Πανεπιστήμιο Πατρών Advanced Topics in Finance Week 6/13 Capital Market Line

1. Πανεπιστήμιο Πατρών Advanced Topics in FinanceWeek 6/13Capital Market Line Vasilios I. Sogiakas

2. Advanced Topics in Finance Capital Market Line • Capital Market Line • Separation Fund Theorem • Capital Asset Pricing Model • Suppose now, that an investor could invest on equity (shares) and a riskless asset, such as government bonds (rf) • The fixed income securities embed no risk which means that their standard deviation as well as their correlation or covariance with other random variables is equal with zero • Now, suppose that there exists a portfolio (portfolio*) consisting of an EPF portfolio (consisting of n shares) with weight wEPF and a riskless asset with weight wrf, such that:

3. Advanced Topics in Finance Capital Market Line Thus, the expected return of the portfolio* is linearly associated with its risk (σportfolio*):

4. Advanced Topics in Finance Capital Market Line This linear relationship is expressed through the Capital Market Line, which is a line the intercept of which is equal with the interest rate rf, and its slope is given by the ratio that expresses the excess return of the EPF portfolio over the risk free asset, per unit of risk: (rEFP-rf)/σEPF . Figure 1 Capital Market Line

5. Advanced Topics in Finance Capital Market Line • When an investor can borrow or lend at rf, his investment opportunity set is expanded, according to a line with intercept Δ (rf) and a slope which depends on the equity portfolio that he is willing to hold, as shown in Figure 6.11 • If the initial equity portfolio does not lie on the EPF, curve ΚΛ, then the final portfolio (portfolio*) would lie on a line such as the ΔΛ. This line ΔΛ, consists of portfolios which are inferior to some others because for the same level of risk they do not offer the maximum return and vice versa. • However, if the initial equity portfolio was chosen among the EPF portfolios (i.e. curve ΚΛ), then the final portfolio would offer the maximum possible slope which is represented by the line ΔΕ, which is tangent to the EPF at T. Figure 2 Capital Market Line and the EPF

6. Advanced Topics in Finance Capital Market Line The ΔΕ line represents the Capital Market Line (CML) according to which the investment opportunity set of investors is expanded: • if the investor lends money (invest on rf) then the efficient portfolio frontier becomes the line ΔΤ (at Δ the investor lends all of his money at rf) • if the investor borrows money then the efficient portfolio frontier becomes the line ΤE (at E the investor invests all of his money as well all of his loan at the equity portfolio of size n) • among all EPF equity portfolios only the T is efficient under the lending/borrowing structure (at T the investor invests all of his money at the equity portfolio of size n) • The introduction of the risk free interest rate expands the investment opportunity set, by readjusting the EPF at a new superior level. Thus, investors could lend or borrow money in order to invest on equities an amount of money which is more than their initial wealth, or to substitute part of their equity investments by a riskless asset. Figure 3 Lending and Borrowing opportunities

7. Advanced Topics in Finance Capital Market Line The expansion of the investment opportunity set, by the inclusion of the riskless asset, would result to an optimum portfolio (A*) which is superior to the EPF optimal portfolio, for the same investor. Thus, the incorporation of the riskless asset derives superior portfolios, in terms of expected utility. Figure 4 Lending and Borrowing opportunities and the Expected Utility

8. Advanced Topics in Finance Capital Market Line • In equilibrium, it can be proved that the point T that corresponds to the portfolio, at which the CML is tangent to the EPF, is not an arbitrary portfolio but the market portfolio M • The market portfolio consists of all shares taking into account the relevant capitalization of the corresponding listed firms of the stock exchange (VW portfolio) • The equilibrium conditions refer to the fact that investors are rational, the information is distributed uniformly and rapidly among them and there exists one riskless asset where investors could invest (lend) or borrow money at rf • Usually, a general index of an exchange is a good proxy for the market portfolio. According to the market portfolio M, the CML could be expressed as follows: • The CML is the line with an intercept at rf and a slope equal to (rm-rf)/σΜ (market risk premium or excess return for each unit of risk - standard deviation or Sharpe Ratio), which expresses the increment of the (required) return as a compensation for the extra risk that the portfolio embeds • More specifically, the slope of the CML represents the Sharpe ratio (expected return in terms of risk units) and as a result, all lines lying below the CML have lower slope and represent not efficient investment choices for rational investors, while all lines lying above the CML are not feasible opportunity choices.

9. Advanced Topics in Finance Separation Fund Theorem According to the Separation Fund Theorem, the Optimum portfolio choice process could be separated in two independent steps: • step 1: choose a portfolio that consists of all risky equities with the appropriate weights (VW), that is the market portfolio M which is common for all investors (tangent of the CML on the EPF) • step 2: construct a portfolio consisting of the M portfolio and the riskless asset (rf), at a weight which is determined by the maximization of this investor’s subjective satisfaction by lending or borrowing money at rf Portfolio Theory Benefits: The principles of modern portfolio theory could be applied for: • the construction of efficient portfolios, • the evaluation of the cost of capital, and • the valuation of financial products