Stochastic Discount Factors and Efficient Portfolio Choice
Summary
The document asks whether investors who understand asset pricing should hold only portfolios on the mean-variance efficient frontier. It presents a relation between an asset’s expected return, its volatility, its correlation with the stochastic discount factor, and the mean and volatility of that factor. The discussion interprets assets on opposite ends of the frontier as having different correlations with the discount factor and different roles in bearing risk or providing insurance.
The text raises the possibility that an investor might combine assets or portfolios that are individually inefficient to achieve an outcome between those extremes, but it does not provide an answer or a derivation. Its claims about frontier assets and correlation are presented as the questioner’s understanding, so readers should treat them as framing assumptions rather than established conclusions. The document is useful as a conceptual prompt about the relationship between pricing, systematic risk, and portfolio efficiency.
Key ideas
- The document links expected returns to volatility and correlation with a stochastic discount factor.
- It frames the efficient frontier in terms of risk bearing and insurance across states.
- It asks whether investors might use inefficient holdings to reach intermediate portfolio outcomes.
- The discussion does not resolve the question or establish the stated interpretations.
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Full text
# Asset Pricing and Stochastic Discount Factor: Do well-informed investors only buy efficient portfolios?
# Asset Pricing and Stochastic Discount Factor: Do well-informed investors only buy efficient portfolios?
I'm currently dealing with the following question:
In Asset Pricing, well-informed investors know about the concept of the efficient frontier. Does this mean that they only invest in portfolios that lie on this efficient frontier? I have that
$E[R^i]=R^f - \rho_{m, R^i} \sigma[R^i]\frac{\sigma[m]}{E[m]}$
where $R^i$ is the return of an asset i, $m$ is the stochastic discount factor therefore $\rho_{m, R^i}$ is the correlation between the stochastic discount factor and asset i, $\sigma[R^i]$ is the standard deviation of asset i, $\sigma[m]$ is the standard dev. of the SDF and $E[m]$ is the expected value of the SDF.
My thoughts: I know that assets that are located on the efficient frontier contain no unsystematic risk and are perfectly correlated with the stochastic discount factor (those on the upper part of the frontier have a correlation of -1 with the SDF and therefore carry maximum returns at maximum risk, those on the lower part of the frontier have a correlation of +1 with the SDF and therefore represent an optimal insurance against fluctuations in consumption. Generally, it would make sense to only invest in assets that lie on the efficient frontier as they are efficient, but on the other hand, I do not fully understand whether it can make sense to invest in an inefficient portfolio in order to achieve something that lies between the "maximum risk" and "perfect insurance" extremes I just described...can anyone help?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.