Introduction
The one of the most important questions for investors is how the expected return of an investment is affected by its risk. The answer to this question gives the Capital Asset Pricing Model (CAPM). It was developed by W. Sharpe, J. Linter, J. Treynor and J. Mossin in 1960s.
The idea of CAPM is that there are a number of risks that have an impact on the asset prices. It is natural to assume that an investor should receive high reruns from the high-risk investments. The price of such assets should be relatively high in order to compensate a significant chance of profit loss. However, there are some risks that do not affect the prices of the assets. Such risks can be diversified by using other investments. The CAPM introduces a mathematical relationship of the relationship between risk and return of the asset.
Body
The problem of risk diversification has been discussed for a very long time. Markovits (1952) showed analytically the relationship between correlation and the benefits of risk diversification. He has considered the correlation between the returns of two assets as a measure of fluctuation of these assets. Assume, there are two risky assets, A and B. The risk of the assets is represented by their standard deviations σA and σB. Denote ρ as correlation between the returns of these assets, let the portion invested in asset A is x and the portion invested in asset B is (1-x). If there is a positive functional relationship between the returns of these assets (the correlation coefficient is equal to 1), the weighted average of the risks determines the portfolio risk as follows:
σP=xσA+(1-x)σB
The more complicated case is when there is no perfect correlation between the returns of the asstes. Then, the relationship between the returns is not linear and the standard deviation of the portfolio is less than the weighted average of σA and σB:
σP2=x2σA2+1-x2σB2+2x(1-x)ρσAσB
or
σP2=xσA+1-xσB2-2x1-x(1-ρ)σAσB
The risks of the assets combine nonlinearly, but the expected returns are linearly associated. The expected return of the portfolio is a usual weighted average of the returns of the assets. Assume that an investor is able to lend and borrow at the risk-free rate, denoted as rf. When x>1, an investor borrows at the risk free rate and when x<1, an investor lends at the risk-free rate. The expected return of the portfolio of assets A and risk free investment is:
1-xrf+xEA=rf+x(EA-rf)
The last equation shows that the risk of the portfolio is proportionally associated with the risk of asset A, because asset A is the only risky asset in the portfolio. The risk premium of the asset A divided by the risk of the asset A is called Sharpe Ratio:
Sharpe Ratio=EA-rfσA
The Sharpe Ratio is an indicator of the efficiency of the risky portfolio. Sharpe Ratio is the ratio of income adjusted for risk. It shows (in quantitative terms), how much revenue (return on equity) you will receive at a given level of risk. The Sharpe Ratio can be improved to get the final version of the capital asset pricing model. There are four assumptions should be declared:
Investors evaluate their portfolios only in terms of expected return and risk and they are risk averse over a single holding period.
The capital markets are perfect (infinitely divisible, the cost of transaction is 0, all investors are able to lend and borrow at the risk free rate).
The investment opportunities are equal for all investors.
The estimates of risk, expected return and correlation between the assets are equal for all investors.
These assumptions are needed in order to obtain the CAPM in its basic form. The portfolio improvement rule states that the risk premium of each asset is such that:
ES-rf=βEM-rf
where ES is the expected return of the assent and EM is the expected return of the market portfolio. The value of beta is a measure of sensitivity of the return on the asset to the return on the market portfolio. The last equation establishes the CAPM in equilibrium. The expected return of an asset is:
ES=βEM-rf+rf
The CAPM shows how to calculate the expected return of a stock: an investor should know the risk premium of the equity market EM-rf and the stock’s beta. There are several significant implications of CAPM. It shows that the expected return of the asset does not depend on its stand-alone risk. Moreover, beta coefficient introduces a method of measuring the undiversified risk. According to the model, the expected return of a stock does not depend on the growth rate of the future cash flows. Hence, there is no need in extensive financial analysis and forecasts when determining the expected return of a company’s shares.
Conclusion
The CAPM is a useful method of estimating the expected risk and return of assets. The CAPM shows that the expected return of the assets are lowered when the assets are located in diversified investors. An investor with undiversified portfolio will more likely to take risks that will not be paid off. Finally, the model provides the answer about how an investor should create his portfolios. Although CAMP model is fairly simple to use, many of its original assumptions are not fully or partially met in the real markets.
References
Fama, E. and French, K. (2004). The Capital Asset Pricing Model: Theory and Evidence. Journal of Economic Perspectives, 18(3), pp.25-46.
Perold, A. (2004). The Capital Asset Pricing Model. Journal of Economic Perspectives, 18(3), pp.3-24.
Shih, Y., Chen, S., Lee, C. and Chen, P. (2013). The evolution of capital asset pricing models. Rev Quant Finan Acc, 42(3), pp.415-448.
