Risk (without charts or algebra)

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Risk is a central concept in finance, yet its definition is very controversial. As well, much of modern finance assumes that higher risk investments deliver, on average, higher returns (e.g., [1][2][3][4]). However, in recent years this relationship has been challenged. We look at each of these questions in turn.

One thing finance never mentioned in the early advanced textbooks on asset pricing, because they never expected this problem, is that risk as a practical matter is insanely subtle.

— E.G. Falkenstein, Risk and Return in General: Theory and Evidence[5]

Definition of risk

Commonly people think of risk as the possibility of a loss.[6] It may be that, when the time comes to sell a security, the price one receives is less than what one originally paid.[7][8] In a slightly different context, if one’s house burns down, one has lost an amount of money equal to what must be spent to restore the house (or, if buying a new house, the price of the new house less the value of the land on which the old house stood).

In many cases, it is possible to identify the various losses that can occur if a certain asset is owned, e.g. a drop in share price because a company’s earnings are lower than expected. It may even be possible to assign probabilities to losses of various magnitudes, e.g. a 1% chance that the loss is greater than half the value. Assigning such probabilities to losses of various magnitudes is the heart of any risk assessment of an investment or other project.

In other cases, we do not have enough information to estimate such probabilities. Indeed, losses may be due to totally unforeseen events, e.g. a meteor strike wipes out the factory in which we own shares. Some people use the word “uncertainty” rather than “risk” to denote the possibility of loss in such circumstances.[9][10] Used in this way, uncertainty is risk that we, as investors, cannot really get a handle on. We should be aware that it exists, and we may even guess that it is greater for certain investments than for others. But since we can’t quantify it in any meaningful way, it is usually left out of formal analyses of investments. We do likewise in what follows. However, just because we don’t explicitly talk about uncertainty doesn’t mean that it isn’t there: investor beware.

Measures of risk

It can be tedious to estimate a large number of probabilities corresponding to various levels of loss. Most discussions of investments use a small number of summary measures, in the hopes that these will capture the essence of risk. In what follows, we outline a number of popular risk measures and their shortcomings. Needless to say, various finance professionals keep themselves busy inventing and advocating new, improved summary risk measures. Most of these are confusing for the individual investor, without adding much value, and we do not discuss them here.

Volatility or variance

The main measure of risk used by the financial community is volatility, i.e. fluctuations in the value of an investment.[11] This can be measured either by changes in the market value (or price) of a security, or changes in its rate of return. These are two different measures, and can give different results. Unfortunately, discussions of risk often do not specify which risk is being referred to.

Volatility reflects both how often fluctuations occur, and how big those fluctuations are when they do occur. The typical investor dislikes large fluctuations more than small ones. Generally, infrequent but very large fluctuations are more distressing than frequent small fluctuations. Unfortunately, the extent to which this is true varies from investor to investor. As a compromise, volatility is often measured by weighting individual fluctuations by the square of their size, and then averaging over all expected fluctuations. The result is referred to as the variance of the anticipated return, and is generally used as a measure of volatility. (Sometimes the square root of the variance, known as the standard deviation, is used.[4][11][12])

Note that the measure of volatility just described, the variance, increases as both positive and negative fluctuations increase in size and magnitude. This is counter-intuitive to many people. After all, risk is associated with losses, or negative fluctuations, not with gains, or positive fluctuations. The answer is that, IF the distribution of fluctuations is symmetric, i.e. positive fluctuations are as likely and as large as negative ones, it doesn’t matter. If you count both positive and negative fluctuations, you will get a result that is twice as large as if you counted just the negative ones. Since counting all the fluctuations is easier than sorting and counting just the negative ones, the variance, which counts all fluctuations, is used because of its convenience. (Note that using the variance depends crucially on the assumption that the distribution of returns is symmetric. If it is not, then some other measure might be preferable, e.g. the semi-variance.)

Systematic versus non-systematic risk

Many financial analysts maintain that, while the volatility (or variance) of an investment captures important aspects of risk, it omits others. There are two ideas using the notion of covariance as a measure of risk. The first is that a loss is more painful if it occurs during bad times than if it occurs during good times. The second is that certain risks can be diversified away, i.e. reduced by pooling different risks, while others cannot be so diversified. We start with the second idea, and then return to the first and show how it is related.

The idea behind diversification is that “bad things” (i.e. losses) happen to some degree at random. For example, not all corporations will lose market share at the same time. Not all technologies will become obsolete at the same time. More vividly, not all corporate management will be shown to be crooks who have been systematically robbing shareholders. While market share loss, obsolescence, dishonesty and other such factors do occur, they do not happen to all investments, and even when they do occur, they do not all occur at the same time. Importantly, we cannot predict where or when they will occur.

One way investors can protect themselves is by buying a little bit of each of many different corporations. While any one corporation may do badly, the investor has only a small portion of his portfolio in that company, and will not suffer any big loss. Extending this, the investor is prudent to spread his money out over as many different opportunities as possible. Thus, having all one’s money in equities (common shares) is riskier than spreading it out over bonds, real estate, commodities, and other investments as well: the different pieces can be smaller. As well, the factors that can cause losses will vary from equities to bonds to real estate to commodities. It is unlikely that all of these factors will be at work at the same time, so it is unlikely that the investor will suffer losses on different pieces of his portfolio at the same time.

It follows that a well-diversified portfolio is one whose individual pieces are affected by different factors, and hence, where the different pieces are unlikely to be affected at the same time. If one invests in many different opportunities, but they all react in the same way to the same factors, very little risk reduction will have been achieved, and the portfolio is not well diversified. If one looks at a given set of assets, there are limits to how much one can diversify. For example, consider the equities of all companies traded on all Canadian stock exchanges. The vast majority of them are sensitive to the business cycle, i.e. their prices will increase when GDP is increasing and their prices will fall or stagnate when GDP is falling. By looking at just the set of equities, one cannot diversify away the risk of a downturn in the economy. Thus this risk is considered non-diversifiable or systematic.

Financial analysts often partition risk into risk that cannot be reduced through diversification (called “systematic risk” or “nondiversifiable risk” or “market risk”) and risk that can be so reduced (called “idiosyncratic risk” or “diversifiable risk” or “systematic risk”).[4][13] To the degree that an investment’s value depends on the business cycle (e.g. through growth in sales), that risk is systematic and cannot be reduced by investing over a large number of companies. To the degree that an investment’s value depends on other factors (e.g. corporate malfeasance and fraud), that risk is non-systematic and can, in theory, be diversified away.

It is often assumed that a portfolio reflecting the composition of the market as a whole (often referred to as a “market portfolio”) will show only systematic risk. This is because, if an investor is holding a portfolio that is proportional to the market, he has already invested in all the assets in the relevant set, and so he cannot diversify any further. This is true only if one limits one’s investment opportunities to the stocks composing that market. If one has access to other opportunities, e.g. real estate, then further diversification might be possible, and the risk of the market as a whole might not all be systematic. In what follows, however, we assume that the market has been defined broadly enough to include all realistic investment opportunities, and so that the market’s risk is wholly systematic.

For an individual stock, some of the fluctuations in value happen at the same time that there are fluctuations in the market portfolio, and some fluctuations that occur at other times. Those fluctuations that occur at other times can, in theory, be largely offset by investments in other stocks; after all, the market as a whole has not fluctuated. Thus the associated risk is diversifiable by adding other stocks to the portfolio, and so is non-systematic. By contrast, fluctuations in the individual stock that happen when the market as a whole fluctuates (in other words, synchronized with market fluctuations) cannot all be diversified away. By definition, there are not enough other stocks in the market to be able to offset these fluctuations. Thus, fluctuations in the value of individual stock that take place when the market portfolio fluctuates are not diversifiable, and so constitute systematic risk.

It remains to measure this systematic risk. The convention is that each individual stock is assigned an index, called “beta” in the jargon.[4] We look at those fluctuations in value that are common both to the individual stock and to the market portfolio (other fluctuations do not contribute to systematic risk). If these fluctuations, on average, are the same in magnitude for the individual stock and for the market portfolio, we assign a beta of one to the stock. If fluctuations of the stock, on average, are half as big as fluctuations in the market portfolio, beta for the stock is 0.5. Similarly, if fluctuations for the individual stock, on average, are twice as large as for the market portfolio, beta is 2. The higher the value of beta, the more risky (in terms of systematic risk) the individual stock.

Note that, by definition, the beta of the market portfolio is one. Note also that one can consider different markets, and so different market portfolios, e.g. all companies trading on U.S. exchanges, all stocks trading on public exchanges anywhere in the world, all stocks trading on public exchanges and over-the-counter, all stocks and bonds, and so on. Since the definition of the beta of an individual stock is with respect to a given market portfolio, one can have different values of beta, depending on the market that is used as a reference point.

Measuring volatility and beta

It is customary to measure beta for an individual stock, or an individual security more generally, using historic data.[4] For the volatility, one calculates the variance of past market values or past returns. For beta, one estimates a linear regression, with the return or value of the individual stock as the dependent variable, and the return or value of the market portfolio as the independent variable. Beta is the estimated coefficient of the independent variable (hence its name).

Note that some people define beta to be the coefficient of such a regression equation. This is incorrect. The regression coefficient is merely one way, albeit the most commonly used, to measure beta. Other approaches to measuring beta are possible. Hence the definition should be independent of the measuring instrument used.

In any case, using historic data to measure volatility or systematic risk (beta) is problematic. To obtain accurate estimates, one would like a lot of observations under different conditions. That requires going back quite far in time.[14] But this in turn risks including data that are too old to be relevant now, e.g. because circumstances have changed.[14] Alternatively, one can use relatively recent data, but then one may not have enough observations to obtain a reliable measure.

Some financial analysts try to solve this trade-off by using data from very short intervals, e.g. weekly, daily, or even hourly. This data, they hope, will be both plentiful and recent. But it is not clear that very short-term volatility or beta are relevant to financial planning that is done in terms of years, e.g. a retirement plan. As well, it is not at all clear that short term patterns are any indication of long term patterns. Thus, picking short time intervals may be helpful to day traders or algorithmic trading programs. It is unlikely to be helpful to long term investors.

A more fundamental problem is that variances and betas vary over time. For example, Fernandez (2017) showed that the beta of single US companies based on 5 years of monthly data versus the S&P500 changed dramatically from one day to the next.[15] A historic measure of either volatility or systematic risk may be a very poor indicator of either of these going forward. In general, the theory postulates a relationship between expected future returns and future risk. It is not clear how relevant measures of actual or experienced returns and volatility are to investors looking at the future.

Maximum expected drawdown

Many retail investors do not perceive either volatility or systematic risk as measured by beta to be particularly relevant to them. Rather, they are interested in the biggest loss that they can reasonably expect (or worry about). It is the very large loss, not the smaller fluctuations, that shake their confidence in their investment strategy. In many cases, when such large losses exist, the retail investor may exit the equity markets completely, missing out on the recovery that follows.

Some advisers will talk about the maximum drawdown, i.e. the largest loss that can be expected with an investment approach. Such a large loss will happen infrequently, say once every twenty years (five per cent of the time), or once every thirty years (i.e. three per cent of the time). The historic record can be examined to discover how big these losses have been. However, as usual, historic experience is an imperfect guide to the future and even bigger losses can occur. Using notions of maximum drawdown, however, an investor can better think about the worst that can happen to his investment.

Risk and return

See also: Risk and return

Until the turn of the twenty-first century it was widely accepted as self-evident that the riskier an investment, the higher the expected return, to compensate the investor for taking on the extra risk (e.g., [4]). It followed that the only way to increase expected returns was to invest in increasingly risky assets. Recently, however, financial analysts and observers have questioned whether this is always true. This questioning has taken two forms. The first holds that price depends on factors other than risk. i.e. the riskiness of an asset, whether measured by volatility or systemic risk, is not enough to explain the price or expected return of the asset. The second version postulates that at least some people like risk in certain circumstances, and are willing to accept lower returns on assets that are very risky. i.e. some investment is driven, at least in part, by speculation, and can be characterized, again in part, as a form of gambling. We discuss each of these in turn.

Other factors

It has long been known that knowledge of a company’s volatility or beta (i.e. systematic risk) by itself does not allow a very good prediction of the company’s return, even on average, and even over extended periods of time. Other factors are at play to help determine the company’s expected return. Two such factors have been popularized by Eugene Fama and Kenneth French.[16] One is a company’s price-to-book ratio, expressed as P/B. The lower P/B for a given company, the higher its expected return, even after taking into account its beta. The intuition is that companies with low P/B ratios are “value” companies, whose prices have been unreasonably beaten down.

Other analysts interpret a low P/B ratio as an indicator that the company in question is undergoing hard times, and that this is an additional risk not captured by beta. This line of reasoning maintains the relationship between return and risk, but at the price of extending the definition of risk in ways that some people may find non-intuitive.

The second additional factor uncovered by Fama and French is a company’s size. The smaller a company the higher is the expected return, even after taking into account its beta and its P/B ratio. A common explanation is that smaller companies may be less liquid and so more difficult for institutions such as pension funds to invest large amounts of money. Another explanation is that smaller companies tend to be overlooked and so tend to be undervalued. Finally, those analysts who persist in maintaining the risk-return link argue that smaller companies are intrinsically riskier, e.g. due to a greater probability of going bankrupt, and that this is yet another additional risk not captured by beta.

Recently a fourth factor has been added to the list of commonly invoked explanations for a company’s expected return. This is momentum, i.e. the tendency, in the short run, for rising share prices to keep rising, and for falling share prices to keep falling.[17][18][19] Inclusion of this leads to the widely used “four-factor model”: expected return of a company’s stock is a linear function of the company’s beta, P/B ratio, size, and momentum. The weights are estimated from historic data, usually through regression analysis. As mentioned above, however, the relevance of weights based on historic data is not clear. What is wanted are the weights which are expected to govern in the future, and history may not be of much help here.


Lately a number of people have noted that the behaviour of many individual investors suggests that they may be motivated at least in part by a desire to speculate, or gamble. This is seen in its most extreme form in purchasers of lottery tickets, who accept a very unfavourable bet (on average, only a quarter to a third of purchasers’ money is returned in the form of prizes). The attraction is that if a lottery ticket pays off, it pays off big. The one or ten million dollars that the purchaser stands to win will fundamentally alter his or her life. By contrast, the one or ten dollar purchase price, which the purchaser will almost certainly lose, is a small amount and its loss will not have any impact on the purchaser’s life style. In such a case, the purchaser actually likes risk and seeks it out, rather than trying to avoid or minimize it.[20]

The same behaviour, in less extreme form, can be seen on the stock market.[21] Some individual investors seek out IPOs (new issues of shares by young companies) even though these are not a particularly good investment, on average.[22][23] The occasional IPO, such as Amazon.com or Google, does extremely well and can make its early purchasers very rich. It is the dream of finding the next Amazon.com that motivates some investors, and drives up the price (and hence drives down future returns) of these issues.

The same phenomenon can be found in many other investments. People may actually favour very volatile stocks, because of the possibility of making a lot of money, even though the probability of doing so is actually low. In such situations, people may actually accept a lower expected return for the (small) chance of making big gains.

Based on systematic studies, some analysts have concluded that there is actually no relationship between expected return and risk (e.g., [5]). Indeed, if there is a relationship, it may be that less risky assets actually have a greater return, although not much. This view leads to an investment strategy of choosing low-volatility portfolios, on the assumption that these will have good returns (as good as the market portfolio), but with much less risk (as expressed by volatility or systematic risk). A number of exchange-traded funds (ETFs) have been launched during the last few years with the objective to invest in low-volatility stocks, or to construct a portfolio with minimum variance.

See also


  1. Chapter 9 in Malkiel, B., 2011, A Random Walk Down Wall Street, W. W. Norton & Company
  2. Chapter 2 in Ang, A., 2014, Asset Management - A Systematic Approach to Factor Investing, Oxford University Press, 720 p. (ISBN 9780199959327)
  3. Ilmanen, A., 2011, Expected Returns. Wiley, 592 p. (ISBN 978-1-119-99072-7)
  4. 4.0 4.1 4.2 4.3 4.4 4.5 Womack, K.L., Zhang, Y., 2003, Understanding Risk and Return, the CAPM, and the Fama-French Three-Factor Model. Tuck School of Business, Tuck Case No. 03-111. Available at SSRN, viewed December 28, 2017
  5. 5.0 5.1 Falkenstein, E.G., 2009, Risk and Return in General: Theory and Evidence. Available at SSRN, viewed December 28, 2017.
  6. Definition of Risk from English Oxford Dictionary: “1.5 The possibility of financial loss”, viewed December 27, 2017.
  7. Definition of Risk from Merriam-Webster Dictionary, “4: the chance that an investment (such as a stock or commodity) will lose value”, viewed December 27, 2017
  8. Definition of risk: “Defined in its most basic way, risk is the possibility of loosing money” -- William Bernstein, 2002, The Four Pillars of Investing, McGraw-Hill, page 11
  9. Federal Reserve Bank of St. Louis, The Stock Market: Risk vs. Uncertainty, Fall 2012, viewed December 27, 2017
  10. Huffington Post, Financial Risk vs. Financial Uncertainty: A Big Distinction That Everyone Needs to Understand, April 2017, viewed December 27 ,2017.
  11. 11.0 11.1 Investopedia definition: Volatility, viewed December 27, 2017
  12. Explanation of standard deviation: see William Bernstein, 2002, The Four Pillars of Investing, McGraw-Hill, page 24, footnote 1
  13. N. Chapurat, H. Huand and M.A. Milevsky, 2012, Strategic financial planning over the lifecycle. Cambridge University Press, p. 221 (ISBN 0521148030)
  14. 14.0 14.1 Bartholdy, J., Peare, P. 2001, The Relative Efficiency of Beta Estimates. Available at SSRN, viewed December 28, 2017
  15. Fernandez, P. (2017) Ten badly explained topics in most Corporate Finance Books. IESE Business School, University of Navarra. Available on SSRN, viewed December 28, 2017
  16. Fama, E.F., French, K.R. 1992, The cross-section of expected stock returns Journal of Finance 47, 427-465 (PDF), viewed December 28, 2017
  17. Jegadeesh, N., Titman, S., 1993, Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency. Journal of Finance, vol. 48, pp. 65-91 (PDF), viewed December 28, 2017.
  18. Asness, C. et al., 2014, Fact, Fiction and Momentum Investing, Journal of Portfolio Management, 40th Anniversary Issue. Available at SSRN, viewed December 28, 2017.
  19. Chapter 10 in Ang, A., 2014, Asset Management - A Systematic Approach to Factor Investing, Oxford University Press, 720 p. (ISBN 9780199959327)
  20. Friedman, M., Savage, L.J., 1948, The Utility Analysis of Choices Involving Risk, Journal of Political Economy, vol. 56, no. 4, 1948, pp. 279–304 (PDF), viewed December 28, 2017.
  21. Chapter 19 in Ilmanen, A., 2011, Expected Returns. Wiley, 592 p. (ISBN 978-1-119-99072-7)
  22. Ritter, J.R., 1991, The long-run performance of initial public offerings, Journal of Finance 47, 3–27 (PDF), viewed December 28, 2017.
  23. Loughran, T., Ritter, J.R., 1995, The new issues puzzle, Journal of Finance 50, 23–51 (PDF), viewed December 28, 2017.

Further reading

Financial Wisdom Forum topic: Risk = ??

External links