Fig. 4 The efficient Frontier
Portfolio Choice with a Risk-free Security.
In the above discussion of the efficient frontier, we assumed that the opportunity set
comprised only risky securities, so that all portfolios had some risk as measured by non-
zero volatility. Let us examine how the portfolio choice would be altered when we
introduce a security with zero volatility. In other words, we bring in a more realistic
scenario, wherein investors have opportunities to invest or borrow at risk-free rate of
return. This security being risk-free, yields a fixed return, and hence has a correlation
coefficient of zero with all other securities available for investment. The government
bonds may be regarded as one such security, the expected return on which is the yield
available on the bonds. Let us examine the impact of introducing this security on the
nature of the efficient set.
The risk-free security F which yields a return of rf has been depicted in Fig. 5. The return
on a portfolio with part investment in the risk-free security and the balance in a set of
risky securities represented by point V would be the weighted average of the two returns,
depicted by point P on the line joining F and V. The s represented by point P will merely
be the proportion of investment in V times the s of V.
For example, if the risk-free rate of return rf is assumed to be 12% (with a s of zero, since
the return on a risk-free security has no volatility by definition) and portfolio V denotes
an expected return and s pf 30% and 20% respectively, and an investor has 40% of his
investment in F and 60% in V, then the combined