML, and 2.5 for SMB. Limiting the portfolio to less than 30 could allow a portfolio to see an exposure profile of up to 4.0 for HML and SMB. In essence, one can easily lever these factors without paying for a margin account or using levered ETFs.

Here's a look at what a portfolio might look like if these factors could be perfectly levered with exposures of 1.5, 2.5, and 2.5 for the market factor, SMB, and HML respectively (the nice, rosy picture of the theory at work)...

The blue line tracks the combined effects, while the other lines track the contribution of each factor when examined alone. There certainly won't be any stocks that perfectly track these indexes (R square of 100), but a well diversified portfolio whose average exposure matches these indexes should eliminate most of the noise. For example, if we could identify 30 stocks whose exposures average 1.5 for RM-RF, 2.5 for SMB, and 2.5 for HML, we would expect the portfolio to move closely in line with the light blue index shown above. This gives the strategy a great deal of flexibility in terms of expectations for the future of each factor. For example, if we had access to regression data for the entire universe of equities, and we expect a market recession to occur in the near future, we could screen for only those stocks, which have been highly uncorrelated to the market factor. In this way, the portfolio would be theoretically immune to the expected downturn in the market.

Consider Arbor Realty Trust, Inc. (NYSE:ABR), which has a traditionally reported beta of 3.75. A multiple regression reveals that there is a 73% chance that Arbor Realty Trust's returns are independent of the market factor. The regression also reveals that Arbor Realty Trust is exposed to the SMB factor by an adjusted magnitude of 4.54 and the HML factor by an adjusted magnitude of 5.35. It's a very real possibility that droves of potential Arbor Realty Trust investors have been turned off by its outrageous traditional beta of 3.75 out of fear that it might doom their portfolios amidst a market downturn. Yet, a little bit of digging reveals that the company may be one of the best plays for weathering a bear market. The real value of the company's exposure is lost in a useless, robotic calculation of traditional beta.

## A Sample Strategy

In keeping with the strategy of detaching from the market factor, the following names would make decent picks...

Each of these stocks has significant exposure to SMB and HML while having insignificant exposure to the market factor (shown by the p-value). The p-value represents the probability that the given stock's returns are not dependent on the given factor. Anything over 0.05 implies statistical insignificance of the factor.

Despite including a total of 12 stocks, average adjusted exposure to SMB and HML remains quite high at 3.55 and 3.70 respectively. That seems extremely desirable, but do not forget that the market factor is possibly the strongest and most frequently positive factor. Detaching from it may help preserve capital during a recession, but remaining detached may mean missing out on bull markets as well. For example, using the above portfolio's average level of exposure, the past 10 years would theoretically have looked like this, all else being equal...

Note that the green line does not track actual returns of the selected portfolio, but rather it tracks what the portfolio's returns would have been if each stock's R square was 1.00. You can see from the table above that this is not the case, but diversifying goes a long way toward closing the gap between actual returns and modeled returns.

Clearly, a single position in the S&P 500 ETF Trust (NYSEARCA:SPY) would have served a portfolio better over the last 10 years. However, note that before the market's exceptional bull run over the last couple of years, the theoretical portfolio remained on top for an overwhelming majority of the time. As the market begins to sober up from its bull run, this may become a more relevant factor profile to employ.

**Final Thoughts** **- "History Repeats Itself"**

My personal risk appetite is such that I prefer maximum exposure to all three factors. The past has shown us in detail that each factor has been convincingly positive for any significant length of time. The funny thing about history is that it tends to repeat itself. This strategy is not asking that the future conform perfectly to what history predicted, but it is telling us what our best guess should be based on billions of data points. Any poker player knows that if you can be right 51% of the time, you've found yourself a winning strategy. We're shooting for much better than 51% here, but the point still stands.

Of course, adjusted exposures to these factors are not reported anywhere, making any factor profile strategy difficult to employ. Even to calculate the monthly values of SMB and HML is no trivial task. Doing so requires the ability to analyze the month's returns for many thousands of tickers and sort them according to size and value breakpoints. I can painstakingly regress individual names until I find what I am looking for, but performing "guess and check" 8000 times a month is certainly not ideal. I'd like to see one of the major financial data providers start to provide these measures, since they are the only parties (outside academics like Fama and French) that have the data and computing power to do so. Until then, a "guess and check" approach is the best we can do.

In the near future, I hope to report some more specific candidates that would be ideal for each factor exposure profile, as well as some ETFs that capture exceptional exposure. In Part 3, I will discuss how buyer psychology and irrationality will ensure that SMB and HML will continue to be positive in the future, and I will also discuss the use of commodities and other systematic risk factors that have flown under the radar.

**Disclosure: **I 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 (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.