## Expected Return vs. Standard Deviation: An Overview

Expected return and standard deviation are two statistical measures that can be used to analyze a portfolio. The expected return of a portfolio is the anticipated amount of returns that a portfolio may generate, whereas the standard deviation of a portfolio measures the amount that the returns deviate from its mean.

### Key Takeaways

• Expected return calculates the mean of an anticipated return based on the weighting of assets in a portfolio and their expected return.
• Standard deviation takes into account the expected mean return, and calculates the deviation from it.
• An investor uses an expected return to forecast, and standard deviation to discover what is performing well and what is not.

## Expected Return

Expected return measures the mean, or expected value, of the probability distribution of investment returns. The expected return of a portfolio is calculated by multiplying the weight of each asset by its expected return and adding the values for each investment.

For example, a portfolio has three investments with weights of 35% in asset A, 25% in asset B, and 40% in asset C. The expected return of asset A is 6%, the expected return of asset B is 7%, and the expected return of asset C is 10%.

Therefore, the expected return of the portfolio is

[(35% * 6%) + (25% * 7%) + (40% * 10%)] = 7.85%

This is commonly seen with hedge fund and mutual fund managers, whose performance on a particular stock isn't as important as their overall return for their portfolio.

## Standard Deviation

Conversely, the standard deviation of a portfolio measures how much the investment returns deviate from the mean of the probability distribution of investments.

The standard deviation of a two-asset portfolio is calculated as:

σP = √(wA2 * σA2 + wB2 * σB2 + 2 * wA * wB * σA * σB * ρAB)

Where:

• σP = portfolio standard deviation
• wA = weight of asset A in the portfolio
• wB = weight of asset B in the portfolio
• σA = standard deviation of asset A
• σB = standard deviation of asset B; and
• ρAB = correlation of asset A and asset B

Expected return is not absolute, as it is a projection and not a realized return.

For example, consider a two-asset portfolio with equal weights, standard deviations of 20% and 30%, respectively, and a correlation of 0.40. Therefore, the portfolio standard deviation is:

[√(0.5² * 0.22 + 0.5² * 0.32 + 2 * 0.5 * 0.5 * 0.2 * 0.3 * 0.4)] = 21.1%

Standard deviation is calculated to judge the realized performance of a portfolio manager. In a large fund with multiple managers with different styles of investing, a CEO or head portfolio manager might calculate the risk of continuing to employ a portfolio manager who deviates too far from the mean in a negative direction. This can go the other way as well, and a portfolio manager who outperforms their colleagues and the market can often expect a hefty bonus for their performance.