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3 Asset pricing

A rational investor would not invest in an asset which does not improve the risk-return characteristics of his existing portfolio. Since a rational investor would hold the market portfolio, the asset in question will be added to the market portfolio. MPT derives the required return for a correctly priced asset in this context.

3.1 Systematic risk and specific risk

Specific risk is the risk associated with individual assets - within a portfolio these risks can be reduced through diversification (specific risks "cancel out"). Systematic risk, or market risk, refers to the risk common to all securities - systematic risk cannot be diversified away (within one market). Within the market portfolio, asset specific risk will be diversified away to the extent possible. Systematic risk is therefore equated with the risk (standard deviation) of the market portfolio.

Since a security will be purchased only if it improves the risk / return characteristics of the market portfolio, the risk of a security will be the risk it adds to the market portfolio. The volatility of the asset, and its correlation with the market portfolio, is historically observed and is therefore a given. The (maximum) price paid for any particular asset (and hence the return it will generate) should also be determined based on its relationship with the market portfolio.

3.2 Capital Asset Pricing Model

The asset return depends on the amount paid for the asset today. The price paid must ensure that the market portfolio's risk / return characteristics improve when the asset is added to it. The CAPM is a model which derives the theoretical required return (i.e. discount rate) for an asset in a market, given the risk free rate available to investors and the risk of the market as a whole.

The CAPM is usually expressed:

Once the expected return, , is calculated using CAPM, the future cash flows of the asset can be discounted to their present value using this rate to establish the correct price for the asset. (Here again, the theory accepts in its assumptions that a parameter based on past data can be combined with a future expectation.)

A more 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. In theory, an asset is correctly priced when its observed price is the same as its value calculated using the CAPM derived discount rate. If the observed price is higher than the valuation, then the asset is overvalued; it is undervalued for a too low price.

Mathematically:

(1) The incremental impact on risk and return when an additional risky asset, a, is added to the market portfolio, m, follows from the formulae for a two asset portfolio. These results are used to derive the asset appropriate discount rate.

Risk =

Hence, risk added to portfolio =
but since the weight of the asset will be relatively low,
i.e. additional risk =

Return =

Hence additional return =

(2) If an asset, a, is correctly priced, the improvement in risk to return achieved by adding it to the market portfolio, m, will at least equal the gains of spending that money on an increased stake in the market portfolio. The assumption is that the investor will purchase the asset with funds borrowed at the risk free rate, rf; this is rational if .

Thus:
i.e. :
i.e. :
is the “beta” , -- the covariance between the asset and the market compared to the variance of the market, i.e. the sensitvity of the asset price to movement in the market portfolio.

3.3 Securities Market Line


The relationship between Beta and required return is plotted on the Securities Market Line (SML) which shows expected return as a function of . The intercept is the risk free rate available for the market, the slope is unit of return per unit of risk , .

4 References

5 See also



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