only in the proportion they invest in it versus in the risk-free asset. What if a risk-free asset is not available? Although T-bills are risk-free assets in nomi- nal terms, their real returns are uncertain. Without a risk-free asset, there is no tangency portfolio that is best for all investors. In this case investors have to choose a portfolio from the efficient frontier of risky assets redrawn in Figure 8.15. Each investor will now choose an optimal risky portfolio by superimposing a personal set of indifference curves on the efficient frontier as in Figure 8.15. An investor with indif- ference curves marked U , U , and U in Figure 8.15 will choose Portfolio P. More risk- averse investors with steeper indifference curves will choose portfolios with lower means and smaller standard deviations such as Portfolio Q, while more risk-tolerant investors will choose portfolios with higher means and greater risk, such as Portfolio S. The common fea- ture of all these investors is that each chooses portfolios on the efficient frontier. Even if virtually risk-free lending opportunities are available, many investors do face borrowing restrictions. They may be unable to borrow altogether, or, more realistically, they may face a borrowing rate that is significantly greater than the lending rate. When a risk-free investment is available, but an investor cannot borrow, a CAL exists but is limited to the line FP as in Figure 8.16. Any investors whose preferences are repre- sented by indifference curves with tangency portfolios on the portion FP of the CAL, such as Portfolio A, are unaffected by the borrowing restriction. Such investors are net lenders at rate rf. Aggressive or more risk-tolerant investors, who would choose Portfolio B in the ab- sence of the borrowing restriction, are affected, however. Such investors will be driven to portfolios such as Portfolio Q, which are on the efficient frontier of risky assets. These in- vestors will not invest in the risk-free asset. In more realistic scenarios, individuals who wish to borrow to invest in a risky portfolio will have to pay an interest rate higher than the T-bill rate. For example, the call money rate charged by brokers on margin accounts is higher than the T-bill rate. Investors who face a borrowing rate greater than the lending rate confront a three-part CAL such as in Figure 8.17. CAL1, which is relevant in the range FP1, represents the efficient II. Portfolio Theory 8. Optimal Risky Portfolio The McGraw−Hill Companies, 2001           CHAPTER 8 Optimal Risky Portfolios 237     Figure 8.15 Portfolio selection without a risk-free asset.