### Calculating Returns using CAPM

Given as assets systematic risk, the expected return can be computed using the capital asset pricing model. The CAPM result is usually used as a first estimate of return and is also used in capital budgeting and the determination of economic feasibility. In addition to providing security expected returns, CAPM can be used for estimating the cost of capital and setting insurance premiums.

## Calculating Expected Returns

The Security Market Line (SML) is the graphical representation of the CAPM with beta reflecting systematic risk on the x-axis and expected return on the y-axis. The SML intersects the y-axis at the risk-free rate and the slope of the line is the market risk premium, Rm – Rf.

The SML is formulated as follows:

$$E(R_i) = R_f + β_i [E(R_m) – R_f]$$

Where $$β_i= \frac{ρ_{ i,m} σ_i } {σ_m}$$.

Therefore, if we have data for the risk-free rate, the market expected return, and the beta of a security or the correlation between the security and the market as well as the standard deviations of the security and the market, we can determine an expected return based on CAPM.

## Question

Assume the risk-free rate is 2%, a security has a correlation of 0.8 with the market index and a standard deviation of 16% while the standard deviation of the market is 12%. If the market expected return is 8%, what is the expected return of the security?

A. 10.56%

B. 5.60%

C. 8.42%

Solution

$$E(R_i) = R_f + β_i [E(R_m) – R_f]$$

Where $$β_i= \frac{ρ_{ i,m} σ_i } {σ_m}$$

Step 1: Find the Beta

β = (0.8 × 0.16) / 0.12 = 1.07

Step 2: Find the expected return

E(Ri) = 2% + 1.07 × (8% – 2%) = 8.42%

Calculate and interpret the expected return of an asset using the CAPM

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