Calculate the expected return and risk (volatility) of a three asset portfolio using each asset’s weight, return, volatility, and correlations.
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3 Asset Portfolio Formula
The expected return of a three asset portfolio is the weighted average of the three asset returns:
E(R_p) = w_1 R_1 + w_2 R_2 + w_3 R_3
The portfolio variance accounts for how the assets move together through their correlations:
Var_p = w_1^2 s_1^2 + w_2^2 s_2^2 + w_3^2 s_3^2 + 2 w_1 w_2 p_12 s_1 s_2 + 2 w_1 w_3 p_13 s_1 s_3 + 2 w_2 w_3 p_23 s_2 s_3
The portfolio volatility (standard deviation) is the square root of the variance:
Volatility_p = sqrt(Var_p)
Where:
E(R_p) is the expected return of the portfolio. w_1, w_2, and w_3 are the weights (fractions) of each asset, which should sum to 1 (100%). R_1, R_2, and R_3 are the expected returns of each asset. s_1, s_2, and s_3 are the volatilities (standard deviations) of each asset. p_12, p_13, and p_23 are the correlation coefficients between each pair of assets, each ranging from -1 to 1. Var_p is the portfolio variance and Volatility_p is the portfolio standard deviation, the most common measure of total portfolio risk.
Each weight scales how much an asset contributes to return and risk. The correlation terms capture diversification: when assets are less than perfectly correlated, the portfolio volatility falls below the simple weighted average of the individual volatilities.
Correlation and Diversification Reference
The correlation you enter between each pair of assets has a large effect on total risk. The table below shows how the relationship between two assets changes the diversification benefit.
| Correlation | Relationship | Effect on Portfolio Risk |
|---|---|---|
| +1.0 | Move together exactly | No diversification; risk equals the weighted average |
| +0.5 | Move together loosely | Some risk reduction |
| 0.0 | Unrelated | Meaningful risk reduction |
| -0.5 | Tend to offset | Strong risk reduction |
| -1.0 | Move opposite exactly | Maximum diversification; risk can fall sharply |
Typical reference ranges for broad asset classes are shown below to help you fill in reasonable inputs.
| Asset Class | Typical Expected Return | Typical Volatility |
|---|---|---|
| Stocks | 7% to 10% | 15% to 20% |
| Bonds | 2% to 5% | 4% to 8% |
| Cash / Money Market | 1% to 4% | 0% to 2% |
Example
Suppose you hold three assets with the following inputs. Asset 1: weight 40%, expected return 8%, volatility 15%. Asset 2: weight 35%, expected return 5%, volatility 10%. Asset 3: weight 25%, expected return 3%, volatility 6%. The correlations are 0.20 between assets 1 and 2, 0.10 between assets 1 and 3, and 0.30 between assets 2 and 3.
The expected return is (0.40)(8) + (0.35)(5) + (0.25)(3) = 5.70%. The portfolio variance works out to 63.85, so the portfolio volatility is the square root of that, about 7.99%. Notice this is well below the simple weighted average volatility of 11.0%, which is the diversification benefit from holding assets that are not perfectly correlated.
FAQ
Why is my portfolio volatility lower than the average of the individual volatilities?
Because the assets are not perfectly correlated. Whenever a correlation is below 1, the gains in one asset partly offset losses in another, which lowers the combined risk below the weighted average. This is the core benefit of diversification.
Do the weights have to add up to 100%?
For a fully invested portfolio, yes. The calculator still computes a result if they do not and shows your total weight so you can correct it. If the weights do not sum to 100%, the return and risk figures will not represent a fully invested position.
What correlation values should I use?
If you do not have estimates, leaving the correlations at 0 assumes the assets are uncorrelated. In practice, stocks within the same market are often correlated around 0.5 to 0.8, while stocks and bonds are frequently near 0 or slightly negative. Use historical data for your specific assets when you can.