Derivation of the CAPM
We know from Markowtiz’ framework concerning two-fund separation that each investor will have a utility-maximizing portfolio that is a combination of the risk free asset and the tangency portfolio. If all investors see the same capital allocation line, they will all have the same linear efficient set called the Capital Market Line (CML). This forms a linear relationship between expected return of the portfolio and the standard deviation. If market equilibrium is to exist we know that the prices of all assets must adjust such that all assets are held by investors, there can be no excess demand. We get the market portfolio, M. Hence, in equilibrium the market portfolio will consist of all marketable assets held in proportion to their value weights.
If we invest a % in a risky asset, i, and (1-a) % in the market portfolio, we get the following mean and standard deviation:
Change in the mean and standard deviation with respect to the percentage of the portfolio, a, invested in asset i is a follows:
However we notice that by the definition of the market portfolio asset i is already hold in the market portfolio according to its market value weight. Therefore the percentage a in the equations is excess demand for i, which in equilibrium must be zero. We elaborate the new information in our equations:
The slope of the risk-return trade-off evaluated at point M in the graph, in market equilibrium, is:
This slope will also be equal to the slope of the CML (known as the Sharpe Ratio) in the point M:
If we rearrange and solve for :
This is the capital asset pricing model, graphically called the security market line.
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