## Expected Return vs. Standard Deviation

### 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 might not be.

### 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 7.85% (35%*6% + 25%*7% + 40%*10%).

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 by squaring the weight of the first asset and multiplying it by the variance of the first asset, added to the square of the weight of the second asset, multiplied by the variance of the second asset.

Then, add this value to 2 multiplied by the weight of the first asset and second asset multiplied by the covariance of the returns between the first and second assets. Finally, take the square root of that value, and the portfolio standard deviation is calculated.

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, variances of 6% and 5%, respectively, and a covariance of 40%. The standard deviation can be found by taking the square root of the variance. Therefore, the portfolio standard deviation is 16.6% (√(0.5²*0.06 + 0.5²*0.05 + 2*0.5*0.5*0.4*0.0224*0.0245)).

Standard deviation is calculated, much like expected return, 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.