Why the Capital Market Line Uses Variance in Mean-Variance Equilibrium

Summary

The document explains why a mean-variance equilibrium relation can express the market risk premium as proportional to market variance, even though the Capital Market Line is usually drawn against standard deviation. For an investor choosing between the market portfolio and a risk-free asset, mean-variance utility yields an optimal market allocation equal to the market risk premium divided by risk aversion times market variance. The square arises from the variance penalty in the investor’s objective.

The answer then aggregates these allocations across investors. In equilibrium, their combined holdings must equal the market portfolio, which produces a risk premium proportional to variance, with aggregate risk aversion as the proportionality term. This distinguishes an equilibrium condition about investor allocations from the CML’s graphical relationship between expected return and standard deviation. The explanation assumes mean-variance preferences and a risk-free asset, and gives a derivation rather than empirical evidence.

Key ideas

• Mean-variance utility penalizes portfolio variance, which introduces a squared risk term.
• An investor’s optimal market allocation depends inversely on risk aversion and market variance.
• Aggregating investor allocations gives an equilibrium premium proportional to market variance.
• The variance-based equilibrium equation differs from the CML plotted against standard deviation.

Tags

# CML equation - from where does the square come from?


# CML equation - from where does the square come from?












In his textbook Asset management Andrew Ang uses the following CML formula (chapter 6)

E(rm) - rf = y * σ^2 Where y is risk aversion factor

What is the source of square? When I look at CML graph there is a straight line of E(rm) versus σ so I would expect equation to be E(rm) - rf = y * σ

## Answer by phdstudent (score 1, accepted)

https://quant.stackexchange.com/a/77234

You are right. We usually plot $E(r_m)$ against $\sigma_m$. But Ang is making a different point there.

Any investor $i$ with Mean-Variance preferences should optimally hold the following fraction of wealth on the market portfolio:

$$w_i^\star = \frac{r_m - r_f}{\gamma_i \sigma^2_m}$$

The formula above comes directly from maximizing:

$$\max_{w_i} E(r_p) - \frac{\gamma_i}{2}\sigma^2(r_p)$$

where $$E(r_p) = w_i E(r_m) + (1-w_i)) r_f$$

Now if you aggregate the first equation across all investors, because in equilibrium and in aggregate we have to hold the market we have $$\sum_{i=1}^I w_i = 1$$ thus:

$$1 = \sum_{i=1}^I \frac{E(r_m) - r_f}{\gamma_i \sigma^2_m}$$

$$\bar{\gamma} \sigma^2_m = E(r_m) - r_f$$

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.