FIN 312 Final Exam Cheatsheet

Cheatsheet Content

### Section A: Weeks 1–2 Foundation #### Core Formulas & Symbol Key * **T-Bill Yields (Canada - BEY):** $BEY = \frac{100 - P}{P} \times \frac{365}{n}$ * $P$: Price of T-bill, $n$: days to maturity * **T-Bill Yields (US - BDY):** $BDY = \frac{100 - P}{100} \times \frac{360}{n}$ * $P$: Price of T-bill, $n$: days to maturity * **Equivalent Annual Yield (EAY):** $EAY = (1 + \frac{BEY \times n}{365})^{365/n} - 1$ or $EAY = (1 + \frac{APR}{m})^m - 1$ * **APR from HPR:** $APR = HPR \times \frac{365}{\text{days}}$ * **EAR from HPR:** $EAR = (1 + HPR)^{\frac{365}{\text{days}}} - 1$ * **Continuous Compounding (EAR):** $EAR = e^{APR} - 1$ * **Continuous Compounding (FV):** $FV = PV \times e^{rT}$ * **Holding Period Return (HPR) with Dividends:** $HPR = \frac{P_1 - P_0 + D_1}{P_0}$ * **Inflation Rate:** $I = \frac{CPI_1 - CPI_0}{CPI_0}$ * **Fisher Equation:** $(1 + R_{nominal}) = (1 + R_{real})(1 + I)$ or $R_{nominal} \approx R_{real} + I$ * **Scenario Mean Return ($E[R]$):** $E[R] = \sum_{i=1}^N p_i R_i$ * **Scenario Variance ($\sigma^2$):** $\sigma^2 = \sum_{i=1}^N p_i (R_i - E[R])^2$ * **Standard Deviation ($\sigma$):** $\sigma = \sqrt{\sigma^2}$ (Units check: $\sigma$ is in % return, $\sigma^2$ is in % return squared) * **Levered Return:** $R_L = \frac{(P_1 - P_0) + (D_1) - \text{Interest on Margin}}{P_0 \times \text{Equity Fraction}}$ * **Margin Call Threshold:** $P_{call} = P_0 \times \frac{1 - \text{Initial Margin}}{1 - \text{Maintenance Margin}}$ #### Recognize the Question * **"Annualize a return"**: Use HPR to APR/EAR conversions. * **"Compare investments with different compounding"**: Convert all to EAR. * **"What is the return given probabilities?"**: Use scenario analysis. * **"Maximum loss for a short sale"**: Unlimited, unless stop-buy. * **"Margin call price"**: Use margin call threshold formula. #### Step-by-Step Templates * **T-bill BEY/BDY Conversions (Canada vs. US):** 1. Identify if it's a Canadian (365 days) or US (360 days) T-bill. 2. Use the appropriate formula for BEY or BDY. 3. Convert to EAY if asked, using the EAY formula. * **Example:** A 90-day Canadian T-bill with a price of $98.50. * $BEY = \frac{100 - 98.50}{98.50} \times \frac{365}{90} = 0.015228 \times 4.0555 = 0.0617$ or 6.17% * $EAY = (1 + \frac{0.0617 \times 90}{365})^{365/90} - 1 = (1 + 0.015228)^{4.0555} - 1 = 1.0628 - 1 = 0.0628$ or 6.28% * **Market-Value Weighted Index Return:** 1. Calculate market cap for each stock at $t=0$ and $t=1$. 2. Sum market caps for total index value at $t=0$ and $t=1$. 3. Return = (Total Value $t=1$ - Total Value $t=0$) / Total Value $t=0$. * **Example:** | Stock | Shares Outstanding | Price t=0 | Price t=1 | |-------|--------------------|-----------|-----------| | A | 100 | $10 | $12 | | B | 200 | $5 | $4 | * Market Cap t=0: A: $100 \times 10 = 1000$, B: $200 \times 5 = 1000$. Total: $2000$. * Market Cap t=1: A: $100 \times 12 = 1200$, B: $200 \times 4 = 800$. Total: $2000$. * Return = $(2000 - 2000) / 2000 = 0\%$. * **Price-Weighted Index Return:** 1. Sum prices of stocks at $t=0$ and $t=1$. 2. Return = (Sum Prices $t=1$ - Sum Prices $t=0$) / Sum Prices $t=0$. * **Example (using above data):** * Sum Prices t=0: $10 + 5 = 15$. * Sum Prices t=1: $12 + 4 = 16$. * Return = $(16 - 15) / 15 = 1/15 = 6.67\%$. * **Equally Weighted Index Return:** 1. Calculate HPR for each individual stock. 2. Average the individual HPRs. * **Example (using above data):** * HPR A = $(12-10)/10 = 20\%$. * HPR B = $(4-5)/5 = -20\%$. * Average HPR = $(20\% + (-20\%)) / 2 = 0\%$. * **Limit Order Book Execution:** 1. Buy orders fill at lowest available Ask price. 2. Sell orders fill at highest available Bid price. 3. Partial fills are common; track remaining order size. * **Example:** Bid: 100 shares @ $10.00, 200 shares @ $9.95. Ask: 150 shares @ $10.05, 50 shares @ $10.10. * A market buy order for 120 shares would fill 120 shares at $10.05. * A market sell order for 250 shares would fill 100 shares at $10.00 and 150 shares at $9.95. * **Margin Ratio Calculation:** 1. Margin Ratio = (Market Value of Assets - Loan Amount) / Market Value of Assets. 2. Compare to Maintenance Margin to check for margin call. * **Example:** Initial stock purchase $10,000 (50% margin). Stock drops to $8,000. * Initial Equity = $5,000, Loan = $5,000. * Current Margin Ratio = ($8,000 - $5,000) / $8,000 = $3,000 / $8,000 = 37.5%. * If maintenance margin is 30%, no margin call yet. * **Short Sale P/L (with Stop-Buy):** 1. Initial gain: $P_{sell} - P_{buy\_back}$. 2. Add dividends received by lender, subtract interest on proceeds. 3. Stop-buy caps loss; if price hits stop, loss is $(P_{stop} - P_{sell})$. * **Example:** Short sell 100 shares at $50. Stock rises to $55, then you cover. No dividends. * Loss = $(50 - 55) \times 100 = -$500. * If a stop-buy was placed at $52, your loss would be $(50 - 52) \times 100 = -$200. #### Common Mistakes / Traps * **360 vs 365 days:** US BDY uses 360, Canadian BEY uses 365. Always check context. * **Price Denominator vs Face Value Denominator:** BDY uses face value, BEY uses price. * **APR vs EAR:** APR is simple annual interest, EAR accounts for compounding. Always convert to EAR for comparison. * **VaR vs CTE:** VaR is the threshold return for the worst X% of outcomes. CTE (Expected Shortfall) is the *average* return within that worst X% tail. * **Short Sale P/L:** Remember to include dividends paid by short seller and interest earned on proceeds (if any). * **PV/FV:** Ensure interest rate and periods match (e.g., if rate is annual, periods must be years). If time is in months/days, convert to years for annual rates. ### Section B: Weeks 3–5 Portfolio Theory Foundation #### Core Formulas & Symbol Key * **Mean-Variance Utility:** $U = E[R] - \frac{1}{2} A \sigma^2$ * $A$: Risk aversion coefficient * **CAL Equation:** $E[R_P] = R_f + \frac{E[R_M] - R_f}{\sigma_M} \sigma_P$ * Slope of CAL is Sharpe Ratio: $SR = \frac{E[R_M] - R_f}{\sigma_M}$ * **Optimal Risky Portfolio Weight (y\*):** $y^* = \frac{E[R_M] - R_f}{A \sigma_M^2}$ * Weight in risky asset $M$ (Tangency portfolio). $1-y^*$ is weight in risk-free asset. * **Complete Portfolio Expected Return:** $E[R_C] = y E[R_P] + (1-y) R_f$ * **Complete Portfolio Standard Deviation:** $\sigma_C = y \sigma_P$ * **Two-Asset Portfolio Expected Return:** $E[R_P] = w_A E[R_A] + w_B E[R_B]$ * **Two-Asset Portfolio Variance:** $\sigma_P^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \text{Cov}(R_A, R_B)$ * **Covariance:** $\text{Cov}(R_A, R_B) = \rho_{AB} \sigma_A \sigma_B$ * $\rho_{AB}$: Correlation coefficient (unit-free, between -1 and 1) * **Minimum Variance Portfolio (MVP) Weights (2 assets):** $w_A = \frac{\sigma_B^2 - \text{Cov}(R_A, R_B)}{\sigma_A^2 + \sigma_B^2 - 2 \text{Cov}(R_A, R_B)}$ $w_B = 1 - w_A$ * **CAPM Expected Return (SML):** $E[R_i] = R_f + \beta_i (E[R_M] - R_f)$ * **Beta ($\beta_i$):** $\beta_i = \frac{\text{Cov}(R_i, R_M)}{\sigma_M^2} = \rho_{iM} \frac{\sigma_i}{\sigma_M}$ * **Alpha ($\alpha_i$):** $\alpha_i = R_i - E[R_i]$ (where $E[R_i]$ is from CAPM) * **Portfolio Beta:** $\beta_P = \sum_{i=1}^N w_i \beta_i$ #### Recognize the Question * **"Which portfolio is preferred by a risk-averse investor?"**: Use mean-variance utility or dominance. * **"Combine risky asset with risk-free asset"**: Think CAL. * **"Find the best mix of risky and risk-free"**: Optimal y*. * **"Risk of a two-asset portfolio"**: Use two-asset variance formula. * **"Impact of correlation on diversification"**: Analyze $\rho$ in variance formula. * **"Market equilibrium expected return"**: CAPM / SML. * **"Under/overpriced asset"**: Compare actual return to CAPM expected return ($\alpha$). #### Step-by-Step Templates * **Choosing Between Assets (Mean-Variance Utility):** 1. Calculate $U$ for each asset/portfolio using given $E[R]$, $\sigma^2$, and $A$. 2. Choose the asset with the highest utility. * **Example:** Investor A=4. Portfolio 1: $E[R]=0.10, \sigma=0.15$. Portfolio 2: $E[R]=0.12, \sigma=0.20$. * $U_1 = 0.10 - 0.5 \times 4 \times (0.15)^2 = 0.10 - 2 \times 0.0225 = 0.10 - 0.045 = 0.055$. * $U_2 = 0.12 - 0.5 \times 4 \times (0.20)^2 = 0.12 - 2 \times 0.04 = 0.12 - 0.08 = 0.04$. * Portfolio 1 is preferred. * **Weak Dominance / Mean-Variance Criterion:** 1. Portfolio A dominates B if $E[R_A] \ge E[R_B]$ AND $\sigma_A \le \sigma_B$, with at least one strict inequality. 2. If A has higher return AND lower risk, it dominates. * **Example:** Port A: $E[R]=10\%, \sigma=15\%$. Port B: $E[R]=8\%, \sigma=15\%$. Port A dominates B. Port C: $E[R]=10\%, \sigma=12\%$. Port C dominates A. * **Complete Portfolio (Risky + Risk-Free):** 1. Calculate $E[R_P]$ and $\sigma_P$ of the risky portfolio. 2. Decide on allocation $y$ to risky portfolio. 3. Use complete portfolio $E[R_C]$ and $\sigma_C$ formulas. * **Example:** $R_f=4\%$. Risky portfolio $P$: $E[R_P]=12\%, \sigma_P=20\%$. Investor allocates $y=70\%$ to $P$. * $E[R_C] = 0.70 \times 0.12 + (1-0.70) \times 0.04 = 0.084 + 0.012 = 0.096$ or 9.6%. * $\sigma_C = 0.70 \times 0.20 = 0.14$ or 14%. * **Optimal y\* using investor risk aversion:** 1. Calculate $E[R_M]$, $R_f$, $\sigma_M^2$, and use investor's $A$. 2. Plug into $y^* = \frac{E[R_M] - R_f}{A \sigma_M^2}$. * **Example:** $E[R_M]=10\%, R_f=3\%, \sigma_M=15\%$. Investor $A=3$. * $y^* = \frac{0.10 - 0.03}{3 \times (0.15)^2} = \frac{0.07}{3 \times 0.0225} = \frac{0.07}{0.0675} = 1.037$. (Investor borrows to invest more than 100% in risky asset). * **Two-Asset Portfolio Variance:** 1. Identify $w_A, w_B, \sigma_A, \sigma_B, \rho_{AB}$. 2. Calculate $\text{Cov}(R_A, R_B) = \rho_{AB} \sigma_A \sigma_B$. 3. Plug all into $\sigma_P^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \text{Cov}(R_A, R_B)$. 4. Take square root for $\sigma_P$. * **Example:** $w_A=0.5, w_B=0.5$. $\sigma_A=0.10, \sigma_B=0.15$. $\rho_{AB}=0.4$. * $\text{Cov}(R_A, R_B) = 0.4 \times 0.10 \times 0.15 = 0.006$. * $\sigma_P^2 = (0.5)^2 (0.10)^2 + (0.5)^2 (0.15)^2 + 2(0.5)(0.5)(0.006)$ * $\sigma_P^2 = 0.25 \times 0.01 + 0.25 \times 0.0225 + 0.5 \times 0.006$ * $\sigma_P^2 = 0.0025 + 0.005625 + 0.003 = 0.011125$. * $\sigma_P = \sqrt{0.011125} = 0.1055$ or 10.55%. * **Effect of $\rho$:** * $\rho = 1$: No diversification benefits, $\sigma_P = w_A \sigma_A + w_B \sigma_B$. * $\rho = -1$: Perfect diversification, $\sigma_P = |w_A \sigma_A - w_B \sigma_B|$. Can achieve zero risk if $w_A \sigma_A = w_B \sigma_B$. * $-1 E[R_i]$, then $\alpha_i > 0$, asset is underpriced (buy). 4. If $R_i E[R_X]$, so Stock X is underpriced. $\alpha_X = 12\% - 11.4\% = 0.6\%$. #### Common Mistakes / Traps * **Variance vs Standard Deviation:** $\sigma^2$ is variance (squared units), $\sigma$ is standard deviation (same units as return). Use $\sigma^2$ in utility/variance formulas, $\sigma$ for risk measure. * **SML vs CML:** CML plots $E[R_P]$ vs $\sigma_P$ for efficient portfolios with risk-free asset. SML plots $E[R_i]$ vs $\beta_i$ for individual assets or portfolios. CML uses total risk ($\sigma$), SML uses systematic risk ($\beta$). * **Beta:** Measures systematic risk, not total risk. * **Alpha:** Mispricing measure relative to CAPM. Not the same as abnormal return in event studies (see Section F). * **Correlation vs Covariance:** Covariance is scaled by standard deviations. Correlation is unit-free. Make sure to use the correct one in formulas. ### Section C: Week 8 Behavioral Finance #### Core Concepts & Distinctions * **Behavioralists vs Rationalists:** Behavioralists argue psychological biases affect decisions, leading to market anomalies. Rationalists believe markets are efficient, and anomalies are measurement errors or compensation for risk. * **Forecasting Errors:** Over-reliance on recent past, extrapolating trends. * **Overconfidence:** Overestimating own abilities, leading to excessive trading. * **Conservatism:** Slow to update beliefs in face of new evidence. * **Sample Size / Representativeness:** Assuming small samples are representative, ignoring base rates. * **Framing:** Decisions affected by how choices are presented. * **Mental Accounting:** Treating money differently based on its source or intended use. * **Regret Avoidance:** Reluctance to sell losing stocks to avoid admitting a mistake. * **Conventional Utility vs Prospect Theory:** * **Conventional:** Utility based on absolute wealth levels, risk-averse for all wealth levels. * **Prospect Theory:** Utility based on *changes* in wealth (gains/losses) relative to a reference point. Risk-averse in gains, risk-seeking in losses. * **Law of One Price vs Limits to Arbitrage:** * **Law of One Price:** Identical assets should trade at identical prices. * **Limits to Arbitrage:** Even if mispricing exists, arbitrage may not be able to correct it due to transaction costs, model risk, or fundamental risk. * **Royal Dutch / Shell & 3Com / Palm:** Classic examples of Law of One Price violations due to limits to arbitrage (e.g., short-selling restrictions, investor sentiment). * **Breadth:** Number of advancing stocks minus number of declining stocks. * **TRIN (Traders' Index):** $\frac{\text{Volume Declining / Number Declining}}{\text{Volume Advancing / Number Advancing}}$. Measures market sentiment. #### Recognize the Question * **"Investor ignoring diversification"**: Mental accounting. * **"Holding onto losing stocks"**: Regret avoidance. * **"Believing past trends will continue"**: Forecasting errors, representativeness. * **"Decision depends on how presented"**: Framing. * **"Why mispricing persists"**: Limits to arbitrage. * **"Comparing Royal Dutch and Shell shares"**: Law of One Price violation. #### Step-by-Step Templates * **Classify Behavior / Bias:** 1. Read investor's action/thought process. 2. Match to definitions: e.g., "not selling a stock below purchase price" = regret avoidance. * **Example:** An investor refuses to sell a stock trading below his purchase price, hoping it will recover, even though fundamental analysis suggests it's a poor investment. This is **Regret Avoidance**. * **Determine Decision Under Prospect Theory:** 1. Identify investor's reference point. 2. Determine if decision is framed as a gain or a loss. 3. **Gain:** Investor will be risk-averse (prefer sure gain over risky larger gain). 4. **Loss:** Investor will be risk-seeking (prefer risky chance to avoid loss over sure smaller loss). * **Example:** An investor has a choice: a sure gain of $500 or a 50% chance to gain $1000 (and 50% chance to gain nothing). She chooses the sure gain. This demonstrates **risk aversion in gains**. * **Identify Gain-vs-Loss Risk Preference:** 1. If choosing between two positive outcomes, preferring the certain one is risk-averse in gains. 2. If choosing between two negative outcomes (or one negative, one zero/positive), preferring the uncertain one is risk-seeking in losses. * **Example:** An investor has a choice: a sure loss of $500 or a 50% chance to lose $1000 (and 50% chance to lose nothing). He chooses the 50/50 gamble. This demonstrates **risk seeking in losses**. * **Law-of-One-Price Intuition:** 1. If two identical cash flow streams or claims on the same underlying assets trade at different prices, a violation exists. 2. Example: A company's shares traded on two different exchanges (Royal Dutch/Shell). * **Example:** Royal Dutch Petroleum shares traded in Amsterdam and Shell Transport shares traded in London represent claims on the same underlying cash flows of Royal Dutch/Shell Group. If their market prices don't reflect the exact ownership split, it's a LoOP violation. * **Explain Why Arbitrage May Fail:** 1. **Fundamental Risk:** The "identical" assets are not perfectly identical (e.g., different voting rights). 2. **Implementation Costs:** Transaction costs, short-selling restrictions. 3. **Model Risk:** Belief that assets are identical might be wrong. 4. **Noise Trader Risk (Sentiment):** Mispricing could worsen before it improves, forcing arbitragers to liquidate at a loss. * **Example (3Com/Palm):** 3Com owned 95% of Palm. When Palm was spun off, Palm's market cap briefly exceeded 3Com's entire market cap, implying 3Com's core business had negative value. Arbitrage was limited because of short-selling restrictions on Palm and the risk that investors' irrational exuberance for Palm could increase. * **Compute / Interpret Breadth:** 1. Breadth = (Number of Advancing Stocks) - (Number of Declining Stocks). 2. Positive breadth suggests bullish market, negative suggests bearish. * **Example:** On a given day, 2000 stocks advanced, 1000 declined. * Breadth = $2000 - 1000 = +1000$. This suggests a strong bullish sentiment. * **Compute / Interpret TRIN:** 1. TRIN = (Volume Declining / Number Declining) / (Volume Advancing / Number Advancing). 2. **TRIN > 1:** More selling pressure/volume in declining stocks (bearish). 3. **TRIN 1, it indicates strong selling pressure in declining stocks, which is a bearish signal. #### Common Mistakes / Traps * **Law of One Price vs Efficient Markets:** LoOP is a condition for no-arbitrage. EMH says prices reflect all information. LoOP can be violated even in semi-strong efficient markets if limits to arbitrage exist. * **Prospect Theory Risk Preference:** Remember it flips: risk-averse for gains, risk-seeking for losses. * **Arbitrage Failure:** It's not that arbitrage *doesn't exist*, but that it's *limited* or risky. ### Section D: Week 9 Index Models #### Core Formulas & Symbol Key * **Single-Factor Model (General):** $R_i = E[R_i] + F + e_i$ * $F$: Common macro factor, $e_i$: Firm-specific surprise * **Single-Index Model (SIM):** $R_i - R_f = \alpha_i + \beta_i (R_M - R_f) + e_i$ * Alternatively: $R_i = \alpha_i + \beta_i R_M + e_i$ (if $R_f$ is zero or ignored) * $R_M$: Market index return, $R_f$: Risk-free rate * **Security Characteristic Line (SCL):** Regression line for SIM. Plots $(R_i - R_f)$ vs $(R_M - R_f)$. * Intercept = $\alpha_i$, Slope = $\beta_i$ * **Total Risk Decomposition:** $\sigma_i^2 = \beta_i^2 \sigma_M^2 + \sigma^2(e_i)$ * $\beta_i^2 \sigma_M^2$: Systematic risk (market risk) * $\sigma^2(e_i)$: Firm-specific (idiosyncratic / unsystematic / residual) risk * **Covariance between two securities (SIM):** $\text{Cov}(R_i, R_j) = \beta_i \beta_j \sigma_M^2$ * **Covariance between stock and market (SIM):** $\text{Cov}(R_i, R_M) = \beta_i \sigma_M^2$ * **Percentage of Variance Explained by Systematic Risk:** $R^2 = \frac{\beta_i^2 \sigma_M^2}{\sigma_i^2}$ * Also, $R^2 = \text{Corr}(R_i, R_M)^2$ * **Well-Diversified Portfolio Variance (SIM):** $\sigma_P^2 = \beta_P^2 \sigma_M^2$ * Firm-specific risk $\sigma^2(e_P)$ approaches 0 for well-diversified portfolios. * **Predicting Next Beta (Blume's Method):** $\beta_{t+1} = 0.35 + 0.65 \beta_t$ (Example coefficients, check class notes for specific values) #### Recognize the Question * **"Decompose total risk into market and unique risk"**: Use total risk decomposition formula. * **"Covariance between two stocks given market info"**: Use $\text{Cov}(R_i, R_j) = \beta_i \beta_j \sigma_M^2$. * **"How much risk is explained by the market?"**: $R^2$. * **"What is the regression equation for a stock?"**: SCL / SIM. * **"Why does diversification reduce risk?"**: Firm-specific risk vanishes. * **"How does beta change over time?"**: Beta prediction models. #### Step-by-Step Templates * **Compute Variance Decomposition:** 1. Given $\sigma_i^2$, $\beta_i$, $\sigma_M^2$. 2. Systematic Risk = $\beta_i^2 \sigma_M^2$. 3. Firm-Specific Risk = $\sigma_i^2 - \beta_i^2 \sigma_M^2$. * **Example:** Stock A has $\sigma_A=20\%$, $\beta_A=1.2$. Market $\sigma_M=15\%$. * Systematic Risk = $(1.2)^2 \times (0.15)^2 = 1.44 \times 0.0225 = 0.0324$ or 3.24%. * Total Variance = $(0.20)^2 = 0.04$ or 4%. * Firm-Specific Risk = $0.04 - 0.0324 = 0.0076$ or 0.76%. * **Compute Covariance using Betas (Two Stocks):** 1. Given $\beta_i, \beta_j, \sigma_M^2$. 2. $\text{Cov}(R_i, R_j) = \beta_i \beta_j \sigma_M^2$. * **Example:** Stock A: $\beta_A=1.2$. Stock B: $\beta_B=0.8$. Market $\sigma_M=15\%$. * $\text{Cov}(R_A, R_B) = 1.2 \times 0.8 \times (0.15)^2 = 0.96 \times 0.0225 = 0.0216$. * **Compute Stock-Market Covariance:** 1. Given $\beta_i, \sigma_M^2$. 2. $\text{Cov}(R_i, R_M) = \beta_i \sigma_M^2$. * **Example:** Stock A: $\beta_A=1.2$. Market $\sigma_M=15\%$. * $\text{Cov}(R_A, R_M) = 1.2 \times (0.15)^2 = 1.2 \times 0.0225 = 0.027$. * **Interpret Regression Output (for SIM/SCL):** 1. **Intercept (alpha):** The stock's excess return if the market excess return is zero. A positive alpha indicates outperformance relative to the market/CAPM. 2. **Slope (beta):** Sensitivity of the stock's excess return to the market's excess return. $\beta > 1$ (aggressive), $\beta ### Section E: Week 10 Factor Models & APT #### Core Formulas & Symbol Key * **Two-Factor Model (Return):** $R_i = E[R_i] + \beta_{i1} F_1 + \beta_{i2} F_2 + e_i$ * $F_k$: Factor surprise (actual factor value - expected factor value). * $\beta_{ik}$: Sensitivity of asset $i$ to factor $k$. * **APT Assumptions:** 1. Asset returns generated by a $k$-factor model. 2. No arbitrage opportunities exist. 3. Well-diversified portfolios can eliminate firm-specific risk. * **APT Expected Return Relation (k-factor):** $E[R_i] = R_f + \beta_{i1} RP_1 + \beta_{i2} RP_2 + ... + \beta_{ik} RP_k$ * $RP_k$: Risk premium for factor $k$. * **Two-Factor Required Return:** $E[R_i] = R_f + \beta_{i1} (E[F_1] - R_f) + \beta_{i2} (E[F_2] - R_f)$ * This assumes factors are excess returns over $R_f$. More generally, $RP_k$ is the risk premium for that factor. (Check course notation for $RP_k$ vs $(E[F_k] - R_f)$). * **Fama-French 3-Factor Model (if covered):** $R_i - R_f = \alpha_i + \beta_M (R_M - R_f) + \beta_{SMB} SMB + \beta_{HML} HML + e_i$ * $SMB$: Small minus Big (size factor). * $HML$: High minus Low (value factor). #### Recognize the Question * **"How does a specific economic surprise affect a stock?"**: Use factor model return change. * **"Arbitrage opportunity with same beta but different returns"**: APT no-arbitrage logic. * **"Required return based on multiple risk factors"**: APT expected return formula. * **"Differences between CAPM and APT"**: Compare assumptions and structure. #### Step-by-Step Templates * **Compute Return Change from Factor Surprise:** 1. Identify the factor surprise $F_k$ (actual - expected). 2. Identify the asset's sensitivity $\beta_{ik}$ to that factor. 3. Change in return $\Delta R_i = \sum \beta_{ik} F_k$. * **Example:** Stock has $\beta_{GDP}=1.5$ and $\beta_{InterestRate}=-0.8$. GDP surprise = +2% (higher than expected). Interest Rate surprise = +0.5% (higher than expected). * $\Delta R_i = (1.5 \times 0.02) + (-0.8 \times 0.005) = 0.03 - 0.004 = 0.026$ or 2.6%. * **Interpret Sign / Magnitude of Factor Loading ($\beta_{ik}$):** 1. **Sign:** Positive $\beta_{ik}$ means asset return moves in same direction as factor. Negative means opposite. 2. **Magnitude:** Larger absolute $\beta_{ik}$ means greater sensitivity to the factor. * **Example:** $\beta_{GDP}=1.5$: Stock return increases by 1.5% for every 1% unexpected increase in GDP. $\beta_{InterestRate}=-0.8$: Stock return decreases by 0.8% for every 1% unexpected increase in interest rates. * **Compute Arbitrage Profit in Same-Beta Mispricing Setup:** 1. Identify two assets with identical factor sensitivities ($\beta_{i1}=\beta_{j1}$, etc.) but different expected returns ($E[R_i] \ne E[R_j]$). 2. **Arbitrage Strategy:** Buy the underpriced asset (higher $E[R]$), sell the overpriced asset (lower $E[R]$). 3. **Profit:** The difference in returns, with no net investment and no risk (for well-diversified portfolios). * **Example:** Asset A: $\beta_{GDP}=1.0, \beta_{IR}=0.5, E[R_A]=12\%$. Asset B: $\beta_{GDP}=1.0, \beta_{IR}=0.5, E[R_B]=10\%$. * Since betas are identical but $E[R_A] > E[R_B]$, Asset A is underpriced relative to B. * Arbitrage: Buy Asset A, Sell Asset B. Profit = $12\% - 10\% = 2\%$. * **Compute Required Return Under 2-Factor APT:** 1. Given $R_f$, $\beta_{i1}, \beta_{i2}$, and factor risk premiums ($RP_1, RP_2$). 2. Plug into $E[R_i] = R_f + \beta_{i1} RP_1 + \beta_{i2} RP_2$. * **Example:** $R_f=4\%$. Stock Z: $\beta_{GDP}=1.2, \beta_{IR}=0.7$. GDP Risk Premium = 5%. IR Risk Premium = 2%. * $E[R_Z] = 0.04 + (1.2 \times 0.05) + (0.7 \times 0.02) = 0.04 + 0.06 + 0.014 = 0.114$ or 11.4%. * **Distinguish APT Assumptions from CAPM Assumptions:** * **APT:** * Assumes a factor structure of returns. * Relies on *no-arbitrage principle* (stronger than mean-variance efficiency). * Does not assume investors are mean-variance optimizers or that the market portfolio is efficient. * Can use *any* set of systematic factors. * **CAPM:** * Assumes mean-variance efficient investors. * Assumes a single systematic risk factor (the market portfolio). * Relies on market equilibrium. * Market portfolio is assumed to be the tangency portfolio. #### Common Mistakes / Traps * **Factor Surprise vs Factor Return:** Factor surprise is the *unexpected* part of the factor. Factor return is the total return. * **Arbitrage vs Alpha:** APT arbitrage is about exploiting mispricing relative to the factor model, with zero risk and zero investment. CAPM alpha is a measure of mispricing relative to the market. * **APT Factors:** APT does not specify *which* factors are relevant, only that they exist. CAPM specifies only one (market). ### Section F: Week 11 Market Efficiency #### Core Concepts & Distinctions * **EMH Intuition:** Prices fully reflect all available information. Competition among investors drives prices to fair value. * **Random Walk Logic:** If prices reflect all information, price changes are unpredictable (random). * **Forms of EMH:** * **Weak Form:** Prices reflect all *past market data* (prices, trading volume). Technical analysis is useless. * **Semi-Strong Form:** Prices reflect all *publicly available information* (financial statements, news, etc.). Fundamental analysis is useless. * **Strong Form:** Prices reflect all *public and private (insider) information*. Even insider trading is not profitable. * **Grossman-Stiglitz Impossibility Theorem:** Information is costly to acquire. If markets were perfectly efficient, no one would pay for information, and thus prices would not reflect it. Therefore, markets must be *somewhat* inefficient to incentivize information gathering. * **Event Study Logic:** Examine stock price reactions to specific information announcements to test semi-strong EMH. * **Abnormal Return (AR):** $AR_t = R_t - E[R_t]$ * $R_t$: Actual return of security at time $t$. * $E[R_t]$: Expected return of security at time $t$ (e.g., from CAPM or index model). * **Cumulative Abnormal Return (CAR):** Sum of ARs over an event window: $CAR_T = \sum_{t=1}^T AR_t$. * **Buy-and-Hold Abnormal Return (BHAR):** Compounded abnormal return over a period. Often used for longer horizons. * **Technical Analysis vs EMH:** Directly contradicts weak-form EMH. * **Fundamental Analysis vs EMH:** Contradicts semi-strong form EMH if it consistently generates superior returns after accounting for risk. * **Anomalies:** Empirical findings that contradict EMH (e.g., post-earnings announcement drift, small-firm effect, value premium). #### Recognize the Question * **"Can past prices predict future prices?"**: Weak-form EMH. * **"Can public news generate abnormal profits?"**: Semi-strong form EMH. * **"Can insider information generate abnormal profits?"**: Strong-form EMH. * **"Measuring impact of an announcement"**: Event study, AR, CAR. * **"Why markets aren't perfectly efficient"**: Grossman-Stiglitz. #### Step-by-Step Templates * **Classify a Claim as Weak / Semi-Strong / Strong EMH:** 1. **Weak:** If claim relies on *historical price/volume patterns*. 2. **Semi-Strong:** If claim relies on *publicly available financial or news data*. 3. **Strong:** If claim relies on *any information, including private/insider*. * **Example:** "A trader consistently profits by analyzing historical stock charts and identifying patterns." This claim violates **Weak-form EMH**. * **Compute AR:** 1. Get actual return $R_t$. 2. Calculate expected return $E[R_t]$ using a model (e.g., CAPM: $E[R_t] = R_f + \beta (R_M - R_f)$). 3. $AR_t = R_t - E[R_t]$. * **Example:** Stock's actual return on announcement day is $2\%$. CAPM expected return for that stock is $1.5\%$. * $AR = 0.02 - 0.015 = 0.005$ or 0.5%. * **Compute CAR:** 1. Calculate $AR_t$ for each day in the event window. 2. Sum the $AR_t$ values. * **Example:** Day -1 AR = 0.2%, Day 0 AR = 0.5%, Day +1 AR = -0.1%. * $CAR_{(-1,+1)} = 0.002 + 0.005 - 0.001 = 0.006$ or 0.6%. * **Compute BHAR (conceptually):** 1. Calculate $(1+R_t)$ for actual return and $(1+E[R_t])$ for expected return over the period. 2. $BHAR = \prod (1+R_t) - \prod (1+E[R_t])$. (More complex in practice, focus on concept). * **Example:** Over 3 months, actual returns are 1.05 and 1.03. Expected returns are 1.02 and 1.01. * $BHAR = (1.05 \times 1.03) - (1.02 \times 1.01) = 1.0815 - 1.0302 = 0.0513$ or 5.13%. * **Interpret Evidence for or Against Efficiency:** 1. **For:** If ARs are consistently zero after an announcement, or if no trading strategy based on information consistently generates positive alphas. 2. **Against:** If significant positive or negative ARs persist after an announcement, or if strategies (technical/fundamental) consistently beat the market. * **Example:** If an event study finds that positive CARs persist for several days *after* a positive earnings announcement, this is evidence **against semi-strong form EMH** (post-earnings announcement drift). * **Distinguish Alpha from Abnormal Return:** * **Alpha ($\alpha$):** A measure of a portfolio's or asset's performance relative to a benchmark (e.g., CAPM or factor model). It's the intercept of the SCL. Generally refers to a persistent outperformance. * **Abnormal Return (AR):** The difference between an asset's actual return and its expected return over a specific, usually short, period, often around an event. It measures the market's reaction to new information. ARs are typically expected to be transient. #### Common Mistakes / Traps * **Alpha vs AR:** While similar in calculation, AR is specific to an event window and measures market reaction, while alpha is a more general measure of persistent outperformance. * **EMH Forms:** Don't confuse the types of information. Public information is semi-strong, past prices are weak. * **Grossman-Stiglitz:** It doesn't say markets are *inefficient*, but that perfect efficiency is impossible; there must be *some* inefficiency to incentivize information. ### Section G: Week 12 Derivative Securities / Options #### Core Concepts & Distinctions * **Call Option:** Right to buy an asset at a specified price (strike) by a certain date (expiration). * **Holder (Long):** Buys the call, hopes price rises. * **Writer (Short):** Sells the call, hopes price falls or stays same. * **Put Option:** Right to sell an asset at a specified price (strike) by a certain date (expiration). * **Holder (Long):** Buys the put, hopes price falls. * **Writer (Short):** Sells the put, hopes price rises or stays same. * **Strike Price (X):** The price at which the underlying asset can be bought/sold. * **Premium:** The price paid for the option. * **Expiration Date (T):** Last day option can be exercised. * **In-the-money (ITM):** * **Call:** $S_T > X$ (profitable to exercise). * **Put:** $S_T X$ (not profitable to exercise). * **At-the-money (ATM):** $S_T = X$. * **Payoff vs Profit:** * **Payoff:** Value of the option at expiration if exercised (before considering premium). * **Profit:** Payoff - Premium Paid (or +Premium Received for writer). * **Exercised vs Profitable:** * An option is **exercised** if it's ITM. * An option is **profitable** if its payoff exceeds the premium paid (for holder) or the premium received exceeds the loss (for writer). * **Option Leverage:** Small change in underlying price can lead to large percentage change in option value. * **Protective Put:** Long stock + Long put. Provides downside protection. * **Covered Call:** Long stock + Short call. Limits upside potential for premium income. * **Straddle:** Long call + Long put (same X, T). Profits from high volatility. * **Bull Spread (Bull Call Spread):** Long call (low X) + Short call (high X). Profits from moderate price increase. * **Lookback Option:** Option whose strike price is determined by the minimum (for call) or maximum (for put) price of the underlying asset over its life. * **Lookback Call:** Payoff at expiration is $S_T - \min(S_t)$ over option life. Guarantees lowest strike. * **Lookback Put:** Payoff at expiration is $\max(S_t) - S_T$ over option life. Guarantees highest strike. * Lookback options are more valuable than standard options. * **Put-Call Parity (European Options, non-dividend paying):** $P + S = C + X e^{-rT}$ * $P$: Put price, $S$: Current stock price, $C$: Call price, $X$: Strike price, $r$: Risk-free rate, $T$: Time to expiration. #### Recognize the Question * **"Right to buy/sell"**: Call/Put. * **"Downside protection"**: Protective put. * **"Income generation, limited upside"**: Covered call. * **"Profits from volatility"**: Straddle. * **"Profits from moderate rise"**: Bull spread. * **"Strike price adjusted to favorable level"**: Lookback option. * **"Arbitrage with options and stock/bond"**: Put-Call Parity violation. #### Step-by-Step Templates * **Call Value and Profit at Expiration:** 1. **Payoff:** $\max(0, S_T - X)$. 2. **Profit (Holder):** Payoff - Premium. 3. **Profit (Writer):** Premium - Payoff. * **Example:** Call option with $X=$50, Premium=$3. Stock price at expiration $S_T=$55. * Payoff = $\max(0, 55 - 50) = 5$. * Holder Profit = $5 - 3 = 2$. * Writer Profit = $3 - 5 = -2$. * **Put Value and Profit at Expiration:** 1. **Payoff:** $\max(0, X - S_T)$. 2. **Profit (Holder):** Payoff - Premium. 3. **Profit (Writer):** Premium - Payoff. * **Example:** Put option with $X=$50, Premium=$2. Stock price at expiration $S_T=$45. * Payoff = $\max(0, 50 - 45) = 5$. * Holder Profit = $5 - 2 = 3$. * Writer Profit = $2 - 5 = -3$. * **Determine if Option is Exercised:** 1. **Call:** Exercised if $S_T > X$. 2. **Put:** Exercised if $S_T X + \text{Premium}$. 2. **Put:** Profitable if $S_T C + X e^{-rT}$: Synthetic call ($P+S$) is overpriced. **Sell synthetic call, Buy actual call.** (Sell put, Sell stock, Buy call, Borrow $X e^{-rT}$). 3. If $P + S