How do you quantify investment risk?

It's obvious that investing in stocks and bonds is not a guaranteed return--there is some risk, and this is well known. People talk about risk averse investors or riskier ones, or that one should pick a strategy that suits their comfort with risks, etc.

The question is: how do you quantify investment risk?

This is easy to do in casino games, based on basic probability such as roulette or slots. But what has been done with the various kinds of investment risks? My point is saying that certain bonds are "low risk" isn't good enough; I'd like some numbers--or at least a range of numbers--and therefore one could calculate expected payoff (in the statistics sense).

Or can it not be done--and if not, why not?

I'm aware that Wall Street is nothing like roulette, but then again there must be some math and heavy economics behind calculating risk for individual investors. This is, after all, what "quants" are paid to do, in part. Is it all voodoo? I suspect some of it is, but not all of it.

I also tend to think that when people point to the last x number of years of stock market performance, that is of less value than they expect. Even going back to 1900 provides "only" 110 years of data, and in my view, complex systems need more data than those 40,500 data points. 10,000 years' worth of data, ok, but not 110.

Any books or articles that address these issues, or your own informed views, would be helfpul.

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I think maybe you should be asking this in quants.stackexchange.com – Chuck van der Linden Aug 1 '11 at 17:52
@ Chuck - I disagree. That site is for finance professionals and academics. I don't think they would welcome a question on the basics of evaluating risk. – BlackJack Aug 2 '11 at 14:57

The question is: how do you quantify investment risk?

As Michael S says, one approach is to treat investment returns as a random variable. Bill Goetzmann (Yale finance professor) told me that if you accept that markets are efficient or that the price of an asset reflects it's underlying value, then changes in price represent changes in value, so standard deviation naturally becomes the appropriate measure for riskiness of an asset. Essentially, the more volatile an asset, the riskier it is.

There is another school of thought that comes from Ben Graham and Warren Buffett, which says that volatility is not inherently risky. Rather, risk should be defined as the permanent loss of capital, so the riskiness of an asset is the probability of a permanent loss of capital invested.

This is easy to do in casino games, based on basic probability such as roulette or slots. But what has been done with the various kinds of investment risks? My point is saying that certain bonds are "low risk" isn't good enough; I'd like some numbers--or at least a range of numbers--and therefore one could calculate expected payoff (in the statistics sense).

Or can it not be done--and if not, why not?

Investing is more art than science. In theory, a Triple-A bond rating means the asset is riskless or nearly riskless, but we saw that this was obviously wrong since several of the AAA mortgage backed securities (MBS) went under prior to the recent US recession. More recently, the current threat of default suggests that bond ratings are not entirely accurate, since US Treasuries are considered riskless assets.

Investors often use bond ratings to evaluate investments - a bond is considered investment grade if it's BBB- or higher. To adequately price bonds and evaluate risk, there are too many factors to simply refer to a chart because things like the issuer, credit quality, liquidity risk, systematic risk, and unsystematic risk all play a factor.

Another factor you have to consider is the overall portfolio. Markowitz showed that adding a riskier asset can actually lower the overall risk of a portfolio because of diversification. This is all under the assumption that risk = variance, which I think is bunk.

I'm aware that Wall Street is nothing like roulette, but then again there must be some math and heavy economics behind calculating risk for individual investors. This is, after all, what "quants" are paid to do, in part. Is it all voodoo? I suspect some of it is, but not all of it.

Quants are often involved in high frequency trading as well, but that's another note. There are complicated risk management products, such as the Aladdin system by BlackRock, which incorporate modern portfolio theory (Markowitz, Fama, Sharpe, Samuelson, etc) and financial formulas to manage risk.

Crouhy's Risk Management covers some of the concepts applied.

I also tend to think that when people point to the last x number of years of stock market performance, that is of less value than they expect. Even going back to 1900 provides "only" 110 years of data, and in my view, complex systems need more data than those 40,500 data points. 10,000 years' worth of data, ok, but not 110.

Any books or articles that address these issues, or your own informed views, would be helfpul.

I fully agree with you here. A lot of work is done in the Santa Fe Institute to study "complex adaptive systems," and we don't have any big, clear theory as of yet. Conventional risk management is based on the ideas of modern portfolio theory, but a lot of that is seen to be wrong. Behavioral finance is introducing new ideas on how investors behave and why the old models are wrong, which is why I cannot suggest you study risk management and risk models because I and many skilled investors consider them to be largely wrong.

There are many good books on investing, the best of which is Benjamin Graham's The Intelligent Investor. Although not a book on risk solely, it provides a different viewpoint on how to invest and covers how to protect investments via a "Margin of Safety."

Lastly, I'd recommend Against the Gods by Peter Bernstein, which covers the history of risk and risk analysis. It's not solely a finance book but rather a fascinating historical view of risk, and it helps but many things in context.

Hope it helps!

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The standard measure of risk is the variance of the asset. The return on investment of the asset is understood as a random variable with a particular distribution. One can make inferences about the underlying distribution using historical data. As you say, this is what the quants do. There are other, more sophisticated measures of risk that allow for such things as skewed distributions and Markov switching.

If you are interested in learning more, I suggest starting with the foundations of Modern Portfolio Theory: "Portfolio Selection" by Harry Markowitz and "Capital Asset Prices" by William Sharpe.

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Although I'm sure many people use it this way, the variance is not really what people are concerned about when they are concerned about risk. They are generally concerned about the likelihood of losing big chunks of money. A random variable with a higher variance but also higher mean could (at least in principle) offer lower "risk" in the sense that I think most people understand the term. – Michael McGowan Aug 4 '11 at 20:56

Another approach would be more personalized, which is to measure the risk of missing your goals, rather than measuring the risk of an investment in some abstract sense.

Financial planners do this for example with Monte Carlo simulation software (see http://en.wikipedia.org/wiki/Monte_Carlo_method). They would put in a goal such as not running out of money before you die, with assumptions such as the longest you might live and how much you'll withdraw every year. You'd also assume an asset allocation.

The Monte Carlo simulation then generates random market movements over the time period, considering historical behavior of your asset allocation, and each run of the simulation would either succeed (you are able to support yourself until death) or fail (you run out of money).

The risk measure is the percentage of simulation runs that fail.

You can do this to plan saving for retirement in addition to planning withdrawals; then your goal would be to have X amount of money in real after-inflation dollars, perhaps, and success is if you end up with it, and failure is if you don't.

The great thing about this risk measure is that it's relevant and personal; "10% chance of being impoverished at age 85," "20% chance of having to work an extra decade because you don't have enough at 65," these kinds of answers. Which is a lot easier to act on than "the variance is 10" or "the beta is 1.5" - would you rather know your plan has a 90% chance of success, or know that you have a variance of 10? Both numbers are probably just guesses, but at least the "chance of success" measure is actionable and relevant.

Some tangential thoughts FWIW:

• I argued in another post that rules of thumb are probably as good as or better than detailed quantification: Saving for retirement: How much is enough?
• my opinion is that one should optimize for chance of success, not for things like max returns per risk or beating the market: http://blog.ometer.com/2010/11/10/take-risks-in-life-for-savings-choose-a-balanced-fund/
• don't underestimate "behavioral risk" which is the risk that you do something bad during a panic or bubble or personal emergency. You can ruin years of diligence with one day of emotional action. (this risk can be minimized by reducing the number of decisions you have to make, actions you have to take, and volatility of the portfolio)
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For a retail investor who isn't a Physics or Math major, the "Beta" of the stock is probably the best way to quantify risk.

Examples:

• IBM - 0.71
• AAPL - 1.31
• ED - 0.31

A Beta of 1 means that a stock moves in line with the market. Over 1 means that you would expect the stock to move up or down faster than the market as a whole. Under 1 means that you would expect the stock to move slower than the market as a whole.

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