## Safety First With Riskless Lending And Borrowing

In the text we discuss the choice of portfolios that satisfies each of the three standard formulations of the safety first criteria, assuming choices are to be made from among risky assets. In this appendix we extend the analysis to include riskless lending and borrowing.

### Roy Criteria

If a riskless lending and borrowing rate exists and returns are normally distributed, then the Roy criteria lead to infinite borrowing or only investment in the riskless asset depending on the relationship between RL and RF.19 Figures 11.10 and 11.11 illustrate two possible patterns.

In both cases the investor wishes to rotate the line passing through RL as much as possible in the counterclockwise direction. In Figure 11.10 this leads to investing in the riskless portfolio. This makes intuitive sense. If the riskless asset yields more than RL (as it does in Figure 11.10), then by investing in the riskless asset the investor has zero chance of obtaining a return below Rv Since returns are assumed normally distributed, any policy involving a risky asset has some probability of a return below RL\ thus 100% investment in the riskless asset is optimal.

IfRp is below Rl, then investment in RF guarantees a return below Rv Examining the figure shows that borrowing an infinite amount at RF and investing in a risky asset is optimal.

### Kataoka Criteria

If lending and borrowing at the riskless rate of interest exists, then using Kataoka's criteria, the optimum policy is to invest 100% in the riskless asset or borrow an infinite amount, depending on the slope of the RL lines in relation to the slope of the Rf line.20 Figure 11.12

18Another limitation of this so-called delta-normal approach is that it fails to account appropriately for sources of risk that arise from changes in volatility or the other characteristics of returns. Phillipe Jorion {55] describes this and other more advanced procedures referred to as delta-gamma approximations which attempt to account for these additional sources of risk.

19There are an infinite number of solutions if RL = Rf.

2l>rhere are an infinite number of solutions if the slope of the RL lines is the same as the slope of the lending-borrowing line.

Figure 11.12 Investment alternatives leading to 100% investment in riskless asset.

shows a situation where the optimum policy is to invest 100% in the riskless asset. (Slope of R{ lines greater than slope of lending-borrowing line.) Figure 11.13 shows a case where the optimum policy is infinite borrowing (slope of lending-borrowing iine greater than slope of Rl line).

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