Efficient Frontier Portfolios and the Capital Market Line

Summary

The document asks whether an efficient-frontier portfolio must also lie on the capital market line (CML), and whether a portfolio’s standard deviation can be inferred from its expected return using the CML equation. It presents a CAPM setup with a risk-free rate, market return and volatility, alongside a portfolio whose expected return is specified as efficient. The question notes that the risky-asset efficient frontier is curved while the CML is a tangent line through the risk-free asset and market portfolio.

The key issue is that the CML relation applies to efficient combinations of the risk-free asset and the market portfolio under the usual CAPM assumptions; a risky portfolio on the efficient frontier need not itself be such a combination. The provided material contains no answer or additional frontier parameters, so it does not establish the portfolio’s standard deviation. Expected return and market volatility alone are insufficient to locate an arbitrary efficient portfolio on the risky-asset frontier.

Key ideas

• The CML describes risk and expected return for combinations of the risk-free asset and the market portfolio.
• A risky-asset efficient frontier is generally curved, while the CML is tangent to it at the market portfolio under CAPM assumptions.
• Being on the efficient frontier does not by itself establish that a portfolio lies on the CML.
• The stated expected return and market volatility do not determine the portfolio’s standard deviation without further frontier information.

Tags

# Does a portfolio on efficient frontier also lie on CML(capital market line)?


# Does a portfolio on efficient frontier also lie on CML(capital market line)?












I am trying to solve this question:

Assume that CAPM is true. The risk-free rate is 3%, the expected return on the market portfolio is 10% and the standard deviation of the return on the market portfolio is 15%. Consider a portfolio P with expected return of 25% and assume that it is on the efficient frontier.

Question: What is the standard deviation of the portfolio?

After I read the question, I immediately think of using CML to calculate the portfolio standard deviation: $$R_p = r_f + \frac{E[R_m] - r_f}{\sigma_m}\sigma_p$$

However, the question states that the portfolio lies on the efficient frontier. My concern is, the efficient frontier has a concave look; on the other hand, the CML is a line connects the point $(0, r_f)$ and tangent to the market portfolio which also lies on the efficient frontier. So wouldn't any portfolio on the efficient frontier necessarily lie below the CML? How should one calculate the standard deviation in this case?

For example, the red point represents the portfolio of interest; clearly, it is below the CML. What approach should I adopt to calculate its standard deviation?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.