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Financial Economics Part I Contents Session 1: Introduction 3 Financial Crisis of 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Asset Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Session 2: Equity Valuation 7 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Intrinsic Value vs Market Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Dividend Growth Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Price Earnings Ratio and Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 P/E Ratios and Stock Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Issues with P/E Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 The Aggregate Stock Market . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Free Cash Flow To the Firm Approach . . . . . . . . . . . . . . . . . . . . . . . . . 12 Session 3: Risk and Return 13 Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Session 4: Portfolio Theory 15 Expected utility framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Quadratic Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Indifference Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Capital Allocation Line (CAL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Portfolio Choice on the Capital Allocation Line . . . . . . . . . . . . . . . . . . . . 19 Session 4.1 Optimal Risky Portfolios 22 The case of a Two-Asset Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Efficient Frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Best Capital Allocation Line . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Markowitz Portfolio Optimization Model . . . . . . . . . . . . . . . . . . . . . . . . 26 Diversification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Session 5: Single Index Model 28 Assumptions of the Single Index Models. . . . . . . . . . . . . . . . . . . . . . . . . 28 Index Model and Diversification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Estimation of the Single Index Models . . . . . . . . . . . . . . . . . . . . . . . . . 30 Single Index Model vs Full Markowitz model . . . . . . . . . . . . . . . . . . 31 Correction Methods for Historical Beta . . . . . . . . . . . . . . . . . . . . . . 31 Session 5.1 Capital Asset Pricing Model (CAPM) 32 1 Assumptions of the CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Market Risk Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 The Security Market Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Comparing SML and CML . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 SML as a Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Single Index Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Summary of the CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Extensions of the CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Zero-Beta CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 CAPM with Labor Income . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Merton Multiperiod Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Testing the CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Early Tests of the CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Modern tests of the CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Session 6: Multifactor Models and Abritrage Pricing Theory 42 Introduction and Differences with CAPM . . . . . . . . . . . . . . . . . . . . . . . 42 Single Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Diversification and APT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Exploiting Arbritrage Opportunities . . . . . . . . . . . . . . . . . . . . . . . 45 SML and APT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Multifactor APT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Replicating Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Factor Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 SML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Summary of APT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 APT and the Cost of Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Construction of Multifactor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Session 7: Efficient Market Hypothesis 53 Event Studies and Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Interpreting Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Session 8: Empirical Tests of Asset Pricing Models 56 Adding non-traded assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Adding consumption hedging and investment opportunities . . . . . . . . . . . . . 57 Fama-French-type Factor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Interpretation of the Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Behavioral Explanations for Value Premium . . . . . . . . . . . . . . . . . . . . . . 58 Liquidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Equity Premium Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2 Money markets are short-term, liquid, low-risk, and often have large denominations. The main sources are federal funds and repurchase agreements, small-denomination time deposits and savings deposits, large-denomination time deposits, treasury bills, and commercial pa- per. Recently accessible by households through money market mutual funds. During the crisis there was no widespread bank run, but there were runs in money market mutual funds and the federal government had to extend insurance to them as well. Money market securities are not free of default risk. Capital market instruments are: • Treasury notes. • Treasury bonds. • Inflation-protected securities. • Federal agency debt. • International bonds. • Municipal bonds. • Corporate bonds. The capital market is long-term, liquid, and low-risk (not as low as money market). Equity can be common stock, preferred stock, or ADRs (American Depository Receipts, certificates traded in US markets that represent ownership in shares of a foreign company). Common stock involves ownership in a company, it is a residual claim and implies limited liability. Basic features of stocks: • Dividend Yield. • Capital Gains. • P/E Ratio. Stock Market Indices: may be price-weighted or market-value weighted. Investors can base their portfolio on an index, by buying an index mutual fund or exchange-traded funds (ETFs). Derivatives are securities that get their value from the value of another asset, such as commodity prices, bond and stock prices, or market index values. Investment companies can be divided in three: • Unit investment trusts. • Managed investment companies, open-end or closed-end, comprising the biggest chunk. • Other investment organizations, most importantly real estate investment funds. The value of a fund is the net asset value, the difference between the market value of assets minus liabilities all divided by shares outstanding. Liabilities in the case of funds tend to reflect operating expenses. Open-end funds are mutual funds. Their shares are not traded in the market and they are priced at net asset value. We can distinguish them depending on their strategy; they may invest in the money market or equity. More importantly index funds replicate market indexes and are very important for passive investments. Tracking error is how we evaluate them, by looking at the deviations from market index. There is debate about whether they are actually passive in their voting behavior. Closed-end funds are closed in the sense that the number of shares outstanding is constant. There is a market for shares and they are piced at premium or discount to NAV, it is not exactly clear why. 5 Hedge funds take an active approach, often they are levered, take short positions, they are generally quite risky and accessible to high-net worth individuals only , and their managers take high fees. Costs of investing in mutual funds are important, as fees play an important role in evaluating performance. There are four types of fees generally: • Operating expenses. • Front-end load. • Back-end load. • 12 b-1 charge (specific to US). Fees may not seem so high but could make a big difference over longer time periods. Exchange Traded Funds allow you to trade the index directly. In a way similar to open-end funds, as they trade at price similar to net value but they trade continuously like stocks like closed end funds. 6 Session 2: Equity Valuation Introduction We would like to come up with some reasonable estimate of the value of a stock. There is a distinction between fundamental analysis and technical analysis. The first identifies mis-priced stocks relative to some measure of ”true” value derived from financial data. The second is based on predicting future stock prices based on past data. Models of equity valuation based on relative valuations: • Balance sheet models. • Dividend discount models (DDM). • Price/earnings ratios. • Free cash flow models. You can compare valuation ratios of a firm to industry averages. We would like to see if the firm is under- or over-priced by looking at the pricing of similar companies. A limitation of looking at book values is that book values are based on historical cost, not actual market values. They are also not good as floors for stock prices as some companies’ book value is higher than market value. Instead, liquidation value may be used. It is found by breaking up the firm into assets and selling them separately. As a cap on the price we may instead use the replacement cost, meaning how much it would cost to purchase all assets of the firm in the market, remaking the firm from scratch. Tobin’s q: ratio of market price to replacement cost. One idea is that it should trend towards 1 as otherwise if replacement cost is lower than market price many more players would enter the market lowering prices. In the US there were large swings in Tobin’s q. Between 1970s and 2000s it was consistently lower than 1. This indicates that the firm as a unique firm was less valuable than the sum of the individual value of its assets. Example of relative valuation: • At first the P/E ratio suggests that Microsoft is undervalued relative to other firms in the industry. At the same time it is a mature company while the industry is relatively young. 7 Dividend Growth Rate To come up with an estimate of the growth rate of dividends we can multiply the return on equity ROE by the retention rate b = 1 − dividend payout rate, which is a measure of the fraction of earnings reinvested in the company. So we have: g = ROE × b Higher dividends now mean lower reinvestment. Depending on the type of the firm a higher retention rate (lower dividends) due to higher reinvestment could translate into higher future dividends. This depends on the kind of investment opportunities the company has. The value of the firm can be represented as the value of assets already in place (no-growth value of the firm) plus the net present value of its future investments (present value of growth opportunities, PVGO). P0 = E1 k + PV GO In the formula we assume that all dividends are distributed, so D1 = E1, so the firm cannot grow and will keep the same earnings. A dividend drop may not necessarily translate into a lower price, as it could signal higher investments and therefore there will be higher dividends in the future. If PV GO is negative it means that the investment opportunities of the firm give a lower expected return than alternative investments available to shareholders that they could enter into if more dividends were paid out instead of reinvesting. Note that these models are more appropriate for mature firms, for others it’s better to use multistage models as younger firms have consistently high growth rates and are less likely to destroy value with reinvestment. In a multistage model we would instead make estimates for cash flows, one for each year, until the year we think the company will enter steady state growth, after which we use the constant growth model. Price Earnings Ratio and Growth P0 = E1k + PV GO, dividing by E1 we get: P0 E1 = 1 K + PV GO E1 = 1 K ( 1 + PV GO E1/K ) Where PV GO is growth opportunities and E1/K is assets in place. It gives an idea of growth opportunities as the higher the ratio of growth opportunities to assets in place, the higher the price earnings ratio will be. If PV GO = 0 then P0 = E1/K. We can represent it in the following way: • D1 = E1(1− b). • g = ROE × b. • P0 = D1k−g substitute D1 and g and divide by E1 to get: P0 E1 = (1− b) k −ROE × b It provides us with rigorous guidance on when reinvesting is value increasing and when it is not: the company should reinvest only if ROE > k. 10 P/E increases: • As ROE increases. • As plowback b increases, if ROE > k. • As plowback b decreases, if ROE < k. • As k decreases. Example with k = 12%: P/E Ratios and Stock Risk When risk is higher, meaning β is higher, then the required rate of return k is also higher, and the P/E is lower: P/E = 1− b k − g The higher risk leads to a higher discount rate, which lowers the present value of the firm. To take an example, startups tend to have a higher k due to their higher risk, but at the same time they can still have high P/E ratios due to their high expected growth rates g. Issues with P/E Analysis The main issue with P/E ratios analysis is that it is based on accounting measures, and therefore there might be issues with earnings management, as well as the fact that we are looking at historical cost. An assumption of the P/E analysis is that we are instead looking at economic cost. For example, in a high inflation environment economic depreciation will deviate from accounting depreciation, inflating earnings. For this reason during periods of high inflation earnings are of ”low quality.” The Aggregate Stock Market If we look at the aggregate stock market rather than individual firms, we need to forecast corporate profits and P/E both at the aggregate level. We multiply the two together, getting the forecast market price. The earnings yield is the inverse of P/E ratio, so Ey = E/P . The earnings yield on treasury bonds tends to move with earnings yields of stocks. For example, we might have a yield to maturity of 2.5%, a spread between earnings yield and yield to maturity of 2.6% (which means estimate of earnings yield is 5.1%). The forecast 11 of earnings per share (of S&P) is 118$. Then we can get an estimate of the S&P500 price by multiplying one over earnings yield by EPS, so P/E×EPS = 1Ey×EPS = 19.61×118 = 2093 Free Cash Flow To the Firm Approach The advantage over the dividend discount model is that this is not based on dividends, it is therefore more suitable for companies that do not distribute dividends or do not pay them regularly as they re-invest their earnings, or redistribute money through stock repurchases. There are two routes that could be followed with this approach, based on the balance sheet structure: 1. We look at free cash flows generated from assets of the firm that are paid to all claim- holders of the firm, both equity- and debt-holders. The present value of free cash flows from assets is the value of assets. 2. We can get free cash flows for equity directly, removing payments to debt. For the first we deal with Free Cash Flows to the Firm (FCFF). The starting point is operating income, you add back depreciation, remove capital expenditures, and remove the increase in net working capital: FCFF = EBIT × (1− t) +Depreciation− CapEx−∆NWC Then we discount it at the Weighted Average Cost of Capital: WACC = RE EE+D + RD D E+D (1− tc) to get the value of assets: Firm Value = T∑ t=1 FCFFt (1 +WACC)t + Vt (1 +WACC)T where Vt = FCFFT +1WACC−g is the terminal value. The idea is the same as the dividend discount model, where for an initial phase we have cash flow estimates discounted at the proper rate, and then we have a terminal value estimate discounted at the proper rate. If we look at free cash flow to equity instead we look at free cash flow, remove after tax interest and add increase in debt: FCFE = FCFF − Interest× (1− t) + ∆Debt We get the value of equity: Equity Value = T∑ t=1 FCFEt (1 + ke)T + Et(1 + ke)T Where Et = FCFET +1kE−g and ke is the cost of equity. If we make the right assumptions and are careful, these different valuation models should give us approximately the same results. 12 Session 4: Portfolio Theory We look at a two step process of portfolio construction: 1. Choice of the composition of the risky portfolio. 2. Composition of the complete portfolio, choosing between risky portfolio and risk-free asset. Investors are willing to consider risk-free assets, and speculative positions with positive risk premiums. Portfolio attractiveness increases with expected return and decreases with risk. Knight (1921) defines risk as the possibility to assign a objective probability to an event (known unknown), whereas he defines uncertainty as the inability to assign such a probability to an event (unknown unknown). Decisions under uncertainty are characterized by: 1. Uncertainty: modeled as the future realization of one among the possible states of the world S = {s1, ..., sl}. 2. Actions: in finance, choice of portfolio weights: x1, ..., xn. 3. Consequences: in finance, this is the final portfolio outcome, the final value of the portfolio in the realized state of the world k. Wk = pk,1x1 + pk,2x2 + ... + pk,nxn. p here stands for price. The investor’s problem is that of choosing the portfolio weights that maximize her utility: MaxxU(W1, ...Wl) = U(W ) Risk preferences and utility functions: The risk lover would pay more than 10 dollars for the lottery (on right side of eq. we have lottery with expected value of 10, on left we have 10 for certain), the risk averse less, and the risk neutral exactly 10. Measures of risk aversion: they are measures of the concavity of the utility function, so they depend on its second derivative. In finance we assume that investors are risk averse, they are willing to take risk only if remunerated for it. 15 Expected utility framework Assumptions: 1. We can assign a probability to each state of the world πk (subjective probability). 2. A Von Neumann-Morgenstern utility U() depending only on consequences exists: U(W1, ...Wl) = π1U(W1) + ...+ πnU(Wn) = E(U(W )) . Investors maximize the expected value of the Von Neumann Morgenstern utility: Max E[U(W )] = π1U(W1) + ...+ πnU(Wn). It’s a normative theory (how investors should behave), but often used as a positive theory (how investors actually behave). Quadratic Utility Quadratic Utility has the following form: U = µp − 1 2Aσ 2 p Where A is a risk aversion coefficient: • If A > 0 we have risk aversion. • If A = 0 we have risk neutrality. • If A < 0 we have risk seeking. Unambiguously the first quadrant is the best. Given some fixed level of risk, you’d choose higher expected returns, given some fixed level of expected returns you’d choose lower risk. In the extreme cases: • When A = 0 in the case of risk neutrality, we would have horizontal lines for indiffer- ence curves so that the investors only cares about going north without caring about risk. • When A → ∞ in the case of highest risk aversion, we would have vertical lines for indifference curves so that the investor only cares about going to the left decreasing risk. 16 The advantage of quadratic utility is that derivatives of order higher than two are zero. The disadvantage is that it has a maximum, beyond which investors prefer less to more. One advantage of this approach is that first we can determine the efficient portfolios, and then the investor’s preferences come into play to determine in which efficient portfolio to invest. Example: We would like to use the utility function to assign a score to each of these portfolios. The utility score can be interpreted as a certainty equivalent rate of return. We notice that as risk aversion increases, utility scores go down. Mean Variance Criterion: portfolio X dominates portfolio Y if E(rx) ≥ E(ry) and σx ≤ σy and at least one of the two inequalities is strict. Indifference Curves Portfolios with the same utility lie on the same indifference curve in the return-volatility plane. Indifference curves are combinations of E(r) and σ which keep utility scores un- changed: To get indifference curves, we start from the equation of the utility function U(r) = E(r)− 1 2Aσ 2(r) and set the variation of the utility to 0 so that dU = 0 and that utility will be at some fixed level c, so we have c = E(r)− 12Aσ2(r). We rearrange to get the equation of the indifference curve: 17 we maximize utility with respect to the weight y of the risky portfolio, subject to constraints on expected return and on variance (which must be the combinations of expected return and variance among the set of feasible portfolios). We get: max y U = µc − 1 2Aσ 2 c Subject to the following constraints: • E(rc) = rf + y × [E(rp)− rf ]. • σ2c = y2 × σ2p. By substituting the constraints in the formula we get: max y rf + y[E(rp)− rf ]− 1 2Ay 2σ2p To find the maximum we take the first order derivative with respect to y and set it equal to 0: δ δy = E(rp)− rf −Ayσ2p = 0 If we solve for y we get: y∗ = E(rp)− rf Aσ2p Notice the presence of both the Sharpe ratio and the risk aversion coefficient. With this problem we are finding the highest indifference curve tangent to the capital allocation line. As the Sharpe ratio goes up, the investor will invest more in the risky portfolio (move to the right on the CAL). As the risk aversion goes up (since we have 1/A in the formula) the investor will invest less in the risky portfolio and more in the risk-free asset (move to the left on the CAL). The higher the risk aversion A, the steeper the indifference curve, and the more to the left the portfolio of choice will be on the CAL. The optimal point is the one where the indifference curve is tangent to the CAL: 20 The Capital Market Line (CML) is a capital allocation line formed from investment in two passive portfolios: 1. Risk-free asset (usually virtually risk-free short-term T-bills, or money market fund). 2. Market portfolio (usually fund of stocks mimicking market index). 21 Session 4.1 Optimal Risky Portfolios How do we construct efficient risky portfolios? The key is diversification: we exploit the fact that securities are not perfectly correlated and their movements (specific to the individual stocks) cancel each other out in a large portfolio, so that we are left only with market risk exposure (or non-diversifiable risk). As we increase the number of stocks in the portfolio risk goes down, until we’re left with market risk only (as in Panel B). The case of a Two-Asset Portfolio We have a portfolio P with two assets: bonds B and equity E. The return of the portfolio is given by: rp = wdrd + were Where wd and we are, respectively, the weights of bonds and equity, and where rd and re are, respectively, the returns of bonds and equity. The expected return of the portfolio is given by: E(rp) = wdE(rd) + weE(re) While the variance of the portfolio is given by: σ2p = w2dσ2d + w2eσ2e + 2wdweCov(rd, re) Alternatively we can write the variance in terms of the correlation coefficient ρ = Cov(rd,re)σdσe : σ2p = w2dσ2d + w2eσ2e + 2wdweρσdσe The value of ρ is important for diversification. When ρ = 1 there is no diversification, as: σ2p = w2dσ2d + w2eσ2e + 2wdweσdσe = (wdσd + weσe)2 22 If we substitute the constraint in the formula we get: min wd w2dσ 2 d + (1− wd)2σ2e + 2wd(1− wd)ρσdσe We can take the first derivative δδwd and set it equal to 0 to get the weight for D in the global minimum variance portfolio: wd(GMV ) = σ2e − σd,e σ2d + σ2e − 2σd,e Best Capital Allocation Line It’s going to be the one that is tangent to the efficient frontier. To get it we need to maximise its slope, which is the Sharpe ratio Sp = E(rp)−rfσp . Finally once we have the best capital allocation line we can take into account the investor’s preferences (her risk aversion) to determine the composition of the complete portfolio. The optimal complete portfolio will be the mix of optimal risky portfolio and risk-free asset with the highest utility, so the one on the indifference curve tangent to the capital allocation line. 25 The higher of the risk aversion of the investor the more he will be invested in the risk-free asset (moving the optimal complete portfolio to the left on the CAL). Markowitz Portfolio Optimization Model We are interested in n assets now. The idea is the same, portfolio choice can be separated in two parts: 1. Determination of the optimal risky portfolio. 2. Allocation between the risk-free asset and the optimal risky portfolio depending on the risk aversion of the individual investor. With many assets we have the efficient frontier: Diversification By definition the variance of the portfolio with n assets is equal to : σ2p = n∑ i=1 n∑ j=1 wiwjCov(ri, rj) Given equal weights so wi = 1n , we can write it as: σ2p = n∑ i=1 1 n2 V ar(ri)+ ∑ i 6=j ∑ j 1 n 1 n Cov(ri, rj) = 1 n ∑ i V ar(ri) n + 1 n 1 n n− 1 n− 1 ∑ i6=j ∑ j Cov(ri, rj) Where ∑n i V ar(ri) n is the average variance which we call σ̄2, which captures the risk of individual stocks. Furthermore, notice that n(n− 1) is the number of off-diagonal elements in the variance-covariance matrix. We have that 1n(n−1) ∑∑ Cov(ri, rj) is the average covariance which we call Covavg, which is the risk common to all stocks in the portfolio. We can then write portfolio variance as: σ2p = 1 n σ2 + n− 1 n Covavg 26 We can see that as the number of assets tends to infinity, so n → ∞, we have that the variance of the portfolio tends to the average covariance so σ2p → Covavg. This level is therefore the limit on the possible benefits of diversification, risk cannot be lowered more than this according to the model. When we have few stocks in the portfolio adding more gives a significant risk reduction. Once there are many stocks adding more gives lower and lower risk reductions. Example: Note: very old returns may not be representative anymore. 27 Estimation of the Single Index Models To estimate the parameters usually OLS regressions are used: βi = σim σ2m = Cov(Ri, Rm) σ2m For time series samples this is equal to:∑n t=1[(Rit − R̄it)(Rmt − R̄mt)]∑n t=1[(Rmt − R̄mt)]2 Furthermore we have: αi = R̄it − βiR̄mt Something to pay attention to is whether there was a structural break in the time series used for the estimation of the β. For example if a company has recently merged it might be misleading to use data from 15 years ago that might not be representative anymore. Example of a regression for the single index model: Regression output: We look at R2, β coefficient, and its p-value. A β of 2 means that there is high exposure to the market pattern. Furthermore, a positive α might suggest an under-valued asset, as it has an expected return higher than the one predicted by the single index model (vice versa 30 for a negative α). In this case there’s a very high p-value for the intercept, and therefore we cannot reject the null hypothesis that the intercept is zero. For the Beta coefficient we instead have a very low p-value, and therefore it is ”statistically significant”. Single Index Model vs Full Markowitz model The inputs needed for the single index model are: • Risk premium on the S&P 500 portfolio. • Estimate of the standard deviation of the S&P 500 portfolio. • n sets of estimates of: – Beta coefficient. – Stock residual variances. – Alpha values. So for a portfolio of 150 assets we would need 1 + 1 + 150 + 150 + 150 = 452 estimates, compared with more than 11000 for the full Markowitz model. In principle the full Markowitz model should be better but: • using full covariance matrix can lead to estimation risk of thousands of terms (and positivity is not guaranteed). • cumulative errors may result in a portfolio that is actually inferior from that derived from the single index model. The single index model is therefore much simpler and more practical. Correction Methods for Historical Beta The problem is that Betas change through time and that they are estimated with error, so we need to mitigate this problem by applying an adjustment method. In more detail some issues with beta estimation are: • Coefficients change with time. • Estimated betas tend to move, on average, around 1. • There is a larger error for smaller portfolios. • The farther away from 1 the coefficient is the higher the probability of high errors. The Blume Adjustment Technique is based on the idea that betas are likely to regress to 1 in the next period. It was found by calculating the beta in two different periods, at time t1, and time t2, and then regressing the beta of the first period against the one of the second period. The formula for the correction is: βi2 = 0.343 + 0.677βi1 If there is a βi1 > 1 the formula will make βi2 smaller and closer to 1. If there is a βi1 < 1 the formula will make βi2 larger and closer to 1. For example, if we have a βi1 = 0.5 we will have a βi2 = 0.343 + 0.677× 0.5 = 0.6815. If instead we have a βi1 = 0.5 we will have a βi2 = 0.343 + 0.677× 2 = 1.697. In both cases the β in the next period moves closer to 1. 31 Session 5.1 Capital Asset Pricing Model (CAPM) It is a general equilibrium model. Portfolio theory is about an individual investor’s optimal selection of portfolio (partial equilibrium), while the CAPM is the equilibrium of all indi- vidual investors (and asset suppliers), so general equilibrium. It is a positive theory, so it’s tsupposed to be a description of how things actually work. Assumptions of the CAPM Assumptions about markets: • Investors are price takers, so there is perfect competition. • The investment horizon is one period, so this is a short-term model. • Investments are limited to publicly traded assets and you can lend and borrow at the risk-free rate. • No taxes and no transaction costs. • Assets are infinitely divisible and there are no restrictions on short-selling. Note that this model is from the investor’s point of view, supply is taken as exogenous and not really part of the model. Assumptions about investors: • information is free and available to all investors. • Investors are rational and are mean-variance optimizers. • Their expectations are homogeneous, they agree on expected returns and variance covariance matrix. They agree on the list of inputs for this analysis. They will agree on the capital allocation line and risky efficient frontier. In summary: 1. Perfect Markets: 1. Frictionless, with perfect information. 2. No imperfections like taxes, regulations, and restrictions to short-selling. 3. All assets are publicly traded and perfectly divisible. 4. Perfect competition, everyone is a price-taker. 2. Investors: 1. Same one-period horizon. 2. Rational, and maximize expected utility over a mean-variance space. 3. Homogeneous beliefs. Several assumptions are unrealistic: • people pay taxes and commissions. • many people look ahead more than one period. • not all investors have the same beliefs about the distribution of returns. 32 We can re-arrange to find that the relationship between risk premium on the stock and risk premium on the market: E[rGE − rf ] = Cov(rGE , rm) σ2m [E(rm)− rf ] Where: Cov(rGE , rm) σ2m = βGE So we have the classic CAPM relation: E[rGE − rf ] = βGE [E(rm)− rf ] The CAPM also holds for portfolios, so: E(rp) = rf + βp[E(rm)− rf ] Which means it also holds for the market portfolio, in which case βm = 1. The Security Market Line The Security Market Line (SML) has expected returns E(r) on the y axis and β on the x axis. It captures equilibrium expected returns of individual securities. The equation is that of the CAPM so E(ri) = rf + βi(E(rm)− rf ) where rf is the intercept and E(rm)− rf is the slope (and it is the market risk premium). 35 In equilibrium individual securities, given their β, should have an excess return that lines them up on the security market line. Therefore, the SML is a benchmark to determine whether security prices reflect equilibrium, and whether a stock is over- or under-valued with respect to the equilibrium expected return. Comparing SML and CML The key distinction is that the security market line concerns single assets within diversified portfolios, where systematic risk in the form of β is the only thing that matters. On the other hand with the capital market line we are concerned with portfolios, and we consider their total risk in the form of total standard deviation, so systematic plus unsystematic (or firm-specific). The key point is that if an individual is not diversified he will bear a risk not remunerated by the market. Another point is that on the SML assets lie on the line, while individual assets alone are inefficient and therefore do not lie on the CML. SML as a Benchmark We can look at the distance between a stock and the SML, which is the stock’s α, and which is performance not explained by the market risk of the stock: • α > 0 means that the stock has higher expected returns than what the stock’s exposure to market risk would justify, meaning it’s undervalued. • α < 0 means the security is overvalued. α can be seen as a measure of risk-adjusted extra performance with respect to the market. 36 Single Index Model The formula for the single index model is: ri − rf = αi + βi(rm − rf ) + ei By taking expectations we get: E(ri − rf ) = αi + βiE(rm − rf ) When we try to estimate this model with a regression the prediction of the CAPM would be that all αi = 0, in that case the optimal risky portfolio is the market portfolio and we are in equilibrium. The single index model is therefore more general than the CAPM as it allows for the existence of remuneration for non-market risk. Summary of the CAPM The CAPM builds on the Markowitz framework in which all investors agree on the distri- bution of individual asset returns, as a consequence they agree on the tangency portfolio which will coincide with the market portfolio. The market risk premium will be proportional to the average risk aversion in the market and the variance of market returns so E(rm)− rf = Āσ2m. The individual asset risk premium will be proportional to the exposure of the individual asset to the market, which reflects its contribution to the market portfolio so E(ri)− rf = βi(E(rm)− rf ). The CAPM is very simple as only one source of risk, market risk, affects expected returns. • Advantages: 1. Simplicity. 2. Good as a benchmark for performance evaluation. 3. It distinguishes between diversifiable and non-diversifiable risk. • Disadvantages: 1. Likely that other sources of risk exist (omitted variable bias in OLS). 2. The market portfolio is unobservable. We use a proxy for the market portfolio but this introduces substantial measurement error (Roll’s critique), which can invalidate all our conclusions.1 Extensions of the CAPM Zero-Beta CAPM In this model we remove the assumption that there is a risk-free asset, and therefore that we can lend and borrow at this rate. It can be shown that in this case for each portfolio on the efficient frontier there is a com- panion portfolio with zero correlation to the original portfolio on the inefficient part of the 1Also any ex-post mean-variance efficient portfolio used as the index will exactly satisfy the SML by construction. 37 The main issue with this method is that there is measurement error in the estimation of the βi which leads to two issues: upward bias in the intercept and downward bias in the slope. A way to address this was to use portfolio βp rather than the βi of individual stocks, which due to aggregation should reduce measurement error. The idea to use portfolios is now common in all tests of asset pricing models. CAPM test of Lintner (1965) Data used: • Monthly returns of 100 firms over 5 years. • Returns on S&P500 over the same period, proxy for market portfolio. • Monthly returns on T-bills, proxy for risk-free rate. First there is the first pass regression: r̂i,t − rf,t = αi + βi(r̂m,t − rf ) + ̂i,t So we run 100 regressions, one for each stock, to estimate the individual βi. Already from the results of these regressions we can make some observations. As the model is cast in terms of excess returns, we should observe an α = 0 according to the CAPM. If we don’t it can be seen as disequilibrium relative to the CAPM. Next we move to the second-pass regression: ri − rf = γ0 + γ1β̂i + γ2σ̂2(i) + ûi Where ri − rf is the average return on security i (proxy for expected returns), β̂i is the esti- mated βi from the first-pass regression, and σ̂2(i) is the variance of the estimated residuals. According to the CAPM we should have that: • γ0 = 0, as the model is in terms of excess returns. • γ1 = rm − rf , so approximately equal to the market risk premium. • γ2 = 0, as if it was different from zero it would mean that there is some firm-specific component that goes into expected returns, while the CAPM predicts that only market risk should be priced. The results were disappointing for the CAPM: • γ0 was estimated to be significant and larger than 0. • γ1 was estimated to be significant and positive, but significantly smaller than the market risk premium. • γ2 was estimated to be significant and quite large. This could be an effective rejection of the CAPM, or there could be econometric problems due to measurement error. CAPM Test of Black-Jensen-Scholes (1972) The main issue is that with 5-10 years of data you get inaccurate estimates of β, but using longer time frames for estimating it is also problematic because β changes over time. A solution is to use portfolios rather than individual stocks. The approach taken by Sharpe and Cooper was to: 1. Estimate βi for each stock. 40 2. Sort stocks in deciles based on the estimate of β̂i from smallest to largest to form ten portfolios. 3. Repeat procedure and re-balance every year so that portfolios have a constant exposure to the market. Once you have the portfolios and their portfolio beta you can do the cross-sectional regression and regress returns on β. What was found was good for the CAPM in terms of R2 and market risk premium, but also a high risk-free rate which is problematic. The test of Black-Jensen-Scholes is similar with one additional thing, after the formation of the portfolio they estimate its beta again. The results are consistent with the CAPM in terms of α as only for two portfolios it was significantly different from zero, and in terms of β as portfolios with higher betas were associated with higher average returns. At the same time the estimated security market line was, again, very flat. Test of Fama-MacBeth (1973) The main difference with this test is that the second- pass regression (the cross-sectional one to estimate the security market line) is performed many times, to get many estimates of the SML and many coefficients. Furthermore, their model specification allows for non-linearity by including (βp)2 and for firm-specific risk by including the variance of the residuals. Four parameters are therefore estimated, for the intercept, slope, non-linearity, and firm- specific risk. We’d expect the first to be 0 (as it is framed in terms of excess returns), the second to be approximately the market risk premium, and the third and fourth to be 0 as according to the CAPM the relation should be linear and firm-specific risk should not be priced. By doing the regression many times they get many estimates for each coefficient, then they average them out and perform statistical tests on these averages. They find positive results for the CAPM, but again with a flatter security market line. The third and fourth coefficients are estimated to be 0, but they found a positive coefficient for the intercept and a small slope, both leading to a flatter SML. This points to the assumptions of CAPM being unrealistic. Modern tests of the CAPM Two main modern approaches have been to: 1. Test jointly the significance of αi, as joint tests are more efficient, so: H0 : α1 = α2 = ... = αn = 0 for all i 1. Add more explanatory variables Zi,t to the CAPM regression: Ri,t − rf = αi + βi(Rm,t − rf ) + δZi,t + i,t Then the test is: H0 : δ = 0 Since according to the CAPM only the market risk premium should explain market returns, the coefficient of additional explanatory variables should be equal to 0 if the model is correct. Generally these tests have had negative findings for the CAPM. 41 Session 6: Multifactor Models and Abritrage Pricing Theory Introduction and Differences with CAPM This model is related to the extension of the CAPM by Merton, which extends it to a multiperiod setting. Doing so creates some hedging demand, due to the goal of the investor which is to optimize their stream of consumption over their lifetime. Because of this the investor becomes sensitive to changes in future investment opportunities and to prices of consumer goods. As a consequence there might be multiple factors beside the market risk one that explain systematic variations in returns and we would like a model flexible enough to deal with this. Arbitrage Pricing Theory (APT) explains the return on a risky asset as a linear com- bination of various macroeconomic factors (not specified by the theory). Like the CAPM it assumes investors are fully diversified and that systematic risk is an influencing factor in the long run. Unlike the CAPM it does not require the use of the market portfolio, instead what is needed is a well-diversified portfolio. Furthermore it is more flexible as it allows for more factors such as inflation, industrial production, and the slope of the term structure of interest rates. The CAPM uses equilibrium conditions: • Investors maximize utility. • There is market clearing (supply of securities equals demand, and aggregate borrowing equals aggregate lending). APT instead only assumes the no-arbitrage principle: that the market is free of arbitrage. An arbitrage opportunity exists when an investor can earn risk-less profits without making a net investment. For example if a stock is sold for different prices on different exchanges. According to the law of one price, if two assets are economically equivalent (same cash flows), they should have the same price. This law is enforced by arbitrageurs, if they observe an arbitrage opportunity they will buy the cheap asset and sell the expensive one until the arbitrage opportunity is eliminated. This is an important difference with the CAPM. In the CAPM fairly priced securities lie on the security market line, stocks above it are under-valued and stocks below it are over- valued. When some stocks do not lie on the SML equilibrium is restored through a risk-return dominance argument: all investors re-balance their portfolio weights until securities reach equilibrium prices. APT is more powerful as only a few investors, arbitrageurs who take large positions in arbitrage opportunities, are needed to restore equilibrium prices. APT does not assume: • Normal distribution of security returns. • Quadratic utility function. • Equilibrium conditions. Furthermore it does not imply the use of the market portfolio. Single Factor Model The single factor model is only based on market risk: Ri = αi + βiRm + i 42 In Panel A we can see well-diversified portfolios who all line up on the line, while in Panel B we see individual securities that do not necessarily line up on the line, this is allowed by the APT model. The idea is that for individual securities there’s small arbitrage oppor- tunities that might not be exploited but they cannot exist for well-diversified portfolios or arbitrageurs would enter into play with large positions to go back to equilibrium Exploiting Arbritrage Opportunities Two portfolios with same Beta Portfolio A and portfolio B have the same β, and therefore according to the rule of one price they should be priced in the same way. Just like when comparing stocks with respect to the security market line, here stocks that have higher returns with same risk are under-priced, while stocks that have lower returns with the same risk are over-priced. Therefore portfolio A is under-priced and portfolio B is over-priced. An arbitrageur could exploit this arbitrage opportunity by taking a long position in A and a short position in B. Arbitrage opportunity are 0 risk and with 0 net investment. So for example with a β = 1 the arbitrageur could short-sell 1 million dollars worth of portfolio B, and buy with the proceeds 1 million worth of portfolio A. The return is going to be equal to (.10 + 1× F )× 1 million− (.08 + 1× F )× 1 million = 20, 000 in net proceeds. Two portfolios with different Beta Imagine now that there are only portfolios A and C (not D) with different betas. How can this arbitrage opportunity be exploited? With synthetic beta. Since the beta of a portfolio is the weighted average of the betas 45 making up the portfolio, we can create a portfolio that is a mix of the risk-free asset and one of the two portfolios to create a new portfolio with the same beta as the other portfolio. For example, if A has a βA = 1 and C has a βC = 0.5, we can create a portfolio P = 0.5rf + 0.5A whose βP = ∑ wiβi = 0.5× 0 + 0.5× 1 = 0.5. Then, we proceed in the same way as in the previous example with two portfolios with same beta. SML and APT Given the result that for a well-diversified portfolio we have the relation: Rp = E(Rp) + βpF We have that there is perfect correlation between the returns on the portfolio Rp and on the macro factor F , as one is a linear combination of the other. Due to this, we can approximate exposure to the macro-factor F with exposure to the well-diversified portfolio, which we take to be the market index. We can therefore get again the security market line: E(rp) = rf + [E(rm)− rf ]βp In this case the market index portfolio does not have to be the true market portfolio as in the case of the CAPM, it just has to be a well-diversified portfolio. The idea in these multi- factor models is that the factor portfolio is a portfolio that approximates the underlying factor by having a β = 1 with respect to that factor and a β = 0 for every other factor, so that it’s a good measure of a security’s exposure to that particular factor. The advantage of using portfolios (over the factor itself) is that portfolios of securities are easier to observe. In this way we have obtained a security market line equivalent to that of the CAPM but without the restrictive assumptions of the CAPM. The APT derivation of the SML needs 3 assumptions: 1. A factor model describing security returns. 2. A sufficient number of securities to form well-diversified portfolios. 3. The absence of arbitrage opportunities. In the APT model the main conclusion of the CAPM, which is the security market line expected return-beta relation, should be approximately valid. APT allows for individual securities to be above or below the SML for a given beta, but the relationship should hold for well-diversified portfolios. The idea is that if only few securities violate the expected-return beta relationship the effect of this violation on well-diversified portfolios will be too small to be taken advantage of by arbitrageurs, so some individual stocks can be mispriced within APT. At the same time if many securities violate the expected return-beta relationship this will have an effect on the well-diversified portfolios creating arbitrage opportunities. Therefore in APT the expected return-beta relationship holds for all securities other than, possibly, a small number of individual securities. Furthermore, the APT does not require that the benchmark SML portfolio is the true market portfolio, and any well-diversified portfolio on the SML can be the benchmark portfolio. Summarizing the main differences between the two models: 46 • CAPM : – Based on the inherently unobservable ”market” portfolio. – Rests on mean-variance efficiency. – Actions of many small investors restore equilibrium. – Equilibrium is for all assets (not just well-diversified portfolios). • APT : – Equilibrium rests on no arbitrage opportunities. – Equilibrium is quickly restored by few investors who recognize arbitrage oppor- tunities and take large positions. – Expected return-beta relationship can be derived without using the ”true” market portfolio. Multifactor APT APT can be extended to multifactor models, for example we might consider as factors GDP and interest rate risk. We have the following model: E(ri) = rf + βiGDPRPGDP + βiIRRPIR Where βiGDP and βiIR are, respectively, the sensitivity to GDP and interest rate risk, while RPGDP and RPIR are, respectively the risk premium for bearing GDP and interest rate risk. Some stocks might have, for example, a βIR > 0 so a positive sensitivity to interest rate risk, so they perform better when interest rates go up, so it is an hedge against interest rate risk. Due to the fact that investors value this hedge there might even be a negative risk premium on interest rate risk so RPIR < 0 as investors are willing to accept a lower expected return for stocks that provide them with some protection on interest rate risk. So we will have that βIR ×RPIR < 0. For a stock with βIR < 0 (as RPIR < 0) we will have that βIR ×RP > 0. So in the first the contribution to the risk premium is negative, as the stock protects us from interest rate risk and we are willing to be remunerated less. For the second the contribution to the risk premium is positive as the stock increases our exposure to interest rate risk and we would like to be remunerated for that. In general the multifactor model can be written as: Ri = E(Ri) + βi1F1 + βi2F2 + ...+ βiKFK + i Here we have K factors, all defined as macro-surprise so zero-mean, and so that the expected return on the asset is E(Ri). The possible factors are left unspecified, they might be interest rate, inflation, etc... The expected return component is captured by E(Ri), which will be explained with APT, and all the risk is captured by the other terms. With this model (and all linear factor models): • Portfolio betas are a weighted average of individual betas. 47 • ∑ βi2xi = 1: weighted sum of the βi2 should be equal to the exposure to the second factor of the portfolio to replicate β2. So we have following system: xA + xB + xC = 1 xA + 3xB + 1.5xC = 2 − 4xA + 2xB + 0xC = 1 Solving it we find for this example that xA = −0.1, xB = 0.3, xC = 0.8. Factor Portfolios A factor portfolio for factor k is a factor that has sensitivity equal to one with that one factor k and zero to all other factors: rk = αk + Fk + εk Where αk = E(rk) and βk = 1. The risk premium is E(rk)− rf = αk − rf = λk. To create a factor portfolio we follow the same procedure. For each factor portfolio we have to solve a system of equations where the weights sum to one, the weighted average for the factor of interest is 1, and the weighted average for all other factors is 0. Example We would like to replicate two factor portfolios with the following three securities: rA = 0.08 + 2F1 + 3F2 rB = 0.10 + 3F1 + 2F2 rC = 0.10 + 3F1 + 5F2 For the first factor portfolio we need to solve the following system: xA + xB + xC = 1 2xA + 3xB + 3xC = 1 3xA + 2xB + 5xC = 0 If we solve it we find the portfolio weights of (2, 1/3,−4/3). For the second factor portfolio we need to solve the following system: xA + xB + xC = 1 2xA + 3xB + 3xC = 0 3xA + 2xB + 5xC = 1 If we solve it we find the portfolio weights of (3,−2/3,−4/3). Given that E(rA) = 0.08, E(rB) = 0.10, E(rC) = 0.10 since E(F1) = 0, E(F2) = 0. We have that the expected returns of the two factor portfolios are: E(Pf1) = 2× 0.08 + 1 3 × 0.10− 4 3 × 0.10 = 0.06 E(Pf2) = 3× 0.08− 2 3 × 0.10− 4 3 × 0.10 = 0.04 50 Given a rf = 0.05, we have the following risk premia for the two factor portfolios: λ1 = E(Pf1)− rf = 0.01 λ2 = E(Pf2)− rf = −0.01 SML Suppose an asset with returns from 2-factor model: ri = αi + βi1F1 + βi2F2 The replicating portfolio for this can be also formed by investing β1 in the first factor portfolio, β2 in the second factor portfolio, and (1 − β1 − β2) in the risk-free asset. By no-arbitrage, the expected return of the portfolio is given by: E(ri) = (1− βi1 − βi2)rf + βi1(λ1 + rf ) + βi2(λ2 + rf ) Where λ1, λ2 are the risk premia on the two factors. It can be re-written as: E(ri) = rf + βi1λ1 + βi2λ2 Which is the security market line. Summary of APT If two investments have the same factor sensitivities β1, β2, ... and they have no residual risk (which happens when we have well-diversified portfolios) then they will be perfectly correlated with the same variance. Two well-diversified portfolios with the same sensitivity factors must have the same expected returns (or there’d be arbitrage opportunities). A portfolio with factor sensitivities β1, β2, ... is the portfolio consisting of the factor portfolios with weights β1, β2... and (1− β1 − β2) in the risk-free asset. Its expected return is: E(ri) = rf + βi1λ1 + βi2λ2 + ...+ βiKλK Given a set of securities we can determine the presence of arbitrage by using a K factor model (using K + 1 securities) to determine risk premia on factor portfolio. Then we can check that the risk premia are consistent with the expected returns on all other securities. If not, there is an arbitrage opportunity. APT and the Cost of Capital We can use APT to determine the cost of capital (k in the DDM). We need estimates of the factor sensitivities and the risk premia. To get the estimates of β we regress the security returns on factors (or on portfolios replicating the factors). To get the estimates of the risk premia we have (rfactor − rf ). For projects, we look at firms with similar economic characteristic to the project. Then the cost of capital is going to be: 51 k = rf + ∑ β̂λ̂ Factors could be, for example: yield spreads (rbond − rtreasury bill), interest rate, exchange rate, real GDP, inflation, market return (to capture omitted systematic factors). Construction of Multifactor Models The issue with constructing models with many factors is in determining which factors to use. APT does not specify any, and leaves us freedom to choose. There’s two general approaches to this, one is purely statistical and focused on finding the factors that explain as much variation as possible, with no economic interpretation. The other is based on pre-specifying factors based on some economic thinking and again there are two approaches. One is to use macroeconomic variables (interest rates etc...), the other is to form portfolios based on past anomalies. Fama and French used firm characteristics that proxy for systematic factors: • SMB: Small Minus Big (firm size). • HML: High Minus Low (book-to-market ratio). rit = αi + βiMRMt + βiSMBSMBt + βiHMLHMLt + eit The SMB portfolio is a 0 net investment portfolio, long on small firms and short on large firms, which tends to generate positive returns. This is not consistent with the CAPM. As both the small-firm part and large-firm part are well-diversified portfolios with many assets they have a βM (with respect to market factor) close to 1 but small-firms outperform large ones, which can’t be explained by the CAPM according to which it should be the same. The same thing is with the HML portfolio, which is long on high book-to-market firms and short on low book-to-market firms. Factor models such as Fama-French find factors that explain systematic patterns but there’s not much grounding in economic theory, so there’s not much agreement on why it’s the case, for example, that high book-to-market ratio firms outperform low book-to-market ratio firms. According to one school of thought there might be some extra-risk associated to these kinds of firms and therefore the higher return is fully rational from investors wanting to be compensated more. Another explanation could be behavioral biases on the part of investors. Leading indicator: variables that anticipate the business cycle. Relevant macroeconomic variables that could be used for factors are P/E ratio, earnings growth, liquidity, etc... 52 • Magnitude issue: it’s difficult statistically to determine whether there are actual dif- ferences in performance and most importantly their size. Even small differences could matter, for example, if a manager is investing billions of dollars. In terms of the weak-form we have two effects that are tested: • Momentum effect: short-run serial correlation in returns, winning stocks keep win- ning, losers keep losing (relatively). • Reversal effect: negative serial correlation in returns from 3-5 years, losing stocks outperform winning stocks. These two might be due to over-reaction, there is over-investment in the winning stocks which pushes their price above intrinsic value in the short-run (vice versa for losing stocks) leading to momentum. As investors adjust in the longer term the prices adjust leading to reversal. In terms of semi-strong tests we have: • Small Firm Anomaly: systematic over-performance of small stocks. This is present after controlling for market β. • Book-to-market Anomaly: systematic over-performance of high book-to-market firms over small book-to-market firms. • Drift Anomaly: post-earnings announcement. The cumulative average returns go up a bit before announcement, there’s a big jump on announcement day, but it continues for quite some time afterwards, which should not happen after the information becomes public according to semi-strong efficiency. Furthermore the bigger the announcement (good or bad) the more this effect is evident. The way this is done is by looking at a cross-section of firms making earnings an- nouncements, and subtracting from them the expected earnings by analysts to get the earnings surprise. Other semi-strong effects are the P/E effect (related to book-to-market though) so low P/E ratio firms outperforming high P/E ratio firms and the neglected firm effect so neglected firms outperforming more visible firms (related to small firm effect though). Also liquidity effect. Tests of strong-form efficiency are focused on insider trading. We should not expect there to be strong-form efficiency also due to the strict regulation against insider trading, which means it’s more difficult to trade on inside information. Interpreting Anomalies We need to keep in mind that these effects or anomalies could also be due to inaccurate risk adjustment but the main ones have been observed in different markets in different times with different and more sophisticated risk-adjustment methods, such as momentum and book- to-market. Still, some anomalies have disappeared. Furthermore exploiting these anomalies eliminates abnormal profits, and studies have shown that over time their effect has reduced. The rational explanation of anomalies for some semi-strong effects such as the small firm and book-to-market anomaly could be due to different exposure to risk that we do not capture in our models, so the excess performance is just remuneration for additional risk. 55 Session 8: Empirical Tests of Asset Pricing Models The idea goes back to the inter-temporal model of Merton which creates hedging demand and leads to some factors that could augment the market risk factor. The hedging could be about non-traded assets (labor income for example), or about consumption or future investment opportunities. Adding non-traded assets In a study Jagannathan and Wang showed two deficiencies in tests of the single-index model: • Many assets are not traded, most importantly human capital, which might be impor- tant in explaining returns. They used changes in labor income per capita to proxy for human capital risk. • Betas are cyclical. They used credit spreads for this (difference between junk bond yields and investment grade bonds yield). First they look at single stocks and estimate market Beta and market capitalization rate. Then they form portfolios (as usual with these tests) that are double-sorted depending on market beta and capitalization rate ME. Then they run the first-pass regression: estimate market beta, beta with respect to labor income, and with respect to credit spread. Then they run the second-pass regression which is cross-sectional (also including size): rp − rf = c0 + cmwβ̂mw + claborβ̂labor + ccreditβ̂credit + csizeln(ME) + ε Here we have the results for different regression specifications: • In the first we see the classic CAPM, there’s a very low R2, a positive and significant α, and an insignificant market risk premium, all against the CAPM. • In the second we see the market factor plus a size component, and the R2 is now sizeable. Furthermore the size component is in line with the small-firm effect which we know: negative relationship between returns and firm size. • In the third there’s the market factor plus the credit factor and the labor income factor, and again the R2 is sizeable, with positive risk premia on both the additional factors. • In the fourth there’s the market factor plus credit and labor income, as well as the size factor. R2 increases. 56 Adding consumption hedging and investment opportunities First were Chen, Roll, and Ross, who try to look at factors that have effects on consumption and investment opportunities, for which investors would look for hedging: • Growth rate in industrial production IP . • Changes in expected inflation EI = ∆T-Bill Rate, changes in t-bill rate. • Unexpected inflation UI = Actual Infl. - Exp. Infl. • Unexpected changes in risk premiums on bonds CG = BAAY TM − T-BondY TM . • Unexpected changes in term premium on bonds TB = T-BondY TM − T-BillY TM . In general the procedure to test multi-factor models is the following: 1. Pre-specify factors. 2. Find hedging portfolios against these factors. 3. Empirical test (like first and second pass procedure). The second step is not necessarily needed as one could just assume that these hedging portfolios exist, and regress returns directly on the factors. They create portfolios sorted by size and market value of equity, then ran the first and second pass and there’s the Fama-Macbeth procedure whereby the risk premia found in the second pass regression are averaged over the sample period. What they found for these factors was that industrial production, risk premium on bonds, and unanticipated inflation were significant. Market index returns were not found to be statistically significant. Fama-French-type Factor Models The Fama-French three factor model incorporates factors for book-to-market and size. We know high book-to-market firms and smaller firms experience higher returns. To induce the highest amount of variations Fama and French create long-short portfolios with zero net investment. Portfolios are double-sorted by size and book-to-market: S stands for small and B for big, while H,M,L stand for high, medium, low. S B H SH BH M SM BM L SL BL Then the small minus big (SMB) portfolio is created by going long on small (S) firms and short on big (B) firms, its returns are: SMB = 13(RSH +RSM +RSL − 1 3(RBH +RBM +RBL) The high minus low (HML) portfolio is created by going long on high book-to-market-ratio (H) firms and short on low book-to-market ratio (L) firms (medium ones are not included). 57