# Capital asset pricing model

The

**Capital Asset Pricing Model** (commonly referred to as

**CAPM**) derives the risk appropriate required rate of return for a given asset in a given market. It was introduced by

William Sharpe, Lintner and Mossin independently, though it is commonly attributed only to the first of them, who published it earliest (in 1964), and subsequently received (jointly with Harry Markowitz and

Merton Miller)

The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel for his contribution to the field of

financial economics.

The risk of a portfolio is comprised of systematic risk and specific risk. Systematic risk refers to the risk common to all securities - i.e. market risk. Specific risk is the risk associated with individual assets. Specific risk can be

diversified away (specific risks "cancel out"); systematic risk (within one market) cannot. Dependent on market, a portfolio of approximately 15 well selected shares (and more) would be sufficiently diversified to leave the portfolio exposed to systematic risk only.

An investor cannot expect to be rewarded for taking on diversifiable risk, (it is not rational to expose one's wealth to more risk than necessary). Therefore, the required return on an asset, that is, the return that compensates for risk taken, must be linked to its riskiness in a portfolio context - i.e. its contribution to overall portfolio riskiness - as opposed to its "stand alone riskiness." In the CAPM context, portfolio risk is represented by higher variance i.e. less predictability.

The CAPM assumes that the risk-return profile of a portfolio can be optimized - an optimal portfolio displays the lowest possible level of risk for its level of return. Additionally, since each additional asset introduced into a portfolio further diversifies the portfolio, the optimal portfolio must comprise every asset, (assuming no trading costs) with each asset value-weighted to achieve the above (assuming that any asset is infinitely divisible). All such optimal portfolios, i.e. one for each level of return, comprise the efficient (Markowitz) frontier.

An investor might choose to invest a proportion of her wealth in a portfolio of risky assets with the remainder in cash - earning interest at the risk free rate (or indeed may borrow money to fund her purchase of risky assets in which case there is a negative cash weighting). Here, the ratio of risky assets to risk free asset determines overall return - this relationship is clearly linear. It is thus possible to achieve a particular return in one of two ways: either 1) by investing all of one’s wealth in a risky portfolio or 2) by investing a proportion in a second portfolio and the remainder in cash (either borrowed or invested). For a given level of return, however, only one of these portfolios will be optimal (in the sense of lowest risk). Since the risk free asset is, by definition, uncorrelated with any other asset, option 2) will generally have the lower variance and hence be the more efficient of the two.

This relationship also holds for portfolios along the efficient frontier: a higher return portfolio plus cash is more efficient than a lower return portfolio alone for that lower level of return. For a given risk free rate, there is only one optimal portfolio which can be combined with cash to achieve the lowest level of risk for any possible return. This is the market portfolio.

The required rate of return for a particular asset in a market is derived based on its sensitivity to the movement of the market portfolio (i.e. the broader market). This sensitivity is known as the asset beta and reflects asset specific risk. The market portfolio, by definition, has a beta of one; a more sensitive (risky) stock will have a higher beta and will be discounted at a higher rate; less sensitive stocks will have lower betas and be discounted at a lower rate. According to the CAPM the required rate of return for a stock is derived by:

r_{s} = β ( r_{m} - r_{f} ) + r_{f}

where:
r_{s} is the required rate of return on a stock
r_{m} is the market portfolio (or proxy) rate of return
r_{f} is the risk free interest rate
β is the beta of the stock - its sensitivity to the movement of the market portfolio

(Fortunately,) the CAPM is consistent with intuition - investors should require a higher return for holding a more risky asset. Betas exceeding one signify more than average riskiness; betas below one indicate lower than average riskiness. Stock market indices are frequently used as local proxies for the market portfolio - and in that case (by definition) have a beta of one. (Most mutual funds portfolios have systematic risk smaller than one.

*?? question truth*')

The CAPM returns the asset appropriate required return or discount rate - i.e. the rate at which future cash flows produced by the asset should be discounted given that asset's relative riskiness. In theory, therefore, an asset is correctly priced when its observed price is the same as its value calculated using the CAPM discount rate. If the observed price is higher than the valuation, then the asset is overvalued (and undervalued for a too low price). Alternatively, one can "solve for discount rate" for the observed price given a particular valuation model and compare that discount rate with the CAPM rate. If the discount rate in the model was lower than the CAPM rate then the asset is overvalued (and undervalued for a too high discount rate).