Reading 83

An examination of the returns and standard deviation of returns for the major investable asset classes supports the idea of a tradeoff between risk and return. Using U.S. data over the period 1926–2017 as an example, shown in Figure 83.1, small-capitalization stocks have had the greatest average returns and greatest risk over the period. T-bills had the lowest average returns and the lowest standard deviation of returns.

Results for other markets around the world are similar: asset classes with the greatest average returns also have the highest standard deviations of returns.

The annual nominal return on U.S. equities has varied greatly from year to year, ranging from losses greater than 40% to gains of more than 50%. We can approximate the real returns over the period by subtracting inflation. The asset class with the least risk, T-bills, had a real return of only approximately 0.5% over the period, while the approximate real return on U.S. large-cap stocks was 7.3%. Because annual inflation fluctuated greatly over the period, real returns have been much more stable than nominal returns.

Evaluating investments using expected return and variance of returns is a simplification because returns do not follow a normal distribution; distributions are negatively skewed, with greater kurtosis (fatter tails) than a normal distribution. The negative skew reflects a tendency towards large downside deviations, while the positive excess kurtosis reflects frequent extreme deviations on both the upside and downside. These non-normal characteristics of skewness (≠ 0) and kurtosis (≠ 3) should be taken into account when analyzing investments.

Liquidity is an additional characteristic to consider when choosing investments because liquidity can affect the price and, therefore, the expected return of a security. Liquidity can be a major concern in emerging markets and for securities that trade infrequently, such as low-quality corporate bonds.

A risk-averse investor is simply one that dislikes risk (i.e., prefers less risk to more risk). Given two investments that have equal expected returns, a risk-averse investor will choose the one with less risk (standard deviation, $\sigma$). Financial models assume all investors are risk averse.

A risk-seeking (risk-loving) investor would actually prefer more risk to less and, given equal expected returns, would prefer the more risky investment. A risk-neutral investor would have no preference regarding risk and would therefore be indifferent between any two investments with equal expected returns.

Consider this gamble: A coin will be flipped; if it comes up heads, you receive $100; if it comes up tails, you receive nothing. The expected payoff is $0.5(\$100) + 0.5(\$0) = \$50$. A risk-averse investor would choose a payment of $50 (a certain outcome) over the gamble. A risk-seeking investor would prefer the gamble to a certain payment of $50. A risk-neutral investor would be indifferent between the gamble and a certain payment of $50.

When the expected returns on two portfolios are equal, a risk-averse investor will always prefer the less risky portfolio. Those who choose high-risk portfolios feel that the increase in expected portfolio returns is adequate compensation for their portfolio's higher risk.

Investors' utility functions represent their preferences regarding the tradeoff between risk and return (i.e., their degrees of risk aversion). For example, a utility function might be expressed as follows:

$$U = E(r) - \tfrac{1}{2}A\sigma^{2}$$

where:
$U$ = utility
$E(r)$ = expected return
$\sigma^{2}$ = variance of returns
$A$ = investor's degree of risk aversion

The more risk-averse an investor is, the greater the $A$ in the investor's utility function. While a risk-neutral investor would have an $A$ equal to zero and a risk-seeking investor would have a negative $A$, the theories of portfolio management we will discuss generally assume all investors are risk-averse ($A$ greater than zero).

An indifference curve is a tool from economics that, in this application, plots combinations of risk (standard deviation) and expected returns among which an investor is indifferent. In constructing indifference curves for portfolios based on only their expected return and standard deviation of returns, we are assuming that these are the only portfolio characteristics that investors care about. In Figure 83.2, we show three indifference curves for an investor. The investor's expected utility is the same for all points (portfolios) along any single indifference curve. Portfolios along indifference curve $I_{1}$ in Figure 83.2 are preferred to all portfolios along $I_{2}$, which are preferred to all portfolios along $I_{3}$.

Indifference curves slope upward for risk-averse investors because they will only take on more risk (standard deviation of returns) if they are compensated with greater expected returns. An investor who is more risk averse requires a greater increase in expected return to compensate for a given increase in risk than a less risk-averse investor. In other words, the indifference curves of a more risk-averse investor will be steeper than those of a less risk-averse investor, reflecting a higher risk aversion coefficient.

In our previous illustration of efficient portfolios available in the market, we included only risky assets. Now we will introduce a risk-free asset into our universe of available assets, and we will examine the risk and return characteristics of a portfolio that combines a portfolio of risky assets and a risk-free asset. As we have seen, we can calculate the expected return and standard deviation of a portfolio with weight $w_{A}$ allocated to risky Asset A and weight $w_{B}$ allocated to risky Asset B using the following formulas:

$$E(R_{portfolio}) = w_{A}E(R_{A}) + w_{B}E(R_{B})$$

$$\sigma_{portfolio} = \sqrt{w_{A}^{2}\sigma_{A}^{2} + w_{B}^{2}\sigma_{B}^{2} + 2w_{A}w_{B}\rho_{AB}\sigma_{A}\sigma_{B}}$$

Allow Asset B to be the risk-free asset and Asset A to be the risky asset portfolio. Because a risk-free asset has zero standard deviation and zero correlation of returns with those of a risky portfolio, this results in the reduced equation:

$$\sigma_{portfolio} = \sqrt{w_{A}^{2}\sigma_{A}^{2}} = w_{A}\sigma_{A}$$

The intuition of this result is straightforward: If we put X% of our portfolio into the risky asset, and the rest into the risk-free asset, our portfolio will have X% of the risk of the risky asset. The relationship between portfolio risk and return for various portfolio allocations is linear, as illustrated in Figure 83.3.

Combining a risky portfolio with a risk-free asset is the process that supports the two-fund separation theorem, which states that all investors' optimal portfolios will be made up of some combination of the optimal portfolio of risky assets and the risk-free asset. The line representing these possible combinations of risk-free assets and the optimal risky asset portfolio is referred to as the capital allocation line.

Point X on the capital allocation line in Figure 83.3 represents a portfolio that is 40% invested in the risky asset portfolio and 60% invested in the risk-free asset. Its expected return will be $0.40 \cdot E(R_{risky\,portfolio}) + 0.60 \cdot R_{f}$, and its standard deviation will be $0.40 \cdot \sigma_{risky\,portfolio}$.

Now that we have constructed a set of the possible efficient portfolios (the capital allocation line), we can combine this with indifference curves representing an individual's preferences for risk and return to illustrate the logic of selecting an optimal portfolio (i.e., one that maximizes the investor's expected utility). In Figure 83.4, we can see that Investor A, with preferences represented by indifference curves $I_{1}$, $I_{2}$, and $I_{3}$, can reach the level of expected utility on $I_{2}$ by selecting Portfolio X. This is the optimal portfolio for this investor, as any portfolio that lies on $I_{2}$ is preferred to all portfolios that lie on $I_{3}$ (and in fact to any portfolios that lie between $I_{2}$ and $I_{3}$). Portfolios on $I_{1}$ are preferred to those on $I_{2}$, but none of the portfolios that lie on $I_{1}$ are available in the market.

The final result of our analysis here is not surprising: investors who are less risk averse will select portfolios with more risk. Recall that the lower an investor's risk aversion, the flatter his indifference curves. As illustrated in Figure 83.5, the flatter indifference curve for Investor B ($I_{B}$) results in an optimal (tangency) portfolio that lies to the right of the one that results from a steeper indifference curve, such as that for Investor A ($I_{A}$). An investor who is less risk averse should optimally choose a portfolio with more invested in the risky asset portfolio and less invested in the risk-free asset.

Variance (Standard Deviation) of Returns for an Individual Security

In finance, the variance and standard deviation of returns are common measures of investment risk. Both of these are measures of the variability of a distribution of returns about its mean or expected value.

We can calculate the population variance, $\sigma^{2}$, when we know the return $R_{t}$ for each period, the total number of periods ($T$), and the mean or expected value of the population's distribution ($\mu$), as follows:

$$\sigma^{2} = \frac{\displaystyle\sum_{t=1}^{T}(R_{t}-\mu)^{2}}{T}$$

In the world of finance, we are typically analyzing only a sample of returns data, rather than the entire population. To calculate sample variance, $s^{2}$, using a sample of $T$ historical returns and the mean, $\overline{R}$, of the observations, we use the following formula:

$$s^{2} = \frac{\displaystyle\sum_{t=1}^{T}(R_{t}-\overline{R})^{2}}{T-1}$$

Covariance and Correlation of Returns for Two Securities

Covariance measures the extent to which two variables move together over time. A positive covariance means that the variables (e.g., rates of return on two stocks) tend to move together. Negative covariance means that the two variables tend to move in opposite directions. A covariance of zero means there is no linear relationship between the two variables. To put it another way, if the covariance of returns between two assets is zero, knowing the return for the next period on one of the assets tells you nothing about the return of the other asset for the period.

Here we will focus on the calculation of the covariance between two assets' returns using historical data. The calculation of the sample covariance is based on the following formula:

$$\mathrm{Cov}_{1,2} = \frac{\displaystyle\sum_{t=1}^{n}(R_{1,t}-\overline{R}_{1})(R_{2,t}-\overline{R}_{2})}{n-1}$$

where:
$R_{1,t}$ = return on Asset 1 in period $t$
$R_{2,t}$ = return on Asset 2 in period $t$
$\overline{R}_{1}$ = mean return on Asset 1
$\overline{R}_{2}$ = mean return on Asset 2
$n$ = number of periods

The magnitude of the covariance depends on the magnitude of the individual stocks' standard deviations and the relationship between their co-movements. Covariance is an absolute measure and is measured in return units squared.

The covariance of the returns of two securities can be standardized by dividing by the product of the standard deviations of the two securities. This standardized measure of co-movement is called correlation and is computed as:

$$\rho_{1,2} = \frac{\mathrm{Cov}_{1,2}}{\sigma_{1}\sigma_{2}}$$

The relation can also be written as: $\mathrm{Cov}_{1,2} = \rho_{1,2}\,\sigma_{1}\sigma_{2}$.

The term $\rho_{1,2}$ is called the correlation coefficient between the returns of securities 1 and 2. The correlation coefficient has no units. It is a pure measure of the co-movement of the two stocks' returns and is bounded by $-1$ and $+1$.

How should you interpret the correlation coefficient?

• A correlation coefficient of $+1$ means that deviations from the mean or expected return are always proportional in the same direction. That is, they are perfectly positively correlated.
• A correlation coefficient of $-1$ means that deviations from the mean or expected return are always proportional in opposite directions. That is, they are perfectly negatively correlated.
• A correlation coefficient of zero means that there is no linear relationship between the two stocks' returns. They are uncorrelated. One way to interpret a correlation (or covariance) of zero is that, in any period, knowing the actual value of one variable tells you nothing about the value of the other.

The variance of returns for a portfolio of two risky assets is calculated as follows:

$$\mathrm{Var}_{portfolio} = w_{1}^{2}\sigma_{1}^{2} + w_{2}^{2}\sigma_{2}^{2} + 2w_{1}w_{2}\,\mathrm{Cov}_{1,2}$$

where $w_{1}$ is the proportion of the portfolio invested in Asset 1, and $w_{2}$ is the proportion of the portfolio invested in Asset 2 (with $w_{2} = 1 - w_{1}$).

Previously, we established that the correlation of returns for two assets is calculated as $\rho_{1,2} = \dfrac{\mathrm{Cov}_{1,2}}{\sigma_{1}\sigma_{2}}$, so that we can also write $\mathrm{Cov}_{1,2} = \rho_{1,2}\sigma_{1}\sigma_{2}$.

Substituting this term for $\mathrm{Cov}_{1,2}$ in the formula for the variance of returns for a portfolio of two risky assets, we have the following:

$$\mathrm{Var}_{portfolio} = w_{1}^{2}\sigma_{1}^{2} + w_{2}^{2}\sigma_{2}^{2} + 2w_{1}w_{2}\rho_{1,2}\sigma_{1}\sigma_{2}$$

Because $\mathrm{Var}_{portfolio} = \sigma_{portfolio}^{2}$, this can also be written as:

$$\sigma_{portfolio} = \sqrt{w_{1}^{2}\sigma_{1}^{2} + w_{2}^{2}\sigma_{2}^{2} + 2w_{1}w_{2}\rho_{1,2}\sigma_{1}\sigma_{2}}$$

Writing the formula in this form allows us to easily see the effect of the correlation of returns between the two assets on portfolio risk.

If two risky asset returns are perfectly positively correlated, $\rho_{1,2} = +1$, then the square root of portfolio variance (the portfolio standard deviation of returns) is equal to:

$$\sigma_{p} = \sqrt{w_{1}^{2}\sigma_{1}^{2} + w_{2}^{2}\sigma_{2}^{2} + 2w_{1}w_{2}\sigma_{1}\sigma_{2}(1)} = w_{1}\sigma_{1} + w_{2}\sigma_{2}$$

In this unique case, with $\rho_{1,2} = 1$, the portfolio standard deviation is simply a weighted average of the standard deviations of the individual asset returns. A portfolio 25% invested in Asset 1 and 75% invested in Asset 2 will have a standard deviation of returns equal to 25% of the standard deviation ($\sigma_{1}$) of Asset 1's return, plus 75% of the standard deviation ($\sigma_{2}$) of Asset 2's return.

Focusing on returns correlation, we can see that the greatest portfolio risk results when the correlation between asset returns is $+1$. For any value of correlation less than $+1$, portfolio variance is reduced. Note that for a correlation of zero, the entire third term in the portfolio variance equation is zero. For negative values of correlation $\rho_{1,2}$, the third term becomes negative and further reduces portfolio variance and standard deviation.

Note that portfolio risk decreases as the correlation between the assets' returns decreases. This is an important result of the analysis of portfolio risk: the lower the correlation of asset returns, the greater the risk reduction (diversification) benefit of combining assets in a portfolio. If asset returns were perfectly negatively correlated, portfolio risk could be eliminated altogether for a specific set of asset weights.

We show these relations graphically in Figure 83.6 by plotting the portfolio risk and return for all portfolios of two risky assets, for specific values of the assets' returns correlation.

From these analyses, the risk reduction benefits of investing in assets with low return correlations should be clear. The desire to reduce risk is what drives investors to invest in not just domestic stocks, but also bonds, foreign stocks, real estate, and other asset classes.

For each level of expected portfolio return, we can vary the portfolio weights on the individual assets to determine the portfolio that has the least risk. These portfolios that have the lowest standard deviation of all portfolios with a given expected return are known as minimum-variance portfolios. Together they make up the minimum-variance frontier.

Assuming that investors are risk averse, investors prefer the portfolio that has the greatest expected return when choosing among portfolios that have the same standard deviation of returns. Those portfolios that have the greatest expected return for each level of risk (standard deviation) make up the efficient frontier. The efficient frontier coincides with the top portion of the minimum-variance frontier.

A risk-averse investor would only choose portfolios that are on the efficient frontier because all available portfolios that are not on the efficient frontier have lower expected returns than an efficient portfolio with the same risk. The portfolio on the efficient frontier that has the least risk is the global minimum-variance portfolio.