When we consider the risk in investing, we’re often thinking about volatility: that is, the sometimes dramatic movements in equity prices. But as Alan Fustey explains in his book, Risk, Financial Markets & You, there’s a big problem with equating volatility with risk.

One of the biggest shortcomings in financial models is the reliance on standard deviation (SD) as a measure of risk. SD is a measure of volatility—or more specifically, how much returns vary around the average. About two thirds of all returns will fall within one standard deviation of the average, and 95% will fall within two standard deviations. In theory, annual returns that vary by three standard deviations should happen only once in a century, while a six-SD event would occur about once in a billion years.

This idea makes more sense when you use some real-world numbers. According to Credit Suisse, from 1900 to 2011 the average annual return on equities was 8.5%, with an SD of 17.7%. That means two years out of three should see returns between –9.2% and 26.2% (the average +/- one SD). In only one year out of 20 would returns be lower than –26.9% or higher than 43.9% (the average +/- two SDs). Looking back to 1970, you would expect two such years in Canada, and that’s what we experienced: a 44.8% gain in 1979, and a –33% loss in 2008.

### One in a billion?

Although SD is a backward-looking measure, it seems a reasonably reliable measure of annual market volatility. Yet as Fustey explains in his book, the probabilities don’t hold up when we look at *daily* returns. If the average daily return on stocks is 0.03% and the SD is 1% (a close enough estimate), then two-thirds of daily returns should be between –0.97% and 1.03%. In about 19 days out of 20, the returns should be between –1.97.% and 2.03%. A daily return that is five SDs from the average (a gain or loss of about 6%) should happen only once in 3.5 million trading days, and a six-SD event is a one-in-a-billion proposition.

But that isn’t borne out by history. Fustey presents data going back to 1927 that shows the S&P 500 has seen 19 days with losses greater than –7%, three of which came in 2008 alone. Black Monday in 1987 was a 22-standard-deviation event, which should have been as likely as throwing a dime off a tall building and having it land on its edge.

With this in mind, Fustey makes a distinction between *risk* and *uncertainty*. “Risk is what you have when you’re playing poker,” he told me in our interview. “There are a known number of outcomes with 52 cards, and the probabilities can be mathematically calculated. You can’t do that with markets. There is an infinite set of possibilities, so it is a completely different animal. *Risk* has the connotation that there is some kind of control there. But even with indexing, there isn’t that control. You get whatever the market gives you.”

Fustey’s takeaway message is that standard deviation can’t model uncertainty. “Anything can happen in financial markets,” he says. “So you’ve got to somehow assess the likelihood of these bizarre, extreme events, both positive and negative.” In practical terms, this means that investors may want to consider some kind of safety net to protect them against sudden, unexpected drawdowns. Next week, I’ll look at a couple of methods that index investors might use to protect their portfolios from crippling losses.

Just a note on why the use of standard deviation fails. Using standard deviation in the way presented above is based on three assumptions, all of which fail for markets. These assumptions are Gaussianity, stationarity, and lack of correlation. Gaussianity is the assumption that each day’s return is a random number drawn from a normal (Gaussian) distribution. Stationarity is the assumption that this distribution is the same each day and never changes. Lack of correlation is the assumption that today’s returns are not correlated with yesterdays returns.

These assumptions are obviously incorrect but people use this model as a first approximation to estimate the probability of given return/loss because it’s by far the mathematically easiest thing you can do.

I find the language Fustey uses to be somewhat strange. I agree with the basic idea that the normal (Gaussian) distribution is a poor model for extreme market events. However, the idea of standard deviation doesn’t belong solely to the normal distribution. Other distributions have standard deviations, variances, etc. So, for me, the statements like “there’s a big problem with equating volatility with risk” and “standard deviation can’t model uncertainty” just don’t have any real meaning.

@Kiyo: Thanks for drilling deeper on this point. The lack of correlation is an interesting point that applies to annual returns as well. Almost all of the biggest one-year gains in the markets have followed huge losses, and this clearly isn’t a coincidence.

When you flip heads four times in a row, you are not more likely to flip tails on the fifth try. But if markets fall 30% one year, it probably does raise the probablity that the following year will see a positive return.

@Michael: Let me stress that those were my words, not Fustey’s, so I take responsibility here. I don’t have any background in mathematics so my language is probably imprecise. I hope the main point is clear: that uncertainty, by definition, cannot be modeled. A Texas Hold’em player can expect to be dealt pocket aces about one time in 220 hands. But what is the probability that an equity investor will suffer a 50% annual loss? We just don’t know.

I read Taleb’s Black Swan and my takeaway was simple. Because uncertainty cannot be modelled I cannot assume any potential for gain or loss from equity correlated in any given year or over 10 years (maybe more).

I just ask myself two basic questions:

How would I feel if there was a 40-50% loss of the equity correlated portion of my portfolio in any given year? (versus the potential to maybe gain 20-25% in equities in any given year)

How will I feel if the equity correlated part of my portfolio has a return of zero for a decade?

I just assume that these outcomes are possible and allocate to equity correlated accordingly.

If one of these outcomes comes to pass (as they just did) I think it could be risky to assume that they cannot happen again in the near future.

I am interested to know the ideas for protection against serious losses in next post.

@Michael, any distribution is going to be a poor indicator of extreme events (aka outliers). It doesn’t matter that there are other distributions we could apply. They would also fail miserably at predicting the markets. Otherwise we’d all take a 2nd year stats course and be millionaires.

@Adam: That’s a bold claim. I know of no evidence to support it. In one of Taleb’s books he talked about how an inverese square-law distribution seemed to model some low-probability events well. Any time we make an investment we are implicitly presuming something about the distribution of returns. The fact that the normal distribution fails so miserably doesn’t mean that we can’t find some other distribution that would do a better job.

I see the point that the markets are erratic and cannot conform to any on statistic or pattern.

However, the standard deviation are better for longer time windows. If you are day trader, this is important.

I assume, and I can be corrected, that 99% of your readership are long term investors. This s a rhetorical statement, but probably “near” correct. For them, this does not change anything.

Stay the course….

Mr. Market told us this a generation ago 🙂

As always, a very thought provoking blog!

How can you model for human psychology? Wasn’t much of the collapse in US real estate caused by reliance on mathematical models?

I think the takeaway point was said by Fustey himself: “You get whatever the market gives you.” I say, if you don’t like it, find some other way to make money. The stock market is just one way. But it’s the laziest way to make money. Oh wait, that’s why it’s called the “couch potato portfolio”! That’s why people like it so much. Then they complain that the returns are “lumpy”. What, you want something that gives you free money exactly when you need it? It’s called a tax farm. Be a government.

@Dan: I agree that we don’t know the exact probability of a 50% loss in our portfolios. But modeling isn’t always about exactness. To act we must make assumptions and those assumptions constitute an approximate model. When I walk across the floor in my employer’s building, I assume that the probability the floor will give way and I will plunge to my death is small enough that I can ignore it. I don’t know this probability with any accuracy, but I’m assuming it is below some threshold. As for the 50% loss, if I believed that this probability were so low that I could ignore it, I’d probably use a little leverage. But I don’t because I think this probability is high enough that it is worth making sure that I wouldn’t be devastated by a 50% loss. By choosing to avoid leverage, I’m using a crude model of the probability of large losses. So, while I agree that mathematical models of Texas Hold’em have an accuracy that cannot be achieved in investing, it is impossible to make a decision without implicitly using a model, even if we’re not aware that we’re doing it.

I totally agree. This business of confusing risk with volatility is long overdue for scrutiny. Bonds have not been at all volatile recently (interest rates constant), but they do look like a highly risky asset class.

Hi, this is my first time posting. I love your blog, thanks for all the time you’ve spent on it. I’m currently a PhD student (in cs) and I wanted to start investing recently and you blog helped me so much. I read a lot about different strategies and obviously the one you promote makes the most sense =) I also learned about the TD e-series funds here which are great for someone just stating out like myself and also makes little money =)

Anyway, I just wanted to point out that it has been known for a long time that using just standard deviation to model risk doesn’t work on a day to day basis. One way to model derivatives is to use diffusion equations. The simplest equation you might have heard of is the Black–Scholes equation (the equation used in the theory to win a Nobel Prize in economics). It is basically the simplest diffusion model you can have (for derivative) and uses just standard deviation to model risk (actually it is really volatility but people equate them). Anyway, situations where this equation fail are for example like what happened after 911. The event caused a negative spike in stock prices. This can be modeled using jump diffusion equations. The idea is you add in some kind of probability that derivative prices will spike at a random times. The problem with this is it is really hard to determine what parameters (like the probability of a spike) are for a particular derivative. I’m guessing over longer periods of time these spikes don’t really have much influence so just using standard deviation works well.

Also you made a comment about how lots of the best gains have come right after large losses. This can also be modeled. It is known as mean reversion. Basically your equation takes into account that if the price of a derivative is different from some magical growing price (the mean price) then there will be some bias to return to the mean price. This models the correlation between years of huge losses where the derivative price is probably much lower than it’s magical mean price and years of large gain where the price is moving back towards the magical mean price.

I should have win the book 🙁

No gift for my birthday then …

Anyway … I’m curious and can’t wait for your next post.

Jason,

Nice comments. I read MANY investing sites and this is by far the best in terms of the posts by Dan and in terms of the comments from readers. I have learned more hear that is useful than anywhere else.

Couple of related things: If you have not read Taleb’s Black Swan I encourage you to do so given your field of study.

If you are using the TD e-series funds in a discount broker account then consider several things. Since you could be more math oriented for periodic contributions over time consider using value averaging approaches instead of dollar cost averaging. Its useful to take advantage of the automatic dividend reinvestment features as well. It may be possible to use a combo of dollar cost and value averaging. You can set it up to automatically invest a certain proportion every month in each fund according to asset allocation and then every six months check where the balances are intervene and rebalance using lump sums based on value paths.

http://www.valueaveraging.ca/

One thing about risk that many people forget to consider is there is a big behavioural difference between how a loss is apprehended versus the feeling obtained from a gain. The dread from a loss is about twice as bad as the joy of a gain in emotional “weight”. Prospect theory talks about this. I learned much about this by studying the papers that family offices put out. It is something that can be considered in asset allocation.

There is also a big difference between apprehension of a loss in percentage terms verses absolute dollar terms because the dollar figure is more concrete.