# 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 portfolios of risky assets paid for in full with the investor’s equity.

Recall that the efficient frontier for portfolios made up of many risky 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 return can be predicted with virtual certainty. Since nuclear holocausts, natural disasters, and revolution are conceivable, the word “virtual” is necessary in the preceding sentence. Nevertheless, most investors have an extraordinarily great confidence that they can predict accurately the rate of return on securities of the federal government for any period which is equal to their maturity. For example, Treasury bills maturing in one year have a precisely predictable rate of return for one 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 by i, and the portfolio of risky assets at the point of tangency by ;, it is easy to see that only the second term of the equation has a positive value. The value of the first term is 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 asset has a standard deviation of zero. It is also true that the variance of the portfolio of risky assets is a parameter which is given. Thus, the variance of the combined portfolio depends exclusively on the risky assets at the point B with the riskless asset, or by levering the portfolio B by borrowing and investing the funds in B. Portfolios on RfBD are preferred to portfolios between A and B and between B and C since they offer greater return for a given level of risk or less risk for a given level of return. The efficient frontier is now linear in its entirety. The line RfBD is Sharpe’s capital market line. It relates the expected return on an efficient portfolio to its risk as measured by the standard deviation.

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

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