# Insights gleaned from our coverage of portfolio and capital market theory

The topics covered during the course of the last couple of Finance 4335 class meetings (portfolio and capital market theory) rank among the most important finance topics; after all, the scientific foundations for these topics won Nobel prizes for Markowitz (portfolio theory) and Sharpe (capital market theory). The following outline pretty much summarizes what we covered in class on Thursday, October 12 and Tuesday, October 17:

• Portfolio Theory
1. Mean-variance efficiency
2. Portfolio Mean-Variance calculations
3. Minimum variance portfolio (n = 2 case)
4. Efficient frontier (n = 2 case under various correlation assumptions)
• Capital Market Theory
1. Efficient frontiers with multiple number (“large” n) of risky assets (aka the “general” case)
2. Portfolio allocation under the general case
• degree of risk aversion/risk tolerance determines how steeply sloped indifference curves are
• indifference curves for investors with high (low) degrees of risk tolerance (aversion) are less steeply sloped than indifference curves than for investors with low (high) degrees of risk tolerance (aversion)).
• Optimal portfolios (i.e., portfolios that maximize expected utility) occur at points of tangency between indifference curves and efficient frontier.
3. Introduction of a risk-free asset simplifies the portfolio selection problem since the efficient frontier is now a straight line rather than an ellipse in $E({r_p}), {\sigma _p}$ space. The same selection principle holds as in the previous point (point 2); i.e., investors determine optimal portfolio by finding the tangency between highest indifference curve and the efficient frontier. The point of tangency occurs on the capital market line (CML) where the Sharpe ratio is maximized; everyone chooses some combination of the risk-free asset and the market portfolio, and risk tolerance determines whether the point of tangency involves either a lending (low risk tolerance) or borrowing (high risk tolerance) allocation strategy.
4. The security market line (SML), aka the CAPM, is deduced by arbitrage arguments. Specifically, it must be the case that all risk-return trade-offs (as measured by the ratio of “excess” return ($E({r_j}) - {r_f}$) from investing in a risky rather than risk-free asset, divided by the risk taken on by the investor (${\sigma _{j,M}}$) are the same. If not, then there will be excess demand for investments with more favorable risk-return trade-offs and excess supply for investments with less favorable risk-return trade-offs). “Equilibrium” occurs when markets clear; i.e., when there is neither excess demand or supply, which is characterized by risk-return ratios being the same for all possible investments. When this occurs, then the CAPM obtains: $E({r_j}) = {r_f} + {\beta _j}(E({r_M}) - {r_f})$.