Lending and Borrowing

Lending and Borrowing

Lending and Borrowing

Lending and Borrowing

A natural extension of the Markowitz analysis was to consider the problem of building portfolios which included riskless assets and portfolios purchased in part with borrowed funds as well as portfoliosof risky assets paid for in full with the investor's equity.

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Recall that the efficient frontier for portfolios made up of manyrisky assets is typically concave from below in the plane whose axes are risk (as measured by the standard deviation) and expected return.For any given period of time, there are assets whose rates of returncan be predicted with virtual certainty. Since nuclear holocausts, natural disasters, and revolution are conceivable, the word "virtual" isnecessary in the preceding sentence. Nevertheless, most investors havean extraordinarily great confidence that they can predict accuratelythe rate of return on securities of the federal government for any period which is equal to their maturity. For example, Treasury billsmaturing in one year have a precisely predictable rate of return forone year.*

The introduction of riskless assets into portfolios has interesting consequences. In the following diagram the return on a risk-free asset

If the riskless" asset is represented byi, and the portfolio of risky assetsat the point of tangency by ;, it is easy to see that only the secondterm of the equation has a positive value. The value of the first termis zero because the return on the riskless asset has zero variance; the third term has a value of zero because the return on the riskless assethas a standard deviation of zero. It is also true that the variance ofthe portfolio of risky assets is a parameter which is given. Thus,the variance of the combined portfolio depends exclusively on therisky assets at the pointB with the riskless asset, or by levering theportfolioB by borrowing and investing the funds inB. PortfoliosonRfBD are preferred to portfolios betweenA andB and betweenB and C since they offer greater return for a given level of risk orless risk for a given level of return. The efficient frontier is now linear in its entirety. The lineRfBD is Sharpe's capital market line.It relates the expected return on an efficient portfolio to its risk asmeasured by the standard deviation.

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In the diagram above, there is only one portfolio ofrisky assets which is optimal, and it is the same for all investors. Since there isonly one portfolio of risky assets which is optimal, it must be the market portfolio. That is, it includes all assets in proportion to their marketvalue. We can now describe the capital market line mathematicallyin terms of the risk-free rate of interest and the return on the marketportfolio.

This says the expected return on an efficient portfolio is a linear function of its risk as measured by the standard deviation. The slope ofthe line has been called the price of risk. It is the additional expectedreturn for each additional unit of risk.

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