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Optimal risky portfolio and optimal investor portfolio

Since many investors have access to risk-free assets (in the form of government bonds), adding these to a portfolio of risky assets determined on the efficient frontier enriches the risk-return profile of an investor.

Capital allocation line and optimal risky portfolio

We can combine any portfolio on the efficient frontier with the risk-free asset. The line possible combinations are given by the capital allocation line, which connects the risk-free asset (represented by a point on the y-axis) with a risky portfolio on the efficient frontier. The best return per unit of risk is offered by a combination of risky portfolio and the risk-free asset which corresponds to the capital allocation line which is tangent to the efficient frontier.

The two-fund separation theorem

The two-fund separation theorem states that all investors will hold a combination of two portfolios, an optimal portfolio of risky assets and a risk-free asset.

Under the separation theorem, we can segregate the investment decision and the financing decision. The optimal risky portfolio is identified from multiple risk portfolios while ignoring investor’s risk preferences. This decision is based on the risk and return profile of the portfolio assets and their correlations. The line connecting the risk-free asset and the optimal risk portfolio is called the capital allocation line. Each investor identifies his allocation between the risk-free asset and the optimal risk portfolio keeping in view his indifference curve (which depends on his risk preferences, etc.).

Optimal investor portfolio

The optimal portfolio for any investor occurs at a point at which the (best) capital allocation line, the one tangent to the efficient frontier at the optimal risky portfolio is also tangent to his highest indifference curve.

Investor preferences and optimal portfolios

An investor can combine the risk-free asset and the risky portfolio in a way that suits their risk preferences.

For example, a very risk-averse investor may invest 100% in the risk-free asset. Hence, his optimal investor portfolio will be close to the y-axis. However, other investors may choose any combination of optimal risky portfolio and the risk-free asset. This framework allows an investor to even borrow at the risk-free rate and invest in the risky portfolio on margin (thereby taking a leveraged position).

by Obaidullah Jan, ACA, CFA on Tue Feb 11 2020

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