Formula for calculating expected return of an asset given its risk and market data:

$ER_A = r_f + \beta_A(ER_M - r_f)$

where:

$ER_A$ is an expected return of investment

$r_f$ is a risk-free rate

$\beta_A$ some "beta" of the investment (explained below)

$ER_M$ is an expected market return

$(ER_M - r_f)$ is an expected market risk premium

Three main messages

1. $ERP_A = \beta_A * ERP_M$ where:

   $ERP_A$ is the Expected Risk Premium of the current asset,

   $ERP_M$ is the Expected Risk Premium of the whole market,

   $\beta_A$ is some coefficient for current asset (comes in the next message)

2. $\beta_A = \dfrac{Cov(r_A, r_M)}{Var(r_M)}$ where:

   $r_A, r_M$ are rates of asset and market

   Hence, the $\beta_A$ shows us a measure of how much risk the investment will add to a portfolio that looks like the market. If a stock is riskier than the market, it will have a beta greater than one. If a stock has a beta of less than one, the formula assumes it will reduce the risk of a portfolio.

3. The expected risk premium of a stock is unrelated to the stocks idiosyncratic (individual risk of asset, not systematic)

Assumptions

• all investors are identical Mean-Variance Portfolio optimizers (Mean-Variance Portfolio).
• all investors share the same homogeneous expectations about the risk-free rate $r_f$, the expected holding period return of all assets $\mu$, and the covariance matrix $\Sigma$

Consequences of the assumptions

• All investors will come up with the same tangency portfolio (see Tangency Portfolio)
• Tangency portfolio must coincide with the aggregate market and hence can be called 'market portfolio'
• Price of each asset adjusts such that the aggregate demand equals its fixed supply
• As the market portfolio is a Markowitz efficient portfolio that invests 100% of wealth in Tangency Portfolio it holds $E[r_M] - r_f = \gamma_M * Var(r_M)$
• The expected risk premium of an individual security is $E[r_A] - r_f = \beta_A * (E[r_M] - r_f)$ for all assets A