Hi again, I was hoping you can help with this investment management assignment. I am attaching the word document. It is 4 problems about asset management. Thank you for your help!

Fin 6310 1. Assume that the returns of individual securities are generated by the following two-factor model: Rit = E(Rit) + i1f1t + i2f2t where Rit is the return for security i at time t, and f1t and f2t are two factors with zero expectation and zero correlation between them. In addition, assume that there is a capital market with four securities, where each one has the following characteristics: Security 1 2 E(Rit) 1 1.0 2.0 20% 2 0.6 2.0 18% 3 1.0 0.8 10% 4 1.5 0.7 10% The capital market is perfect, in the sense that there are no transactions costs and short sales can take place. (a) Construct a portfolio p (with weights that sum to 1) containing securities 1 and 2 such that the return does not depend on the market factor f1t in any way. Compute the expected return and 2p coefficient for this portfolio. (b) Similarly, construct a portfolio q containing securities 3 and 4 such that the return does not depend on f1t. Compute the expected return and 2q coefficient for this portfolio 2. In the AT&T and TCI merger example of lecture note 8, after announcing the deal, AT&T stock price went down $5.375 to $60 while TCI (class A) stock price went up $1.0625 to $39.75. According to the term of merge, 0.7757 shares of AT&T stock will be paid for each TCI class A share. If the betas for AT&T and TCI are both 0.8, and ignore the risk-free rate (i.e. let it be zero), answer the following questions (a) If the market had climbed during the announcement day by 1.2%, what are the dollar amount changes in both AT&T and TCI stocks due to purely the merger news? Can we say that the merger announcement represented bad news for AT&T and good news for TCI? (b) If the market had fully anticipated that the merger will go through on the announcement day and given the $60 closing price was correct for AT&T, what would be the total return for TCI during that day due to purely the merger news? (c) If you were confident at that time that the merger will go through, how would you design an investment strategy that would profit from it given the actual stock prices on that day and the term of merger? 3. John is a fund manager and achieved the following performance in a recent month. Assets Actual Actual Benchmark Index Return Return(%) Weight(%) Weight Stocks 2 75 65 2.5 Bonds 1 20 25 1.2 Cash 0.5 5 10 0.5 (a) What was John's relative return performance as a fund manager in the month with respect to the benchmark portfolio? (b) What was John's contribution of security selection to the relative performance? (c) What was John's contribution of asset allocation to the relative performance? 4. Here are some data on four mutual funds (A,B,C,D), market index (M), and riskfree asset fund (F): Fund Average return(%) Standard Deviation(%) Beta A B C D Market Risk-free 14 12 10 20 16 3 6 4 6 10 8 0 1.5 0.5 0.5 2.0 1.0 0.0 (a) Calculate the Sharpe ratio for each mutual fund and rank them. (b) How much would the return on fund B have to change to reverse the ranking relative to A? (c) Calculate the Treynor ratio for each mutual fund and rank them. (d) Calculate the Jensen's for each mutual fund and rank them. Lecture 7 The Arbitrage Pricing Theory and Multifactor Models School of Management The University of Texas at Dallas 1 Outline Factor Models Why additional factors? Single-factor models Multi-factor models Arbitrage Pricing Theory Factor portfolio The pricing relationship Applications Fama and French 3-factor model Examples of Factors Used by Money Managers A comparison of CAPM and APT 2 Why Additional Factors? CAPM built on strong assumptions Poor cross-sectional performance Mean excess return vs. beta Size- and B/M-sorted portfolios Monthly mean excess return (percent) 0.012 0.01 0.008 0.006 0.004 0.002 0 0 0.3 0.6 0.9 1.2 1.5 Beta 3 Why Additional Factors? Average investor has three sources of wealth Investment portfolio Employment Entrepreneurial wealth Market movements and economic cycles not perfectly correlated Correlation between economic cycles and bull/bear markets less than 0.4 Simple Example Compare two stocks with same market sensitivity () One does well in recessions, the other poorly Most investors would prefer the first since it provides cushion to other sources of wealth 4 What Are Factor Models? Factor models are descriptive models of security returns. A factor model assumes that there is more than one source of systematic risks It states that individual security returns is a linear function of known factor returns Factor models are not equilibrium asset pricing models in general Example: A Single Factor Model The market return model is a one-factor model where the factor is the return on a market index, Rm,t We can also have a single factor model, where the factor is GDP growth To generalize, the factor could be whatever you think is appropriate 5 A Two-factor Model Security i's return is related to two factors, F1,t and F2,t Ri,t = i + bi,1f1,t + bi,2f2,t + i,t f1,t = F1,t - E(F1,t) and f2,t = F2,t - E(F2,t) are surprises in factors bi,1 and bi,2 are called factor loadings and measure the sensitivities of security i's return to the two factors, respectively i,t is security i's specific (idiosyncratic) return It has zero expected return, E(i,t) = 0 It is uncorrelated with the factors, Cov(i,t, f1,t) = Cov(i,t, f2,t) =0 Idiosyncratic returns are uncorrelated with each other among different securities i should equal to the expected return of security i, i.e., i = E(Ri) 6 Example Suppose the stock of GE are determined by two risk factorsGDP growth and the inflation rate: RGE = E(RGE) + bGDP SurpriseGDP + bI SurpriseI + GE We are given that: E(RGE) = 10%, bGDP = 2, and bI = -1 The expected GDP growth this year is 2%, and the expected inflation rate is 4% If actual numbers come in as: GDP growth = 3% Inflation = 5% We would expect GE's return to be: RGE = 10% + 2(3%-2%) - 1(5%-4%) + GE = 11% + GE 7 Possible Factors In practice, the following factors are used: Growth in GDP Change in industrial production Oil prices Growth in labor income Rate of inflation Spread between long-term and short-term interest-rates Market factor Industry factor Size factor (market capitalization) \"Value-growth\" factor (typically measured by BV/MV) 8 k-Factor Models Security i's return is related to k factors, F1,t , F2,t , ..., and Fk,t Ri,t = i + bi,1f1,t + bi,2f2,t + ...+ bi,kfk,t + i,t i should equal the expected return of security i, i.e., i = E(Ri) fj,t = Fj,t - E(Fj,t) is surprise in the jth factor bi,j is security i's sensitivity to the jth factor fj,t Each security has k sensitivities, bi,1, ..., bi,k i,t is security i's specific (idiosyncratic) return It has zero expected return, E(i,t) = 0 It is uncorrelated with any of the factors Idiosyncratic returns are uncorrelated with each other among different securities Estimation: use multivariate regression analysis 9 Application of Multi-factor Models Compute an efficient portfolio from a large number of securities Difficulty in estimating expected returns, variances and covariance matrix, and numerical difficulties Simplifying the estimation by using the factor structure Construct a portfolio according to your expectation in a multifactor world Achieve better performance by incorporate additional information on the direction of the factors \"Tailor\" the portfolio's sensitivities to the factors based on what you think will be happening to the factors For example: suppose that the factors are GDP and inflation and you believe that GDP growth will be negative inflation will be low What should you do? Set the sensitivity of your portfolio to GDP to be low Set the sensitivity of your portfolio to inflation to be high 10 The Arbitrage Pricing Theory The APT is based on two basic assumptions: Security returns are generated by a k-factor model: Ri,t = E(Ri) + bi,1f1,t + bi,2f2,t + ...+ bi,kfk,t + i,t bi,j is security i's sensitivity to the jth factor fj,t fj,t = Fj,t - j, where j is the expected return of factor j, i.e., E[Fj,t] = j Absence of arbitrage opportunities A factor structure for asset returns The fact that all arbitrage opportunities are exploited (i.e. prices adjust) Can E(Ri) be anything? The number of assets in the economy is very large. Idiosyncratic risk can be diversified away 11 Factor Portfolios Assume there is no idiosyncratic risk ( ) Factors can be traded or non-traded When factors are non-traded, we can construct factor portfolios A portfolio that is only sensitive to one factor Approach Assume there are k sources of risks Also assume that we have at least k stocks with zero idiosyncratic risk Ri,t = E(Ri) + bi,1f1,t + bi,2f2,t +...+ bi,kfk,t Construct a portfolio (i.e., finding a set of weights) using the k stocks that is only sensitive to one factor 12 Example Suppose there are two sources of risk and two securities with the following return structure R1 = E(R1) + 2f1 + 1f2 R2 = E(R2) + 1f1 + 3f2 In order to find the factor portfolios, we can construct a portfolio with weight w on the first security and (1-w) on the second security Rp = w R1 + (1-w) R2 = [wE(R1) +(1- w)E(R2)] + [2w+(1-w)]f1 + [w+3(1-w)] f2 = E(Rp) + (w+1)f1 + (3-2w) f2 The first factor portfolio Let the sensitivity to the second factor be zero, i.e. (3-2w)=0, or w=1.5 Rp1 = E(Rp1) + 2.5 f1 The second factor portfolio Let the sensitivity to the first factor be zero, i.e. (w+1)=0, or w=-1 Rp2 = E(Rp2) + 5 f2 13 The APT Model Since it is easy to construct factor portfolios, we assume all factors are traded There is a risk-free asset with return Rf We can mimic the return for the ith stock by forming a portfolio using factors and with weights equal to the corresponding sensitivity Such a mimicking portfolio will expose to the same risks as that of the ith stock If there is no arbitrage, they should have the same expected return, which leads to E(Ri) = Rf + bi,1 [E(Rf1)- Rf ] + bi,2 [E(Rf2)- Rf ] +...+ bi,k [E(Rfk)- Rf ] This is the APT model 14 Example Assume there are two sources of risks, GDP and inflation, and two securities. The risk-free rate is 7%. Security A Security B bGDP 1 2 bInf 1 0.5 E(R) 8% 12% Find the expected return for each risk factor 8% = 7% + 1 [E(RGDP) - 7%] + 1.0 [E(RInf) - 7%] 12% = 7% + 2 [E(RGDP) - 7%] + 0.5 [E(RInf) - 7%] the first factor risk premium: E(R GDP) - Rf = 3% the second factor risk premium: E(RInf) - Rf = -2% Assume that firm C has factor loadings bGDP=1 and bInf=-1. The expected return for firm C is then E(Rc) = Rf+1 3% - 1 (-2%) = 12% 15 Why is this called the arbitrage pricing theory? If the expected return on C were, say, 11%, you could create an arbitragea strategy that guarantees a profit with no risk and cost How would you create the arbitrage: buy low, sell high in order to generate positive profits with zero cash flow today, Sell $2.5 of A Buy $2.0 of B Sell $1.5 of C Buy $2.0 of Rf This portfolio has no sensitivity to either factor: E(R) = 8% E(R) = 12% E(R) = 13% E(R) = 7% Sensitivity to the GDP factor -2.5ba,GDP+2bb,GDP -1.5bc,GDP=-2.51+22-1.51=0 Sensitivity to the Inflation factor -2.5ba,Inf+2bb,Inf -1.5bc,Inf=-2.51+2.5+1.51=0 But the expected return is: E(Rp) = -2.58%+212%-1.511%+27%=1.5% for sure. This is an arbitrage. 16 About Factors in APT APT doesn't tell us which factors should be priced, nor by how much We have to rely on our economic intuition: anything that may affect discount rates and general cash flows cannot be diversified away Therefore It may not be an accurate description of the economy Doesn't explain why a factor deserves compensation. Example of \"factors\" that should get positive risk premia The market factor Business cycles Examples of \"factors\" that should get negative risk premia Inflation Market volatility Interest rates 17 Fama and French Three-Factor Model Fama and French (1992, 1993) have proposed a three-factor model that includes the following factors: market factor (for example, the return on the S&P 500) size factor as measured by market capitalization a value-growth factor as measured by the book-value to market-value ratio. E Ri R f i i E Rm R f SMB ,i SMB HML,i HML where, SMB: Return of small-cap minus large-cap stocks HML: Return of high B/M (value) minus low B/M (growth) stocks SMB andHML are expected return on SMB and HML SMB and HML can be considered as proxies for nondiversifiable risk factors 18 Estimating Fama and French Model for IBM stock Running the F-F regression (monthly 1995-1999): Coefficient t Stat P-value Intercept 0.0136 0.0116 1.1683 0.2476 Market 1.1408 0.3048 3.7419 0.0004 SMB 0.0694 0.3527 0.1968 0.8446 HML St.Error 0.2894 0.3683 0.7858 0.4352 In summary: M,IBM =1.14 S,IBM = 0.07 H,IBM = 0.29 Rf = 6% E[RM] -Rf = 6.0% E[SMB]= 5.2% E[HML] = 4.8% Using the F-F Model we estimate the expected return of IBM to be: E[RIBM] = 6%+ 1.146.0% + 0.075.2% + 0.294.8% =14.60% 19 Interpreting Fama and French Factors SMB and HML have been linked to future GDP growth rates Reflect premiums for holding: Positive correlation Above and beyond the market return True for many countries besides US Macroeconomic risks associated with the business cycle Assets which do poorly in times of financial distress Value stocks Those that appear to be undervalued for reasons other than earnings growth potential Usually identified by analysts as having low P/E ratios or low price to book value ratios Typically have a price driven down due to financial distress In the event of a credit/liquidity crunch, stocks in financial distress will do badly Investors demand premium 20 Morgan Stanley's Macro Proxy Model Factors: GDP growth long-term interest rates foreign exchange (Yen, Euro, Pound basket) market factor commodities or oil price index Factor-mimicking portfolios (\"Macro Proxy\") Step 1: Regress individual stocks (S&P 500) on macro factors Step 2: Create long-short portfolio of the most and least sensitive stocks (100 each) Macro-proxy returns predict macro factors 21 Morgan Stanley's Macro-Proxy Portfolio left bar = actual GDP growth right bar = GDP mimicking portfolio Procedure: 1. Regress each stock on GDP and market factor 2. Sort stocks based on GDP sensitivity 3. Create factor mimicking portfolio long the top quintile (stocks reacting most positively to GDP) and short the bottom quintile (stocks reacting most negatively to GDP) 22 Examples of Factors Used by Money Managers Piper Capital Management Long-term government bonds Spread in returns on long-term corporate bonds and government bonds Inflation rate Interest rates Industrial production Oil prices Market return 23 Examples of Factors Used by Money Managers Salomon Smith Barney Growth in GDP Inflation rate Interest rates Oil prices Growth in defense spending Roll and Ross Asset Management Industrial production Long-term government bonds Spread in returns on low grade bonds and high grade corporate bonds Inflation rate 24 Usefulness of the APT Model Particularly useful for risk management or asset allocation. Significantly diminishes the number of input variables Provides motivation for analyst specialization Only need to understand a stock's sensitivity to fundamental factors Problem: ignores intra-industry correlations Continues to provide intuitive definition of systematic and idiosyncratic risk Only need mild assumptions about risk and no-arbitrage 25 A comparison of CAPM and APT CAPM: Easy to see where it comes fromyou see exactly what portfolios people hold and the role of diversification Relies a lot on the market portfolio of assets that everybody holds, which we know isn't actually true Everything collapses into this single factor the market return APT: Allows multiple factors Doesn't tell us what they are It's harder to explain 26 Market Efficiency Firm Foundation Theory (Fundamental Analysis) Castle-in-the-Air Theory (Technical Analysis) Random Walk Hypothesis (Three Versions) Market Efficiency Empirical Evidence 1 The View from the Mid-50's No real academic study of financial markets Investors need \"professional\" advice to form investment portfolios. These professionals will exploit their \"professional knowledge\" study individual companies in depth tailor portfolio to your individual needs Example: Young, aggressive, long-run investors should invest in \"risky\" or \"growth\" stocks Old and conservative investors should invest largely in bonds These views can be embodied in two distinct philosophies: Firm-Foundation Theory Castle-in-the-Air Theory 2 The Firm-Foundation Theory Each asset has a key element called 'intrinsic value' Intrinsic value can be determined by careful analysis of present conditions and future prospects When market prices rise above or fall below this benchmark, it creates a selling or buying opportunity Let P denote the market price of a particular security and V denote its intrinsic value If P V, then that security is overpriced Problem: How do we calculate intrinsic value? 3 Determinants of Intrinsic Value Expected Growth Rate of Dividends Implications for asset prices: Investors should be willing to pay a higher price for an asset the higher the growth rate of dividends the longer the growth rate continues The Expected Dividend Payment Implications for asset prices: price should be high when the expected dividend payment is large Degree of Risk Quality of a stock is associated with low risk Implications for asset prices: A rational investor should be willing to pay more for a stock with a lower risk Market Interest Rates Implications for asset prices: A rational investor should be willing to pay more for a stock when the interest rate is low 4 The Firm-Foundation Theory (Cont'd) Problems with using fundamentals to calculate Intrinsic Value: Difficult to predict the future from the present Dividend growth can't continue forever Difficult to predict long-term growth rates Hard to get precise figures from imprecise data (GIGO) The above four determinants do loosely correlate with asset prices: Higher P/E (price/earnings) ratios are associated with higher expected long-term growth The other fundamentals show a correlation with prices as well 5 The Castle-in-the-Air Theory Concentrates on psychic values and investor expectations Mood of the investment community is important, not intrinsic value John M. Keynes: \"Animal spirits,\" no logical rules Success of the investor depends upon trying to predict which situations are most likely to promote \"castle building\" and buying before the crowd moves Problem: Purely psychic support for prices is not dependable (not unlike a pyramid scheme) Investors need some type of benchmark (buy and sell) 6 Possible Predictors Those who subscribe to this philosophy will often invest according to various factors that they believe help predicting the future direction of the market: Patterns of stock prices Sunspots Hemlines Outcome of the Super Bowl The election year Flight patterns of migrating Canadian geese, etc. 7 Investment Strategies Different investment philosophies suggest different investment strategies \"Firm foundation theory\" suggests fundamental analysis \"Castle-in-the-Air Theory\" calls for technical analysis Fundamental analysts: Believe the market is logical Seek to determine the \"fair\" (intrinsic) value of a stock through the 'present value' of the cash flows to the holder of the stock must try to predict: Sales Operating costs Profits Dividends Earnings, etc. 8 Why might fundamental analysis fail? Information, projections and/or analysis may be incorrect The analysts' estimate of value may be wrong Analysts may have got the correct information but used the wrong formula or weights \"As Stocks Trample Price Measure, Analysts Stretch to Justify Buying,\" WSJ. Market may not 'correct the mistake' Cost of acquiring the information may be greater than the value of using it 9 Technical analysis Technical analysts (chartists): Look at various past price patterns for clues to the future direction of the market. There are two major principles to technical analysis: All information about a stock is reflected in past market prices Either Trends in stock prices tend to persist-if the price of a stock is rising it is most likely to continue to rise, etc The market overreacts-if the price rises today it is most likely to fall in the future, (and vice versa) The technical analyst looks for trends People generally subscribe to technical analysis because they believe that there is: A crowd instinct of mass psychology An uneven flow of information (first to insiders, then large institutional investors and then the average investors) 10 Examples of Technical Analysis Some popular applications of technical analysis: Herding or following the crowd Filter rules New highs, new lows Advance/decline lines Volume Dow Theory: attempts to identify long-term trends Elliot Wave Theory: Prices can be described by a set of wave patterns Resistance and support levels Non-price pattern indicators: Super Bowl, Elections, etc. 11 Why might technical analysis fail? Changes in trends are difficult to predict They can be sudden and/or very large If the scheme could generate unusual profits, popularity would kill it The way to make a profit under this theory is to buy when you know the price is going to rise However, if everyone is trying to do this, then the price will rise instantaneously and no profits will be made What we tend to see is that information gets included in the price of a stock very quickly Trading costs might overwhelm any potential profits Factors that are correlated with security prices do not necessarily cause security prices 12 Random Walk Hypothesis First introduced by Louis Bachelier (1900) and rekindled by academic studies in the late 50's and early 60's The Random Walk Hypothesis (RWH) Definition: persistent patterns in stock prices occur no more frequently than runs of luck in gambling Example: flipping a coinconstant odds All current and past information is captured in today's price New information is, by definition, unpredictable Adjust for upward trend in stock pricesrandom walk with drift Under this view, no system, chart analysis, study of fundamentals, can predict which way price will go Sometimes the Random Walk Hypothesis is known as the Efficient Market Hypothesis (EMH) 13 A Random Walk Series versus Dow Jones Index (1930.12004.6) Heads Heads $260.00 $250 Tails Heads Tails $242.50 Tails $270.40 $252.20 $252.20 $235.23 14 Random Walk and Security Returns Implications for asset prices: Since history does not repeat itself, it is impossible to predict the future Hence we cannot use current information to improve our forecast Competition among analysts is the driving force Favorable future prospects will be reflected as higher prices today Do not confuse random price changes with random or irrational price levels The term random walk is often used in a loose sense that price changes are essentially unpredictable If price is unpredictable, so is the return A more careful statement: Returns should be unforecastable Return = interest rate + risk premium + unforecastable change Again, by saying unforecastable, we mean something similar to a coin toss 15 Three Forms of the RWH Weak: Future stock prices cannot be predicted on the basis of past prices (It = pt, pt-1, ,...) or other readily available trading data such as volume, etc Semi-strong: Printed or published information cannot be utilized to predict future prices (It = all public ally available information as of time t) Strong: Nothing, not even unpublished developments, can be of use in predicting prices Everything that is known or is knowable is reflected in prices (It = everything that is known or knowable as of time t) The weak form has direct implications for technical analysis The semi-strong and strong forms attack fundamental analysis 16 Market Efficiency A random walk would be the natural result of prices reflecting all available information Financial markets are said to be efficient if security prices rapidly reflect all relevant information about asset values, and all securities are fairly priced in light of the available information The different forms of market efficiency are related to the different forms of the random walk (weak, semi-strong, strong) Strong-form: Semistrong-form: Weak-form: Past prices & volume Public information Public and private information 17 Evidence on Weak-Form Efficiency Most technical analysts do no better than other investors Returns are close to uncorrelated over time There is no evidence of consistently successful market timing Studies tend to find that investors are better off with a buy-and-hold strategy (vs. timing the market) because of upward trend Bull market gains tend to be more than bear market losses You would have to be correct 70% of the time to beat the buy-and-hold strategy 18 Testing Semi-strong Form Efficiency When markets are semi-strong efficient, stock prices should respond rapidly to public release of information Price Days Examples: announcements of earnings and dividends or forecasts of these by analysts changes in accounting practices disclosures of stock splits 19 Mini Case: The AT&T - TCI Merger The merger agreement between AT&T and TCI was announced on June 24, 1998; The merger plan includes the issuance of 440 million new AT&T shares: 0.7757 shares of AT&T stock (NYSE-T) will be paid for each TCI class A share (NDQ-TCOMA) class A TCI shares are widely held and have one vote per share 0.8533 shares of common AT&T stock will be paid for each TCI class B share (NDQ-TCOMB) class B TCI shares are closely held and have ten votes per share At the time of the announcement, the merger was expected to be completed by the end of June 1999 20 $56 AT&T TCI 01-Jul-98 30-Jun-98 29-Jun-98 26-Jun-98 25-Jun-98 24 -Jun-98 23-Jun-98 22-Jun-98 19-Jun-98 18-Jun-98 17-Jun-98 ATT Stock Price $66 $62 $38 $60 $36 TCI Stock Price An Example: the AT&T - TCI Merger $40 $64 $58 $34 21 Mini Case: the AT&T - TCI Merger On June 24, 1998, AT&T stock price went down $5.375 to $60; TCI (class A) stock price went up $1.0625 to $39.75 With the information above can we say that the merger announcement represented bad news for AT&T and good news for TCI? We need to analyze how the stock market performed on that day; On that day, the S&P500 climbed 1.2%; Suppose one is sure that the AT&T-TCI merger will be completed on June 30, 1999, following the terms specified on June 24, 1998. What is the value of one class A share of TCI? Did TCI's stock price climb to that value on June 24, 1998? Why? This is called a risk arbitrage opportunity. Why? Suppose that the deal will not go through, what would probably happen to the stock prices of AT&T and TCI? 22 Event study The important idea is that the stock prices should reflect the release of information at time t How to test the semi-strong form? Event study An event study is a research methodology designed to measure the impact of an event of interest on stock prices after the event date Since prices are different at different event, we examine return instead Return can change due to overall market movement use the abnormal return of a stock on a given day t. 23 Event study (continued) When market is efficient: Before event at time Event at time t After event at time t-1 t+1 ARt ARt-1 = 0 ARt+1 = 0 Testing methodology: CAR - cumulative abnormal return If semi-strong efficient CAR should be constant after the event The evidence: The classic event study Jensen, and Roll (1969), stock splits CAR is by Fama, Fisher, who studied 940 different time (in days) event date 24 What if markets aren't efficient? The classic forms of inefficient response to new information are: delayed response overreaction Price Price time after event Delayed Response time after event Over-reaction 25 Problems with the RWH The RWH rests on some key assumptions: 1. News travels instantaneously What about small companies not followed closely by Wall Street? Even if the market has the news, it may be difficult to translate known information into value 2. Presupposes no individual/institution possesses monopolistic control over the market 3. Presupposes no large scale buying or selling based on unfounded recommendations 4. Investors behave rationally and therefore stocks sell at the best, rational estimates of intrinsic value Market hysteria may sometimes cloud this issue Example: OCT. 1987 People may use simple rules of thumb 26 Predictability The Random Walk Hypothesis implies that Returns should be unforecastable There should be no: Positive autocorrelation (momentum) Negative correlation (overreaction or reversal) between successive returns How does it work? Answer: Run a regression rt+1 - rft+1 = 0.005 + 0.04 (rt - rft) + errort+l; (3.3) (1.0) rt = stock return (NYSE portfolio) (Monthly 1947-1995) rf = interest rate (Treasury bill) R2 = explained variance / total variance = .002 Evidence of momentum, but statistically insignificant No potential profitable trading opportunities 27 Predictability (continued) Try two price-based variables rt+1 - rft = a + 0.15 (Dt / Pt) + 0.37 Termt + errort+1; (3.3) (2.9) Dt / Pt = dividend - price ratio Termt = 10 year bond rate - 3 month bond rate R2 = 0.03 Statistically significant ability to predict direction of stock returns! But still discouragingly low R2 Many academic studies found no reliable rules, systems, gimmicks, tricks that forecast stock returns for short horizons Returns behave just like coin tosses Filter rules also generate no trading profits 28 Some Puzzles Calendar effect: January effect Monday effect Holiday effect Outperformance of some groups of stocks Value stocks Small stocks Some drift in stock prices after earnings announcements IPO underperformance The crash of 1987 The close-end fund puzzle 29 Summary Fundamental Analysis Castle-in-the-Air Theory Technical Analysis Random Walk Hypothesis Forms of Market Efficiency Empirical Evidence 30 This sheet answers question 3. Assets Stocks Bonds Cash Actual Return Actual Weight 2% 1% 0.50% 75% 20% 5% Benchmark Weight Index Return Return difference 65% 25% 10% 2.50% 1.20% 0.50% -0.500% -0.200% 0.000% Benchmark portfolio return 1.975% ...sum of the product of the benchmark weight x Index return. Actual portfolio return 1.725% ...sum of the product of the actual weight x actual return. John's relative return performance -0.250% ...actual portfolio return - benchmark portfolio return. Contribution from stock selection -0.415% .... Sum of the product of return difference x actual weight. Return difference = Actual re Contribution from asset allocation 0.165% .... Relative return performance - contribution from stock selection. ual return - benchmark return. This sheet answers question 3. Assets Stocks Bonds Cash Actual Return Actual Weight 2% 1% 0.50% 75% 20% 5% Benchmark Weight Index Return Return difference 65% 25% 10% 2.50% 1.20% 0.50% -0.500% -0.200% 0.000% Benchmark portfolio return 1.975% ...sum of the product of the benchmark weight x Index return. Actual portfolio return 1.725% ...sum of the product of the actual weight x actual return. John's relative return performance -0.250% ...actual portfolio return - benchmark portfolio return. Contribution from stock selection -0.415% .... Sum of the product of return difference x actual weight. Return difference = Actual re Contribution from asset allocation 0.165% .... Relative return performance - contribution from stock selection. ual return - benchmark return. This sheet answers question 4. Fund Average return Standard Deviation Beta A B C D Market Risk-free 14% 12% 10% 20% 16% 3% 6% 4% 6% 10% 8% 0% 1.5 0.5 0.5 2 1 0 Sharpe ratio 1.83 2.25 1.17 1.70 Sharpe ratio Treynor ratio ranking 2 0.07 1 0.18 4 0.14 3 0.09 Return on fund B required to equate the Sharpe ratio of fund A to that of fund B = (Sharpe ratio for fund A) x (Standard deviation on fund B) + (Risk free rate) = 10.33% Therefore change required = -1.67% Sharpe ratio = Treynor ratio = Portfolio return - Risk free rate Portfolio standard deviation of return Portfolio return - Risk free rate Portfolio beta Jensen's = Portfolio return - Risk free rate + Portfolio beta Return on market - risk free rate Treynor ratio ranking 4 1 2 3 Jensen's Jensen's alpha alpha ranking -0.09 3 0.03 1 0.01 2 -0.09 4 Question 1 a The return from security 1 can be expressed as follows: Equation 1 R1 0.20 f1 2 f 2 The return from security 2 can be expressed as follows: Equation 2 R2 0.18 0.6 f1 2 f 2 Let w denote the weight for security 1 in portfolio p. Since the weights sum to 1, it follows that security 2's weight is (1 - w). The return on portfolio p is thus: Equation 3 RP wR1 1 w R2 w 0.20 f1 2 f 2 1 w 0.18 0.6 f1 2 f 2 0.20 w 0.18 1 w w 1 w 0.6 f1 2 w 2 1 w f 2 Let the return on portfolio p not depend on f1 . That is: w 1 w 0.6 w 0.6 0.6w 0.4 w 0.6 0 Equation 4 0.6 1.5 0.4 1 w 1 1.5 2.5 w In summary, we assign a weight of -150% on security 1 and 250% in security 2. The expected return on the portfolio: = The 2 P 0.2 w 0.18 1 w 0.2 1.5 0.18 2.5 15% Equation 5 coefficient is: 2 P 2 w 2 1 w 2 1.5 2 1 1.5 2 Equation 6 Question 1 a The return from security 1 can be expressed as follows: Equation 1 R1 0.20 f1 2 f 2 The return from security 2 can be expressed as follows: Equation 2 R2 0.18 0.6 f1 2 f 2 Let w denote the weight for security 1 in portfolio p. Since the weights sum to 1, it follows that security 2's weight is (1 - w). The return on portfolio p is thus: Equation 3 RP wR1 1 w R2 w 0.20 f1 2 f 2 1 w 0.18 0.6 f1 2 f 2 0.20 w 0.18 1 w w 1 w 0.6 f1 2 w 2 1 w f 2 Let the return on portfolio p not depend on f1 . That is: w 1 w 0.6 w 0.6 0.6w 0.4 w 0.6 0 Equation 4 0.6 1.5 0.4 1 w 1 1.5 2.5 w In summary, we assign a weight of -150% on security 1 and 250% in security 2. The expected return on the portfolio: = The 2 P 0.2 w 0.18 1 w 0.2 1.5 0.18 2.5 15% Equation 5 coefficient is: 2 P 2 w 2 1 w 2 1.5 2 1 1.5 2 b) The return from security 3 can be expressed as follows: Equation 6 Equation 7 R3 0.10 f1 0.8 f 2 The returns from security 4 can be expressed as follows: Equation 8 R4 0.10 1.5 f1 0.7 f 2 Let w denote the weight for security 3 in portfolio q. Since the weights sum to 1, it follows that security 4's weight is (1 - w). The return on portfolio q is thus: Equation 9 Rq wR3 1 w R4 w 0.10 f1 0.8 f 2 1 w 0.10 1.5 f1 0.7 f 2 0.10 w 0.10 1 w w 1 w 1.5 f1 0.8w 0.7 1 w f 2 Let the return on portfolio q not depend on f1 . That is: w 1 w 1.5 w 1.5 1.5w 0.5w 1.5 0 Equation 10 1.5 3 0.5 1 w 1 3 2 w In summary, we assign a weight of 300% on security 3 and -200% in security 4. The expected return on the portfolio: = The 2 P 0.1w 0.1 1 w 0.1 3 0.10 2 10% Equation 11 coefficient is: 2 P 0.8w 0.7 1 w 0.8 3 0.7 2 1 Equation 12 This sheet answers question 3. Assets Stocks Bonds Cash Actual Return Actual Weight 2% 1% 0.50% 75% 20% 5% Benchmark Weight 65% 25% 10% Index Return 2.50% 1.20% 0.50% Return difference -0.500% -0.200% 0.000% Benchmark portfolio return 1.975% ...sum of the product of the benchmark weight x Index return. Actual portfolio return 1.725% ...sum of the product of the actual weight x actual return. John's relative return performance -0.250% ...actual portfolio return - benchmark portfolio return. Contribution from stock selection -0.415% .... Sum of the product of return difference x actual weight. Return difference = Actual re Contribution from asset allocation 0.165% .... Relative return performance - contribution from stock selection. ual return - benchmark return. This sheet answers question 4. Fund Average return Standard Deviation Beta A B C D Market Risk-free 14% 12% 10% 20% 16% 3% 6% 4% 6% 10% 8% 0% 1.5 0.5 0.5 2 1 0 Sharpe ratio 1.83 2.25 1.17 1.70 Sharpe ratio Treynor ratio ranking 2 0.07 1 0.18 4 0.14 3 0.09 Return on fund B required to equate the Sharpe ratio of fund A to that of fund B = (Sharpe ratio for fund A) x (Standard deviation on fund B) + (Risk free rate) = 10.33% Therefore change required = -1.67% Sharpe ratio = Treynor ratio = Portfolio return - Risk free rate Portfolio standard deviation of return Portfolio return - Risk free rate Portfolio beta Jensen's = Portfolio return - Risk free rate + Portfolio beta Return on market - risk free rate Treynor ratio ranking 4 1 2 3 Jensen's Jensen's alpha alpha ranking -0.09 3 0.03 1 0.01 2 -0.09 4 Question 1 a The return from security 1 can be expressed as follows: Equation 1 R1 0.20 f1 2 f 2 The return from security 2 can be expressed as follows: Equation 2 R2 0.18 0.6 f1 2 f 2 Let w denote the weight for security 1 in portfolio p. Since the weights sum to 1, it follows that security 2's weight is (1 - w). The return on portfolio p is thus: Equation 3 RP wR1 1 w R2 w 0.20 f1 2 f 2 1 w 0.18 0.6 f1 2 f 2 0.20 w 0.18 1 w w 1 w 0.6 f1 2 w 2 1 w f 2 Let the return on portfolio p not depend on f1 . That is: w 1 w 0.6 w 0.6 0.6w 0.4 w 0.6 0 Equation 4 0.6 1.5 0.4 1 w 1 1.5 2.5 w In summary, we assign a weight of -150% on security 1 and 250% in security 2. The expected return on the portfolio: = The 2 P 0.2 w 0.18 1 w 0.2 1.5 0.18 2.5 15% Equation 5 coefficient is: 2 P 2 w 2 1 w 2 1.5 2 1 1.5 2 b) The return from security 3 can be expressed as follows: Equation 6 Equation 7 R3 0.10 f1 0.8 f 2 The returns from security 4 can be expressed as follows: Equation 8 R4 0.10 1.5 f1 0.7 f 2 Let w denote the weight for security 3 in portfolio q. Since the weights sum to 1, it follows that security 4's weight is (1 - w). The return on portfolio q is thus: Equation 9 Rq wR3 1 w R4 w 0.10 f1 0.8 f 2 1 w 0.10 1.5 f1 0.7 f 2 0.10 w 0.10 1 w w 1 w 1.5 f1 0.8w 0.7 1 w f 2 Let the return on portfolio q not depend on f1 . That is: w 1 w 1.5 w 1.5 1.5w 0.5w 1.5 0 Equation 10 1.5 3 0.5 1 w 1 3 2 w In summary, we assign a weight of 300% on security 3 and -200% in security 4. The expected return on the portfolio: = The 2 P 0.1w 0.1 1 w 0.1 3 0.10 2 10% Equation 11 coefficient is: 2 P 0.8w 0.7 1 w 0.8 3 0.7 2 1 Equation 12 Question 2 a) Given a beta of 0.8, the expected movement in the share price attributed to the market is: = beta x Stock market movement = 0.8 x 1.2% = 0.96%. The actual percentage change in the two stocks attributable to news is shown in the table below. Stock Price before Change Price after Percentage change % change due to market % attributable to news AT&T 65.375 -5.375 60 -8.22% 0.96% -9.18% TCI 38.6875 1.0625 39.75 2.75% 0.96% 1.79% Note: (% attributable to news) = Stock percentage change - (% change due to market). The news represented good news to TCI as it contributed positively to total percentage change. In contrast, the news contributed negatively to AT&T as (% attributable to news) is negative. Question 1 a The return from security 1 can be expressed as follows: Equation 1 R1 0.20 f1 2 f 2 The return from security 2 can be expressed as follows: Equation 2 R2 0.18 0.6 f1 2 f 2 Let w denote the weight for security 1 in portfolio p. Since the weights sum to 1, it follows that security 2's weight is (1 - w). The return on portfolio p is thus: Equation 3 RP wR1 1 w R2 w 0.20 f1 2 f 2 1 w 0.18 0.6 f1 2 f 2 0.20 w 0.18 1 w w 1 w 0.6 f1 2 w 2 1 w f 2 Let the return on portfolio p not depend on f1 . That is: w 1 w 0.6 w 0.6 0.6w 0.4 w 0.6 0 Equation 4 0.6 1.5 0.4 1 w 1 1.5 2.5 w In summary, we assign a weight of -150% on security 1 and 250% in security 2. The expected return on the portfolio: = The 2 P 0.2 w 0.18 1 w 0.2 1.5 0.18 2.5 15% Equation 5 coefficient is: 2 P 2 w 2 1 w 2 1.5 2 1 1.5 2 b) The return from security 3 can be expressed as follows: Equation 6 Equation 7 R3 0.10 f1 0.8 f 2 The returns from security 4 can be expressed as follows: Equation 8 R4 0.10 1.5 f1 0.7 f 2 Let w denote the weight for security 3 in portfolio q. Since the weights sum to 1, it follows that security 4's weight is (1 - w). The return on portfolio q is thus: Equation 9 Rq wR3 1 w R4 w 0.10 f1 0.8 f 2 1 w 0.10 1.5 f1 0.7 f 2 0.10 w 0.10 1 w w 1 w 1.5 f1 0.8w 0.7 1 w f 2 Let the return on portfolio q not depend on f1 . That is: w 1 w 1.5 w 1.5 1.5w 0.5w 1.5 0 Equation 10 1.5 3 0.5 1 w 1 3 2 w In summary, we assign a weight of 300% on security 3 and -200% in security 4. The expected return on the portfolio: = The 2 P 0.1w 0.1 1 w 0.1 3 0.10 2 10% Equation 11 coefficient is: 2 P 0.8w 0.7 1 w 0.8 3 0.7 2 1 Equation 12 Question 2 a) Given a beta of 0.8, the expected movement in the share price attributed to the market is: = beta x Stock market movement = 0.8 x 1.2% = 0.96%. The actual percentage change in the two stocks attributable to news is shown in the table below. Stock Price before Change Price after Percentage change % change due to market % attributable to news AT&T 65.375 -5.375 60 -8.22% 0.96% -9.18% TCI 38.6875 1.0625 39.75 2.75% 0.96% 1.79% Note: (% attributable to news) = Stock percentage change - (% change due to market). The news represented good news to TCI as it contributed positively to total percentage change. In contrast, the news contributed negatively to AT&T as (% attributable to news) is negative. b) The TCI share, if correctly AT&T is correctly priced, is: 0.7757 x $60 = $46.542 This corresponds to a return of 20.3% (ie, 46.542/38.6875-1). Question 1 a The return from security 1 can be expressed as follows: Equation 1 R1 0.20 f1 2 f 2 The return from security 2 can be expressed as follows: Equation 2 R2 0.18 0.6 f1 2 f 2 Let w denote the weight for security 1 in portfolio p. Since the weights sum to 1, it follows that security 2's weight is (1 - w). The return on portfolio p is thus: Equation 3 RP wR1 1 w R2 w 0.20 f1 2 f 2 1 w 0.18 0.6 f1 2 f 2 0.20 w 0.18 1 w w 1 w 0.6 f1 2 w 2 1 w f 2 Let the return on portfolio p not depend on f1 . That is: w 1 w 0.6 w 0.6 0.6w 0.4 w 0.6 0 Equation 4 0.6 1.5 0.4 1 w 1 1.5 2.5 w In summary, we assign a weight of -150% on security 1 and 250% in security 2. The expected return on the portfolio: = The 2 P 0.2 w 0.18 1 w 0.2 1.5 0.18 2.5 15% Equation 5 coefficient is: 2 P 2 w 2 1 w 2 1.5 2 1 1.5 2 b) The return from security 3 can be expressed as follows: Equation 6 Equation 7 R3 0.10 f1 0.8 f 2 The returns from security 4 can be expressed as follows: Equation 8 R4 0.10 1.5 f1 0.7 f 2 Let w denote the weight for security 3 in portfolio q. Since the weights sum to 1, it follows that security 4's weight is (1 - w). The return on portfolio q is thus: Equation 9 Rq wR3 1 w R4 w 0.10 f1 0.8 f 2 1 w 0.10 1.5 f1 0.7 f 2 0.10 w 0.10 1 w w 1 w 1.5 f1 0.8w 0.7 1 w f 2 Let the return on portfolio q not depend on f1 . That is: w 1 w 1.5 w 1.5 1.5w 0.5w 1.5 0 Equation 10 1.5 3 0.5 1 w 1 3 2 w In summary, we assign a weight of 300% on security 3 and -200% in security 4. The expected return on the portfolio: = The 2 P 0.1w 0.1 1 w 0.1 3 0.10 2 10% Equation 11 coefficient is: 2 P 0.8w 0.7 1 w 0.8 3 0.7 2 1 Equation 12 Question 2 a) Given a beta of 0.8, the expected movement in the share price attributed to the market is: = beta x Stock market movement = 0.8 x 1.2% = 0.96%. The actual percentage change in the two stocks attributable to news is shown in the table below. Stock Price before Change Price after Percentage change % change due to market % attributable to news AT&T 65.375 -5.375 60 -8.22% 0.96% -9.18% TCI 38.6875 1.0625 39.75 2.75% 0.96% 1.79% Note: (% attributable to news) = Stock percentage change - (% change due to market). The news represented good news to TCI as it contributed positively to total percentage change. In contrast, the news contributed negatively to AT&T as (% attributable to news) is negative. b) The TCI share, if correctly AT&T is correctly priced, is: 0.7757 x $60 = $46.542 This corresponds to a return of 20.3% (ie, 46.542/38.6875-1). c) I would go short on the offerer (AT&T) and long on the target (TCI). Stock Price before Change Price after Percentage change % change due to market % attributable to news AT&T 65.375 -5.375 60 -8.22% 0.96% -9.18% TCI 38.6875 1.0625 39.75 2.75% 0.96% 1.79% Market movement Beta Change due to market movement 1.20% 0.8 0.96% This sheet answers question 3. Assets Stocks Bonds Cash Actual Return Actual Weight 2% 1% 0.50% 75% 20% 5% Benchmark Weight 65% 25% 10% Index Return 2.50% 1.20% 0.50% Return difference -0.500% -0.200% 0.000% Benchmark portfolio return 1.975% ...sum of the product of the benchmark weight x Index return. Actual portfolio return 1.725% ...sum of the product of the actual weight x actual return. John's relative return performance -0.250% ...actual portfolio return - benchmark portfolio return. Contribution from stock selection -0.415% .... Sum of the product of return difference x actual weight. Return difference = Actual re Contribution from asset allocation 0.165% .... Relative return performance - contribution from stock selection. ual return - benchmark return. This sheet answers question 4. Fund Average return Standard Deviation Beta A B C D Market Risk-free 14% 12% 10% 20% 16% 3% 6% 4% 6% 10% 8% 0% 1.5 0.5 0.5 2 1 0 Sharpe ratio 1.83 2.25 1.17 1.70 Sharpe ratio Treynor ratio ranking 2 0.07 1 0.18 4 0.14 3 0.09 Return on fund B required to equate the Sharpe ratio of fund A to that of fund B = (Sharpe ratio for fund A) x (Standard deviation on fund B) + (Risk free rate) = 10.33% Therefore change required = -1.67% Sharpe ratio = Treynor ratio = Portfolio return - Risk free rate Portfolio standard deviation of return Portfolio return - Risk free rate Portfolio beta Jensen's = Portfolio return - Risk free rate + Portfolio beta Return on market - risk free rate Treynor ratio ranking 4 1 2 3 Jensen's Jensen's alpha alpha ranking -0.09 3 0.03 1 0.01 2 -0.09 4