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I am currently looking through some actuarial study materials (CM2, formerly CT8) in which models of asset returns are being discussed. One such model is the market model (A.K.A The single-index model), which is defined to be of the form $$ R_i = \alpha_i + \beta_i R_M + \varepsilon_i $$ where

  • $R_i$ is the return on asset $i$ (being modelled)
  • $R_M$ is the return of the market
  • $\alpha_i$ and $\beta_i$ are constants (to be found through regression analysis)
  • $\varepsilon_i$ is a random variable representing the return from asset $i$ that is independent of market returns.

Within this context, it goes on to say that only the systematic risk, measured by $\beta_i$ should be expected to be rewarded by increased returns since this risk is non-diversifiable).

Can anyone explain why the unsystematic risk (i.e. the variance of $\varepsilon_i$) would not be expected to be rewarded with greater returns?

For example, suppose you have two companies within the same industry. If one company is a long-standing big player (HSBC, say) and another is a smaller up-and-coming company (Monzo) then although they would likely be correlated with the same underlying index, you would expect a greater return on the smaller player due to the greater market risk associated with it.

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  • $\begingroup$ By "greater market risk" do you mean that during a sharp economic downturn Monzo has greater chance of going out of business that HSBC? If so this risk should in theory be captured by a higher beta for Monzo. $\endgroup$
    – noob2
    Jul 28 '20 at 18:14
  • $\begingroup$ Those $R$'s are excess returns. $\endgroup$
    – kurtosis
    Jul 28 '20 at 18:25
  • $\begingroup$ @noob2 What about internal issues such as governance, which affect companies independently? It might be argued that smaller companies are at greater risk of cyberattacks, unknowingly violating regulation (i.e. GDPR), or running into cashflow problems that are not a result of broader economic circumstances? $\endgroup$
    – John Smith
    Jul 28 '20 at 18:56
  • $\begingroup$ @noob2 Actually, isn't the specific risk rewarded through the $\alpha$ term (which may have a non-zero expectation) as this is the return on the asset above (or below) market returns? $\endgroup$
    – John Smith
    Jul 28 '20 at 19:21
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Equal-weighted Portfolio

The typical way to answer this is to consider an equally-weighted portfolio of $n$ assets. An asset $i$ thus has portfolio weight $w_i=\frac{1}{n}$ and excess returns $R_i$ with all assets having the same volatility $\sigma_i=\sigma$ and correlation $\rho_{ij}=\rho$ with another asset $j$.

The portfolio variance ${\rm var}(R_P)$ (the risk measure we consider in mean-variance optimization) is given by: $${\rm var}(R_P) = \sum_{i=1}^n \sum_{j=1}^n w_i w_j {\rm cov}(R_i,R_j).$$ If we expand this out and remember that ${\rm cov}(R_i,R_i)={\rm var}(R_i)=\sigma^2$, we can rewrite the portfolio variance as: $$\begin{align} {\rm var}(R_P) &= \frac{1}{n^2} \left[\sum_{i=1}^n \sigma^2 + \sum_{i=1}^n \sum_{j=1,j\neq i}^n {\rm cov}(R_i,R_j)\right], \\ &= \frac{n\sigma^2}{n^2} + \frac{1}{n^2}\sum_{i=1}^n \sum_{j=1,j\neq i}^n \rho \sigma^2, \\ &= \frac{n\sigma^2}{n^2} + \frac{n^2-n}{n^2}\rho \sigma^2 = \frac{\sigma^2}{n} + \left(1-\frac{1}{n}\right)\rho \sigma^2 . \end{align}$$

If we hold a very-well-diversified portfolio, we can see that this is like taking the limit of the portfolio variance as $n$ gets large. $$\lim_{n\to\infty} \left(\frac{\sigma^2}{n} + \left(1-\frac{1}{n}\right)\rho \sigma^2\right) = \rho\sigma^2.$$

Non-equally-weighted Portfolio

We could also extend this by allowing small variations from $w_i=\frac{1}{n}$ in the portfolio weights and allowing for some differences from $\sigma_i^2=\sigma^2$ in variances. (In that case, the CAPM would say $\sigma_i^2=\beta_i^2\sigma_M^2 + \sigma_{\epsilon_i}^2$.) So long as those differences were not "too large" (and defining that would get very technical), the limit would still exist, the first term would still disappear in the limit, and the limit would still be related to the correlation among assets and some average level of variance.

Source of Systematic Risk

If we use the CAPM, the source of the common correlation among stocks is their sensitivity $\beta$ to "the market."

Aside from the CAPM, what causes a correlation among all assets? The macroeconomy is a good guess. Since we use a forward-looking measure of stock returns to assess the macroeconomy, even if we do not impose the CAPM, it is a sensible model.

Finally, that common correlation term is the systematic risk in portfolio $P$. Hence only the systematic risk matters in the limit as $n\to\infty$ (i.e. as our portfolio becomes increasingly diversified).

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Nobody is saying that diversifiable risks are not rewarded (or punished) in practice.

The question is whether it's reasonable to expect compensation for taking this kind of risk. This would then implicitly assume that someone else should reasonably expect to take a loss on the other side of the same risk that they could hedge away, or diversify down.

You might expect your idiosyncratic risk to pay off; and someone else with the opposite idiosyncratic risk probably expects their's to pay off for them too. Neither of you would be positioned thus if you both didn't. However, the rest of us without your biases will just expect a zero-sum game between the two of you. If you are long the market plus 1% Apple; and he's long the market less 1% Apple and your combined performance is anything different to the market, something very wrong has been very badly mismeasured!!!

Of course, one of the two of you will be an Apple winner; and the other an Apple loser... but why should anyone else without either of your opposed biases believe that there was any special reason that Apple should deliver expected positive or negative excess returns to the market?

For you to expect to get paid for your idiosyncratic risks, you have to believe there's a "patsy" who is willing to pay you to hold your positions, for reasons that transcend them maximising their own profits. That's the crux of the problem.

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