Return and Risk Lecture 2 Calculation of Covariance Calculation of Beta Calculation of coefficient of correlation Concept of Efficient Frontier CML Relationship between risk and expected return CAPM Lecture 1 Concept and calculation of Expected return Concept and calculation of Risk – SD / Variance Concept of Diversifiable and Non Diversifiable Risk Concept of Correlation Coefficient Concept of Beta

Expected Return, Variance, and Covariance Consider the following two risky asset . There is a 1/3 chance of each state of the economy and the only assets are a stock fund and a bond fund.

Expected Return, Variance, and Covariance

Expected Return, Variance, and Covariance

Expected Return, Variance, and Covariance

Expected Return, Variance, and Covariance

Expected Return, Variance, and Covariance

Expected Return, Variance, and Covariance

The Return and Risk for Portfolios Note that stocks have a higher expected return than bonds and higher risk. Let us turn now to the risk-return tradeoff of a portfolio that is 50% invested in bonds and 50% invested in stocks.

The Return and Risk for Portfolios The rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio:

The Return and Risk for Portfolios The rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio:

The Return and Risk for Portfolios The rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio:

The Return and Risk for Portfolios The expected rate of return on the portfolio is a weighted average of the expected returns on the securities in the portfolio.

The Return and Risk for Portfolios The variance of the rate of return on the two risky assets portfolio is 0.001 = ((9%-5%)^2*33.33%+(9%-9.5%)^2*33.33%+(9%-12.5%)^2*33.33%)) where BS is the correlation coefficient between the returns on the stock and bond funds.

The Return and Risk for Portfolios Observe the decrease in risk that diversification offers. An equally weighted portfolio (50% in stocks and 50% in bonds) has less risk than stocks or bonds held in isolation.

11.4 The Efficient Set for Two Assets 100% stocks 100% bonds We can consider other portfolio weights besides 50% in stocks and 50% in bonds …

11.4 The Efficient Set for Two Assets 100% stocks 100% bonds We can consider other portfolio weights besides 50% in stocks and 50% in bonds …

11.4 The Efficient Set for Two Assets 100% stocks 100% bonds Note that some portfolios are “better” than others. They have higher returns for the same level of risk or less. These compromise the efficient frontier.

How would you find the efficient frontier? Map out the return and risk on the graph Find the min risk portfolio The efficient frontier starts from the min risk portfolio To maximize utility, an investor would prefer to hold a risky portfolio on the highest indifference curve. This means that the investor will prefer to hold the portfolio that is represented by the point of tangency of an indifference curve with the feasible set. This, of course, is a portfolio on the efficient frontier. There is little reason to believe, however, that different investors will hold the same portfolio on the efficient frontier. Our typical investor chooses a portfolio that we will call D.

11.5 The Efficient Set for Many Securities return Individual Assets P Consider a world with many risky assets; we can still identify the opportunity set of risk-return combinations of various portfolios.

11.5 The Efficient Set for Many Securities return minimum variance portfolio Individual Assets P Given the opportunity set we can identify the minimum variance portfolio.

11.5 The Efficient Set for Many Securities return efficient frontier minimum variance portfolio Individual Assets P The section of the opportunity set above the minimum variance portfolio is the efficient frontier.

The Efficient Set for Many Securities return efficient frontier minimum variance portfolio Individual Assets P The section of the opportunity set above the minimum variance portfolio is the efficient frontier.

Riskless Borrowing and Lending return 100% stocks rf 100% bonds  In addition to stocks and bonds, consider a world that also has risk-free securities like T-bills

Portfolios Involving Riskless Borrowing and Lending return CML efficient frontier rf P With a risk-free asset available and the efficient frontier identified, we choose the capital allocation line with the steepest slope

Market Equilibrium return CML efficient frontier M rf P With the capital allocation line identified, all investors choose a point along the line—some combination of the risk-free asset and the market portfolio M.

The Separation Property return CML efficient frontier M rf P The Separation Property states that the market portfolio, M, is the same for all investors—they can separate their risk aversion from their choice of the market portfolio.

Three Important Funds The riskless asset has a standard deviation of zero The minimum variance portfolio lies on the boundary of the feasible set at a point where variance is minimum The market portfolio lies on the feasible set and on a tangent from the riskfree asset Suppose we allow investors to hold combinations of a riskfree asset and a risky portfolio. To understand the decision problem in this case we need to define three funds (one asset and two portfolios) that are of critical importance. The first fund is the riskless asset and it lies on the y-axis because it has zero standard deviation. The second fund is the minimum variance portfolio and it lies on the boundary of the feasible set at a point where variance is minimum. You can locate this fund by finding the point of tangency between a vertical straight line and the feasible set. Finally, if you draw a line from the riskless asset toward the feasible set and rotate it until it is tangent to the feasible set, the point of tangency is a fund known as the market portfolio.

Definition of Risk When Investors Hold the Market Portfolio Beta measures the responsiveness of a security to movements in the market portfolio.

Estimates of b for Selected Stocks Beta Research in Motion 3.04 Nortel Networks 3.61 Bank of Nova Scotia 0.28 Bombardier 1.48 Investors Group. 0.36 Maple Leaf Foods 0.25 Roger Communications 1.17 Canadian Utilities 0.08 TransCanada Power

Calculation of Beta and Covariance Given below are the expected returns of Hindalco and Indal along with the indicative market returns for different scenarios: Risk free rate is 4%

Step 1 – Calculation of Expected return and variance for Market

Calculation of Covariance between Indal and Market

Calculation of Beta for Indal (x) Beta = Covariance of Indal and Market / Variance of Market Beta (x) = 138.92 / 111.56 Beta (x) = 1.25

Calculation of Covariance between Hindalco and Market

Calculation of Beta for Hindalco (y) Beta = Covariance of Hindalco and Market / Variance of Market Beta (y) = 106.36 / 111.56 Beta (x) = .95

Relationship between Risk and Expected Return (CAPM) Expected Return on the Market: Expected return on an individual security: Market Risk Premium This applies to individual securities held within well-diversified portfolios.

Expected Return on an Individual Security This formula is called the Capital Asset Pricing Model (CAPM) Expected return on a security = Risk-free rate + Beta of the security × Market risk premium Assume bi = 0, then the expected return is RF. Assume bi = 1, then

Relationship Between Risk & Expected Return – Security Market Line 1.0

The Capital Asset Pricing Model CAPM serves as a benchmark Against which actual returns are compared Against which other asset pricing models are compared

CAPM Assumptions No transactions costs No taxes Infinitely divisible assets Perfect competition No individual can affect prices Only expected returns and variances matter Quadratic utility or Normally distributed returns Unlimited short sales and borrowing and lending at the risk free rate of return Homogeneous expectations

Return for Indal as per CAPM Model Rm = 10.8% Rf = 4% Beta for Indal = 1.25% R = Rf + Beta * ( Rm – Rf) R = 4% + 1.25 ( 10.8% – 4% ) R = 4% + 8.5% Risk Premium R = 12.5 %

Is the share underpriced or overpriced ?? The expected return of Indal = 15.6 Return as per the CAPM model = 12.5% Alfa = Expected Return – CAPM Return Alfa = 15.6% - 12.5 % = 3.1% When Alfa is positive share is underpriced.

Relationship Between Risk & Expected Return 1.0

Return for Hindalco as per CAPM Model Rm = 10.8% Rf = 4% Beta for Hindalco = .95% R = Rf + Beta * ( Rm – Rf) R = 4% + .95 ( 10.8% – 4% ) R = 4% + 6.46% Risk Premium R = 10.46 %

Is the share underpriced or overpriced ?? The expected return of Hindalco = 12.3 Return as per the CAPM model = 10.46% Alfa = Expected Return – CAPM Return Alfa = 12.3% - 10.46 % = 1.84% When Alfa is positive share is underpriced.

Relationship Between Risk & Expected Return 1.0

If the market return is below the risk free rate, then stocks which posses high systematic risk give return which ----- as compared to stock which have low systematic risk Are lower Are Higher Are same Cannot Determine None of the above Ans 1

Securities which are plotted above the SML line are Underpriced Those whose intrinsic value is equal to the market value Overpriced Favourable Investment Both 1 and 4 Ans 5 as those securities that plot above the SML generates above normal returns for the risk

Consider the below data and calculate the return expected Rf = 6 Consider the below data and calculate the return expected Rf = 6.5% Beta = 1.24 Market Return = 11.5% 11.7% 12.7% 13.7% 14.7% 15.7% Ans 6.5 + 1.24 ( 11.5 – 6.5 ) = 12.7%

Covariance between the market and stock is 33 Covariance between the market and stock is 33.56 and variance of market is 19.15 . Beta ?? 1.55 1.75 1.85 1.95 2.05 Ans - 2

The beta of a stock is 1.12 and its covariance with market is 220 the SD of the stock ?? 16% 14% 12% 11.3% 8% Ans 2 = (220/1.12)^(1/2)

Consider a stock with a require return of 11% the market return is 10% and risk free return is 5%. The stock can be classified as ?? Defensive Aggressive Market Mirror Risk Free Insufficient information Ans 2 , 11 = 5 + Beta ( 10-5 ) , 6 = Beta * 5 , Beta = 1.2

The Capital Market Line depicts the risk return relationship for Aggressive securities Zero beta portfolios Efficient Portfolios Securities which are nethier less nor more volatile than the market None of the above Ans 3

Which of the following statements are true .. Securities lying above the SML are overpriced Securities lying on SML are zero risk securities Alfa of a security is intercept of SML on Y axis SML consist of only individual securities The market portfolio always lies on SML Ans 5

Which of the following is not an application of SML ? Evaluating the performance of portfolio manager Testing asset pricing theories Identifying undervalued securities Determining the relation, price to risk implicit in the current market prices Determining the relation, liquidity risk implicitly of the current market Ans 5

If the expected rate of return on the stock of Natco Ltd is 23% and required rate of return as per SML is 25% the alfa of the stock Natco is?? +5% +2% 0% -2% -5% Ans 4

If the expected rate of return on the market index is 20%, the beta of the stock of XYZ Ltd is 1.3 and risk free rate of return is 12% the required rate of return is .. ?? 12% 14% 16% 18.4% 22.4% Ans 5

End of Lecture 2