# Portfolios with two risky investments

Author:
Mike May
This applet looks at combining two risky investments, S and B, in a risky portfolio, P, where each investment, S and B, has an expected rate of return, E(r), and a level of risk, . Risky asset S can be thought of as a portfolio of all stock in the economy and riskyasset B can be thought of as a portfolio of all risky bonds in the economy. Besides the expected return and risk of the investments, we must also know the correlation, (or covariance, ) between the returns of the two risky assets and the investment weighting each risky asset is given in the portfolio, P. We then compute the rate of return and the level of risk for P, the risky portfolio. Changing the weight in risky asset S, ws, moves P along the risk-return curve. Changing the features (i.e. expected returns, levels of risk, and correlation coefficient of the returns for the two risky assets) of S and B moves the risk-return curve itself. Pay special attention to the impact changing the correlation coefficient has on the shape of the risk-return curve.

## The math behind the chart

The basic assumption is that portfolio P is achieved by investing in each risky asset, S and B, such that the weights of investment sum to 1.  Then the expected return for portfolio P is a weighted average of the rates of return on S and B (i.e. E(rp) = wS*E(rS)+(1-wS)*E(rB)). If S is the risk associated with S, and is the risk associated with B, and r is the correlation coefficient of the returns on the two risky assets, then:  Note: the above equation gives then variance of returns on the risky portfolio, P. To get the standard deviation, , we simply take the square root of variance. Notice that changing w moves portfolioP along the risk-return curve, while changing the parameters of the two risky assets (including the correlation coefficient) moves the curve itself.

## Special cases

Consider what happens if E(rS)= 0.20,  = 0.30, E(rB) = 0.10, and = 0.15. Now consider weights of 50% in each asset. Notice that a simple weighted average for the riskiness of portfolio P would be = 0.5* 0.30 + 0.50*0.15 = 0.225. As long as is less than 1, the riskiness of P is less than 0.225. This is diversification – the elimination of risk! When  =1 this suggests the returns of S and B are perfectly correlated. The resulting risk-return curve is a simple line segment, suggesting zero benefit from diversification. If = -1 this suggests there is a perfect correlation between the upside of one investment and the downside of the other. In other words, the risky assets are perfectly negatively correlated. In this scenario, there exists a zero-risk portfolio with an abnormal return (i.e. the portfolio that intersects the y-axis).