Risk, Volatility, and Investment Return (Revisited)

**lzhang** Wed Jul 11, 2007 11:09 pm

Risk, Volatility, and Investment Return (Revisited)
Risk, Volatility, and Investment Return (Revisited)

Following last post, Risk, Volatility, Investment Return, and Risk Management, maybe it is worthwhile to revisit the issue now, since lots of related discussions are going on in the forum. To be clear, we need to have a clear definition of risk. For investment purpose, risk is often defined as a potential negative impact to an asset (value) that may arise from some present process or from some future event, but as a general term, from pure statistical point of view, risk is not just possible loss. To have less confusion, we want to use uncertainty whenever risk is used. So statistically, risk is just uncertainty, which is a neutral term. It not only includes possible loss, but also includes possible gain.

How to estimate the risk, or uncertainty? For historical data (sample), we can estimate the uncertainty as the standard deviation of the sample. Or to be general, given a sample A(a1, a2, ..., an), we can estimate the uncertainty, the risk, or the dispersion, etc. of the statistical population the sample represents with sample standard deviation.

However, if we know an a priori distribution about a variable, we do not need any historical data sample to estimate the uncertainty or risk, we can calculate those numbers directly from the distribution.

Let's look at the issue through a few questions.

Is seasonal fluctuation a risk, or uncertainty?

Probably not, since the fluctuation is very predictable. If a variable is very predictable, the uncertainty on the variable could be very small, i.e., the risk is small. For example, if the Bee's Candy sale for holiday season is $300M, while that for the rest of the year is $100M, there is no uncertainty on those numbers, so no risk. One can easily project all the sales for many years, and get the earnings, PV of the earnings, etc, so the seasonal fluctuation is not risk. To get a little bit reality, the holiday sale is not constant at $300M, but it itself can fluctuate from $280M to $320M, and nobody knows the exact numbers. That is uncertainty, and that is risk. The same thing for regular season sale, it might fluctuate from $80M to $120M, and that is uncertainty, and risk as well. So it is a general practice to make seasonal adjustment for business with seasonal fluctuations. It would be naive to combine all the sales from different seasons together and use uncertainty, i.e. risk, calculated that way as the uncertainty of the business.

Now we know that not all changes can be called as risk, or uncertainty. If the change is fully expected and is for sure to happen, there is no uncertainty on that.

Is oil exploration a risky business?

Probably yes, because you never know the outcome for sure. Even with modern science and technology, it is still very possible that the well you draw fails to produce any oil and multi-million investment in the project ends up with nothing. Even if the well does produce oil, it is still very possible that the production is far below the expectation. On the other hand, it is also very possible that you hit the jackpot and the well is very very productive, far above the expectation.

Is risk or uncertainty good or bad?

As a neutral term, it should be neutral, it just describes the random nature of future events. In most cases, we can not totally remove the uncertainty of future events, even we may want to reduce the uncertainty as much as we can.

OK, to be specific, is risk or uncertainty good or bad for business or investment? It depends, and once again, it should be neutral as a general term.It may sound reasonable to require high return for high risk position, but if you can really have high return is another question. Take the gamble business as an example, suppose we have a game with two outcomes for a gambler vs slot machine: for 9% of time he can bet right, and he wins 10 tokens; for the rest of 91% of time, he will bet wrong, and he loses 1 token. The uncertainty of this game is √((10+0.01)^2×0.09+(–1+0.01)^2×0.91)=3.148 for the gambler, while this is also true for the slot machine! √((–10–0.01)^2×0.09+(1–0.01)^2×0.91)=3.148. So they both face the same kind of uncertainty, or risk, but for each bet, the gambler will lose –0.01 on average, while the slot machine will win +0.01 on average. The risk of 3.148 may sound big compared with 0.01 profit per bet for the slot machine, but the slot machine is making `sure' money for the gambling industry!

Take another example of insurance company. Suppose an auto policy will cover $60K total loss if an accident happens and the driver pays a premium of $620 per year. The prob of accident is 1% per year per driver. The insurance company will make –(60000–620)×.01+620×.99=$20 per year per driver, which might sound small compared to the uncertainty of √((–59380–20)^2×0.01+(620–20)^2×0.99)=$5970.

But here the uncertainty or risk is definitely good for gambling and insurance industries! If it is for sure that the gambler could not have large one time gain, nobody would want to gamble and the slot machine would be out of business. If it is for sure that there would be no accidents, no hurricanes, no disasters, etc., the insurance companies would be out of business, including most insurance businesses of Berkshire Hathaway, Mr. Buffett's holding company.

So far, we should realize that risk is not necessarily good or bad for business. In most cases, it is just a matter of fact, and we have to live with it, the uncertainty of future events. In some cases, where the business is directly related to risk or uncertainty, the risk or uncertainty is definitely good to business. But high risk or uncertainty does not always mean high return.

Then, what to do to reduce the risk, or uncertainty in those `high risk' businesses?

There are mainly two approaches, which are very simple and effective. The first one is so simple that you do not need to do anything, just stay put. Bee's Candy may have seasonal fluctuation of fat profit and small loss, not uncertainty though as we have discussed, but it may still have profit fluctuation within each season. However, if the Bee's Candy company stays in the business for long enough, the fluctuation of its profits per year will be averaged out with very stable outcome at the end. Statistical principle tells us if we have μ and σ as mean and uncertainty of rate of return for 1 period, after T periods, we will have Tμ and (√T)σ as the mean and uncertainty of total return for the whole period (assuming continuous compounding), where we can see the uncertainty improve (√T) times with respect to the mean. Staying in the game is especially important for the slot machine. The gambler may have large gain occasionally, but if he stays in the game longer, the odds are more against him and to the slot machine. Over the life time of a slot machine, with millions of bets, it is making `sure' money, i.e., with very small uncertainty, for the gambling companies.

Another simple approach is diversification. With millions of independent auto insurance customers, the insurance companies make pretty stable profit from its auto insurance policies. Statistical principle tells us if we have μ and σ as mean and uncertainty of rate of return for 1 instance, with N independent and identical instances, we will have μ and σ/(√N) as the mean and uncertainty of average rate of return for all of them, where we can see the uncertainty has bee greatly reduced. If a gambling company has hundred of slot machines operating simultaneously and independently, the company can have a very stable income even on a daily basis.

We see the two approaches have some similarities. The most important one is the assumption that each instance's or each period's return is independent and identical. This year's sale fluctuation is independent and identical to the next year's. Customer A's profile is independent and identical to customer B's. Slot machine A's operation is independent and identical to slot machine B's. So in practice, we need to check if those assumptions or approximations are correct or not, and make necessary adjustments.

Can I use historical volatility of a stock to measure its risk?

In finance, people give standard deviation a fancy name, volatility. Can we use volatility (standard deviation) of historical price (or return) as the measurement of risk, or uncertainty of a stock? It sounds obvious at its face, but we need to look deeper: what we have and what we want to know. We have historical price (or return) data sample, we want to know the risk of the investment, which relates to future uncertainty. Suppose σ0 is the uncertainty or risk of investment in stock X, we have returns R(r1, r2, ..., rn) with μ and σ as return mean and standard deviation for the past n years for X. We are asking a question: Is σ a good estimate of σ0? Suppose μ0 is the expected return of investment in stock X, another equally important question is: Is μ a good estimate of μ0?

Wait a second, what is a `good' estimate? In statistics, a `good' estimate is unbiased, efficient and consistent. Unbiased means the expected value of the estimator equals the parameter it is intended to estimate. Efficient means you can not find a better estimator for the parameter. Consistent means the accuracy of the estimate increases as sample size increases. On its face, sample mean and standard deviation are unbiased, efficient, and consistent estimates of the underlying population mean and standard deviation, but if we can claim that μ is a good estimate of μ0 and σ a good estimate of σ0 here in investment really relies on more assumptions, which are called EMH, efficient (capital) market hypothesis. Under those assumptions, capital market is called informationally efficient. The main assumptions, among others, are:

1. there are a large number of profit-maximizing, independent participants;
participants are homogeneous.
2. new information comes randomly and independently;

3. security price adjusts to all new information and reflect all information available at any time;

If those assumptions are correct, obviously we do not have any better estimators out there other than the μ and σ for our purposes. The problem is those assumptions are not always true as granted in real life investment. When any of those assumptions are not (always) true for any security, using historical sample estimators to estimate future expectations will end up with erroneous conclusion.

To our understanding, the main danger of using historical (return) data to estimate future return comes from two sources. The first one is that the overall business environment might have changed and historical sample does not recognize those changes, so the future return and uncertainty can be substantially different from those historical numbers. To use those estimates is just like to cut a mark on the side of one's boat to indicate the place where one's sword has dropped into the river while the boat is moving (刻舟求剑).

The second problem is that the historical price (return) data may or may not reflect the nature of the underlying business. To use historical price (return) data is just a wrong method, like climbing a tree to catch fish (缘木求鱼). To be specific, historical return mean and standard deviation are just two numbers, while they might have sophisticated components, which might have made the nature of the underlying business hard to see. For example, is the high past return really due to business expansion, or PE expansion? Is the high standard deviation really due to high business risk, or people's perception?

All of the possible problems indicate that EMH is just an approximation which can not be over-stretched. The biggest problem of EMH assumptions is about the participants. To oversimplify that each participant in the game has the same expectation, the same rationality, the same knowledge, the same vision, etc., is just way too much. Peculiar to individual cases, particular participant's knowledge and experience play an important role to achieve above average performance. So for business-like value investment, the best way to estimate investment return and uncertainty is not from the historical price (return) data, but from the business itself, which is more thorough and complicated by its nature. For example, if Mr. Buffett sees the return and uncertainty of company A as μB and σB where μB is relatively high and σB is relatively low, so he thinks company A as a wonderful company to hold. Mr. Buffett knows the business of company A very well and we think his view is more or less unbiased and accurate. Unfortunately, company A has some temporary difficulties and the rest participants, or the market as a whole, think the return and uncertainty as μM and σM where μM is relatively low and σM is relatively high. In EMH, Mr. Buffett does not exist, but in reality, there is Mr. Buffett, and many others alike!

The above argument is essential to say that the return and uncertainty of an investment are not universal numbers, which could be as simple as mean and sigma of historical return data sample, rather, they are numbers subject to each participant's own knowledge, experience, and business vision. Over long time, on average of many cases, the assumptions of EMH might be true, because much of the difference will be averaged out, but particular to individual cases and for a particular time frame, participants are different. A simple example in insurance business will illustrate this point. Drivers are not identical, so different drivers have different risk profile and insurance companies fully understand the importance of it. `Good' drivers have lower prob to have accidents, so that insurance companies are willing to give those `good' drivers very low premiums. On the other hand, `bad' drivers tend to have accidents, so they have to pay much higher premiums to the insurance companies. The same thing happens to investment. Investors within their `circle of competence' tend to make more accurate predictions of related investments so they enjoy above average returns from those investments.

Can I use historical volatility of a hedge fund to measure its risk?

This is a similar question like previous one, but they are not exactly the same. A more clearly specified question is to ask: Are historical return and uncertainty of a hedge fund good estimators of its future return and uncertainty? Under some assumptions, we believe historical return and uncertainty of a hedge fund are good estimators of its future return and uncertainty. Those assumptions are:

The fund's investment strategy, etc., remain unchanged
The historical sample is large enough, or in other words, the time frame is long enough

The first assumption is crucial. If the fund's investment strategy, style, key managers, etc., have changed, you can not expect its historical data will be representative to its future return and uncertainty. Large sample size is required to make any decisive conclusion. In our opinion, it is required to have at least 5 years' historical returns to make any meaningful conclusion for most cases.

A natural question is to ask: why do you think it is reasonable to use historical return and uncertainty to measure a hedge fund's future return and uncertainty while it is sometime very wrong to use historical return and uncertainty of a stock to measure its future return and uncertainty? To be clear, it is not always true, but statistically, we believe that is true if time frame is long enough for a hedge fund and the fund's underlying investment philosophy, etc., remain unchanged. The reason is simple, this historical return and uncertainty are what the hedge fund has expected. If the hedge fund has made correct investment decisions, statistically, its return will be about the same as its expectation, which, by definition, will beats the market. Take the above Buffett example again. If Mr. Buffett has made correct prediction on the future return and uncertainty, which are μB and σB , respectively, when the prediction becomes history, the now historical return on the investment becomes μB instead of μM, as Mr. Buffett has expected, so the historical return of Mr. Buffett's portfolio will reflect Mr. Buffett's investment performance. If his investment expectation is far from reality, the historical performance will be far below the general market performance as he is more likely to make the wrong predictions. If his investment expectation is more or less reflect the true reality, the historical performance will be above the general market performance as he is more likely to make the right predictions. But how about the uncertainty part, σB? Even if he has made the mean part right, there is no way to guess the uncertainty part right, (if you want to use the historical price (return) sample to measure it) because that is really subject to the wildness of the market, which is totally off the control of anybody. The ride might be very bumpy even if your goal is correctly recognized. There is no doubt on that. But as we have discussed, the uncertainty, estimated as standard deviation of a sample, will be greatly reduced if we have several independent investments all together. So, if properly diversified, a hedge fund's return uncertainty, measured as standard deviation of historical return sample, will properly reflect the hedge fund's expected return uncertainty, or its risk.

There are so-called `concentrated' investments, which have only a few stocks in their portfolios. The uncertainty, measured as standard deviation of historical sample, is usually very high. Now we need to be very clear on the question, is that uncertainty really its risk. If the business vision on those investments is right, the expected return will be realized at the end, so the strategy is to have very keen vision here and take the ride until the expectation is materialized, i.e., the first approach to handle high uncertainty, or risk, stay put. Unlike a small subsidiary, e.g., See's Candy for Mr. Buffett, if the portfolio is relative large part of one's whole net worth, staying put can not always reduce the risk, because you can never rule out that the initial expectation is totally wrong for a small number of investments (bets). Suppose I have only 10 tokens in my slot machine and that is all I have, one lucky gambler will blow me out of my business, even statistically I will win for sure.

To our observation, compared to the general market, a superior hedge fund or mutual fund usually has relatively high mean and low sigma on its investment return, based on historical sample.

References

Risk, Volatility, Investment Return, and Risk Management

刻舟求剑

Cut a mark on the side of one's boat to indicate the place where one's sword has dropped into the river; take measures without regard to changes in circumstances.

《吕氏春秋·察今》楚人有涉江者，其剑自舟中坠于水，遂契其舟曰：“是吾剑之所以坠。”舟至，从其契者入水求之。舟已行矣，求剑者此，不亦惑乎？

缘木求鱼

Climb a tree to catch fish; a fruitless approach.

《孟子·梁惠王上》 “... 以若所为，求若所欲，犹缘木而求鱼也。”