# Does CAPM hold for markets with two risky assets?

Presentations of the CAPM often include statements similar to this:

While idiosyncratic risk can be "diversified away", systematic risk cannot, which is also expressed in the CAPM, which states that in equilibrium, asset returns are fully determined by systematic risk.

However, it seems that the CAPM can be derived even in a market with only two risky assets:

The crucial step, which proves the equilibrium — i.e. that it is optimal for any market participant not to change their portfolio — does not rely on the number of assets being large or the possibility to eliminate idiosyncratic risks.

Can this be?

It would imply that most texts (esp. ones aimed at non-mathematical/business audience) give a wrong impression of the connection between CAPM and diversification in portfolio theory...

• I do not understand where you see the contradiction. Obviously if you have a market with only one asset diversification is meaningless. In this case only systemic risk exists, no ideosyncratic risk. But already with two assets you can diversify by being invested in both assets instead of only one. You can also define idiosyncratic and systemic risk. – Ami44 Feb 26 '17 at 12:52
• The reduction of idiosyncratic risks by diversification and an asymptotic elimination in the limit of many assets plays no direct role in CAPM, yet I have seen it in most texts explaining the CAPM. (You know, like the typical example for uncorellated assets: $\sigma_P = \sigma/\sqrt{n} \rightarrow 0$) – Johannes Gerer Feb 26 '17 at 14:40

The answer depends on whether you think that the risk free rate is zero or unknown.

If it is unknown, then there is no way to determine the optimal portfolio from an efficient frontier of risky assets. Technically, any tangency point on that frontier (with slope $\ge 0$) can be considered efficient, so none is optimal.

If, however, you assume that the risk free rate is 0, then the equation reduces to a linear regression model with an intercept equal to 0. I will explain.

The CAPM is a linear regression which is intended to infer an asset's expected return. It is guided by the intuition that investors prefer returns which are commensurate with risk. In this case, CAPM assumes a quadratic utility function in which investors only care about the first and second moments of the joint distribution of returns (i.e., mean = expected returns; portfolio variance = risk proxy). Solving the utility function gives a slope which is equal to "beta" and a y-intercept which is equal to the risk free rate.

The formula for a linear regression with a zero intercept is readily tractable. Whereas the beta coefficient in a normal regression is found by minimizing the covariance, this is easily done in a zero intercept model by taking the average rise over average run.

I..e., The CAPM without a risk free asset would simply say that the expected return on a riskless asset is zero, and the required return of a risky asset is proportional to its variance$\ge 0$.

I am not totally sure about this, but I guess the General Equilibrium derivation of CAPM with a representative agent needs no more than two assets, i.e. a risky asset perfectly correlated with consumption and a risk-free asset. The first order condition of the representative investor states that the expected return of any asset $i$ depends on the covariance with $R^C_{t+1}$, i.e. the return on the claim on aggregate consumption: $$E_{t}[R^i_{t+1}-R^f]=-R^fCov_t(R^i_{t+1},R^C_{t+1})$$ If you interpret $C$ as the market M, then you obtain: $$E_{t}[R^M_{t+1}-R^f]=-R^fCov_t(R^C_{t+1},R^M_{t+1})=-R^fVar_t(R^M_{t+1})$$ taking a ratio between the two equations, it collapses to CAPM: $$E_{t}[R^i_{t+1}-R^f]=\frac{Cov_t(R^i_{t+1},R^C_{t+1})}{Var_t(R^M_{t+1})}E_{t}[R^M_{t+1}-R^f]=\beta_{t}^iE_{t}[R^M_{t+1}-R^f]$$ The whole derivation requires a representative agent and two assets.

• The 'market' usually is understood to not include the risk-free asset, but I updated the question to make it clear. So you show that CAPM works with one risky asset and (suposedly) also with 2 risky assets, so you agree with me. (?) – Johannes Gerer Feb 26 '17 at 0:10