Calculating Expected Return and Risk of a Portfolio

Introduction

Investing in a diversified portfolio is a cornerstone of modern finance, providing a strategic way to optimize returns while mitigating risk. The calculation of expected return and risk (variance or standard deviation) of a portfolio is a fundamental concept in portfolio management, enabling investors to make informed decisions. In this article, we delve into the formulas, components, and techniques required to calculate both the expected return and the associated risk of a portfolio.

1. Understanding Expected Return

The expected return of a portfolio is essentially the weighted average of the expected returns of the individual assets within the portfolio. It gives investors a sense of the potential profit that they can earn on their investment.

Formula:

Where:

• $E\left(R_p\right)$E(Rp​) = Expected return of the portfolio
• $w_i$wi​ = Weight of asset $i$i in the portfolio
• $E\left(R_i\right)$E(Ri​) = Expected return of asset $i$i
• $n$n = Number of assets in the portfolio

The weight $w_i$wi​ represents the proportion of total investment in each asset. For instance, if you invest 60% in Asset A and 40% in Asset B, the respective weights would be 0.6 and 0.4.

2. Understanding Portfolio Risk

Risk in the context of a portfolio refers to the variability of returns, commonly measured by standard deviation or variance. When constructing a portfolio, calculating its overall risk is complex because it depends not only on the individual risks of the assets but also on how these assets interact with each other, i.e., their correlation.

Formula for Variance:

Where:

• $w_i$wi​ and $w_j$wj​ = Weights of assets $i$i and $j$j
• $\text{Cov}(R_i,R_j)$Cov(Ri​,Rj​) = Covariance between returns of assets $i$i and $j$j

3. Covariance and Correlation

To understand portfolio risk, it’s crucial to explore covariance and correlation. Covariance measures how two assets move in relation to each other. Positive covariance means that the assets tend to move in the same direction, while negative covariance implies they move in opposite directions.

Formula for Covariance:

Where:

• $R_{i,t}$Ri,t​ = Return of asset $i$i at time $t$t
• $E\left(R_i\right)$E(Ri​) = Expected return of asset $i$i
• $T$T = Number of periods

The correlation coefficient, denoted by ${\rho }_{i,j}$ρi,j​, standardizes the covariance and provides a measure that ranges between -1 and +1.

Formula for Correlation:

Where:

• ${\sigma }_i$σi​ = Standard deviation of asset $i$i
• ${\sigma }_j$σj​ = Standard deviation of asset $j$j

4. Calculating Portfolio Risk (Standard Deviation)

While variance measures risk in terms of squared deviations from the mean, standard deviation provides a more interpretable measure of risk as it is expressed in the same units as the expected return.

Formula for Portfolio Standard Deviation:

Where:

• ${\sigma }_p$σp​ = Standard deviation of the portfolio
• ${\sigma }_p^2$σp2​ = Variance of the portfolio

5. Example: Calculating Expected Return and Risk

To make the concepts clearer, let's consider a simple example of a portfolio with two assets: Asset A and Asset B.

Assumptions:

• Expected Return of Asset A: 8%
• Expected Return of Asset B: 12%
• Weight of Asset A: 50%
• Weight of Asset B: 50%
• Standard Deviation of Asset A: 10%
• Standard Deviation of Asset B: 15%
• Correlation between Asset A and Asset B: 0.3

Step 1: Calculate the Expected Return

Using the formula for expected return:

Step 2: Calculate the Portfolio Variance

Using the formula for variance:

Step 3: Calculate the Portfolio Standard Deviation

Thus, the expected return of the portfolio is 10%, and the portfolio's standard deviation (risk) is approximately 10.19%.

6. Impact of Diversification

One of the primary benefits of holding a diversified portfolio is that diversification reduces risk. By investing in assets that have low or negative correlations, an investor can lower the portfolio’s overall risk without necessarily sacrificing expected returns.

The principle of diversification is rooted in Modern Portfolio Theory (MPT), which was introduced by Harry Markowitz in 1952. According to MPT, an investor can construct an "efficient frontier" of optimal portfolios offering the highest expected return for a defined level of risk.

7. Efficient Frontier and Sharpe Ratio

The efficient frontier is a graphical representation of a set of optimal portfolios that offer the maximum expected return for a given level of risk. The Sharpe ratio is often used to assess the performance of a portfolio relative to its risk.

Formula for Sharpe Ratio:

Where:

• $R_f$Rf​ = Risk-free rate
• $E\left(R_p\right)$E(Rp​) = Expected return of the portfolio
• ${\sigma }_p$σp​ = Standard deviation of the portfolio

A higher Sharpe ratio indicates that a portfolio is offering better risk-adjusted returns.

8. Conclusion

Calculating the expected return and risk of a portfolio is a multi-faceted process that requires understanding both the individual asset returns and their interrelationships. Diversification, by lowering the correlation between assets, reduces the overall risk while maintaining or even improving expected returns. Portfolio theory thus empowers investors to build portfolios that align with their risk tolerance and return objectives.

In conclusion, while the formulas may seem complex, the practical application of calculating expected return and risk enables investors to create balanced portfolios that maximize potential returns for a given risk level. By incorporating these techniques, investors can move closer to achieving their long-term financial goals.