Risk and Return are fundamental concepts in portfolio management. Investors aim to maximize returns while minimizing risk through diversification. By combining assets with different characteristics, portfolio risk can be reduced without compromising expected returns. Markowitz Portfolio Theory provides a framework to calculate the risk and return of multi-asset portfolios.
Risk and Return of a Two-Asset Portfolio:
1. Expected Return of a Two-Asset Portfolio
The expected return of a two-asset portfolio is calculated as the weighted average of the individual asset returns. Mathematically, it is expressed as:
E(Rp) = w1*E(R1) + w2*E(R2)
Where:
- E(Rp) = Expected return of the portfolio
- w1, w2 = Weights (proportion of total investment) in Asset 1 and Asset 2
- E(R1), E(R2) = Expected returns of Asset 1 and Asset 2
This equation shows that the portfolio return is a simple combination of individual asset returns based on their proportions in the portfolio.
2. Risk (Standard Deviation) of a Two-Asset Portfolio
Portfolio risk is not a simple weighted average of individual risks. It depends on the correlation between assets. The formula for portfolio standard deviation (σp) is:Where:
- σp = Standard deviation (risk) of the portfolio
- σ1,σ2 = Standard deviations of Asset 1 and Asset 2
- ρ12 = Correlation coefficient between Asset 1 and Asset 2
If the assets have low or negative correlation, portfolio risk is reduced due to diversification. If ρ12 = −1 (perfect negative correlation), risk can be completely eliminated.
Diversification Benefit in a Two-Asset Portfolio:
The primary advantage of holding a two-asset portfolio is risk reduction through diversification. If the assets have low or negative correlation, the combined risk is lower than the risk of individual assets. This is why investors combine stocks and bonds—bonds often move opposite to stocks, balancing the portfolio’s overall risk.
Risk and Return of a Three-Asset Portfolio:
1. Expected Return of a Three-Asset Portfolio
With three assets, the portfolio return is still calculated as the weighted average of individual asset returns:
E(Rp) = w1*E(R1) + w2*E(R2) + w3*E(R3)
Where:
- = Weights of Assets 1, 2, and 3
- E(R1), E(R2), E(R3) = Expected returns of each asset
By adding a third asset, investors gain more flexibility in adjusting risk and return, allowing for better diversification.
2. Risk (Standard Deviation) of a Three-Asset Portfolio
The standard deviation of a three-asset portfolio is more complex, considering the covariances between all three assets:
Where:
- ρ12, ρ13, ρ23 = Correlation coefficients between the three asset pairs
By including a third asset with low correlation to the other two, risk is further reduced.
Diversification Benefit in a Three-Asset Portfolio
The three-asset portfolio provides more diversification than the two-asset portfolio. Adding an asset with a low or negative correlation with the existing portfolio reduces overall risk. Investors can include assets from different sectors, industries, or asset classes (e.g., stocks, bonds, commodities) to optimize risk-adjusted returns.
For example:
- A two-asset portfolio might have stocks and bonds.
- A three-asset portfolio might include stocks, bonds, and gold (gold often moves opposite to stock markets).
This diversification further minimizes risk while maintaining expected returns.
Comparison: Two-Asset vs. Three-Asset Portfolios
| Feature | Two-Asset Portfolio | Three-Asset Portfolio |
|---|---|---|
| Expected Return | Weighted average of two assets | Weighted average of three assets |
| Risk (Standard Deviation) | Includes variance & correlation between two assets | Includes variance & correlation between three assets |
| Diversification | Limited, based on two asset correlations | Higher, with more asset combinations |
| Portfolio Stability | Less stable due to limited asset selection | More stable due to better risk spread |
| Example | Stocks & Bonds | Stocks, Bonds & Gold |