Risk and return

Risk is a complex topic. There are many types of risk, and many ways to evaluate and measure risk. In the theory and practice of investing, a widely used definition of risk is:

Even though this definition of risk sounds simple, questions arise:
 * “Risk is the uncertainty that an investment will earn its expected rate of return.”


 * Why is risk defined in terms of the uncertainty of investment return?
 * The definition above does not distinguish between gain and loss. Isn't risk just the possibility that the investor will lose money on the investment?
 * What is the relationship between risk and return?
 * How is risk measured?
 * What is "expected rate of return", and how is it evaluated?
 * What can be done to manage risk?
 * How does an investor determine how much risk and what types of risk to take?
 * Are there differences between short-term risk and long-term risk?

To make wise investment decisions, an investor must spend some time studying the answers to these questions.

Risk as the uncertainty of returns
There is very high certainty in the return (short for "rate of return") that will be earned on an investment in a 30-day Treasury bill (T-Bill) or short-term Certificate of Deposit (CD). Similarly, there is fairly high certainty in the return that will be earned over a short period in a money market fund.

Even over longer time periods, the returns earned by T-Bills, CDs and money market funds fall into a relatively narrow range.



As seen in the top chart in Figure 1 (covering the years 1928 through 2011), the tallest bar shows that annual returns (horizontal axis) on 3-month T-Bills have fallen in the range of 0% to 5% in 59 years (vertical axis). Returns have been between 0% and 10% in most years (tallest two bars), and between 0% and 15% in all years (all three bars).

T-Bills and CDs are among the investments referred to collectively as money market securities. For an individual investor, a federally-insured bank account also provides a high degree of certainty in the short-term return. The term cash often is used to refer to money market securities and money in bank accounts. Vanguard refers to these types of assets as short-term reserves.

The return on bonds is less certain, and the return on stocks is even more uncertain. Thus, bonds are considered riskier than money market securities (cash), and stocks are considered riskier than bonds.

The middle chart in Figure 1 shows the range of annual returns on 10-Year Treasury Bonds from 1928 through 2011. Note the larger range (dispersion) of returns--from about -11% to +33%.

The bottom chart in Figure 1 shows the range of annual returns on stocks in the S&P 500 from 1928 through 2011. Note the much larger dispersion of returns--from about -44% to +53%.

Risk as the possibility of loss
Note that the definition of risk as the uncertainty of investment return does not distinguish between loss and gain. Typically, individual investors think of risk as the possibility that their investments could lose money. They are likely to be quite happy with an investment return that is greater than expected (a “positive surprise”). However, since risky assets generate negative surprises as well as positive surprises, defining risk as the uncertainty of the return is reasonable. Greater uncertainty results in greater likelihood that the investment will generate gains as well as losses. The long-term consequence is that there is greater uncertainty in the value of accumulated wealth. This is referred to as terminal wealth dispersion.

In financial planning, the investment goal must be considered in defining risk. Once a goal is established, for example the goal of providing retirement income, an investment portfolio is constructed to generate an expected return that is sufficient to meet the investment goal. But because there is uncertainty that the portfolio will earn its expected long-term return, the long-term realized return may fall short of the expected return. This raises the possibility that the investor's available funds could fall short of the need, and that the investor might outlive the investment portfolio. This is an example of "shortfall risk". The magnitude and consequences of potential shortfall deserve special consideration from investors. However, since the uncertainty of return could result in a realized return that is higher than the expected return, the investment portfolio might "outlive" the investor. Therefore, considering risk as the uncertainty of investment return subsumes considerations of shortfall risk.

Some financial theorists have proposed using measures of risk that only account for the risk of loss (e.g., semi-variance instead of variance, to be discussed later). However, most academic investment theory, especially portfolio theory, is based simply on the uncertainty of return, considering both positive and negative outcomes relative to expectations. Justifications include the observation that long-run investment returns are reasonably symmetrical, and that long-run returns of broadly diversified investment portfolios are reasonably normally distributed. (add references). Although these observations have been challenged (according to Wikipedia article on MPT, etc.), this article is based on standard investment theory based on the stated observations.

Relationship between risk and return
Investors are risk averse; i.e., given the same expected return, they will choose the investment for which that return is more certain. Therefore, investors demand a higher expected return for riskier assets. Note that a higher expected return does not guarantee a higher realized return. Because by definition returns on risky assets are uncertain, an investment may not earn its expected return.

Although the charts in Figure 1 show historical (realized) returns rather than expected (future) returns, they are useful to demonstrate the relationship between risk and return. Note that the mean (average) annual return increases as the dispersion of returns increases.

This demonstrates one of the most fundamental axioms of investing: ''Risk and return are inextricably related. Higher returns generally can be achieved only by taking more risk, but because the risk exists, the higher expected returns may not result in higher realized returns.''



"Risk free" assets


Money market securities are often referred to as risk-free assets, especially the shorter-maturity securities such as 30-day T-Bills. This is because the short-term return is known with relative certainty at the time the investment is made. There is absolute certainty in the nominal return of a T-Bill (assuming the U.S. government does not default on its obligations), and it is unlikely that unexpected inflation will have significant impact on the real return over a short time period.

If longer time periods are considered, even money market securities have some risk. This is because the effect of unexpected inflation on returns is uncertain over longer time periods. Although money market security rates usually respond relatively quickly to changes in inflation, this is not always the case.

Figure 2 illustrates the longer-term uncertainty of real returns on 90-day T-Bills. Also, note that the relative certainty of return does not mean that the real return necessarily is positive. It may be known with certainty that a 90-day T-Bill will earn a nominal annualized return of 1% over its 90-day term (i.e., if held to maturity). However if inflation over the 90-day term is expected to be 3% annualized, the relatively certain, expected real return is -2% annualized. T-bill data source comes from Annual Returns on Stock, T.Bonds and T.Bills: 1928 - Current, Damodaran Online; inflation data comes from Consumer Price Index 1913 -, Minnesota Federal Reserve]. Retrieved 1 April 2012

Uncertainty in real returns can be eliminated by investing in inflation-indexed securities, such as Treasury Inflation Protected Securities (TIPS) and Series I Savings Bonds (I Bonds). Of course in return for this reduction in uncertainty, investors must accept lower expected returns. Marketable inflation-indexed securities also have other risks, such as interest rate risk (i.e., prices decline when interest rates rise) and liquidity risk, as was made evident in late 2008 (September 12 - October 31) when the Vanguard Inflation-Protected Securities fund declined in value by almost 14%. During this same time period other U.S. treasury securities increased in value.

Measuring risk: historical returns
The risk of an investment is related to the uncertainty of the investment returns. This was illustrated graphically in Figure 1, which showed that dispersion of returns can be used to estimate the riskiness of various investments. Most investors like to summarize return and risk with numbers that quantify the average return and the dispersion of returns around the average. This facilitates comparing the return and risk of different investments.

Calculating the average of a set of historical annual returns is straightforward: simply add the annual returns and divide by the number of annual returns. The technical term for this type of average is the arithmetic mean, usually referred to simply as the mean. Statisticians use the term expected value to refer to the mean. In finance theory the term expected value of the return, or just expected return is commonly used in referring to the average of a set of returns.

What about calculating a number to quantify the uncertainty or dispersion of returns? Basic statistics and probability theory provide two measures to quantify the dispersion of a set of numbers: variance and standard deviation. Starting in the early 1950s, finance academics began using these statistical measures to quantify risk.

Historical returns are commonly used as a starting point in evaluating the relative riskiness of different assets. A subset of the historical data shown graphically in Figure 1 provides a simple example to begin developing an understanding of standard deviation as a measure of risk.

Consider the S&P 500 stock returns for 2008-2011:


 * 2008: -36.55%
 * 2009: +25.94%
 * 2010: +14.82%
 * 2011: +2.07%

The mean, or expected value, E(r), of this set of returns is calculated below (for simplicity, the percent signs are dropped):


 * $$ \operatorname{E}(r) = \frac{-36.55 + 25.94 + 14.82 + 2.07}{4} = 1.57 $$

The expected value of the annual return is 1.57%.

An alternate way to calculate expected value is to multiply each value by the probability of that value occurring, then sum the results. This formulation will be useful when calculating the expected value of estimated future returns. When evaluating historical returns, each return is considered to have equal probability, so the calculation using this method is:



\operatorname{E}(r) = \frac 14(-36.55)+\frac{1}{4}(25.94)+\frac{1}{4}(14.82) + \frac{1}{4}(2.07) = 1.57 $$

The average of the annual returns is useful information, but it doesn't indicate anything about the dispersion of returns; i.e., how the returns are distributed around the mean return.

It is intuitive that a useful measure of dispersion of returns around the average return would involve calculating the differences (deviations) between the individual returns and the average return, and perhaps averaging these deviations. For example the deviation for the 2009 return is 25.94 - 1.57 = 24.37, whereas the deviation for the much smaller 2011 return is only 2.07 - 1.57 = 0.50.

The problem with this is that there will be both positive and negative deviations that will tend to cancel each other out, resulting in a misleading value that understates the dispersion of returns around the mean; in fact, the average of the deviations from the mean will always be 0. . This can be verified by calculating the average of the deviations from the mean for all four years in the example. Therefore, this value indicates nothing about dispersion.

This problem is resolved by squaring the deviations from the mean (which results in all positive numbers), then calculating the average of the squared deviations. The term variance is used to describe the average of the squared deviations. Here is the calculation of the variance of the annual returns using the returns from 2008-2011:



\operatorname{Var}(r) = \sigma^2 = \frac{ (-36.55-1.57)^2+(25.94-1.57)^2+(14.82 - 1.57)^2+(2.07-1.57)^2 } {4} = 556 $$

Note that the symbol &sigma;2 is commonly used to represent variance, so Var(r) = &sigma;2 = 556.

This formula can be generalized as:


 * $$\operatorname{Var}(r) = \sigma^2 = \frac{1}{n} \sum_{i=1}^n (r_i - \operatorname{E}(r))^2$$

(When calculating the variance of a sample of the full population of values (as in the example here), statisticians often substitute $$\tfrac{1}{n-1}$$ for $$\tfrac{1}{n}$$ in the above formula. There are arguments for and against this. In this article this substitution will not be made.)

To get a value with the same units as the rate of return (percent), the square root of the variance is calculated, resulting in the standard deviation of the annual returns:


 * $$ \operatorname{SD}(r) = \sigma = \sqrt{556} = 23.6 $$

Note that the symbol &sigma; is commonly used to represent standard deviation, so SD(r) = &sigma; = 23.6 pp (technically, the units of the standard deviation of returns are percentage points (pp), but standard deviation of returns often is displayed using the % symbol, e.g., 23.6%).

Note that the resulting value of 23.6 pp (or 23.6%) for the standard deviation seems somewhat reasonable as an indicator of the dispersion of returns around the mean value of 1.57%, considering that the range of values is between -36.55% and +25.94%.

Standard deviation has some standard statistical interpretations for a large number of values with a normal distribution (the ubiquitous bell curve). For example, about 68% of all values fall within +/- one standard deviation, and about 95% of all values fall within +/- two standard deviations.

There are too few values in our simple example for these statistical attributes to apply, but consider the 84 annual returns for 10-Year treasury bonds represented in the second chart in Figure 1.

First note that the data somewhat resembles the bell-shape curve characteristic of a normal distribution. The mean value of the 84 annual returns for 10-Year treasury bonds is about 5%, and the standard deviation is about 8% (pp). Two standard deviations is about 2 x 8 = 16, so we'd expect most of the annual returns to fall within the range of about 5% +/- 16 pp or between about -11% and +21%. Eyeballing the chart, this looks about right.

For the third chart in Figure 1, representing 84 annual returns for the stocks in the S&P 500, the mean is about 11% and the standard deviation is about 20%, so we'd expect most values to fall within a range of about 11% +/- 40 pp, or between about -29% and + 51%. Again, this looks about right.

So standard deviation seems to be a reasonable measure for the dispersion of annual returns, and the standard statistical interpretation seems to apply pretty well.

Expected return
Expected return has been defined as a synonym for the average of a set of returns. In investing, expected return more often refers to future returns than to historical returns. From here on this will be the context in which the term is used. In this context, expected return can be thought of informally as the return investors expect to receive on an investment. Indeed, the term often is used this way by authors of investment books (as seems to be the case, for example, in Expected Returns by Ilmanen. ) However, in finance theory expected return has a more precise definition.

An investment's expected return, E(r), is calculated as follows:


 * 1) Various economic scenarios are defined.
 * 2) Estimates are developed for the probability, p(s), of each scenario occurring and the return, r(s), for that scenario.
 * 3) The probability and return for each scenario are multiplied together: p(s) x r(s)
 * 4) The results are summed across all scenarios: p(s1) x r(s1) + p(s2) x r(s2) + ... + p(sn) x r(sn)

Thus, expected return is the weighted average of returns across all possible scenarios.

First consider an investment in a 1-year T-Bill. Since the nominal return is unaffected by economic factors, the nominal return is certain; i.e., the probability, p(s), is 100%, and the return, r(s), is the rate of return on the bill. If the rate of return is 1%, the expected nominal return is calculated as:


 * E(r) = p(s) x r(s) = 100% x 1% = 1.0 x 0.01 = 0.01 = 1%

As a slightly more complex, but still simple, scenario analysis, consider a one-year investment in a total stock market index fund, with three possible economic scenarios for the year: recession, normal growth, and strong growth. Assume the following estimates for the three scenarios:


 * Recession: probability, p(s1) = 20%, estimated return, r(s1) = -10%
 * Normal growth: probability, p(s2) = 60%, estimated return, r(s2) = 5%
 * Strong growth: probability, p(s3) = 20%, estimated return, r(s3) = 10%

With these scenario estimates, the expected return, E(r), is calculated as:


 * E(r) = p(s1) x r(s1) + p(s2) x r(s2) + p(s3) x r(s3)
 * E(r) = 0.20 x -0.10 + 0.60 x 0.05 + 0.20 x 0.10 = 0.03 = 3%

Note that the expected return of 3% is not equal to the most probable return of 5%.

Extensive research has demonstrated that there are no good forecasters. It follows that scenario analysis itself is fraught with uncertainty. To put it bluntly, expected returns are not directly observable; they can only be estimated. Nevertheless, financial academics and practitioners develop and publish their estimates of expected returns.

Development notes on external links

Various links to GMO's latest 7-year forecast can be found by searching "GMO 7-Year Asset Class Forecasts (February 2012)". Not sure we want to publicize one of these sites though.

Latest GMO 7-Year Asset Class Return Forecasts is available directly on GMO website (https://www.gmo.com/America/Research/), but registration (free) is required.

(Wiki article Historical and Expected Returns includes the 2011 Ferri estimates, but updated estimates for 2012 are available via link; the Bernstein estimates are way dated (2002).)

Measuring risk: expected returns
(This section needs rewriting, since much of it now appears above in the section on measuring risk of historical returns)

(Moved text from here to Measuring risk: historical returns)

(Text below needs to be reframed in context with what has been written in Measuring risk: historical returns)

Variance of returns, Var(r), is calculated as follows:


 * 1) Subtract the expected return, E(r), from the estimated return, r(s), for the scenario: [r(s) - E(r)]
 * 2) Square the result: [r(s) - E(r)]2
 * 3) Multiply the result by the probability, p(s), of the scenario: p(s) x [r(s) - E(r)]2
 * 4) Sum the resulting values across all scenarios:


 * Var(r) = {p(s1) x [r(s1) - E(r)]2} + {p(s2) x [r(s2) - E(r)]2} + ... + {p(sn) x [r(sn) - E(r)]2}

To develop an intuitive understanding of the calculation of variance, consider the purpose of each step in the calculation.


 * 1) Step 1 yields a number that is larger for a scenario return that is further from the expected return.
 * 2) Step 2 eliminates negative values, so that negative values do not offset positive values; i.e., the measure should increase whether a scenario return is less than or greater than the expected return.
 * 3) Step 3 gives a greater weight to a higher-probability scenario.
 * 4) Step 4 adds the individual, probability-weighted, squared differences to yield a single number.

In general, the variance of a variable has units that are the square of the units of the variable. For example, a variable measured in meters will have a variance measured in square meters. To derive a measure that has the same units as the variable, standard deviation of variable x, SD(x), is calculated as the square root of the variance:


 * SD(x) = SQRT[Var(x)]

Technically, rate of return has no units, since it is the ratio of two numbers with the same units. Nevertheless, return typically is expressed as a percentage, so here percent will be considered the units of return. So it can be said that variance of returns, Var(r), has units of percent squared, and the standard deviation of returns, SD(r), has units of percent.

More precisely, standard deviation of returns has units of percentage points (pp), since the difference between two percentages has units of percentage points, and the [r(s) - E(r)] part of the variance calculation is the difference between two percentages. Nevertheless, standard deviation often is shown as percent (%) rather than as percentage points.

For the case of the 1-year T-Bill with a certain (p(s) = 1) nominal return of 1%, the variance is:


 * Var(r) = p(s) x [r(s) - E(r)]2
 * Var(r) = 1 x [0.01 - 0.01]2 = 0

Since SQRT(0) = 0, the standard deviation is 0 pp. Note the intuitive sense of this: a standard deviation of 0 indicates zero uncertainty in the return.

Recall the three-scenario case discussed earlier:


 * Recession: probability, p(s1) = 20%, estimated return, r(s1) = -10%
 * Normal growth: probability, p(s2) = 60%, estimated return, r(s2) = 5%
 * Strong growth: probability, p(s3) = 20%, estimated return, r(s3) = 10%

The expected return, E(r), was calculated as 3%.

The variance is calculated as (using decimal values instead of percentages and converting to pp at the end):


 * Var(r) = {p(s1) x [r(s1) - E(r)]2} + {p(s2) x [r(s2) - E(r)]2} + {p(s3) x [r(s3) - E(r)]2}


 * Var(r) = {0.20 x [-0.10 - 0.03]2} + {0.60 x [0.05 - 0.03]2} + {0.20 x [0.10 - 0.03]2}


 * Var(r) = {0.20 x [-0.13]2} + {0.60 x [0.02]2 + {0.20 x [0.07]2


 * Var(r) = 0.0034 + 0.00024 + 0.00098 = 0.0046


 * SD(r) = SQRT(0.0046) = 0.07 = 7 pp (or 7%)

Link to good article that covers this in even more detail: Risk and Return: Variance and Standard Deviation

(The following paragraph doesn't really fit here, but holding here temporarily)

The use of historical returns to estimate expected returns was more prevalent prior to the early 1980s. The use of valuation measures (e.g., P/E ratios) to estimate expected returns has increased since then. However, historical returns still are used to help estimate the expected returns.

Managing risk
Already covered at a basic level in Risk and return: an introduction. Perhaps here we dig into portfolio theory, which is the theoretical basis for diversification and asset allocation.

Determining risk exposure
The purpose of this section is to present different frameworks the investor can use to determine the appropriate amount of risk to take. Below are ideas and references for this section. Feel free to flesh it out. The intention is to avoid bias and specific recommendations, but to simply present the frameworks objectively based on the references.

Some authors present risk evaluation as inextricably linked to portfolio construction. Others present it more as a standalone topic.

Following are some references for different frameworks. Consider combining the ones that have a lot of overlap, and/or explaining the commonality and differences; e.g., Swedroe's "willingness" is what most refer to as "risk tolerance".

Boglehead's Guide To Investing, pp. 95-97: Goals, Time Frame, Risk Tolerance, Personal Financial Situation. Age in bonds.

Swedroe: Ability, willingness and need to take risk.

Ferri, All About AA, Chapter 12: Early savers, Mid-life accumulators, Preretirees and active retirees, Mature retirees. Assets, future liabilities, risk tolerance (questionnaires, AA stress test). Risk avoidance.

CFA Institute (Bodie et all, 2008, chapter 21): Investors and Objectives, Investor constraints (Liquidity, Investment Horizon, Regulations, Tax Considerations, Unique Needs.

Bernstein (2010), The Investor's Manifesto, pp. 75-80: Age (human capital considerations), risk tolerance (equipoise point).

Short term risk vs. long term risk
Mentioned in "Risk as the possibility of loss" section, perhaps here we go into more detail on terminal wealth dispersion, how small differences in annualized return can result in large differences in terminal wealth, stocks are risky in the long run, etc.. This is covered in Bodie et al, 2008.

See Bernstein's "The 15 Stock Myth" for more on terminal wealth dispersion, but there he focuses on the danger of a portfolio with only a small number of stocks. He does emphasize that such portfolios, although they may have low standard deviation of monthly returns, also have a large dispersion of terminal wealth.