4. CAPM: the price of systematic risk

Level 6: Regression and Pricing Models

Every finance student should know one of the most important topics in the discipline: the capital asset pricing model, the CAPM (*Capital Asset Pricing Model*).

The CAPM describes the relationship between systematic risk and the expected return of assets, especially of shares. It is used everywhere in finance to value risky securities and to generate expected returns given those assets' risk and the cost of capital.

The formula for an asset's expected return given its risk is written like this in code:

# r_i = r_f + beta_i * (r_m - r_f)
retorno_esperado = rf + beta * (rm - rf)

Where:

• r_i: the expected return on the investment in asset i. It can be the return of any share (Apple, Google, Tesla) or of an investment portfolio (any investment fund or hedge fund).
• r_f: the return of the risk-free asset. Usually the 3-month US Treasury bill is taken. The US government is assumed not to default on a short-term government security, so that bill is treated as risk-free by convention in finance.
• r_m: the market return. It is normally approximated by the return of the S&P 500, as the largest index in the world.
• (r_m - r_f): the market risk premium.

The logic is straightforward: investors expect to be compensated for risk and for the time value of money. The risk-free rate in the formula covers the time value of money. The rest of the formula covers the fact that the investor is taking on additional risk.

The beta of a potential investment measures how much risk that investment will add to a portfolio that looks like the market. If a share is riskier than the market, it will have a beta greater than one. If it has a beta below one, the formula assumes it will reduce the portfolio's risk.

The beta coefficient measures the volatility of an individual share compared with the systematic risk of the whole market. In statistical terms, beta is the slope of the line you get from regressing one set of points on another. In finance, each of those points represents the return of a share against the return of the market as a whole. In other words: beta is not a figure that exists anywhere, it is a coefficient estimated with a regression.

The share's beta is then multiplied by the market risk premium, which is the expected market return above the risk-free rate. The risk-free rate is added to that product. The result should give the investor the required return, or the discount rate, which they can use to find the asset's value.

The aim of the CAPM is to assess whether a share is fairly valued when its risk and the time value of money are compared with its expected return.