Using Portfolio Efficiency to Bound the Risk-Free Rate

Summary

The document considers what can be inferred about a risk-free return when one risky investment is said to be efficient under CAPM assumptions. The answer offers a geometric argument: compare the candidate investment and the other listed portfolios in risk-return space, then consider whether a line from the risk-free return would make any of them attractive alternatives. If another portfolio lies above that line, the claim that the candidate is the only efficient choice may fail.

This is an informal pointer rather than a complete derivation. It suggests a lower-bound intuition using the other investments and proposes checking the line through the candidate and another portfolio, but it does not carefully define efficiency, derive a valid interval for the risk-free rate, or resolve whether the stated efficiency is unique. The figures in the question are inputs to the exercise, not evidence from observed markets, so the conclusion should be checked against the course’s precise assumptions.

Key ideas

• Efficiency can be assessed by comparing portfolios in expected return and standard deviation space.
• A risk-free return defines a line against which risky portfolios can be compared.
• If another portfolio lies above that line, it may challenge a claim that a given portfolio is uniquely efficient.
• The reply offers an intuition rather than a complete derivation of the allowed risk-free rate.

Tags

# What do the existence and parameters of an efficient investment tell you about the value of a risk-free return?


# What do the existence and parameters of an efficient investment tell you about the value of a risk-free return?












I'm working on an unassessed course problem,

> Consider the following risky investments \begin{matrix} \text{name} & \text{expected return} & \text{standard deviation of return} \\ A & 9\% & 21\% \\ B & 5\% & 7\% \\ C & 15\% & 36\% \\ D & 12\% & 15\% \end{matrix} Suppose there is a risk-free return $R$ and you are told that $C$ is efficient. What can you say about the value of $R$?

There are other questions about finding the market portfolio, so I'm guessing something different is intended here, but I can't think what. Could someone give me a pointer? (CAPM assumptions apply.)

## Answer by T123 (score 1)

https://quant.stackexchange.com/a/77370

I have given it a bit of a thought and here are my 50 ct:

If C is the only efficient portfolio, then your riskfree return shouldn't be lower than 10 %, otherwise A,B or D are above the line and thus not inefficient anymore.. Given C and D, you can construct a line and check its value at $\sigma=0$, then add a bit on the riskfree return such that D becomes inefficient.

.. just a thought. Happy to discuss!

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