Explain How the Zero Beta Version of CAPM Is Derived

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The Answer is The second assumption of the CAPM is that investors can lend and borrow unlimited sums of money at the riskless rate of interest.In the absence of such a situation,which is more representative of the real world,an alternative capital asset pricing model is required.
The systematic risk (Beta)of a portfolio is an appropriate measure of risk in case of a large diversified portfolio.This is because the unsystematic risk of a large diversified portfolio is essentially zero.Thus,even if an individual asset had a great deal of unsystematic risk,it would have little impact on portfolio risk and,therefore,unsystematic risk would not require a higher return.
If securities or portfolios happen to lie on a straight line in expected return Beta space,all combinations of securities or portfolios would lie on the straight line.A straight line can be described by any two points.One convenient point is the market portfolio with a Beta of 1.The second convenient point is where the straight line cuts the vertical axis,where the Beta equals zero.The equation of the straight line is:
Expected return =

a + b

(Beta)
For a portfolio with zero Beta,letting

\bar { R } _ { Z }

be the expected return on a portfolio,we get,

\bar { R } _ { z } = a + b ( 0 )

or

\bar { R } _ { z } = \alpha

The equation must also hold for the market portfolio.If

\bar { R } _ { M }

is the expected return on the market and,recalling that the Beta for the market portfolio is one,we have

\bar { R } _ { M } = \bar { R } _ { z } + b ( 1 )

or

b = \bar { R } _ { M } - \bar { R } _ { z }

.
Putting the two equations together and letting

\bar { R } _ { i }

be the expected return and

\beta _ { i }

be the Beta on an asset or portfolio,the equation for the expected return on any security or portfolio becomes:

\bar { R } _ { i } = \bar { R } _ { z } + \left( \bar { R } _ { M } - \bar { R } z \right) \beta _ { i }

This is called the zero-Beta version of CAPM where there is no riskless lending or borrowing.A graph depicting this model is plotted below. $The second assumption of the CAPM is that investors can lend and borrow unlimited sums of money at the riskless rate of interest.In the absence of such a situation,which is more representative of the real world,an alternative capital asset pricing model is required. The systematic risk (Beta)of a portfolio is an appropriate measure of risk in case of a large diversified portfolio.This is because the unsystematic risk of a large diversified portfolio is essentially zero.Thus,even if an individual asset had a great deal of unsystematic risk,it would have little impact on portfolio risk and,therefore,unsystematic risk would not require a higher return. If securities or portfolios happen to lie on a straight line in expected return Beta space,all combinations of securities or portfolios would lie on the straight line.A straight line can be described by any two points.One convenient point is the market portfolio with a Beta of 1.The second convenient point is where the straight line cuts the vertical axis,where the Beta equals zero.The equation of the straight line is: Expected return = a + b (Beta) For a portfolio with zero Beta,letting \bar { R } _ { Z } be the expected return on a portfolio,we get, \bar { R } _ { z } = a + b ( 0 ) or \bar { R } _ { z } = \alpha The equation must also hold for the market portfolio.If \bar { R } _ { M } is the expected return on the market and,recalling that the Beta for the market portfolio is one,we have \bar { R } _ { M } = \bar { R } _ { z } + b ( 1 ) or b = \bar { R } _ { M } - \bar { R } _ { z } . Putting the two equations together and letting \bar { R } _ { i } be the expected return and \beta _ { i } be the Beta on an asset or portfolio,the equation for the expected return on any security or portfolio becomes: \bar { R } _ { i } = \bar { R } _ { z } + \left( \bar { R } _ { M } - \bar { R } z \right) \beta _ { i } This is called the zero-Beta version of CAPM where there is no riskless lending or borrowing.A graph depicting this model is plotted below.$

Explain How the Zero Beta Version of CAPM Is Derived

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