Risk and return

The relationship between risk and return is a fundamental concept in finance theory, and is one of the most important concepts for investors to understand. A widely used definition of investment risk, both in theory and practice, is the uncertainty that an investment will earn its expected rate of return. ("return" and "rate of return" are used interchangeably in finance literature).

A rational investor will not seek to take more risk without the expectation of a higher return. Since the early 1950s, an enormous amount of theoretical and empirical research has been done to characterize and quantify the relationship between risk and return.

Portfolio Theory, developed initially by Harry Markowitz in the early 1950s, was the first serious theoretical attempt to quantify the relationship between risk and return. Portfolio theory characterizes risk as the uncertainty of returns, and uses standard statistical techniques to quantify the relationship between risk and return. These techniques include the application of statistical measures such as variance and standard deviation to quantify the uncertainty of returns.

Note that "uncertainty of returns" includes not only the possibility of loss, but also the possibility of positive surprises relative to expectations. Since investors typically are more concerned about negative surprises than positive surprises, some theoretical work has been done using measures that consider only negative deviations relative to expectations. However, risk as the uncertainty of returns, including positive as well as negative deviations from expectations, continues to be a predominant risk model used in both the theory and practice of investing.

Portfolio theory was extended by William Sharpe and others to develop asset valuation models, such as the Capital Asset Pricing Model, or CAPM. CAPM introduced the concept of mixing a risk-free asset with a risky portfolio of assets to achieve an efficient balance of risk and return. It also enabled a more straightforward way to quantify the risk of individual assets by comparing the relationship of their returns with the return of a broad market portfolio.

Additional theoretical and empirical work was later done by Eugene Fama, Kenneth French, and others to develop more sophisticated models to evaluate risk and return. Whereas CAPM used a single "factor" to measure the relationship of an asset's return to that of a market portfolio, Fama and French showed that a multifactor model provided a better empirical fit to the data. Rather than comparing the returns of an asset or portfolio of assets only to a broad market portfolio, they also compared them to portfolios of small-cap stocks and value stocks (technically small minus big or SmB, and high book value minus low book value or HmB). This often is referred to as the Fama-French three-factor model.

The finance theory discussed here provides the theoretical foundations for index investing and asset allocation. Expected rates of returns for various asset classes are evaluated and compared to the risk of the asset classes. The goals and risk tolerance of the investor determine the balance between the expected return and risk of the portfolio. Risk is managed by designing a portfolio that includes an appropriate mix of lower-risk and higher-risk asset classes.

Determining risk tolerance is a critical step in designing a portfolio. A number of approaches have been developed to aid the investor in assessing risk tolerance.

Both short-term risk and long-term risk are considered, relative to the investment horizon. Short-term risk is evident in the short-term fluctuations in portfolio value. Longer-term, risk is related more to the uncertainty of the cumulative portfolio value.

Risk as the uncertainty of returns
There is very high certainty in the 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). T-Bills and CDs are among the investments referred to collectively as money market securities. A money market fund provides a convenient way for an investor to own money market securities. There also is a high degree of certainty in the short-term return on deposits in federally-insured bank or credit union accounts (the term cash often is used to refer to money market securities and money in deposit accounts; Vanguard refers to these types of assets as short-term reserves).

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



The top chart in Figure 1 is a histogram showing the frequency distribution of returns for 3-month T-Bills from 1928 through 2011. The height of each bar represents the number of years in which annual returns have fallen within a given 5% range or "bin" (e.g., 0% to 5%, 5% to 10%, etc.). The tallest bar shows that annual returns have been between 0% and 5% in 59 years. Returns have been between 5% and 10% in 22 years (second tallest bar), and between 10% and 15% in three years (shortest bar).

The return on bonds is less certain than the return on money market securities, so bonds are considered riskier than money market securities (cash). The middle chart in Figure 1 shows the frequency distribution of annual returns for 10-Year Treasury Bonds from 1928 through 2011 (the same vertical scale for number of years is used in all charts in Figure 1, and the same 5% bins are used on the horizontal axes). Note that the range of returns is larger, from about -11% to +33%, and that the number of years in which returns fall within any given 5% bin generally is smaller. In other words, the shape of the return distribution is more "spread out" and flatter. The shape of the return distribution represents the dispersion of returns, also referred to as the deviation of returns.

The return on stocks is more uncertain than the return on bonds, so stocks are considered riskier than bonds. The bottom chart in Figure 1 shows the frequency distribution of annual returns on the stocks of large US companies (the S&P 500) from 1928 through 2011. Note the much larger dispersion of returns; the shape of the distribution is much flatter and more spread out, with returns ranging from about -44% to +53%, and none of the 5% bins including returns for more than 10 years (by contrast, recall that one 5% bin included the T-bill returns for 59 years).

Standard deviation is a measure commonly used to quantify the deviation (or dispersion) of returns. The standard deviation is listed for each chart in Figure 1. Note that the standard deviation increases as the dispersion of returns increases (i.e., as the shape of the distribution becomes flatter and more spread out). Standard deviation will be defined in more detail in subsequent sections.

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 positive surprises as well as negative surprises, defining risk as the uncertainty of the return seems reasonable. Greater uncertainty results in greater likelihood that the investment will generate gains as well as losses.

Variance and standard deviation, discussed in detail in subsequent sections, are measures that quantify the negative and positive deviation of returns around the average return (the terms average return, mean return, and expected return are used interchangeably, as are the terms deviation of returns and dispersion of returns). Some theoretical portfolio work has been done using measures that consider only risk of loss; semivariance, which considers only deviations below the average, is one example.

However, to the extent deviation of returns is symmetrical, variance provides as good an indicator of downside risk as semivariance. If returns are symmetrical, the the probabilities and magnitudes of positive and negative surprises are roughly the same. It is reasonable to assume a symmetrical distribution of returns for a well diversified portfolio, and most assets have returns that are reasonably symmetrical. Therefore in portfolio theory, as well as in practice, variance and standard deviation are the most commonly used measures of dispersion of returns. In other words, even though investors may be primarily concerned with negative surprises, investment risk can be adequately estimated with measures that include both positive and negative surprises.

A major concern of investors is that their investment portfolios will not generate returns sufficient to meet their goals. This possibility is referred to as "shortfall risk", and deserves serious consideration. Nevertheless, using uncertainty of returns can be a valid approach in estimating the riskiness of long-term returns as well as short-term returns, as long as the appropriate variance or standard deviation measures are used.

For long-term returns, it is more appropriate to estimate the dispersion of accumulated wealth (also referred to as terminal wealth dispersion) rather than the deviation of returns over relatively short time periods, such as annually. This is because relatively small differences in annualized returns can make large differences in cumulative returns over periods of many years. Note that long-term dispersion of returns accounts for the magnitude as well as the probability of possible losses.

"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. 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

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.

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), and holding them to maturity. 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.

In reality, whether or not an asset is risk free depends on the context of the investment; e.g., the purpose and time horizon of the investment. For example, to meet a nominal liability at a specified future date, a zero-coupon US government bond maturing on that date is essentially risk free.

Nevertheless, short-term T-bills usually are considered to be risk-free assets in portfolio theory, and in practice investors treat a broader range of money market securities as risk free.

Risk aversion and risk premiums
Why would an investor invest in a risky asset class such as stocks rather than in a relatively risk-free asset such as T-bills? If the investor is not simply gambling, there must be some financial incentive, such as the expectation of a higher rate of return. This logic leads to the conclusion that an investment in a risky asset depends on the investor's expectation of a higher rate of return as well as his or her level of risk aversion.

Rational 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.

The amount by which a risky asset is expected to provide a higher rate of return than the risk-free rate is the asset's risk premium (the risk-free rate is the rate of return on a risk-free asset, such as a T-bill). If the risk premium of stocks were zero, then a rational, risk-averse investor would have no incentive to invest in them. On the other hand, a gambler might "invest" in something with a zero or even negative risk premium for entertainment value, or in the irrational expectation of hitting a jackpot.

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, and the risk premiums of bonds and stocks relative to the relatively risk-free returns on T-bills. Note that the mean (average) annual return increases as the dispersion of returns (uncertainty or risk) increases.

Assuming an investor selects an investment portfolio based on expected return and risk, the equation below is used to quantify the relationship between risk premium, risk aversion, A, and risk σ2, where the risk premium is the expected return, E(r), of the risky investment minus the risk-free rate, rf:


 * $$ \operatorname{Risk} \operatorname{Premium} = \operatorname{E}(r) - r_f = \frac12 A \sigma^2$$

In this equation risk is measured by variance, σ2, which quantifies the dispersion of an investment's returns. Variance is discussed in depth in another section. The factor of $$\tfrac12$$ is a scaling factor used by convention, and the use of this equation requires expressing all returns as decimals rather than percentages.

Note that this equation indicates that the risk premium demanded by investors will be higher for individual investors with greater risk aversion, A, and for investments with higher risk, σ2.

Studies have shown that investors' risk aversion probably is in the range of 2-4. This implies that for an increase of portfolio variance of 0.01, investors will require a risk premium that is higher by 0.01-0.02 (1%-2%).

Measuring risk: historical returns
Historical returns often are used as a starting point in estimating expected returns and uncertainty of returns (risk). Although evaluating historical returns may not enable an accurate or comprehensive estimation of risk, evaluating the dispersion of historical returns provides a convenient way to develop an understanding of commonly used risk measures.

The risk of an investment is related to the uncertainty of its investment return. This was illustrated graphically in Figure 1, which showed that dispersion of returns is a way to characterize the riskiness of an investment. Many 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
Although in finance theory, expected return is used 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 investment theory expected return has a more precise definition.

An investment's expected return, $$\operatorname{E}(r)$$, is calculated as follows:


 * 1) Various economic scenarios, $$s$$, are defined.
 * 2) Estimates are developed for the probability, $$\operatorname{p}(s)$$, of each scenario occurring and the return, $$\operatorname{r}(s)$$, for that scenario.
 * 3) The probability and return for each scenario are multiplied together:  $$\operatorname{p}(s)\operatorname{r}(s)$$
 * 4) The results are summed across all scenarios: $$\operatorname{p}(s_1)\operatorname{r}(s_1) + \operatorname{p}(s_2)\operatorname{r}(s_2) + \cdots + \operatorname{p}(s_n)\operatorname{r}(s_n)$$

Or using summation notation:


 * $$\operatorname{E}(r) = \sum_{i=1}^n \operatorname{p}(s_i)\operatorname{r}(s_i)$$

Thus, expected return is the probability-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, $$\operatorname{p}(s)$$ = 1, and the return, $$\operatorname{r}(s)$$, is the rate of return on the bill. If the rate of return is 1%, the expected nominal return is calculated as:



\operatorname{E}(r) = \operatorname{p}(s)\operatorname{r}(s) = (1.0)(0.1) = 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 $$\operatorname{p}(s_1)=0.20$$, estimated return $$\operatorname{r}(s_1)= -10%$$
 * Normal growth: probability $$\operatorname{p}(s_2) = 0.60$$, estimated return $$\operatorname{r}(s_2) = 5%$$
 * Strong growth: probability $$\operatorname{p}(s_3) = 0.20$$, estimated return $$\operatorname{r}(s_3) = 10%$$

With these scenario estimates, the expected return,$$\operatorname{E}(r)$$, is calculated as:



\operatorname{E}(r) = \operatorname{p}(s_1)\operatorname{r}(s_1) + \operatorname{p}(s_2)\operatorname{r}(s_2) + \operatorname{p}(s_3)\operatorname{r}(s_3) $$



\operatorname{E}(r) = (0.20)(-0.10)+(0.60)(0.05)+(0.20)(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.

A concept that will be useful in subsequent sections is the expected return of a portfolio, which is simply the weighted average of the expected returns for the individual assets in the portfolio. This is expressed mathematically as:


 * $$\operatorname{E}(r_p) = \sum_{i=1}^n (w_i)\operatorname{E}(r_i)$$

where wi is the percent of the portfolio in asset i, and E(ri) is the expected return of asset i.

Measuring risk: estimated returns
Variance of returns was illustrated using historical returns in a prior section. The calculation of the variance of estimated future returns is similar, except that the probabilities of the returns are not equal, but are the probabilities assigned to various scenarios.

Variance of estimated future 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:



\operatorname{Var}(r) = \sigma^2 = \operatorname{p}(s_1)[\operatorname{r}(s_1) - \operatorname{E}(r)]^2 + \operatorname{p}(s_2)[\operatorname{r}(s_2) - \operatorname{E}(r)]^2 + \cdots + \operatorname{p}(s_n)[\operatorname{r}(s_n) - \operatorname{E}(r)]^2 $$

Or using summation notation:


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

To deepen the intuitive understanding of the calculation of variance of estimated returns, 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.

Note that the purposes of steps 1, 2 and 4 are basically the same as for the calculation of the variance of historical returns. Step 3 introduces probability weighting.

As with historical returns, the standard deviation of estimated returns is calculated as the square root of the variance of the estimated returns.

For the case of the nominal return on a 1-year T-Bill yielding 1%, the expected return, E(r), = 1%, and there is only one scenario, for which the nominal return is a certain (p(s) = 1). Therefore, the the variance is calculated as:



\operatorname{Var}(r) = \sigma^2 = \operatorname{p}(s_1)[\operatorname{r}(s_1) - \operatorname{E}(r)]^2 $$


 * $$ \operatorname{Var}(r) = \sigma^2 = (1.0)(0.01 - 0.01)^2 $$


 * $$ \operatorname{Var}(r) = \sigma^2 = 0 $$

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) = 0.20; estimated return, r(s1) = -10%
 * Normal growth: probability, p(s2) = 0.60; estimated return, r(s2) = 5%
 * Strong growth: probability, p(s3) = 0.20; estimated return, r(s3) = 10%

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



\operatorname{Var}(r) = \sigma^2 = \operatorname{p}(s_1)[\operatorname{r}(s_1) - \operatorname{E}(r)]^2 + \operatorname{p}(s_2)[\operatorname{r}(s_2) - \operatorname{E}(r)]^2 + \operatorname{p}(s_3)[\operatorname{r}(s_3) - \operatorname{E}(r)]^2 $$



\operatorname{Var}(r) = \sigma^2 = (0.20)(-0.10 - 0.03)^2 + (0.60)(0.05 - 0.03)^2 + (0.20)(0.10 - 0.03)^2 $$



\operatorname{Var}(r) = \sigma^2 = (0.20)(-0.13)^2 + (0.60)(0.02)^2 + (0.20)(0.07)^2 $$


 * $$ \operatorname{Var}(r) = \sigma^2 = 0.0046 $$

And the standard deviation is:


 * $$ \operatorname{SD}(r) = \sigma = \sqrt{0.0046} $$


 * $$ \operatorname{SD}(r) = \sigma = 0.07 = 7% $$

Although three samples is not enough to apply the statistical implications of standard deviation, note that two of the returns fall within E(r) +/- one standard deviation (3% +/- 7 pp), and all returns fall within E(r) +/- two standard deviations (3% +/- 14 pp). Again, standard deviation seems to provide a reasonable measure of the dispersion of returns (risk).

Managing portfolio risk
A portfolio is the investor's collection of financial assets, e.g., stocks, bonds and cash. It is intuitive that portfolio risk is proportional to the amount of risky assets in the portfolio. A critical step in managing portfolio risk is deciding on an appropriate mix of higher-risk assets and lower-risk assets.

A simple way to reduce the risk of a portfolio of risky assets is to add a risk-free asset to the portfolio. The portfolio risk will be proportional to the proportion of risky assets, and inversely proportional to the proportion of the risk-free asset. Although less intuitive, portfolio risk also can be reduced by combining different risky assets. Each of these approaches of managing portfolio risk will be covered in the next two sections.

Portfolio of risky assets combined with a risk-free asset
Risky assets include bonds as well as stocks, but for now it will be assumed that the risky portfolio is a total stock market index fund. As discussed in previous sections, there is no truly risk-free asset, but T-bills often are considered the risk-free asset in portfolio theory. The risk of T-bills and other money market securities is so much lower than the risk of stocks that this is a reasonable approach.

Both the expected return and the risk of a portfolio must be determined to evaluate the risk-return trade-off of combining a portfolio of risky assets with a risk free asset. The following steps develop an equation that relates the expected return of a such a portfolio to its risk, where risk is measured by the standard deviation of portfolio returns.

Portfolio of risky assets combined with a risk-free asset: Expected return 

The expected return of a portfolio of assets is the the weighted average of the expected returns of the individual assets.


 * $$ \operatorname{E}(r_p) = w_1 \operatorname{E}(r_1) + w_2 \operatorname{E}(r_2)

+ \cdots + w_n \operatorname{E}(r_n) $$

Note that the weight of an asset in a portfolio refers to the fraction of the portfolio invested in that asset; e.g., if w1 = $$\tfrac14$$, then one fourth of the portfolio is invested in the asset with E(r1).

For two assets this reduces to:


 * $$ \operatorname{E}(r_p) = w_1 \operatorname{E}(r_1) + w_2 \operatorname{E}(r_2)$$
 * $$ \operatorname{E}(r_p) = w_1 \operatorname{E}(r_1) + (1 - w_1) \operatorname{E}(r_2)$$

Let one asset be the risky portfolio consisting of a total stock market index fund, with expected return E(rs) = 6%, and with the standard deviation of annual returns = 20% (these values are very close to the values for the historical returns of the Vanguard Total Stock Market Index fund from 1998 through 2011). Let the other asset be a risk-free asset with return rf = 1% (since rf is known with certainty, E(rf) = rf); the standard deviation of the risk-free asset is 0% by definition. Applying the above equation to this portfolio:


 * $$ \operatorname{E}(r_p) = w_s \operatorname{E}(r_s) + (1 - w_s)(r_f)$$
 * $$ \operatorname{E}(r_p) = r_f + w_s [\operatorname{E}(r_s) - r_f]$$
 * $$ \operatorname{E}(r_p) = 0.01 + w_s(0.06 - 0.01) $$
 * $$ \operatorname{E}(r_p) = 0.01 + 0.05w_s $$

E(rs) - rf = 0.05 = 5% is the risk premium of the risky portfolio. A risk premium is the excess expected return compared to the risk-free asset. Since the risky portfolio here is a stock fund, its risk premium is referred to as the equity risk premium or ERP (equities is synonymous with stocks). This equation indicates that a portfolio of any expected return between rf = 1% and E(rs) = 6% can be constructed by combining the risky portfolio and risk-free asset in the desired proportions.

If ws = 0, the portfolio consists only of the risk-free asset, and the expected return of the portfolio is simply the risk-free rate:


 * $$ \operatorname{E}(r_p) = 0.01 + (0.05)(0) $$
 * $$ \operatorname{E}(r_p) = 0.01 = 1% $$

If ws = 1, the total portfolio consists entirely of the risky portfolio, and the expected return of the total portfolio is the expected return of the risky portfolio:


 * $$ \operatorname{E}(r_p) = 0.01 + (0.05)(1) $$
 * $$ \operatorname{E}(r_p) = 0.06 = 6% $$

If the portfolio is divided evenly between the risky portfolio and the risk-free asset, ws = 0.5, and the expected return of the portfolio is:


 * $$ \operatorname{E}(r_p) = 0.01 + (0.05)(0.5) $$
 * $$ \operatorname{E}(r_p) = 0.01 + (0.025) $$
 * $$ \operatorname{E}(r_p) = 0.035 = 3.5% $$

Note that in the case of a portfolio evenly divided between the risky portfolio and the risk-free asset, ws = wf = $$\tfrac12$$, and the expected return is simply the average of the returns of the two assets:


 * $$ \operatorname{E}(r_p) = \frac {0.01 + 0.06}{2} = 0.035 = 3.5% $$

Portfolio of risky assets combined with a risk-free asset: Risk 

Since the standard deviation of a risk-free asset is 0%, the standard deviation of a portfolio formed by combining a portfolio of risky assets with a risk-free asset is simply the standard deviation of the risky portfolio times the proportional weight of the risky portfolio:


 * $$ \operatorname{SD}(p) = \sigma_p = w_s \sigma_s $$

So the risk of the total portfolio is linearly proportional to the weight of the risky portfolio. Given the standard deviation of annual returns of the risky portfolio of stocks = 20%, a portfolio can be constructed with a standard deviation between 0% and 20%. A portfolio evenly divided between the risky portfolio and the risk-free asset would have a standard deviation of 10%.

Portfolio of risky assets combined with a risk-free asset: expected return in terms of risk 

The above equations can be combined to show the expected return of the total portfolio as a function of its standard deviation, thus relating the expected return of the portfolio to its risk. Rearranging the equation for the standard deviation of the total portfolio gives ws = $$ \tfrac{\sigma_p}{\sigma_s} $$. Therefore,


 * $$ \operatorname{E}(r_p) = r_f + w_s [\operatorname{E}(r_s) - r_f]$$
 * $$ \operatorname{E}(r_p) = r_f + \frac{\sigma_p}{\sigma_s}[\operatorname{E}(r_s) - r_f]$$
 * $$ \operatorname{E}(r_p) = r_f + \frac{[\operatorname{E}(r_s) - r_f]}{\sigma_s} \sigma_p$$

This is a linear equation that gives portfolio expected return, $$\operatorname{E}(r_p)$$, as a function of portfolio risk, $$\sigma_p$$, with intercept rf and slope $$ \frac{[\operatorname{E}(r_s) - r_f]}{\sigma_s} $$. Substituting in the example numbers of rf = 0.01, E(rs) = 0.06, and σs = 20% = 0.20:


 * $$ \operatorname{E}(r_p) = r_f + \frac{[\operatorname{E}(r_s) - r_f]}{\sigma_s} \sigma_p$$
 * $$ \operatorname{E}(r_p) = 0.01 + \frac{0.06 - 0.01}{0.20} \sigma_p$$
 * $$ \operatorname{E}(r_p) = 0.01 + 0.25 \sigma_p$$

So a portfolio of any expected return between 1% and 6% with corresponding standard deviation between 0% and 20% can be constructed by adding the appropriate portion of the risky portfolio to the risk-free asset. With a slope of 0.25, portfolio expected return increases by 1 pp (percentage point) for each increase of 4 pp in portfolio standard deviation (σp); i.e., solving this equation for values of σp = 0%, 4%, 8%, 12%, 16% and 20% gives values for E(rp) of 1%, 2%, 3%, 4%, 5% and 6%.

INSERT GRAPH SHOWING CAPITAL ALLOCATION LINE

Portfolio of two risky assets
This section develops the theory that shows the risk-return characteristics for a portfolio consisting of two risky assets. It will be extended to more than two risky assets in the following section. This is the theory that underlies the practice of combining multiple risky assets into a portfolio that has less risk than the sum of the risks of the individual assets. Examples are a mutual fund that includes many individual securities, and a portfolio of mutual funds that includes different asset classes; e.g., domestic stocks, foreign stocks, and bonds.

In considering two risky assets, it is useful to consider two mutual funds, but the same theory applies to any two assets; e.g., two individual stocks. In the following section in which portfolios of many assets are considered, it will be more useful to think of individual securities.

The statistical measures covariance and correlation are fundamental to calculating the risk-return characteristics of a portfolio consisting of multiple risky assets.

Covariance is a measure of how the values of two dependent variables move together, or co-vary, as the value of a common independent variable changes. The dependent variables of interest in portfolio theory are the returns of the assets in the portfolio, and the independent variable could be a scenario (e.g., the state of the economy) or a time period.

Consider the deviations of returns for two assets, stock fund s and bond fund b, from their expected values for a particular scenario (or during a certain time period):



[r_s - \operatorname{E}(r_s)][r_b - \operatorname{E}(r_b)] $$

If the returns of the two assets are both above their expected values (both deviations are positive), or the returns of the two assets are both below their expected values (both deviations are negative), the product of the deviations will be positive. If the return of one asset is above its expected value (positive deviation), and the return of the other asset is below its expected value (negative deviation), the product of the deviations will be negative. Thus the sign of the product of the deviations indicates whether or not the returns have deviated in the same direction or opposite directions, and the magnitude of the product of the deviations indicates how much the returns have deviated from their expected values. The covariance of returns is the expected value of the products of the deviations across all scenarios:



\operatorname{Cov}(r_s,r_b) = \sum_{i=1}^n p(i)[(r_s(i) - \operatorname{E}(r_s)][r_b(i) - \operatorname{E}(r_b)] $$

where p(i) is the probability for scenario i, and rs(i) and rb(i) are the respective returns for the stock and bond fund for scenario i.

Often a more convenient measure to work with is the correlation coefficient, which is computed by dividing the covariance by the product of the standard deviations of the two assets:



\rho_{s,b} = \frac{\operatorname{Cov}(r_s,r_b)}{\sigma_s \sigma_b} $$

The correlation coefficient varies from -1 for two perfectly uncorrelated variables to 1 for perfectly correlated variables. A correlation coefficient of 0 indicates no correlation between the two variables.

The variance of the rate of return of a portfolio consisting of two risky assets, in this case stock fund s and bond fund b, is given by this equation:



\sigma_p^2 = (w_s \sigma_s)^2 + (w_b \sigma_b)^2 + 2(w_s \sigma_s)(w_b \sigma_b)\rho_{s,b} $$

or equivalently



\sigma_p^2 = (w_s \sigma_s)^2 + (w_b \sigma_b)^2 + 2w_s w_b \operatorname{Cov}(r_s,r_b) $$

where ws and wb represent the respective proportions (weights) of the portfolio invested in the stock and bond funds, and σs and σb are the respective standard deviations of the stock and bond funds.

Note that the portfolio variance depends on the weighted standard deviations of the assets plus a term that depends on the covariance (or correlation) of the assets. The smaller the correlation between the assets the lower the portfolio variance.

For a correlation coefficient of 1 (perfectly correlated assets), the equation for portfolio variance is:



\sigma_p^2 = (w_s \sigma_s)^2 + (w_b \sigma_b)^2 + 2(w_s \sigma_s)(w_b \sigma_b) $$ which reduces to:

\sigma_p^2 = (w_s \sigma_s + w_b \sigma_b)^2 $$

and the standard deviation is:



\operatorname{SD}(p) = \sigma_p = (w_s \sigma_s + w_b \sigma_b) $$

So for two perfectly, positively correlated assets, the portfolio standard deviation is simply the weighted average of the standard deviations of the two assets. Any correlation less than 1 will lower the portfolio standard deviation.

Since the portfolio expected return is the weighted average of the returns of the individual assets (as shown in previous sections), but the standard deviation is less than the weighted average of the assets' standard deviations, a portfolio of assets that are not perfectly correlated always provides a better risk-return opportunity than the individual assets on their own.

As a simple illustration of this important concept, consider a portfolio of two assets that have the same expected value and standard deviation, but that are not perfectly correlated (correlation coefficient less than 1). The expected value of the portfolio will be the same as the expected value of each asset, but the variance (and standard deviation) of the portfolio will be less than the variance (and standard deviation) of each asset (this can be verified with the above equations). This demonstrates how combining risky assets can result in a portfolio that is less risky than either asset on its own.

DEVELOPMENT NOTE: Consider plugging in some numbers for stock fund and bond fund to illustrate formulas with actual calculations.

Portfolio of many risky assets
The general equation for the variance of a portfolio of risky assets is:


 * $$ \sigma_p^2 = \sum_{i=1}^n w_i^2 \sigma_{i}^2

+ \sum_{i=1}^n \sum_{j=1 j \neq i}^n w_i w_j \sigma_i \sigma_j \rho_{ij}, $$

Recalling that $$\operatorname{Cov}(i,j) = \sigma_i \sigma_j \rho_{ij} $$, the equation can be rewritten more succinctly as:


 * $$ \sigma_p^2 = \sum_{i=1}^n w_i^2 \sigma_{i}^2

+ \sum_{i=1}^n \sum_{j=1 j \neq i}^n w_i w_j \operatorname{Cov}(i,j). $$

The terms summed on the left are the variances of the individual assets multiplied by the squared portfolio weights of the assets. The double summation on the right represents the summation across all assets of the covariance of each pair of assets multiplied by the portfolio weights of the assets in each pair.

Expanding this equation for a portfolio of two risky assets gives:



\sigma_p^2 = (w_1 \sigma_1)^2 + (w_2 \sigma_2)^2 + w_1 w_2 \operatorname{Cov}(1,2) + w_2 w_1 \operatorname{Cov}(2,1). $$

There are two variance terms and two covariance terms. Since $$\operatorname{Cov}(1,2) = \operatorname{Cov}(2,1)$$, the equation can be simplified to:



\sigma_p^2 = (w_1 \sigma_1)^2 + (w_2 \sigma_2)^2 + 2 w_1 w_2 \operatorname{Cov}(1,2). $$

Although the two covariance terms now appear as one term with a multiplier of 2, there are essentially two covariance terms. Note that this is the equation that was used in the previous section.

For three risky assets the equation expands to:



\sigma_p^2 = (w_1 \sigma_1)^2 + (w_2 \sigma_2)^2 + 2 w_1 w_2 \operatorname{Cov}(1,2) + 2 w_1 w_3 \operatorname{Cov}(1,3) + 2 w_2 w_3 \operatorname{Cov}(2,3). $$

There are three variance terms and six covariance terms. As more assets are added to the portfolio, the number of covariance terms becomes much larger than the number of variance terms. Specifically, there are n variance terms and n(n-1) covariance terms. It is intuitive that for a large number of assets the covariance terms will dominate the result; i.e., the covariance between asset pairs will have more impact on overall portfolio variation than the variances of the individual assets.

To further illustrate the power of diversification, consider the case in which a portfolio is constructed of many assets held in equal proportions. In this case wi = $$\tfrac1n$$ for each asset, and the equation for portfolio variance becomes:


 * $$ \sigma_p^2 =

\sum_{i=1}^n (\frac1n)^2 \sigma_{i}^2 + \sum_{i=1}^n \sum_{j=1 j \neq i}^n (\frac1n)(\frac1n) \operatorname{Cov}(i,j). $$

With appropriate factoring, this equation can be rewritten as:


 * $$ \sigma_p^2 =

\frac1n \sum_{i=1}^n \frac1n \sigma_{i}^2 + \frac{n-1}{n} \sum_{i=1}^n \sum_{j=1 j \neq i}^n \frac {\operatorname{Cov}(i,j)}{n(n-1)}. $$

The summation of terms on the left is the average variance of the assets. As n becomes large $$\tfrac1n$$ approaches 0, so the term on the left approaches 0. This term represents the nonsystematic risk of the individual assets that can be diversified away by including a sufficiently large number of assets in the portfolio. In other words, in a portfolio of many assets, the variances of the individual assets have little impact on the variance of the portfolio.

The double summation of terms on the right is the average covariance of the assets. As n becomes large, $$\tfrac{n-1}{n}$$ approaches 1, so the term on the right approaches the average covariance. This term represents the systematic risk of the portfolio that cannot be diversified away.

This further demonstrates that it is the covariance between the assets in a portfolio of many assets that dominates the overall portfolio risk, and that the variances of the individual assets have little impact on the overall portfolio risk.

As additional assets are added to a portfolio of many assets, the number of covariances that must be calculated to evaluate the impact of adding the new assets becomes problematic. Factor models, such as the Capital Asset Pricing Model, or CAPM, provide a solution to this problem.

In a single-factor model, it is assumed that there is a common factor that affects the systematic risk of all assets under consideration. Nonsystematic risk still exists for the individual assets, but as seen above, nonsystematic risk can be eliminated by including enough assets in the portfolio.

Using a single-factor model, an asset's systematic component of risk is characterized by its covariance with the market portfolio. In theory the market portfolio consists of all risky assets; e.g., stocks, bonds, real estate, commodities, collectibles, etc. In practice, a proxy for the market portfolio is used; a commonly used proxy in evaluating the risk of US stocks is the S&P 500 stock index.

The sources of systematic risk are the economic factors that affect the the overall market. For example, the level of economic growth affects all companies to some extent, and therefore affects the price movements and returns of all stocks to some extent. The term beta often is used to describe the extent to which an individual asset's return is related to the market portfolio's return.

The portion of an asset's return variation that is not related to the market portfolio is referred to as unsystematic risk or nonsystematic risk. The sources of unsystematic risk are the firm-specific risks of individual companies, as opposed to the broader economic factors that have some effect on all companies.

Unsystematic risks can be eliminated, or "diversified away", by owning the entire market portfolio. So one way to manage risk is to reduce or eliminate unsystematic risk by holding a large number of securities--ultimately by holding all securities in the market. This is the theoretical foundation for owning a total market mutual fund, such as a total stock market or total bond market index fund.

Covariance and correlation of returns. Reilly, p.245-250

Standard deviation of a portfolio. Reilly, pp. 250-258

The efficient frontier. Reilly, pp. 258-261

Assessing risk tolerance
This section presents several frameworks to aid in assessing risk tolerance. The term risk tolerance is used here to describe both the investor's capacity to bear risk and attitude toward risk. Note however that some authors use the term risk tolerance to describe only the investor's attitude toward risk; i.e., the psychological and emotional aspects related to taking risk. To avoid confusion, the phrase "determining appropriate risk exposure" will be used instead of "assessing risk tolerance" for frameworks that use the term risk tolerance to refer only to the investor's attitude toward risk.

Assessing risk tolerance is a critical step in determining the appropriate trade off between the risk and expected return of a portfolio. Selecting a mix of risky and risk-free assets (or higher-risk and lower-risk assets) is one of the most important decisions in designing a portfolio. The investor must develop a rational assessment of risk tolerance to make rational decisions about portfolio design.

Most approaches to assessing risk tolerance consider these criteria in one way or another:


 * Attitude toward risk
 * Investment time horizon
 * Other factors that affect ability to compensate for investment losses; e.g., net worth, stability of income, future liabilities, flexibility of goals, etc.

Boglehead's Guide To Investing
The Bogleheads' Guide To Investing discusses determining appropriate risk exposure in the chapter on Asset Allocation. The context is designing an efficient portfolio and "staying the course".

Four areas are explored:


 * Goals
 * Time frame
 * Risk tolerance
 * Personal financial situation

Examples of goals are saving for a home, a child's education, or retirement.

Stocks are suitable for long time frames, not short time frames (less than five years)

Risk tolerance is about the psychological and emotional ability to stick with the investment plan during large market declines. Whether or not one thinks one could sleep well at night (the sleep test) with one's current asset allocation (proportions of stocks, bonds and cash) is one indicator of risk tolerance.

The investor's personal financial situation affects how much risk and what types of risk are appropriate. Stability of income and net worth affect the need to take risk. Individuals with higher guaranteed retirement incomes (e.g., a pension or social security) or high net worth do not need to take as much risk, e.g., by investing in stocks.

Three "tools, along with the investor's own experience, are recommended to help factor risk into the one's asset allocation:
 * John Bogle, founder of Vanguard, suggests that a rough guide is to hold one's age in bonds.
 * Consider the maximum decline to expect with various stock/bond ratios (the book presents maximum declines in the 2000 to 2002 bear market, but the maximum declines in the 2008/2009 bear market were worse; a common rule of thumb is to be prepared for a 50% loss in the stock portion of one's portfolio).
 * Vanguard provides an online questionnaire and suggested asset allocations. The book includes a version of the questionnaire in Appendix IV, but see the Vanguard website for the latest version.

Various sample portfolios are presented based on these "stages in life":


 * Young investor
 * Middle-aged investor
 * Investor in early retirement
 * Investor in late retirement

Larry Swedroe: ability, willingness, and need to take risk
Bogleheads author Larry Swedroe suggests that investors evaluate their risk tolerance by considering their ability, willingness and need to take risk.

Ability to take risk involves investment time horizon, liquidity needs, stability of earned income, and the flexibility to adapt if the portfolio does not achieve its expected returns (i.e., "plan B").

Willingness to take risk is characterized as the eat well/sleep well trade-off. Taking more risk is required to enable the possibility of higher expected returns (eat well). However, if investors take more risk than they are emotionally able to handle, then it is likely that they will abandon their investment plans if their portfolios suffer sufficiently severe losses. So it is unwise for investors to take so much risk that they will be unable to sleep well during the inevitable stock market downturns.

Need to take risk is related to the investment goal. If the goal requires a higher expected return, the investor needs to take more risk with the expectation that doing so will result in a higher return. Perhaps more importantly, once the investor has "won the game" by accumulating sufficient wealth, it is unwise to take more risk than is needed, since the value of additional gains is much less important than the consequences of severe losses.

Rick Ferri: lifecycle model
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
CFA Institute (Bodie, Kane, Marcus, 2008, chapter 21): Investors and Objectives, Investor constraints (Liquidity, Investment Horizon, Regulations, Tax Considerations, Unique Needs.

William Bernstein
Bernstein (2010), The Investor's Manifesto, pp. 75-80: Age (human capital considerations), risk tolerance (equipoise point). Bernstein used "risk tolerance" in the sense of willingness to take risk.

Daniel Solin
Daniel R. Solin is the author of The Smartest Investment Book You'll Ever Read, listed in the Bogleheads Books: Recommendations and Reviews, and the subject of this Taylor's Gem.

Chapter 29 in another one of Solin's books, The Smartest Money Book You'll Ever Read, is titled Assessing Your Risk Capacity. Solin presents five major factors for assessing risk capacity:


 * Time horizon and liquidity needs: Longer time horizon and lower shorter-term liquidity needs increase risk capacity.
 * Income and savings rate: Higher income and savings rate increase risk capacity.
 * Net worth: Higher net worth increases risk capacity.
 * Attitude toward risk: This is what most people refer to as risk tolerance, and is the emotional component of handling losses.
 * Investment knowledge: The better your understanding of portfolio theory and the risk-return trade off, the higher your risk capacity.

Solin provides a Risk Capacity Survey on his website. There is a quick survey with five questions, a 25-question "complete" survey, and a 19-question survey for 401k plans.

Managing financial risk
Investment risk, or financial-asset risk, is only one type of financial risk facing households. Other major categories of risk are:


 * Sickness, disability, and death.
 * Unemployment risk.
 * Consumer-durable asset risk: e.g., the risk of loss from fire, theft or obsolescence related to owning a car, house, or other consumer-durable asset.
 * Liability risk: e.g., the possibility of getting sued.

Managing these types of risks, as well as managing investment risk, from a broader financial planning perspective is the topic of this section.

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, Kane, Marcus, 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.