4 Risk and Return

Learn how investment returns are measured, how volatility and correlation shape portfolio risk, and how CAPM relates systematic risk to expected return.

Investment returns

Investors compare an investment’s —the income and change in value it produces—with the risk that its actual will differ from what they expect. Risk means outcomes are less certain, including the possibility of loss; it does not guarantee a higher payoff. Finance models describe a trade-off between risk and expected , not a promise that riskier investments will earn more in every period.

Calculating a period’s

A includes cash received and the change in the investment’s value:

R = \frac{P_1-P_0+D}{P_0}

Here, $P_0$ is the beginning price, $P_1$ is the ending price, and $D$ is income such as dividends or interest received during the period. For example, a share bought for $\$50$ , paying a $\$2$ dividend and ending at $\$54$ , has a of $\frac{54-50+2}{50}=12\%$ . The can be negative if the price loss exceeds the income received.

Averages and compounding

The of periodic returns is their sum divided by the number of periods. It summarizes the average one-period :

R_{\text{arith}}=\frac{1}{n}\sum_{t=1}^{n}R_t

The is the constant per-period rate that would produce the same compounded ending value:

R_{\text{geo}} = \left[\prod_{t=1}^{n}(1+R_t)\right]^{1/n}-1

Because investment gains and losses compound, the is generally more informative for long-run compound growth. For example, a $50\%$ gain followed by a $50\%$ loss turns $\$100$ into $\$75$ , not back into $\$100$ .

Measuring risk

is a common introductory measure of investment risk: it describes how widely returns vary around their average. For possible outcomes with probabilities $p_i$ , expected and variance are:

E(R)=\sum_i p_iR_i, \qquad \sigma^2=\sum_i p_i\big(R_i-E(R)\big)^2

The , $\sigma$ , is the square root of variance and is expressed in the same units as returns, usually percent. With historical data, sample variance is commonly estimated as:

s^2=\frac{\sum_{t=1}^{n}(R_t-\bar R)^2}{n-1}

A larger means returns fluctuated more in the measured period; it does not indicate whether future returns will be positive or negative. is useful but incomplete: it does not by itself describe every kind of risk or the likelihood of extreme losses.

risk and

A is a collection of investments. Its is the weighted average of the assets’ returns:

R_p=\sum_{i=1}^{n}w_iR_i

Here, $w_i$ is the share of the invested in asset $i$ , and the weights sum to $1$ . risk depends not only on each asset’s but also on how the assets’ returns move together. For two assets:

\sigma_p^2=w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2\rho_{12}\sigma_1\sigma_2

Here, $\rho_{12}$ is the correlation between the assets’ returns. Correlation ranges from $-1$ , when returns move perfectly in opposite directions, to $+1$ , when they move perfectly together. When correlation is below $+1$ , combining assets can reduce ; all else equal, lower correlation generally creates more potential for risk reduction.

Holding companies in different industries can reduce exposure to a problem affecting just one firm. Holding many similar companies may provide less protection because their returns may move together. reduces some risks, but it cannot eliminate losses or all market-wide risk.

Two kinds of investment risk

comes from events concentrated in a company or a narrow industry, such as a product failure or management problem. can reduce much of this risk. arises from broad forces that affect many investments, such as recessions, inflation, or economy-wide changes in interest rates. A diversified remains exposed to systematic risk.

Expected and CAPM

Investors may require a : expected above the on a relatively low-risk benchmark, as compensation for bearing risk. In the , the risk relevant to expected is systematic risk, measured by :

\beta_i=\frac{\operatorname{Cov}(R_i,R_m)}{\operatorname{Var}(R_m)}, \qquad E(R_i)=R_f+\beta_i\big(E(R_m)-R_f\big)

Here, $R_f$ is the risk-free rate used in the model, $R_m$ is the market ’s , and $E(R_m)-R_f$ is the market . measures how an asset’s returns have tended to move with the market. A of $1$ indicates market-level sensitivity, a above $1$ indicates greater sensitivity, and a below $1$ indicates lower sensitivity. is not a measure of an asset’s total .

For example, if the risk-free rate is $3\%$ , the expected market is $8\%$ , and an asset’s is $1.2$ , CAPM estimates its expected as $3\%+1.2(8\%-3\%)=9\%$ . The represents the model relationship between and expected .

CAPM is an introductory model, not a guarantee or a complete description of actual returns. Its estimate depends on assumptions and inputs, and other asset-pricing models consider additional sources of risk.