Module 3: Portfolio Risk and Return: Part II

Definitions/Components: Return Generating Models [Part 1/2] 4 models ( last model is next card) (Easy, scan through, identify)

1) Macroeconomic factor model: [It uses economy] use economic factors (e.g., economic growth rates, interest rates, and inflation rates) 2) Fundamental factor models: [It uses accounting] use relationships between security returns and underlying fundamentals (e.g., earnings, earnings growth, and cash flow growth) to estimate returns. 3) Statistical factor models [It uses past data] use historical and cross-sectional returns data to identify factors that explain returns and use an asset's sensitivity to those factors to project future returns.

Quick Recap/Calculation/Definitions: Capital Asset Pricing Model (CAPM) [Part 1/2] (quickly scan over, common sense) (Important, MUST MEMORIZE)

-> E(Ri) = Rf + βi*[E(Rm) − Rf] (recap) + Capital Asset Pricing Model (CAPM) is a single-index model that is widely used to estimate returns, given security betas. (recap) + Assumptions of the CAPM: (New info, Quickly scan over) 1) Investors are utility-maximizing, risk-averse, rational individuals. ** 2) Markets are frictionless and there are no transaction costs and taxes. * 3) All investors have the same single-period investment horizon. 4) Investors have homogenous expectations and therefore arrive at the same valuation for any given asset. 5) All investments are infinitely divisible. 6) Investors are price-takers. No investor is large enough to influence security prices.

Example 3.6 Optimal Investor Portfolio with Heterogeneous Beliefs (Pg 59) An investor gathers the following information regarding three stocks, which are not in the market portfolio: Stocks Expected Return Standard Deviation Beta A 16% 29% 1.7 B 20% 24% 1.4 C 18% 21% 1.2 Given that the return on the market portfolio is 13% with a standard deviation of 15%, and the risk-free rate of return is 5%, answer the following questions: Calculate Jensen's alpha for Stocks A, B, and C. Calculate nonsystematic variance for A, B, and C. If an investor holds the market portfolio, should she add any of these three stocks to her portfolio? If so, which stock should have the highest weight in the portfolio? (We doubt this will be on the test. We believe this is beyond LV1. This is LV2 material. Scan through, don't get stuck here!) (quickly scan over, move on)

1) αA= 16% − [5%+ 1.7(13%− 5%)] = −0.026 αB= 20% − [5%+ 1.4(13%− 5%)] = 0.038 αC= 18%− [5%+ 1.2(13%− 5%)] = 0.034 2) Nonsystematic variance = Total variance − Systematic variance σ^2ei = σ^2i − β^2i* σ^2m A's nonsystematic variance =(0.29)^2 − (1.7^2× 0.15^2) = 0.0191 B's nonsystematic variance =(0.24)^2 − (1.4^2× 0.15^2) = 0.0135 C's nonsystematic variance =(0.21)^2 − (1.2^2× 0.15^2) = 0.0117 3) Stock A should not be included in the portfolio as it has a negative alpha. It should only be included if the investor can short the stock. Stocks B and C have positive alphas and should be included in the portfolio. Their weights are determined as follows: Weight of Stock B = 0.038 / 0.0135 = 2.815 Weight of Stock C = 0.034 / 0.0117 = 2.906 In relative terms, the weight of Stock C will be greater than that of Stock B by 3.23% (2.906 / 2.815 − 1).

Example 3.1 Using the SML to Compute Expected Returns Assume that the risk-free rate in an economy is 5% and the return on the market portfolio is expected to be 10%. Compute the expected rates of return for the following 5 securities: Stock Beta A 0.75 B 1.00 C 1.15 D 1.5 E −0.25 (Pay attention to the explanation, you're good to go!!)

E(Ri) = Rf + βi*[E(Rm) − Rf] A = 0.05 + 0.75(0.10 − 0.05) = 8.75% B = 0.05 + 1.00(0.10 − 0.05) = 10% C = 0.05 + 1.15(0.10 − 0.05) = 10.75% D = 0.05 + 1.5(0.10 − 0.05) = 12.5% E = 0.05 + (−0.25)(0.10 − 0.05) = 3.75% + Security C and D's betas are greater than market beta (1). ... higher level of systematic risk than the market, meaning that they are more risky than the market portfolio. The expected return on these assets is therefore greater than the return on the market. + Security A and E's betas are lower than market beta. ... have a lower level of systematic risk than the market, meaning that they are less volatile than the market portfolio. Therefore, the expected return on these assets is lower than the return on the market. + Stock B's beta equals market beta, so its expected return equals the rate of return expected from the market (10%).

Definitions: Calculation and Interpretation of Beta [Part 2/2] (Job on the exam is to identify, not memorize, but be very familiar) (quickly scan over, common sense)

Important Points Regarding Beta: *** 1) Beta captures an asset's systematic risk or nondiversifiable risk. ** 2) A positive beta suggests that the return on the asset follows the overall trend in the market. ** 3) A negative beta indicates that the return on the asset generally follows a trend that is opposite to that of the current market trend. ** 4) A beta of zero means that the return on the asset is uncorrelated with market movements. 5) market has a beta of 1. Therefore, the average beta of stocks in the market also equals 1. Add-on: From Googel: A beta of 1 indicates that the security's price tends to move with the market. A beta greater than 1 indicates that the security's price tends to be more volatile than the market. A beta of less than 1 means it tends to be less volatile than the market.

Test bank

M3L1: (Haven't done it} M3L2: (Haven't done it} M3L3: (Haven't done it}

Calculation/Definition: Portfolio Performance Evaluation [Part 3/4] **M^2 (Important, MUST MEMORIZE) (quickly scan over, common sense)

M^2: M^2= Sharpe R × σm + Rf 1) M2 can be thought of as a rescaling of the Sharpe ratio that allows for easier comparisons among different portfolios. 2) M2 and Sharpe ratios rank portfolios identically because both (1) the risk-free rate and (2) market volatility are constant across all comparisons.

Calculation/Definition: Portfolio Performance Evaluation [Part 1/4] Sharpe ratio (Important, MUST MEMORIZE) (quickly scan over, common sense)

Sharpe ratio: = [Rp − Rf] / σp 1) (recap) Sharpe ratio = slope of CAL 2) A portfolio with higher Sharpe ratio is preferred then lower given a "positive Sharpe ratio" * 3) Negative Sharpe ratio is not useful. ** 4) 2 drawback: + it uses total risk as a measure of risk even though only systematic risk is priced, + and that the ratio itself is not informative.

Example 3.2. [Great example] Portfolio Beta and Return Allison invests 25% of her money in the risk-free asset, 35% in the market portfolio, and 40% in Alpha Corp, a U.S. stock that has a beta of 1.5. Given that the risk-free rate and the expected return on the market are 5% and 14% respectively, calculate the portfolio's beta and expected return. (Easy, scan through, identify) (recap) (I doubt this will be on the test)

The beta of the risk-free asset and that of the market equal 0 and 1 respectively. The beta of the portfolio is calculated as: βPortfolio = w1β1 + w2β2 + w3β3 βPortfolio=(0.25 × 0) + (0.35 × 1) + (0.4 × 1.5)=0.95 Expected return of the portfolio = 5% + 0.95 (14% − 5%) = 13.55%

Calculation: Calculation and Interpretation of Beta [Part 1/2] (Important, MUST MEMORIZE)

βi = Cov(Ri,Rm) / σ^2m =[ρi,m * σi* σm ]/ σ^2m =[ρi,m* σi] / σm Beta is a measure of the sensitivity of an asset's return to the market's return. 1) It is computed as the covariance of the return on the asset and the return on the market divided by the variance of the market.

Definitions/Quick Recap: Systematic risk Nonsystematic risk (it's recap not from prior lesson, but I personally know it before)

+ Systematic risk: It's risk caused by macroeconomic variables. It cannot be eliminated by diversification [Undiversifiable Risk or Market Risk] + Unsystematic risk: It's risk caused by microeconomic variables such as company specific risk. It's diversifiable. [Unique, Diversifiable, or Firm-specific Risk] 1) Total Risk (standard deviation of expected return) = Systematic risk + Nonsystematic risk 2) Complete diversification of a portfolio requires the elimination of all nonsystematic or diversifiable risk. Once nonsystematic risk has been eliminated and only systematic risk remains, a completely diversified portfolio would correlate perfectly with the market. [12-30 different stocks can diversify away 90% of unsystematic risk. Second, we can attain lower level of systematic risk by diversifying globally vs only diversifying within a country] Note: (very easy) All we did in M2 was to eliminate "unsystematic risk" .We cannot eliminate "systematic risk" or "macroeconomics factors".

Definitions: Security Characteristic Line (I doubt this will be on the test)

- Basically, Security Characteristic Line = Jensen's Alpha Security characteristic line (SCL) plots the excess returns of a security against the excess returns on the market. -> Jensen's Alpha = Ri − [Rf + β(Rm − Rf)] -> Security Characteristic Line = Ri−Rf = αi + βi(Rm−Rf) α = y-intercept β = Slope

Summarization/Calculation: [Part 1/2] Security Selection: Identifying Mispriced Securities (The instructor said this will be on the test! I believe it will be on the test about mispricing) (Important, , identify )

- In simpler word: determine whether a security is undervalued or overvalued, we compare the return that the security offers [Expected Return [(P1 + D1)/P0] − 1] to CAPM (given a level of beta risk) [Required rate of return, Intrinsic value]. {Data are given on the textbook pg. 57) For Example: - Stock A offers a return of 9.24%. Based on the CAPM, the return required from Stock A to compensate for its beta risk is 8.75%. Stock A's current price is too low, which is why its expected return is higher than the required return - forecasted return for Stock C (10%) is lower than the return required by investors for investing in a stock with a beta of 1.15 (10.75%). Stock C's current price is too high. Therefore, investors should sell the stock based on the given forecasts. - if the expected return on a security is less than the return required from the security, to compensate an investor for the systematic risk that it brings to the portfolio, what do you think you should do? This asset is overvalued. You want to sell it. *** (Back in derivative or equity analysis, mispriced is when we calculate the "Intrinsic value" and compare to the market price. If IV > MP, then it's undervalued. If IV < MP, then it's overvalued. In this case with required rate of return and expected return (above), if required rate of return < expected return, it's undervalued. If required rate of return > expected return, it's overvalued.

Summarization/Calculation: Constructing A Portfolio (We doubt this will be on the test. We believe this is beyond LV1. This is LV2 material. Scan through, don't get stuck here!) (quickly scan over, move on)

- Weight of each nonmarket security: (not in the market portfolio) = αi / σ^2ei αi = Jensen's alpha σ^2ei = Nonsystematic variance of the security σ^2ei = Nonsystematic variance of the security -> σ^2ei = σ^2i - β^2*σ^2m " ... decision regarding whether the particular security should be included in our portfolio depends on the αi of the security (based on the CAPM and the S&P 500 as the market portfolio). Positive αi securities (even if they are correctly priced) should be added to the portfolio. ... weight of undervalued securities should be increased and that of overvalued securities should be reduced"

Definitions: The Security Market Line (SML) [Part 2/2] (Not recap! CAPM is recap, SML is new) [I actually don't really get CAL and CML explanation] (Important) (Job on the exam is to identify, not memorize)

1) SML illustrates the CAPM equation. y-intercept = risk free rate. Slope = Risk premium [Rm - RFR] ** 2) (recap) CAL and the CML are only applied to efficient portfolios, not to individual assets or inefficient portfolios They used total risk on the x-axis, and since only systematic risk is priced because efficient portfolio have maximum diversification benefits. Therefore, their total risk equals their systematic risk, and all unsystematic risk is eliminated. The only risk that is left in efficient portfolios is systematic risk or beta risk. (those with no nonsystematic risk and whose total risk therefore was the same as their systematic risk). [Efficient portfolios are those that offer the highest return for each level of risk.] ** 3) The SML and the CAPM, on the other hand, apply to any security or portfolio, regardless of whether it is efficient. This is because they are based only on a security's systematic risk, not total risk. Thus, you can use inefficient portfolios or individual assets and securities that are obviously not efficient, because they're not portfolios and haven't reaped in the diversification benefits 4) (I added myself) In CAL or CML, their y-axis is "E(R)" and x-axis is "σ port, total risk" In SML and CAPM line, their y-axis is "E(R)" and x-axis is "βi"

Example 2.2 Calculating Asset Beta Given that the standard deviation of the returns on the market is 18%, calculate beta for the following assets: - Asset A, which has a standard deviation twice that of the market and zero correlation with the market. - Asset B, which has a standard deviation of 24% and its correlation of returns with the market equals −0.2. - Asset C, which has a standard deviation of 20% and its covariance of returns with the market is 0.035. (Important)

1) Since the correlation of Asset A with the market equals zero, its beta also equals zero. 2) Asset B's beta = (−0.2 × 0.24) / 0.18 = −0.267 or = (−0.2 × 0.24 * 0.18) / 0.18^2 = −0.267 3) Asset C's beta = 0.035 / 0.18^2 = 1.08

Definitions/Calculation: [Return Generating Models] [Part 2/2] 4) Market Model [with an example] (I believe I encounter this somewhere, thus recap) Example 2.1 Using the Market Model to Calculate Abnormal Returns A regression of ABC Stock's historical monthly returns against the return on the S&P 500 gives an αi of 0.002 and a βi of 1.05. Given that ABC Stock rises by 3% during a month in which the market rose 1.25%, calculate the abnormal return on ABC Stock. (Important, MUST MEMORIZE)

=> Ri = αi + βi *Rm + ei [ intercept αi and slope coefficient βi are estimated using historical asset and market returns ] [Rm = market return] + Market model is an example of a single-index return-generation model. It is used to estimate beta risk and to compute abnormal returns. Example Answer: ABC Stock's expected return: Ri = 0.002 + 1.05 * 0.0125 Ri = 0.015125 ABC company specific abnormal return: => 0.03 − (0.015125) = 0.014875 or 1.49%

Example 1.1 Risk and Return of a Leveraged Portfolio Sasha Miles is evaluating how to allocate funds between the risk-free asset and the market portfolio. She gathers the following information: Risk-free rate of return = 6% Expected return on the market portfolio = 14% Standard deviation of returns of the market portfolio = 23% Calculate the expected risk and return of a portfolio that is: 75% invested in the market portfolio. 140% invested in the market portfolio. (Literally the same example Instructor Olinto showed us in M2L1 saying "this may be the hardest they can go with math, rest are concepts") (Olioto went over this in M2)

A. E(Rp)= w1*Rf + (1−w1)* E(Rm) E(Rp)= (0.25×0.06) + (0.75×0.14) E(Rp)= 12% σp = W risky * σ of market portfolio σp = (0.75×0.23) = 17.25% B. E(Rp) = (−0.4×0.06) + (1.4×0.14) E(Rp) = 17.2% σp = W risky * σ of market portfolio σp = (1.4 × 0.23) = 32.2% Note: Recall that risk free asset has 0 σ, 0 covariance

Summarization/Calculation: [Part 2/2] (More important) Security Selection: Identifying Mispriced Securities (The instructor said this will be on the test! I believe it will be on the test about mispricing) (Important, , identify )

Again: 1) If the expected return [using price and dividend forecasts] is "higher" than the investor's required return [CAPM, given the systematic risk in the security] the security is undervalued and the investor should buy it. 2) Vice versa, If the expected return [using price and dividend forecasts] is "lower" than the investor's required return [CAPM, given the systematic risk in the security] the security is overvalued and the investor should buy it. 3) {Looking at the graph pg 58} + Stocks A and D above SML, their expected return is greater than their required return. Securities that plot above the SML are undervalued. [should be purchased] + Stocks B and E are on SML. Their expected return = their required return. These stocks are fairly valued. + Stock C plots below the SML. The expected return is smaller than their required return. ... Overvalued

Calculation/Quick recap: Portfolio Beta (Easy, scan through, identify) (recap) (I doubt this will be on the test)

CAPM can be used for Portfolio Beta: E(Rp) = Rf + βp*[E(Rm) − Rf] [Nothing change from regular CAPM. Instead of using security beta, we use portfolio beta] 1) beta of a portfolio equals the weighted average of the betas of the securities in the portfolio. 2) Portfolio's expected return can be computed using the CAPM

Calculation: Application of CAPM Example 3.3 Application of the CAPM to Capital Budgeting The directors of Mercury Inc. are considering investing in a new project. The project requires an initial investment of $550 million in one year. The probability of success is 60%. If it is successful, the project will provide an income of $350 million at the end of Year 2, but will also require a further investment of $200 million. Further, it will generate net income of $250 million in each of Years 3 and 4. At the end of Year 4, the company will sell the project for $300 million. If the project is unsuccessful, the company will not earn anything. Given that the market return is 14%, risk-free rate is 4%, and beta of the project is 1.5, answer the following questions: Calculate the annual cash flows using the probability of success. Calculate the expected return. Calculate the net present value. (Easy, scan through, identify) (recap) (I doubt this will be on the test)

Expected rate of return computed from the CAPM is used by investors to value stocks, bonds, real estate, and other assets. In capital budgeting, CAPM is used to compute the required rate of return and NPV is sued to make investing decisions. 1) Year 1 = −$550m Year 2 = 0.6 × ($350m − $200m) = $90m Year 3 = 0.6 × $250m = $150m Year 4 = 0.6 × ($250m + $300m) = $330m 2) Using the CAPM, the expected or required rate of return is calculated as: Required return = 4% + [1.5 × (14% − 4%)] = 19% 3) [CF] [2ND] [CE|C] [↓] 550 [+/−] [ENTER] [↓][↓] 90 [ENTER] [↓][↓] 150 [ENTER] [↓][↓] 330 [ENTER] [NPV] 19 [ENTER] [↓] [CPT] NPV = −$145.06 million

Calculation/Definition: Portfolio Performance Evaluation [Part 4/4] ** Jensen's Alpha (Important, MUST MEMORIZE) (quickly scan over, common sense)

Jensen's Alpha: (literally the same about market model's example: finding abnormal return) αp = Rp − [Rf + βp*(Rm − Rf)] or (Ri − Rf) * (σm/σi) + Rf or αp = Return of portfolio - CAPM 1) Jensen's alpha is based on systematic risk (like the Treynor ratio). 1 - estimates a portfolio's beta risk using the market mode. 2 - uses the CAPM to determine the required return from the investment (given its beta risk). 3 - difference between the portfolio's actual return and the required return (as predicted by the CAPM) is called Jensen's alpha ** 2) Jensen's alpha for the market equals zero 3) higher the Jensen's alpha for a portfolio, the better its risk-adjusted performance 4) Jensen's alpha is the maximum that an investor should be willing to pay the portfolio manager.

Definitions/Components/Calculation: Beyond the CAPM + Limitation of CAPM + Extensions of the CAPM (LV2 Material) + Practical Models (LV2 Material) (know the limitation of CAPM, Job on the exam is to identify, not memorize) (We doubt this will be on the test. We believe this is beyond LV1. This is LV2 material. Scan through, don't get stuck here!) (quickly scan over, move on)

Limitation of CAPM: + Theoretical Limitations: 1) Single-factor model: Systematic risk or beta risk is the only risk that is priced in the CAPM. Basically the model asserts that no other investment characteristics should be considered in estimating returns. 2) Single-period model: CAPM is a single-period model that does not consider multi-period implications or future period. + Practical Limitations 1) Market portfolio: CAPM is not testable because the true market portfolio (one that includes all assets, financial and nonfinancial, such as human capital) is unobservable. 2) Proxy for a market portfolio: In the absence of a true market portfolio, market participants generally use proxies, but these proxies vary across investors. Extensions of the CAPM : {LV2 Material} 1) Theoretical models include arbitrage pricing theory (APT) E(Rp) = RF + λ1*βp,1 +...+ λk*βp,k (Skip the formula and know the concepts) Just know: 1) APT allows numerous risk factors. It incorporates the risk-free rate. And these risk factors across assets can be different, but the risk-free rate is the same across all assets. 2) APT is superior to the CAPM, but it isn't commonly used in practice because it does not specify what these risk factors are. Practical Models: {LV2 Material} (I completely skipped, pg. 61 if you want to review)

Example 3.5 Portfolio Performance Evaluation Manager Return σ β A 13% 20% 0.6 B 11% 15% 1.1 C 12% 10% 0.8 Market (M) 10% 16% Risk-free rate (Rf) 4% Calculate the following for each of the investment managers: Expected return Sharpe ratio Treynor ratio M^2 Jensen's alpha Comment on your answers and rank the managers' performance. (Check textbook pg 56) (Important)

Manager A: Expected return = Rf + β*(Rm−Rf) = 4% + [0.6 × (10%−4%)] = 7.6% Sharpe ratio =(RA − Rf) / σA = (13% − 4%) / 20% = 0.45 Treynor ratio =(RA−Rf) / βA = (13% − 4%) / 0.6 = 0.15 M^2 = 0.45 * 16% + 4% = 11.2% Jensen's alpha: = RA − [Rf + β(Rm − Rf)] = 13% − [4% + 0.6(10%−4%)]=5.4% 2) When considering total risk (relevant when the portfolio is not fully diversified), we look at the Sharpe ratio and M2. C performs the best as she has both the highest Sharpe ratio (0.8) and the highest M2 (6.8). When we consider systematic risk (relevant when the portfolio is well diversified), we look at the Treynor ratio and Jensen's alpha. Manager A performs the best as she has both the highest Treynor ratio (0.15) and the highest Jensen's alpha (5.4%).

Calculation/Definition: Portfolio Performance Evaluation [Part 2/4] **Treynor ratio (Important, MUST MEMORIZE) (quickly scan over, common sense)

Treynor ratio: (basically replaces total risk in the Sharpe ratio with systematic risk (beta) = [Rp−Rf] / βp For the Treynor ratio to offer meaningful results: 1) Both the numerator and the denominator must be positive 2) Neither the Sharpe nor the Treynor ratio offer any information about the significance of the differences between the ratios for portfolios.