1. Chapter 5 CAPITAL MARKET THEORY

3. EFFICIENT FRONTIER FOR A TWO-SECURITY CASE Security A Security B Expected return 12% 20% Standard deviation 20% 40% Coefficient of correlation -0.2 Portfolio Proportion of A w A Proportion of B w B Expected return E (R p ) Standard deviation  p 1 (A) 1.00 0.00 12.00% 20.00% 2 0.90 0.10 12.80% 17.64% 3 0.759 0.241 13.93% 16.27% 4 0.50 0.50 16.00% 20.49% 5 0.25 0.75 18.00% 29.41% 6 (B) 0.00 1.00 20.00% 40.00%

4. EFFICIENT FRONTIER FOR THE n-SECURITY CASE Standard deviation,  p Expected return , E ( R p ) A O N M X F • • • • • B D Z • • •

6. ASSUMPTIONS OF MARKOWITZ MODEL The Markowitz portfolio theory is based on a few assumptions i. Investors are risk-averse and thus have a preference for expected return and dislike for risk. ii. Investors act as if they make investment decisions on the basis of the expected return and the variance (or standard deviation) about security return distributions.

10. RISK-FREE ASSETS In simple words, a risky asset is one which gives uncertain future returns whereas a risk-free asset whose expected return is fully certain and thus the standard deviation of such expected returns comes to zero, i.e., σf=0. Thus the rate of return earned on such assets should be the risk-free rate of return (rf).

11. COVARIANCE OF RISK-FREE ASSET WITH A RISKY ASSET The covariance between two sets of returns, A and B where asset A is a risk free asset. n CovAB = Σ[rA - E(rA)] [ rB - E(rB)]/n A=1 The uncertainty for a risk-free asset is known, so σA=0, which implies that rA = E(rA) for all the periods. Thus, rA - E(rA) =0, which further leads to the facts that the product of any other expressions with this expression will be zero. This will result in the covariance of the risk-free asset with any risky asset or portfolio to be also zero. Similarly, the correlation between any risky asset and risk-free asset, will be zero as rAB = CovA,B / σA σB.

12. COMBINING A RISK-FREE ASSET WITH A RISKY PORTFOLIO Any portfolio that combines a risk free asset with any risky asset, the standard deviation is the linear proportion of the standard deviation of the risky asset portfolio.

13. RISK-RETURN POSSIBILITIES WITH LEVERAGE An investor always wants to increase his expected returns. Say, a person has borrowed an amount which is 50 percent of his original wealth, the effect of this on the expected return for the portfolio would be: E(ri) = Wf (rf) + (1 - Wf) E(rk) where k is the risky assets portfolio = – 0.50 (rf) + [1 – (–0.50)] E(rk) = – 0.50 (rf) + 1.50 E(rk) Continued

14. Thus, we see that the return increases in a linear fashion along the line of risk-free rate (rf) and ‘k’. Now, suppose E(rf) = 0.10 And E(rk) = 0.24 E(ri) = = – 0.50 (0.10) + 1.5 (0.24) = 0.31 or 31% Similar is the effect of standard deviation of the leveraged portfolio. E(σi) = (1 – Wf) σk = [1 – (–0.5)] σk = 1.50 σk

15. Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier RFR M C A B D

17. Lending and Borrowing at the Riskfree rate The portfolio expected return for any portfolio i that combines f and M is E(ri) = WfRf + (1 – Wf) E (rM) Where, Wf = The percentage of the portfolio invested in the riskless security f 1 – Wf = The percentage of the portfolio invested in the risky portfolio M. The portfolio variance for portfolio i is: σi² =W²f σ²f + (1 - Wf)² σ²M + 2 Wf (1 - Wf) σf, M By definition, σ²f = 0. Thus, σ²i = (1 - Wf)² σ²M (or) σi = (1 - Wf) σM

18. Borrowing And Lending At Riskfee Rate Rf And Investing In The Risky Portfolio Dominant Portfolio “M” RFR M CML Borrowing Lending

19. • • • • • • A O N M X Standard deviation,  p Expected return , E ( R p ) • • • • • • • • R f u D E V G B C F Y I II • S RISKLESS LENDING AND BORROWING OPPORTUNITY Thus, with the opportunity of lending and borrowing, the efficient frontier changes. It is no longer AFX. Rather, it becomes R f SG as it domniates AFX.

24. CAPITAL MARKET LINE EXPECTED RETURN, E ( R p ) Z • L M • • K R f STANDARD DEVIATION,  p E ( R j ) = R f + λ σ j E ( R M ) - R f λ = σ M

25. THE CAPITAL ASSET PRICING MODEL (CAPM) A model that describes the relationship between risk and expected return and that is used in the pricing of risky securities. The general idea behind CAPM is that investors need to be compensated in two ways: time value of money and risk. The time value of money is represented by the risk-free (rf) rate in the formula and compensates the investors for placing money in any investment over a period of time. The other half of the formula represents risk and calculates the amount of compensation the investor needs for taking on additional risk. This is calculated by taking a risk measure (beta) that compares the returns of the asset to the market over a period of time and to the market premium (Rm-rf).

27. SECURITY MARKET LINE (SML) The SML essentially graphs the results from the capital asset pricing model (CAPM) formula. The x-axis represents the risk (beta), and the y-axis represents the expected return. The market risk premium is determined from the slope of the SML. E ( R i ) = R f + [ E ( R M ) - R f ] β i

28. Security Market Line

29. RELATIONSHIP BETWEEN SML AND CML SML E ( R M ) - R f E ( R i ) = R f + σ iM σ M 2 SINCE σ iM =  iM σ i σ M E ( R M ) - R f E ( R i ) = R f +  iM σ i σ M IF i AND M ARE PERFECTLY CORRELATED  iM = 1. SO E ( R M ) - R f E ( R i ) = R f + σ i σ M THUS CML IS A SPECIAL CASE OF SML

32. TESTS OF ASSET PRICING THEORIES The CAPM pricing model is given by the equation: E(ri) = rf + [E(rM) – rf] βi According to the theory, the expected return on security i, E(ri), is related to the risk-free rate, rf, plus a risk premium, [E(rM) – rf] βi ,which includes the expected return on the market portfolio.

33. TESTS OF MARKET EFFICIENCY SML can also be used for testing market efficiency. As we know, when markets are efficient, the scope for abnormal return will not be there and returns on all securities will be commensurate with the underlying risk. That is, all assets are correctly priced and provide a normal return for their level of risk and the difference between return earned on the asset and required rate of return on the asset should be statistically insignificant if markets are efficient. IDENTIFYING MISPRICED SECURITIES The ex ante SML can be used for identifying mispriced (under and overvalued) securities.