Risk, Expected Returns, and Diversification Under the Present-Value Model

Summary

The document asks whether investors can build a diversified portfolio with higher expected returns by combining risky investments whose returns are uncorrelated. It frames the question with an asset-pricing relation in which price depends on expected future cash flows and the expected return used to discount them. It also connects the argument to the efficient market hypothesis, interpreting market expectations as statistically correct beliefs about future outcomes.

The text does not resolve the question or supply data, calculations, or a portfolio construction method. It is useful as a prompt to distinguish an asset’s expected return from the effect of combining assets: low correlation can reduce portfolio risk, but does not by itself establish that risky assets offer higher expected returns or that their risks are adequately compensated. The author acknowledges that the equations and interpretation may be imprecise. The discussion also leaves unspecified what “risk” means, how expected returns are estimated, and whether the assumed market belief matches the true probability distribution.

Key ideas

• The document asks whether uncorrelated risky investments can raise a diversified portfolio’s expected return.
• It presents price as expected future cash flows discounted using an expected return.
• It invokes market efficiency as a link between market beliefs and the true distribution of outcomes.
• It provides no empirical evidence or solution and flags possible imprecision in its assumptions.

Tags

# Higher risk = high reward?


# Higher risk = high reward?












Some theory (in my understanding) suggests that price is the expectation of future cash flows discounted by expected return: $$p_t=\frac{\mathbb{E}^m_t[c_{t+1}+p_{t+1}]}{1+\mathbb{E}_t^m[r_t]}$$ where $c_{t+1}$ is the cash flow at time $t+1$, $p_{t+1}$ is the resale price at time $t+1$, and $r_t$ is the realized return of period $t$, known only at time $t+1$. Here the expectation is taken under a measure that represents the market aggregate belief, which takes into account all public information at time $t$. The efficient market hypothesis suggests that this belief indeed is the statistically correct distribution of the real world, so $$p_t=\frac{\mathbb{E}[c_{t+1}+p_{t+1}]}{1+\mathbb{E}[r_t]}$$ where the expectation is taken under the actual probability distribution that characterizes the real world dynamics. Here is my question: if what's said above is true, then does it mean if I find a bunch of investments with high risk (so discounted more heavily) that are uncorrelated with each other, then I can achieve a higher expected return of the portfolio in a diversified way? Some of the expressions above might be imprecise (so is my understanding) and I would love to be corrected.

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