# 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.)

• It seems a little trivial but because of risk aversion, R must be less attractive than risky efficient investment C so $R < 15$ percent Nov 14, 2023 at 19:30
• @nbbo2 Thanks, and agreed, but I think the question wants something more substantial (probably with working).
– mjc
Nov 15, 2023 at 11:11

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!

• ...and not be above 15%, I'd say. Nov 15, 2023 at 17:47
• Well spotted 🤣👍
– T123
Nov 15, 2023 at 17:58
• I follow that train of thought. Yet, geometrically, I truly have a hard time drawing an efficient frontier (without risk free asset) in this example. Nov 15, 2023 at 18:40
• Yes, I was also struggling with it. Perhaps C And D are nearly perfectly positively correlated? The problem doesn't tell us anything about it unfortunately..
– T123
Nov 15, 2023 at 18:43
• If they were perfectly correlated, intuitively, I’d assume that we would have to reflect the lower half of the frontier at the corresponding point on the y-axis. Then, asset B would be outside the frontier. Maybe the assets have not been selected sufficiently carefully ? Nov 15, 2023 at 19:02