Final answer:
The portfolio beta, when 20% is invested in the market portfolio, 30% in an asset with twice the market risk, and 50% in a risk-free asset, is calculated as the weighted average of the assets' betas, resulting in a portfolio beta of 0.80.
Step-by-step explanation:
To calculate the portfolio beta, one must account for the proportional investments in each asset type and their respective betas. The beta of the market portfolio is typically considered to be 1. An asset with twice the risk of the market portfolio has a beta of 2. The risk-free asset, by definition, has a beta of 0 since it is not subject to market risk.
The portfolio beta is thus calculated as the weighted sum of the betas of the individual investments. Since 20% is invested in the market portfolio, 30% in the asset with twice as much risk as the market portfolio, and the remainder (which is 50%) in a risk-free asset, the formula would be:
Portfolio Beta = (0.20 × 1) + (0.30 × 2) + (0.50 × 0)
Portfolio Beta = 0.20 + 0.60 + 0.00
Portfolio Beta = 0.80
Therefore, the portfolio beta is 0.80.