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# The Mathematical Derivation of the CAPM

Given the perfect market assumptions of the single period-index CAPM, consider an investor who initially places nearly all their funds in a portfolio reflecting the composition of the market. They subsequently invest the balance in security j. Using sequential numbering from previous equations, let us define R(P) the expected return on the revised portfolio as the weighted average of the expected returns of the individual components. This is given by adapting Equation (1) the basic formula for portfolio return from Chapter Two (remember?).

Where:

x = an extremely small proportion,

rj = expected rate of return on security j,

rm = expected rate of return on the market portfolio.

Subject to the original model's non-negativity constraints and requirements that sources of funds equal uses, the portfolio variance is also based on Equation (2) from Chapter Two:

The portfolio will be efficient if it has the lowest degree of risk for the highest expected return, given by the objective functions:

MAX: R(P), given VAR(P)

MIN: VAR(P), given R(P)

But note what has happened. By introducing security j into the market portfolio, the investor has altered the risk-return characteristics of their original portfolio. According to Sharpe and others, the marginal return per unit of risk is derived by:

1) Differentiating R(P) with respect to the investment in security j; A R(P)/ Ax,

2) Differentiating VAR(P) with respect to the investment in security j; AVAR(P)/ A x.

Since (iii) above simplifies to AR(P)/ AVAR(P) as x tends to zero, the incremental return per unit of risk is therefore given by:

However, you will recall from our explanation of the SML that an investor can either borrow or lend at the risk-free rate of interest (rf) with a beta value of zero. So, by incorporating a risk-free investment or a liability (if x is negative) the incremental rate of return given by Equation (39) is established by substituting rj = rf and bj = 0 into the equation such that:

In a perfectly competitive capital market, the incremental risk-return trade-off must be the same for all investors. So, Equations (39) and (40) are identical:

Now, multiplying both sides of Equation (41) by the denominator on the left hand side and rearranging terms, Sharpe's one period, single factor Capital Asset Pricing Model (CAPM) for individual investments (explained earlier) is confirmed as follows:

And because systematic betas apply to a portfolio, as well as an individual investment we can define R(P) using our earlier notation

Remember, the CAPM is a one period model because the independent variables, rf, rm and bj are assumed to remain constant over the time horizon. It is also a single factor model because systematic risk is prescribed entirely by the beta factor.

Equation (34) represents the expected rate of return on security j, which comprises a risk free return plus a premium for accepting market risk (the market rate minus the risk free rate), assuming that all correctly priced securities will lie on the SML. The market portfolio offers a premium (rm - rf) bj over the risk-free rate, rf , which may differ from the jth security's risk premium measured by the beta factor bj.

Thus, Sharpe's CAPM (like the others mentioned earlier, Lintner et. al) enables an investor to establish whether individual securities (or portfolios) are under or over-priced, since the linear relationship between their expected rates of return and beta factors (systematic risk) can be compared with the SML (the market index).

# The Relationship between the CAPM and SML

Figure 6.1: The CAPM and SML

Activity 1

Take a look at Figure 6.1. This is a reproduction of Figure 5.3, the Security Market Line (SML) explained in Chapter Five. At one extreme we have the expected return on risk-free investment (rf) with a beta value of zero. At the other, portfolio B is a borrowing or leveraged portfolio with a beta of 1.5, which contains securities purchased by borrowing at the risk-free rate of interest. However, superimposed on the new graph are other beta values associated with expected returns, one of which is defined by the point X.

Explain its portfolio implications for rational, risk-averse investors.

Suppose we are considering investing in the security denoted by X on the graph with an expected return of 8 per cent and a beta coefficient of 0.5. We can see that the return is too low for the risk involved and that the security is overpriced because X is located below the SML. Consequently, rational investors wishing to sell their holdings would need to drop their price and increase the return (yield) until it impinges upon the SML at point A.

Given the slope of the SML defined by a risk free rate of 6 per cent and a market return of 16 per cent from a risky balanced portfolio, Figure 6.1 illustrates why the new equilibrium rate of return A with a beta value of 0.5 should be 11%. You can confirm this using the CAPM model:

where the expected return equals the risk-free rate, plus the market rate minus the risk-free rate, multiplied by the beta factor.

11% = 6% + (16% - 6%).0.5

It is also clear from Figure 6.1 why investing in a security such as Y is beneficial. Stocks above the line will be in great demand, so they will rise in price causing a fall in yield.

From our examination of the data we can therefore draw the following conclusions.

In theoretical efficient capital markets in equilibrium that assimilate all information concerning a security into its price, all securities (or portfolios) will lie on the SML.

Individual investors need not conform to the market portfolio. They need only determine how much systematic risk they wish to assume, leaving market forces to ensure that any security can be expected to yield the appropriate return for its beta.

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