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The CAPM Model
A. Portfolio Consisting of A riskfree asset and a risky asset
Let = the rate of return on the portfolio. Let = the rate of return on the risk free asset.
Let = the rate of return on the risky asset. Let = the proportion of the portfolio invested in the risky asset Let = the standard deviation of the rate of return on the risky asset.
B. Portfolio Consisting of two risky assets
Let be respectively the rates of return on the portfolio, asset A and asset B and the standard deviations. Let w = the proportion of the portfolio invested in asset A.
Where =the proportion of asset A in the minimum variance portfolio.
C. A Portfolio Consisting of Two Risky Assets and the riskfree asset.
Consider combining a portfolio consisting of the two risky assets A and B with a riskfree asset.. We assume that A has a higher expected return and a higher variance than asset B. The rational investor risk-averse investor will choose to hold a part or all of his or her wealth in the riskfree asset and the rest in a optimally designed portfolio consisting of specific proportions of assets A and B. The optimal portfolio will be at the point of tangency. ( This will be explained in the class.)
Let the proportion of the portfolio invested in asset A at the point of tangency.
D. Defining the Beta
Consider a portfolio made up of n assets. Let = the rate of return on the portfolio, let ri = the
rate of return on the ith asset. Let the standard deviation of the rate of return on the ith
asset. Then
the risky assets as its proportion in the optimal portfolio. A portfolio which holds assets in the same proportions as they are held in the market is called a market portfolio. The implication of the CAPM model is that the Market portfolio is the optimal portfolio.
We now redefine the Beta of each asset in relations to the Market Portfolio.
We can estimate the for asset k by estimating the following regression.
where is the rate of return on the market portfolio. (As a practical matter we usually use a
broadly based market index such as the Standard and Poor’s 500 as a proxy for the market portfolio.) The beta of a stock measures the degree to which the stock moves with the market and the magnitude of those movements. A beta of one means the stock moves with the market: in other words when the market moves up 10% the stock moves up 10%. A stock with a beta between 0 and 1 moves with the market but to a lessor degree. Such a stock is called a conservative investment. A stock with a beta greater than 1 is a stock which moves in the same direction as the market but it moves more than the market moves. Such a stock is referred to as aggressive. A stock which has a negative beta moves in the opposite direction as the market.
The Capital Market Line
The equation of the capital market line is
The Security Market Line
This simple says the risk premium on the kth asset is simply that asset’s beta times the risk premium on the market portfolio.
