Investment Analysis Master Flashcards

Comprehensive vocabulary flashcards covering Modern Portfolio Theory, CAPM, Fama–MacBeth and Fama–French models, Treynor–Black portfolio optimization, Equity Valuation, Bond Valuation and Duration, Behavioral Finance, Derivatives, and Portfolio Performance Measurement.

Mean-variance utility

An objective function where investors maximize expected complete-portfolio return minus a variance penalty: $U = E(r_C) - \frac{1}{2} A \sigma_C^2$, where $A$ is the risk-aversion coefficient, complete return is $E(r_C) = r_f + \mathbf{w}' \boldsymbol{\mu}^e$, and complete variance is $\sigma_C^2 = \mathbf{w}' \boldsymbol{\Sigma} \mathbf{w}$. The first-order condition yields optimal risky weights $\mathbf{w}^* = \frac{1}{A} \boldsymbol{\Sigma}^{-1} \boldsymbol{\mu}^e$ and risk-free weight $w_{rf}^* = 1 - \mathbf{1}' \mathbf{w}^*$.

Mean-variance frontier, GMV weights

The set of fully invested risky portfolios that minimize variance for each target expected return by solving $\min_{\mathbf{w}} \mathbf{w}' \boldsymbol{\Sigma} \mathbf{w}$ subject to $\mathbf{w}' \boldsymbol{\mu} = \mu_{\text{target}}$ and $\iota^{\prime}\mathbf{w}=1$ . The upper branch above the global minimum-variance (GMV) portfolio, whose weights are $\mathbf{w}_{\text{GMV}}=\frac{\bm{\Sigma}^{-1}\iota}{\iota^{\prime}\bm{\Sigma}^{-1}\iota}$ , constitutes the efficient frontier.

Diversification and covariance

The principle that portfolio variance depends on individual asset variances and pairwise covariances. For two risky assets under budget constraint $w_2 = 1 - w_1$, portfolio return is $E(r_P) = w_1 \mu_1 + w_2 \mu_2$ and variance is $\sigma_P^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \sigma_{12}$, where $\sigma_{12} = \rho_{12} \sigma_1 \sigma_2$. For $K$ assets, $E(r_P) = \mathbf{w}' \boldsymbol{\mu}$ and $\sigma_P^2 = \mathbf{w}' \boldsymbol{\Sigma} \mathbf{w}$.

Tangency portfolio

The fully invested risky portfolio that maximizes the Sharpe ratio, where the optimal capital allocation line (CAL) touches the efficient risky frontier. Its portfolio weights are $\mathbf{w}_{T}=\frac{\bm{\Sigma}^{-1}\bm{\mu}^{e}}{\iota^{\prime}\bm{\Sigma}^{-1}\bm{\mu}^{e}}$ , expected return is $E(r_T) = \mathbf{w}_T' \boldsymbol{\mu}$, variance is $\sigma_T^2 = \mathbf{w}_T' \boldsymbol{\Sigma} \mathbf{w}_T$

Two-fund separation

The property that mean-variance investors combine the same tangency risky portfolio with the risk-free asset, using risk aversion solely to determine the mixture. The optimal allocation to the tangency portfolio is $y^* = \frac{E(r_T) - r_f}{A \sigma_T^2}$, individual risky weights are $\mathbf{w}^* = y^* \mathbf{w}_T$, and the risk-free weight is $w_{rf}^* = 1 - y^*$.

Complete portfolio

A portfolio combining risky assets with a risk-free asset whose risky weights need not sum to one. Its expected return is given by $E(r_{C})=r_{f}+\mathbf{w}^{\prime}\bm{\mu}^{e}$ , or in terms of tangency allocation as $E(r_C) = r_f + y [E(r_T) - r_f]$, or via the CAL shortcut as $E(r_C) = r_f + \text{SR}_T \sigma_C$.

Portfolio volatility

The standard deviation of complete-portfolio returns, given by $\sigma_C = |y| \sigma_T$ with variance $\sigma_C^2 = \mathbf{w}' \boldsymbol{\Sigma} \mathbf{w} = y^2 \sigma_T^2$, reflecting that the risk-free asset contributes zero variance and zero covariance. For positive $y$, it satisfies $\sigma_C = \frac{E(r_C) - r_f}{\text{SR}_T}$.

Sharpe ratio

The expected excess return per unit of total volatility, defined as $\text{SR}_P = \frac{E(r_P) - r_f}{\sigma_P}$, which represents the slope of the Capital Allocation Line. For positive risky allocations, the optimal CAL equation is $E(r_C) = r_f + \text{SR}_T \sigma_C$, yielding optimal volatility shortcut $\sigma_C^* = \frac{\text{SR}_T}{A}$.

Risk aversion coefficient ($A$)

The parameter measuring the penalty placed on variance in mean-variance utility $U = E(r_C) - \frac{1}{2} A \sigma_C^2$. It can be inferred from the optimal tangency allocation via $A = \frac{E(r_T) - r_f}{y^* \sigma_T^2}$ or at positive-$y$ optimum via $A = \frac{\text{SR}_T}{\sigma_C^*}$, where higher $A$ implies a smaller risky allocation $y^*$.

Exogenous (background) risk

A non-traded exposure already held by an investor (such as labor income or a private collection) that covaries with tradable asset returns. Total tradable demand combines speculative demand $\mathbf{w}_{\text{spec}} = \frac{1}{A} \boldsymbol{\Sigma}^{-1} \boldsymbol{\mu}^e$ and hedging demand $\mathbf{w}_{\text{hedge}} = q \boldsymbol{\Sigma}^{-1} \boldsymbol{\sigma}_{rq}$ to yield $\mathbf{w}_q^* = \frac{1}{A} \boldsymbol{\Sigma}^{-1} \boldsymbol{\mu}^e - q \boldsymbol{\Sigma}^{-1} \boldsymbol{\sigma}_{rq}$.

Modern Portfolio Theory notation

Standard lecture conventions where prime ($'$) denotes transpose, $\iota$ is the $K \times 1$ ones vector, $\mathbf{w}$ are risky weights relative to wealth, $\mathbf{w}^*$ are optimal risky weights, $\mathbf{w}_T$ are normalized tangency weights, $\bm{\mu}^{e}=\bm{\mu}-r_{f}\iota$ is the expected excess return vector, and $y^*$ is the optimal share in the tangency portfolio.

Capital Asset Pricing Model (CAPM)

An equilibrium asset-pricing model stating that expected return is determined solely by systematic risk: $E(r_i) = r_f + \beta_i [E(r_M) - r_f]$, where the market portfolio $M$ contains all risky assets in market-value proportions, acts as the tangency portfolio, and prices risk at the market price of risk $\frac{E(r_M) - r_f}{\sigma_M^2}$.

Beta ($\beta$)

A standardized measure of an asset's sensitivity to market portfolio movements and systematic risk, expressed as $\beta_i = \frac{\text{Cov}(r_i, r_M)}{\sigma_M^2}$ or $\beta_i = \frac{\rho_{iM} \sigma_i}{\sigma_M}$, where $\beta = 1$ denotes market-level sensitivity, $\beta > 1$ indicates aggressive sensitivity, and $\beta = 0$ indicates no market covariance.

Portfolio beta ($\beta_P$)

The weighted average of constituent asset betas: $\beta_P = \sum_{i=1}^K w_i \beta_i$. For a complete portfolio combining the market portfolio ($\beta_M = 1$) with the risk-free asset ($\beta_f = 0$), $\beta_C = y \beta_M = y$, such that leveraging with $y = 1.5$ yields $\beta_C = 1.5$.

CAPM required return

The expected return required to compensate an investor for systematic risk under CAPM equilibrium: $E(r_i) = r_f + \beta_i [E(r_M) - r_f]$, which solves for beta as $\beta_i = \frac{E(r_i) - r_f}{E(r_M) - r_f}$ and implies that an asset with $\beta = 0$ earns expected return $r_f$.

Security Market Line (SML)

The graphical depiction of the CAPM expressing expected return as a function of systematic risk ($\beta$) for individual assets and portfolios, having intercept $r_f$ and slope equal to the market risk premium $E(r_M) - r_f$.

Capital Market Line (CML)

The linear relation plotting expected return against total risk ($\sigma$) strictly for efficient portfolios combining the risk-free asset and the market portfolio: $E(r_C) = r_f + \left[\frac{E(r_M) - r_f}{\sigma_M}\right] \sigma_C$, with slope equal to the market Sharpe ratio.

Alpha ($\alpha$)

The abnormal return measuring the difference between an independent expected-return forecast and the CAPM required return: $\alpha_i = E(r_i) - \{r_f + \beta_i [E(r_M) - r_f]\}$, where $\alpha > 0$ indicates underpricing (plots above the theoretical SML) and $\alpha < 0$ indicates overpricing.

Time-series and cross-sectional CAPM tests

Empirical methodologies assessing CAPM validity through time-series regressions ($r_{it} - r_{ft} = \alpha_i + \beta_i(r_{Mt} - r_{ft}) + \epsilon_{it}$ with individual null $H_0: \alpha_i = 0$ and joint GRS null $H_0: \alpha_1 = \dots = \alpha_N = 0$) and cross-sectional regressions ($E(r_i) = r_f + \beta_i [E(r_M) - r_f]$).

Empirical limitations of CAPM

Empirical failures of the single-factor CAPM, including a flatter empirical SML than theory predicts, the low-beta anomaly (outperformance of low-beta stocks relative to required returns), instability of historical betas over time, and the explanatory power of firm characteristics like size and value.

Fama–MacBeth two-pass regression

A two-stage asset pricing procedure where Stage 1 estimates factor loadings ($\beta_i$) via time-series regressions, and Stage 2 runs cross-sectional regressions at each period $t$ ($r_{it} = \gamma_{0t} + \gamma_{1t} \hat{\beta}_i + \gamma_{2t} x_{i,t-1} + \epsilon_{it}$), with estimated coefficients averaged across time as $\hat{\gamma}_j = \frac{1}{T} \sum_{t=1}^T \hat{\gamma}_{jt}$.

Factor risk premium ($\gamma_j$) versus factor loading ($\beta_{ij}$)

In multi-factor pricing models, a factor loading ($\beta_{ij}$) quantifies an asset's sensitivity to factor $j$, while the factor risk premium ($\gamma_j$) represents the expected compensation per unit of exposure, giving an expected return contribution of $\beta_{ij} \hat{\gamma}_j$.

Cross-sectional CAPM restrictions

Parametric hypotheses tested within cross-sectional regressions ($r_{it} = \gamma_{0t} + \gamma_{1t} \hat{\beta}_i + \gamma_{2t} x_{i,t-1} + \epsilon_{it}$): the intercept must equal the risk-free rate ($E(\gamma_0) = r_f$), the beta premium must equal the market risk premium ($E(\gamma_1) = E(r_M) - r_f$), and non-beta characteristics must have zero price ($E(\gamma_2) = 0$).

Fama–French (1992) empirical findings

Empirical results showing that market beta exhibited a weak relation to average stock returns after controlling for size, while market capitalization (firm size) was negatively related to average returns and book-to-market equity was positively related to average returns.

Fama–French three-factor model

A multifactor model specifying excess returns as a function of the market factor, size factor (SMB), and value factor (HML): $r_{it} - r_{ft} = \alpha_i + \beta_{iM}(r_{Mt} - r_{ft}) + s_i \text{SMB}_t + h_i \text{HML}_t + \epsilon_{it}$, with expected excess return $E(r_i) - r_f = \beta_{iM} E(r_M - r_f) + s_i E(\text{SMB}) + h_i E(\text{HML})$ when $\alpha_i = 0$.

Empirical asset-pricing tests and limitations

Methodological challenges in empirical tests including time-varying betas ($\beta_{it} = \frac{\text{Cov}_t(r_i, r_M)}{\text{Var}_t(r_M)}$), beta estimation error ($\hat{\beta}_i = \beta_i + \text{error}$) attenuating cross-sectional slopes, omitted risk factors, and difficulty separating risk premiums from behavioral mispricing.

Treynor–Black model

An active portfolio management framework that integrates an actively managed portfolio of mispriced assets having nonzero alphas ($\alpha_i$) and residual variances ($\hat{\sigma}_{\epsilon,i}^2 = \frac{\sum_t \hat{\epsilon}_{it}^2}{T-2}$) with a passively held market index.

TB active portfolio weights

The asset allocations within the Treynor–Black active portfolio, determined by scaling each asset's alpha ($\sigma_{\epsilon i}^2=\sigma_{i}^2-\beta_{^{}i}^2\sigma_{M}^2$) by its residual variance: unnormalized weight $z_i = \frac{\alpha_i}{\sigma_{\epsilon,i}^2}$, normalized weight $w_i^A = \frac{z_i}{\sum_{j=1}^N z_j}$, where $\sum_i w_i^A = 1$.

Active alpha, beta and residual variance

Portfolio parameters of the Treynor–Black active portfolio: active alpha $\alpha_A = \sum_i w_i^A \alpha_i$, active beta $\beta_A = \sum_i w_i^A \beta_i$, active residual variance $\sigma_{\epsilon,A}^2 = \sum_i (w_i^A)^2 \sigma_{\epsilon,i}^2$ (assuming zero cross-residual covariances), and residual risk $\sigma_{\epsilon,A} = \sqrt{\sigma_{\epsilon,A}^2}$.

Initial and beta-adjusted active weights

In Treynor–Black optimization, the preliminary active weight is $w_A^0 = \frac{\alpha_A / \sigma_{\epsilon,A}^2}{[E(r_M) - r_f] / \sigma_M^2}$, which is adjusted for systematic risk to give optimal active weight $w_A^* = \frac{w_A^0}{1 + (1 - \beta_A) w_A^0}$ and passive weight $w_M^* = 1 - w_A^*$.

Treynor-Black final portfolio characteristics

Expected return: $E\left(r_{P}\right)=r_{f}+\beta_{P}\left\lbrack E\left(r_{M}\right)-r_{f}\right\rbrack+\alpha_{P}$

Portfolio variance: $\sigma_{P}^2=\beta_{P}^2\sigma_{M}^2+w_{A}^2\sigma_{\epsilon A}^2$

Portfolio beta: $\beta_{P}=w_{A}\beta_{A}+w_{M}$

Portfolio alpha: $\alpha_{P}=w_{A}\alpha_{A}$

Performance improvement from active management

The enhancement in total risk-adjusted return from adding the Treynor–Black active portfolio, quantified by the information ratio $\text{IR}_A = \frac{\alpha_A}{\sigma_{\epsilon,A}}$ such that the combined Sharpe ratio satisfies $\text{SR}_P = \sqrt{\text{SR}_M^2 + \text{IR}_A^2}$ and $\text{SR}_P^2 - \text{SR}_M^2 = \text{IR}_A^2$.

Treynor–Black assumptions and limitations

Key assumptions of the Treynor–Black model: residual security returns are uncorrelated ($\sigma_{\epsilon,A}^2 = \sum_i (w_i^A)^2 \sigma_{\epsilon,i}^2$), unrestricted short selling is allowed, and alpha and beta estimates are treated as known with certainty despite real-world estimation risk.

Treynor–Black calculation procedure

A 6-step procedure: (1) estimate single-index models; (2) compute active weights $z_i = \frac{\alpha_i}{\sigma_{\epsilon,i}^2}$ and $w_i^A = \frac{z_i}{\sum_j z_j}$; (3) compute $\alpha_A$, $\beta_A$, and $\sigma_{\epsilon,A}^2$; (4) calculate $w_A^0$; (5) compute beta-adjusted $w_A^*$ and $w_M^*$; (6) evaluate performance via $\text{IR}_A$ and $\text{SR}_P = \sqrt{\text{SR}_M^2 + \text{IR}_A^2}$.

Dividend Discount Model (DDM)

An equity valuation method establishing intrinsic value as the present value of all anticipated future dividends discounted at required return $k$: $V_0 = \sum_{t=1}^\infty \frac{E(D_t)}{(1+k)^t}$, or over finite horizon $H$ as $V_0 = \sum_{t=1}^H \frac{E(D_t)}{(1+k)^t} + \frac{E(P_H)}{(1+k)^H}$.

Gordon Growth Model

A constant-growth DDM valuing equity whose dividends grow forever at constant rate $g < k$, defined by $P_0=\frac{D_1}{k - g}$ where $D_1 = D_0 (1+g)$

Sustainable earnings and dividend growth

The long-run growth rate financed solely by retained earnings, formulated as $g = b \times \text{ROE}$ where retention ratio is $b = 1 - \text{payout ratio}$.

Payout Ratio = D/EPS

Future earnings and dividends evolve via $\text{EPS}_{t+1} = \text{EPS}_t (1+g)$ and $D_{t+1} = \text{EPS}_{t+1}(1 - b)$.

Optimal earnings retention

The managerial decision rule to retain earnings ($b > 0$) if and only if reinvestment profitability exceeds the equity required return ($\text{ROE} > k$); if $\text{ROE} = k$, retention does not affect value, whereas if $\text{ROE} < k$, retention destroys value.

Price-to-earnings ratio (P/E)

A valuation metric relating price to earnings per share. Trailing P/E is $\frac{P_0}{\text{EPS}_0}$, leading P/E is $\frac{P_0}{\text{EPS}_1}$, justified leading P/E is $\frac{1 - b}{k - g}$, and justified trailing P/E is $\frac{(1 - b)(1+g)}{k - g}$.

Present Value of Growth Opportunities (PVGO)

The component of share price attributable to future investments earning returns in excess of the cost of capital: $V_0 = \frac{\text{EPS}_1}{k} + \text{PVGO}$, or $\text{PVGO} = V_0 - \frac{\text{EPS}_1}{k}$, where $\frac{\text{EPS}_1}{k}$ represents the no-growth firm value.

Two-stage Dividend Discount Model

A valuation model accommodating an initial high-growth stage followed by perpetual stable growth $g_2 = b_2 \times \text{ROE}_2$, where terminal price is $P_T = \frac{D_{T+1}}{k - g_2}$ and intrinsic value is $V_0 = \sum_{t=1}^T \frac{D_t}{(1+k)^t} + \frac{P_T}{(1+k)^T}$.

Dividend expectations and valuation ratios

Formulas relating stock valuation to dividend expectations: forward price-dividend ratio $\frac{P_0}{D_1} = \frac{1}{k - g}$, forward dividend yield $\frac{D_1}{P_0} = k - g$, and expected holding-period return $E(\text{HPR}) = \frac{E(D_1) + E(P_1) - P_0}{P_0}$.

Bond valuation

Discounting promised cash flows by nominal annual yield to maturity $y$: for annual coupons, $P_0 = \sum_{t=1}^T \frac{C}{(1+y)^t} + \frac{F}{(1+y)^T}$; for $m$ payments per year, $P_0 = \sum_{t=1}^{mT} \frac{C/m}{(1+y/m)^t} + \frac{F}{(1+y/m)^{mT}}$; for zero coupons, $P_0 = \frac{F}{(1+y)^T}$; and for a perpetuity, $P_0 = \frac{C}{y}$.

YTM, current yield and HPR

Bond return measures: Yield to Maturity (YTM, $y$) is the internal discount rate equating price to promised payments; current yield is $\frac{C}{P_0}$; and holding-period return is $\text{HPR} = \frac{\text{coupons received} + P_1 - P_0}{P_0}$ (or $\frac{P_1 - P_0}{P_0}$ for zero coupons).

Treasury Inflation-Protected Securities (TIPS)

Inflation-indexed bonds where principal adjusts via $F_t = F_{t-1}(1 + \pi_t)$ and coupon is $C_t = c F_t$, yielding nominal return $r_t^N = \frac{C_t + F_t - F_{t-1}}{F_{t-1}}$ and real return $r_t^R = \frac{1 + r_t^N}{1 + \pi_t} - 1$.

Credit risk and embedded options

Bond valuation features where corporate yield spread is $s = y_{\text{corporate}} - y_{\text{benchmark}}$ with price sensitivity $\frac{\Delta P}{P} \approx -D^* \Delta s$. Callable features permit issuer redemption (reducing bond value), whereas puttable and convertible features permit investor actions (increasing bond value).

Macaulay duration ($D$)

The present-value-weighted average maturity of cash flows received from a bond: $D = \sum_{t=1}^T \frac{t \times \text{PV}(\text{CF}_t)}{P} = \frac{1}{P} \sum_{t=1}^T \frac{t \times \text{CF}_t}{(1+y)^t}$, which strictly equals maturity $T$ for a zero-coupon bond.

Modified duration ($D^*$)

A direct measure of bond percentage price sensitivity to yield changes: $D^* = \frac{D}{1+y}$ (or $D^* = \frac{D}{1+y/m}$ for $m$ payments per year), providing first-order price change $\frac{\Delta P}{P} \approx -D^* \Delta y$ and yield change $\Delta y \approx -\frac{\Delta P / P}{D^*}$.

Bond convexity ($\mathcal{C}$)

A measure of the curvature of the bond price-yield curve defined by $\mathcal{C}=\frac{1}{P}\sum_{t=1}^{T}\frac{t(t+1)\text{CF}_t}{(1+y)^{t+2}}$ , refining price sensitivity to $\frac{\Delta P}{P} \approx -D^* \Delta y + \frac{1}{2} \mathcal{C} (\Delta y)^2$.

Negative convexity

The characteristic where price gains are limited as interest rates drop, typically caused by embedded prepayment or call options in callable bonds and mortgage-backed securities; price risk is evaluated using effective duration $D_{\text{eff}} = \frac{P_- - P_+}{2 P_0 \Delta y}$.

Spot rates and term structure

Maturity-matched yields on zero-coupon bonds ($s_t$) used to discount individual cash flows: zero-coupon price $P_0 = \frac{F}{(1+s_T)^T}$ and coupon bond price $P_0 = \sum_{t=1}^T \frac{\text{CF}_t}{(1+s_t)^t}$, capturing variation across maturities that a single YTM averages out.

Forward interest rates

Interest rates agreed upon today for loans extending over future periods, governed by no-arbitrage relationships: $(1 + f_n) = \frac{(1 + s_n)^n}{(1 + s_{n-1})^{n-1}}$ for a one-year loan starting at $n-1$, and $(1 + s_n)^n = (1 + s_1) \prod_{t=2}^n (1 + f_t)$.

Expectations and liquidity premium theories

Theories of term structure: Pure Expectations Theory states forward rates equal expected future short rates ($f_t \approx E(r_t)$), while Liquidity Premium Theory states $f_t \approx E(r_t) + \text{LP}_t$, where $\text{LP}_t = f_t - E(r_t)$ compensates investors for holding longer-term bonds.

Interest-rate swap (IRS)

An Interest Rate Swap (IRS) is a derivative contract in which two parties exchange interest payments, typically a fixed interest rate for a floating interest rate, on an agreed notional principal.

The principal itself is normally not exchanged.

Complete bond valuation procedure

A multi-step calculation:

(1) price $P_0$;

(2) compute Macaulay duration $D=\frac{\sum_{t=1}^{T}t\cdot\left(\frac{CF_{t}}{\left(1+y\right)^{t}}\right)}{P_0}$ ;

(3) compute $D^{*}=\frac{D}{1+y}$ ;

(4) apply $\frac{\Delta P}{P} \approx -D^* \Delta y$;

(5) solve $\Delta y$;

(6) compute convexity $\mathcal{C}$;

(7) apply $\frac{\Delta P}{P} \approx -D^* \Delta y + \frac{1}{2} \mathcal{C} (\Delta y)^2$.

Behavioral finance

The field investigating how psychological biases and limits to arbitrage cause persistent deviations from rational expectations and efficient markets, modeled in subjective valuation as $\frac{P_t}{D_t} = \frac{1 + E_t^S[g_{t+1}]}{E_t^S[r_{t+1}] - E_t^S[g_{t+1}]}$.

Where:

• P/D: Current price-dividend ratio.

• E^S[g]: Subjectively expected dividend growth.

• E^S[r}: Subjectively expected return.

• S: Subjective beliefs rather than necessarily rational expectations.

This is a Gordon growth-style valuation formula using subjective expectations, assuming constant expected growth and returns, with expected return exceeding expected growth.

Rational-expectations calibration test

An empirical regression testing whether subjective expectations are unbiased predictors of future realizations: $g_{t+1} = a + b E_t^S[g_{t+1}] + \epsilon_{t+1}$, evaluated against the rational-expectations benchmark of joint null $a = 0$ and $b = 1$.

• a: Intercept, measuring systematic bias when the forecast is zero.

• b: Slope, measuring how strongly actual outcomes respond to expected outcomes. b<1 overreaction

Return extrapolation and Degree of Extrapolation (DOX)

A belief-formation model where investors project past returns into the future: $E_t^S[R_{t+1}] = b \sum_{j=1}^\infty w_j R_{t-j+1}$

with decaying weights $w_{j}=\frac{\lambda^{j-1}}{\sum_{k=1}^{\infty}\lambda^{k-1}}=(1-\lambda)\lambda^{j-1}$ , memory parameter $\lambda\in(0,1)$ and degree of extrapolation $\text{DOX}=1-\lambda$, high DOX indicates short memory

k is the number of periods used

Diagnostic expectations

A belief distortion where agents overweight outcomes that become more representative after recent shocks: for growth process $g_t = c + \rho g_{t-1} + \epsilon_t$, rational expectations update by $\rho \epsilon_t$, and higher persistence $\rho$ amplifies overreaction, whereas $\rho = 0$ produces zero overreaction.

Behavioral heuristics and biases

Behavioral heuristics are mental shortcuts investors use to make decisions. Behavioral biases are systematic errors in judgment that can result from these shortcuts or from emotions and motivated beliefs.

For your Investment Analysis exam, the four key concepts are:

Concept	Explanation	Investment example
1. Representativeness	Investors judge probabilities based on how closely something resembles a familiar pattern, rather than using all relevant information.	Assuming a company with strong recent growth will continue growing rapidly.
2. Availability	Investors give excessive weight to information that is easy to recall or particularly salient.	Overestimating the probability of a stock market crash after seeing extensive news coverage of one.
3. Overconfidence	Investors overestimate the accuracy of their information, forecasts or investment abilities.	Trading excessively because they believe they can consistently outperform the market.
4. Optimism	Investors systematically overestimate the likelihood of favorable outcomes and underestimate unfavorable ones.	Expecting unusually high returns from their own investments.

Important distinction

• Heuristics: Shortcuts used to process information, such as representativeness and availability.

• Biases: Systematic distortions in beliefs or judgments, such as overconfidence and optimism

Limits to arbitrage

Limits to arbitrage are risks and constraints that prevent rational investors from correcting mispriced securities, allowing mispricing to persist.

Limit	Explanation
1. Fundamental risk	The fundamental value of an asset may change unexpectedly while an arbitrageur holds a position.
2. Noise-trader risk	Irrational investors may push prices even further away from fundamental values before they correct.
3. Implementation costs	Transaction costs, short-selling restrictions, borrowing fees and margin requirements make arbitrage costly or impossible.
4. Asset managers' short-term incentives	Fund managers are evaluated over relatively short periods and may avoid profitable long-term strategies because of interim losses or investor withdrawals.
5. Synchronization risk	Arbitrageurs may recognize mispricing but cannot coordinate when others will trade, making the timing of price correction uncertain.

Survival of irrational investors and persistent mispricing

The theoretical rationale explaining why noise traders persist in financial markets: noise-trader risk discourages rational arbitrageurs, irrational traders can earn higher returns by bearing excess risk, rational traders face short horizons, and imperfect learning allows biases to recur.

Behavioral explanations of return predictability

The framework where investor belief distortions generate predictable return reversals in predictive regressions $R_{t+k+1} = a + b X_t + \epsilon_{t+k+1}$, where predictability strengthens with higher extrapolator share ($1 - \omega$) and higher degree of extrapolation ($\text{DOX}=1-\lambda$ ).

Behavioral explanations of cross-sectional anomalies

The hypothesis that persistent regression alphas in multi-factor models ($r_{it} - r_{ft} = \alpha_i + \sum_{j=1}^K \beta_{ij} f_{jt} + \epsilon_{it}$) stem from systematic investor cognitive biases and incomplete arbitrage correction rather than missing risk factors.

Call and put options

Contracts granting the right, but not the obligation, to buy (call) or sell (put) an asset at strike price $K$ on or before expiration $T$: call payoff is $C_T = \max(S_T - K, 0)$, put payoff is $P_T = \max(K - S_T, 0)$, call profit is $\pi_{\text{call}} = \max(S_T - K, 0) - c_0$, and put profit is $\pi_{\text{put}} = \max(K - S_T, 0) - p_0$.

Option payoffs and combinations

Derivative terminal payoff diagrams and strategies, including protective puts ($\text{long stock} + \text{long put}$), covered calls ($\text{long stock} + \text{short call}$), and long straddles ($\text{long call} + \text{long put}$ at the same strike $K$ and expiration), where combined payoff equals the sum of component payoffs.

Investment fund structures

Institutional vehicles for pooling investor capital: mutual funds are regulated open-end investment vehicles with retail access, daily liquidity at $\text{NAV} = \frac{\text{Assets} - \text{Liabilities}}{\text{Shares outstanding}}$, and long-only strategies; hedge funds cater to qualified investors using short sales, leverage, derivatives, and performance fees.

Sharpe ratio; Modigliani–Modigliani measure ($M^2$)

Total-risk performance metrics: Sharpe ratio is $S_p = \frac{E(R_p) - r_f}{\sigma_p}$, and $M^2$ equates portfolio volatility to market volatility ($w_p = \frac{\sigma_M}{\sigma_p}$) to calculate adjusted return $E(R_p^*) = r_f + w_p [E(R_p) - r_f]$ and excess return $M^2 = E(R_p^*) - E(R_M) = \sigma_M (S_p - S_M)$.

Treynor ratio; $T^2$ measure

Systematic-risk performance metrics: Treynor ratio is $T_p = \frac{E(R_p) - r_f}{\beta_p}$, and $T^2$ scales portfolio beta to market beta ($w_p = \frac{1}{\beta_p}$) to evaluate return $E(R_p^*) = r_f + w_p [E(R_p) - r_f]$ and excess return $T^2 = E(R_p^*) - E(R_M) = T_p - T_M$, where $T_M = E(R_M) - r_f$.

T² adjusts the portfolio to have the same systematic risk as the market

Jensen's alpha

The abnormal return relative to the CAPM required return: $\alpha_p = E(R_p) - \{r_f + \beta_p [E(R_M) - r_f]\}$, measuring portfolio outperformance after controlling for systematic risk ($\beta$).

Dollar-weighted and time-weighted returns

Alternative portfolio return measures: Dollar-weighted return is the internal rate of return (IRR) solving $\sum_{t=0}^T \frac{\text{CF}_t}{(1 + \text{IRR})^t} = 0$; Time-weighted return compounds subperiod returns via $R_{\text{TW}} = \prod_{t=1}^T (1 + R_t) - 1$, eliminating distortions from external cash flow timing.

Market timing and performance attribution

The process of identifying sources of portfolio returns and testing dynamic market exposure via the Treynor–Mazuy model: $R_{pt} - r_{ft} = \alpha_p + \beta_p(R_{Mt} - r_{ft}) + \gamma_p(R_{Mt} - r_{ft})^2 + \epsilon_{pt}$, where a statistically positive quadratic coefficient $\gamma_p > 0$ indicates successful market timing.