Chapter 5 Capital Assets Pricing Models PDF

Fall 2024 Chapter 5 Capital Assets Pricing Models Dr. Dina Qamar 1. Markowitz Theory. 2. The Assumptions of Markowitz Theory. 3. The Calculation of Risk and Return. 4. The Conditions of Portfolio Theory. Efficient Frontier....

Fall 2024 Chapter 5 Capital Assets Pricing Models Dr. Dina Qamar 1. Markowitz Theory. 2. The Assumptions of Markowitz Theory. 3. The Calculation of Risk and Return. 4. The Conditions of Portfolio Theory. Efficient Frontier. Optimal Allocation. Correlation between Each Pair of the Individual Investments 5. Capital Market Theory 6. The Assumptions of Capital Market Theory. 7. Development of Capital Market Theory. CML SML 𝜷 8. The Arbitrage Pricing Theory (APT). 9. The Assumptions of Arbitrage Pricing Theory. 10. Three Factor Model. 1 Markowitz Portfolio Theory: In the early 1960s, the investment community talked about risk, but there was no specific measure for the term. To build a portfolio model, however, investors had to quantify their risk variable. The basic portfolio model was developed by Harry Markowitz (1952, 1959), who derived the expected rate of return for a portfolio of assets and an expected risk measure. Markowitz showed that the variance of the rate of return was a meaningful measure of portfolio risk under a reasonable set of assumptions. More important, he derived the formula for computing the variance of a portfolio. This portfolio variance formula not only indicated the importance of diversifying investments to reduce the total risk of a portfolio but also showed how to effectively diversify. The Assumptions of Markowitz Theory: 1. Investor’s objective is to maximize the utility of terminal wealth at the end of specified period. 2. Investors consider each investment alternative represented by a probability distribution of expected returns over period of time. 3. Investors estimate the risk of the portfolio on the basis of the variability of expected returns. 4. Investors base decisions solely on expected return and risk, so their utility curves are a function of expected return and the expected variance (or standard deviation) of returns only. 5. For a given risk level, investors prefer higher returns to lower returns. Similarly, for a given level of expected return, investors prefer less risk to more risk. 2 The Calculation of Risk and Return: 1) 𝑬(𝑹𝒑𝒐𝒓𝒕 ) = ∑ 𝑾𝒊 𝑹𝒊 2) 𝝈𝟐𝒑𝒐𝒓𝒕= 𝒘𝟐 𝝈𝟐 +𝒘𝟐 𝝈𝟐 +𝟐𝒘 𝒘 𝝈 𝝈 𝒓 𝒊 𝒊 𝒋 𝒋 𝒊 𝒋 𝒊 𝒋 𝒊𝒋 3) 𝝈𝒑𝒐𝒓𝒕 = √𝝈𝟐 𝒄𝒐𝒗𝒊𝒋 4) 𝒓𝒊𝒋 = 𝝈𝒊 𝝈𝒋 5) 𝒄𝒐𝒗𝒊𝒋 = 𝒓𝒊𝒋 𝝈𝒊 𝝈𝒋 Remember that, these equation are explained in details in chapter 4. The Conditions of Portfolio Theory: First Condition: Efficient Frontier: The efficient frontier represents that set of portfolios that has the maximum rate of return for every given level of risk or the minimum risk for every level of return. Every portfolio that lies on the efficient frontier has either a higher rate of return for equal risk or lower risk for an equal rate of return than some portfolio beneath the frontier. Therefore, we would say that Portfolio A dominates Portfolio C because it has an equal rate of return but substantially less risk. Similarly, Portfolio B dominates Portfolio C because it has equal risk but a higher expected rate of return. 3 Figure (6.1): The Efficient Frontier: As an investor, you will target a point along the efficient frontier based on your utility function, which reflects your attitude toward risk. No portfolio on the efficient frontier can dominate any other portfolio on the efficient frontier. All of these portfolios have different return and risk measures, with expected rates of return that increase with higher risk. Second Condition: Optimal Allocation: Investor or trader can get the optimal asset allocation by using mathematical equations to get the best weight for each stock. Third Condition: Correlation between Each Pair of the Individual Investments: The most important condition. The more negative the correlation, the higher the diversification efficiency. Where the high returns of some investments cover the low returns of others, and thus the portfolio’s risk will be decreased. This theory depends on covariance and the coefficient of correlation to measure the correlation between each pair of the individual investments. 4  Note: before Markowitz the diversification was naive which means that it does not lead to reduce risk because this naive diversification does not take into its calculation the covariance and the coefficient of correlation. Following the development of portfolio theory by Markowitz, two major theories have been derived for the valuation of risky assets. 1. The Capital Asset Pricing Model (CAPM). 2. The Arbitrage Pricing Theory (APT). Capital Market Theory: Capital market theory extends portfolio theory by developing a model for pricing all risky assets. The CAPM, will allow you to determine the required rate of return for any risky asset. This development depends critically on: 1. The existence of RFR 2. There is a linear relationship between risk and return. 3. Divided risk to systematic risk and unsystematic risk. 4. Using the market index as a proxy for market portfolio. Assumptions of Capital Market Theory and (CAPM): 1. All investors are Markowitz-efficient in that they seek to invest in tangent points on the efficient frontier. 2. Investors can borrow or lend any amount of money at the risk-free rate of return (RFR). 3. All investors have homogeneous expectations; that is, they estimate identical probability distributions for future rates of return. 4. All investors have the same one-period time horizon, such as one month or one year. 5  5. All investments are infinitely divisible, so it is possible to buy or sell fractional shares of any asset or portfolio. 6. There are no taxes or transaction costs involved in buying or selling assets. 7. There is no inflation or any change in interest rates, or inflation is fully anticipated. 8. Capital markets are in equilibrium. Investments priced in line with their risk. NOTE: Remember that capital market theory consists of portfolio theory, and CAPM. Development of Capital Market Theory: The major factor that allowed Markowitz portfolio theory to develop into capital market theory is the concept of a risk-free asset, that is, an asset with zero variance. Such an asset would have zero correlation with all other risky assets and would provide the risk-free rate of return (RFR). Risky asset as one from which future returns are uncertain, and we have measured this uncertainty by the standard deviation of expected returns. Sharp (1964), Noble Prize. CML SML Capital market line: CML connects between RFR and market portfolio or any other risky portfolio with linear relationship. Investments that fall on CML outperform investments which fall on Markowitz efficient frontier 6  Investor can invest in RFR and Market portfolio or any other risky portfolio Figure (6.2): The Capital Market Line (CMI): It is clear that portfolio E is better that portfolio D because both portfolios E and D have the same risk, but portfolio E has higher return compared to portfolio D. Also, portfolio C is better than portfolios A and B because it has a higher return and lower risk than portfolio A and B. The expected rate of return of portfolio can be measured by: 𝑬(𝑹𝑴 ) − 𝑹𝑭𝑹 𝑬(𝑹𝒑𝒐𝒓𝒕 ) = 𝑹𝑭𝑹 + 𝝈𝒑𝒐𝒓𝒕 [ ] 𝝈𝑴 Example: the following table indicates investment characteristics for portfolios of risky assets. 7 Table (6.1): Portfolios of Risky Assets: The last column in the table calculates the ratio of the expected risk premium (E(R) − RFR) to volatility (σ) for each portfolio. This ratio is the amount of compensation that investors can expect for each unit of risk they assume in a particular portfolio. For example, Portfolio 2 offers investors 0.429 (= [7 − 4]/7) units of compensation per unit of risk while the comparable ratio for Portfolio 6 is lower at 0.393 (= [15 − 4]/28), despite promising a much higher overall return. Portfolio 3 offers investors the best combination of risk and return. No other feasible collection of risky assets in this comparison can match the 0.500 units of expected risk premium per unit of risk. Consequently, Portfolio 3 should be considered as the market portfolio. 𝑬(𝑹𝑴 )−𝑹𝑭𝑹 𝟎.𝟎𝟗−𝟎.𝟎𝟒 Portfolio 3: = = 𝟎. 𝟓𝟎 𝝈𝑴 𝟎.𝟏𝟎 Capital market theory would recommend that you only consider two alternatives when investing your funds: 1. lending or borrowing in the riskless security at 4% 2. Buying Portfolio 3. 8 Example: assume that you can bear a standard deviation of 8.5%. how should you go about investing your money according to CML. 𝑬(𝑹𝑴 ) − 𝑹𝑭𝑹 𝑬(𝑹𝒑𝒐𝒓𝒕 ) = 𝑹𝑭𝑹 + 𝝈𝒑𝒐𝒓𝒕 [ ] 𝝈𝑴 𝟎. 𝟎𝟗 − 𝟎. 𝟎𝟒 𝑬(𝑹𝒑𝒐𝒓𝒕 ) = 𝟎. 𝟎𝟒 + 𝟎. 𝟎𝟖𝟓 [ ] = 𝟖. 𝟐𝟓% 𝟎. 𝟏𝟎 Security Market Line (SML): SML connects between RFR and any individual asset or collection of assets with linear relationship. (Regression) There are two important differences between the CML and the SML. First, the CML measures risk by the standard deviation (total risk) of the investment while the SML considers only the systematic risk. Second, as a consequence of the first point, the CML can be applied only to portfolio holdings that are already fully diversified, whereas the SML can be applied to any individual asset or collection of assets (portfolios). Figure (6.3): The Security Market Line (SML): 9  1) 𝑬(𝑹𝒊 ) = 𝑹𝑭𝑹 + 𝜷{𝑬(𝑹𝑴 ) − 𝑹𝑭𝑹} 𝒄𝒐𝒗( 𝑹𝒊 ,𝑹𝒎 ) 2) 𝜷 = 𝝈𝟐𝒎 3) Where, 𝒄𝒐𝒗𝒊𝒎 = 𝒓𝒊𝒎 𝝈𝒊 𝝈𝒎 𝝈𝒊 4) So, 𝜷 = 𝐫𝐢𝐦 𝝈𝒎 The core of CAPM is that the RRR is determined according to the degree of systematic risk which is measured by Beta coefficient. Beta Coefficient is the sensitivity of the individual investment’s return to the change in the market return. It is the slope of SML. Beta for market portfolio =1 If Beta of any individual investment is > 1, then this investment has a level of risk greater than the average market risk. If Beta of any individual investment is