IA 12-15

Comprehensive vocabulary flashcards covering empirical asset pricing, multi-factor models, performance evaluation metrics, and the Treynor-Black portfolio optimization framework.

Goal of an empirical CAPM test

To test whether expected returns are linearly related to market beta as predicted by CAPM, and whether variables other than beta help explain differences in expected returns.

Three key steps of an empirical asset-pricing test

1. Estimate exposures $\beta$.

2. Estimate risk prices $\gamma$.

3. Test whether model restrictions hold.

Time-series CAPM test

For each asset $i$, regress its excess return on the market excess return: $r_{it} - r_{ft} = \alpha_i + \beta_i (r_{mt} - r_{ft}) + \epsilon_{it}$ and test whether $\alpha = 0$. Beta represents exposure to market risk, while alpha represents abnormal return unexplained by CAPM.

Cross-sectional CAPM test

Tests whether assets with higher beta actually earn higher average returns as CAPM predicts, evaluating the restriction that average returns increase linearly with beta and no other variable explains expected returns once beta is included. In regressions on beta and residual variance, CAPM predicts intercept $\gamma_0 = 0$, beta coefficient $\gamma_1 = E(r_M) - r_f$, and residual-variance coefficient $\gamma_2 = 0$.

Fama-MacBeth test: goal

To test whether estimated risk exposures $\beta$ are priced in the cross-section of returns.

How does the Fama–MacBeth procedure test the CAPM

Stage 1 — Time-series regressions: Estimate each stock's market beta:

$$r_{it}-r_{\operatorname{ft}}=\alpha_{i}+\beta_{i}\left(r_{Mt}-r_{\operatorname{ft}}\right)+\epsilon_{it}$$

Run one regression for each of the N stocks.

Stage 2 — Cross-sectional regressions: For each of the T periods, regress stock excess returns on estimated betas:

$$r_{i}-r_{\operatorname{f}}=\lambda_0+\lambda_1\beta_{i}^{pred}+\epsilon_{i}$$

• $\lambda_0^{pred}$: Estimated intercept; should be zero under the CAPM.

• $\lambda_1^{pred}$: Estimated price of market risk; should equal the average market risk premium under the CAPM.

• $\beta_{i}^{pred}$: Estimated market beta from Stage 1

Total number of regressions:

N + T

For 100 stocks and 100 periods: 200 regressions

CAPM prediction for the Security Market Line

Expected return increases linearly with beta, having an intercept equal to $r_f$ and a slope equal to the market risk premium. Empirically, the SML has been flatter, meaning high-beta stocks earn less and low-beta stocks earn more than standard CAPM predicts.

Positive-alpha and competition

A positive-alpha stock earns a return above that predicted by its beta under the asset-pricing model. Competition should eliminate positive alpha over time: buying leads to higher price, which leads to lower expected return, driving alpha toward $0$.

Betting Against Beta (BAB): basic idea

Exploits the empirically flatter SML by buying low-beta assets, shorting high-beta assets, and adjusting leverage to make risk exposures comparable. Leverage constraints cause investors to overpay for high-beta assets and underprice low-beta assets.

Gibbons-Ross-Shanken test

The Gibbons–Ross–Shanken (GRS) test examines whether an asset pricing model (such as CAPM or Fama–French) explains the returns of multiple portfolios simultaneously.

- Null hypothesis: All portfolios have zero abnormal returns (alphas).

H₀: $\alpha_1 = \alpha_2 = \dots = \alpha_N = 0$

• Alternative hypothesis: At least one portfolio has a nonzero alpha.

• Reject H₀: At least one alpha differs significantly from zero, meaning the model fails to fully explain portfolio returns.

• Do not reject H₀: Insufficient evidence against the model; this does not prove it is correct.

The test accounts for correlation between portfolio residuals.

Important: When regressing raw returns on market excess returns, the null hypothesis is that all intercepts equal r_f, rather than zero (assuming a constant risk-free rate)

Risk as comovement

Common movements across many stocks reveal systematic sources of risk that cannot be diversified away, motivating looking beyond firm-specific volatility toward common factors.

Arbitrage Pricing Theory (APT)

An asset-pricing framework where a small number of common factors explain systematic variation in asset returns, allowing multiple sources of systematic risk to matter beyond a single market factor.

Data-driven/statistical risk factor

Common factors extracted directly from return data without specifying the economic factor beforehand. Its advantage is identifying important common return variation, but its drawback is that resulting factors may lack a clear economic interpretation.

Why look for additional sources of risk?

If market beta does not fully explain differences in returns, other systematic risks may command risk premia, motivating the search for risk factors beyond the market factor.

Factor-mimicking portfolio (FMP)

A tradable portfolio constructed so that its return tracks exposure to a non-traded economic factor, needed because many macroeconomic risks are not directly tradable.

Constructing a factor-mimicking portfolio

1. Estimate stocks' sensitivities to the economic factor.

2. Sort stocks by their sensitivities.

3. Construct a high-minus-low long-short portfolio whose return serves as a tradable representation of the underlying factor.

Labor-market risk factor

A factor-mimicking portfolio whose return tracks changes in aggregate labor-market conditions, converting non-traded labor-market risk into a portfolio return usable in asset-pricing tests.

Labor-market risk exposure and hedging

To acquire exposure, hold assets whose returns covary positively with adverse labor-market conditions. To hedge labor-market risk, hold assets that perform well when labor conditions deteriorate; because assets hedging bad states provide insurance, investors accept lower expected returns on them.

Risk premium and hedging ability

An asset that performs well in bad economic states provides insurance. Because investors value this hedge, they are willing to hold it at a lower expected return: a better hedge implies a lower required risk premium.

Pricing climate risk

A 5-step process:

1. Collect stock returns and a climate variable (e.g., temperature changes).

2. Estimate stocks' climate sensitivities.

3. Sort stocks by sensitivity to construct a high-minus-low climate FMP.

4. Estimate stocks' betas on the climate FMP.

5. Use Fama-MacBeth regressions to test whether climate betas are priced.

Climate risk hedging premium

If exposure to a climate factor pays off during bad states, it acts as a hedge; investors accept a lower expected return, resulting in a lower or negative risk premium for the hedging factor.

Fama-French (1992): main finding

Size and book-to-market help explain the cross-section of average stock returns beyond market beta: smaller firms earned higher average returns than larger firms, and high book-to-market stocks earned higher average returns than low book-to-market stocks (size coefficient is negative, book-to-market coefficient is positive).

From characteristics to factors

The reasoning that if firm characteristics (such as size or book-to-market) predict returns because they represent systematic risk, there must be common return factors associated with them, motivating the transition from characteristics to risk factors.

Multi-factor model

An asset-pricing model where expected returns depend on exposures to several systematic risk factors rather than market beta alone, with each factor having its own factor exposure $\beta$ and factor risk premium.

Testing a multi-factor model Fama-French

Can test using 2-step FMB test

Stage 1 — Time-series regression: Estimate each stock's factor betas:

$r_{it}-r_{\operatorname{ft}}=\alpha_{i}+\beta_{i,CAPM}\left(r_{mt}-r_{\operatorname{ft}}\right)+\beta_{i,SMB}\left(r_{SMBt}\right)+\beta_{i,HML}\left(r_{HMLt}\right)+\epsilon_{it}$Stage 2 — Cross-sectional regression: Estimate factor risk prices:

$r_{i}^{av}-r_{f}^{av}=\lambda_0+\lambda_{CAPM}\beta_{i,CAPM}^{pred}+\lambda_{SMB}\beta_{i,SMB}^{pred}+\lambda_{HML}\beta_{i,HML}^{pred}+\epsilon_{i}$ Factor interpretation:

• Market beta: Exposure to general market movements.

• SMB beta: Positive indicates small-cap-like exposure; negative indicates large-cap-like exposure.

• HML beta: Positive indicates value-like exposure; negative indicates growth-like exposure.

A small-cap stock would generally be expected to have positive SMB exposure, although its actual beta must be estimated.

Equity premium puzzle

The historical observation that equity returns have exceeded safe asset returns by a margin too large to be explained by standard consumption-based models unless investors possess implausibly high risk aversion.

DELA: responsible investing

The integration of ESG factors to protect long-term investment value and performance using four instruments: 1. Exclusions, 2. Voting & engagement, 3. ESG integration, and 4. Impact investing (e.g., green bonds, renewable energy, farmland, sustainable timberland).

DELA: investment approach

Long-term investment framework emphasizing that systematic risk earns a long-run risk premium, diversification reduces risk, market timing is not useful, active management should be used selectively due to difficulty in beating benchmarks, and responsible investing must be integrated.

DELA: portfolio optimisation structure

A three-level hierarchy for portfolio construction: Asset classes $\rightarrow$ Allocation within asset classes $\rightarrow$ Individual securities/managers.

DELA: regional allocation

Constructs a market-neutral baseline using market capitalization, GDP, and regional company revenues, then tactically deviates based on growth, diversification, valuations, market efficiency, ESG, and hedging costs.

DELA's manager-selection framework

Framework emphasizing repeatable investment process over past returns alone, evaluating managers across eight dimensions: Philosophy, Process, Portfolio, Performance, People, Price, Parent, Planet.

Dollar-weighted return vs time-weighted return

Dollar-weighted return is the investor's IRR accounting for the size and timing of external cash flows; time-weighted return is the fund's return that eliminates external cash-flow effects to measure the investment manager's performance.

Time-weighted return calculation

Calculates subperiod returns between cash flows and geometrically links them: $1 + r_G = [(1 + r_1)(1 + r_2)\dots(1 + r_n)]^{1/n}$, weighting each subperiod equally regardless of capital under management.

Peer and index benchmarking

Peer benchmarking measures performance against comparable funds ($R_t - R_{peers,t}$), whereas index benchmarking evaluates performance against an investable reference index ($R_t - R_{benchmark,t}$).

Problem with relative performance assessment

Complications arise when funds have non-comparable investment universes, undergo style drift, benchmark against mismatched indexes, or have apparent performance distorted by arbitrary benchmark/peer selection.

Goal of risk-adjusted performance evaluation

Determining whether returns adequately compensate for risk, selecting appropriate metrics based on the portfolio's role: total risk (Sharpe, $M^2$), systematic risk (Treynor, $T^2$, Jensen's alpha), or active risk (Information Ratio).

Sharpe ratio and M²

$S_P = \frac{\bar{r}_P - r_f}{\sigma_P}$. Measures excess return per unit of total risk (volatility). Appropriate when the portfolio represents the investor's entire risky investment.

M² converts the Sharpe ratio into percentage return units by adjusting the portfolio to the market's volatility.

1. Calculate the weight invested in the risky portfolio:

$$w_{P}=\frac{\sigma_{M}}{\sigma_{P}}$$

2. Calculate the volatility-adjusted return:

$$r_{P}^{\star}=r_{f}+w_{P}\left(r_{P}^{av}-r_{f}\right)$$

3. Calculate M² relative to the market:

$$M_{P}^2=r_{P}^{\star}-r_{M}^{av}=\sigma_{M}\left(S_{P}-S_{M}\right)$$

Interpretation:

• Higher Sharpe and M² indicate better performance per unit of total risk.

• M²: Outperforms the market after adjusting for total risk.

• w_P>1: Requires borrowing; w_P <0: Partly invested in the risk-free asset.

• Appropriate when the portfolio represents the investor's entire risky investment.

Treynor measure and T²

$T_P = \frac{\bar{r}_P - r_f}{\beta_P}$. Measures excess return per unit of systematic risk. Appropriate when evaluating a portfolio that serves as one sub-component of a well-diversified overall portfolio.

1. Calculate the weight invested in the risky portfolio:

$$w_{P}=\frac{\beta_{M}}{\beta_{P}}$$

2. Calculate the beta-adjusted return:

$$r_{P}^{^{\star}}=r_{f}+w_{P}\left(r_{P}^{av}-r_{f}\right)$$

3. Calculate T² relative to the market:

$$T_{P}^2=r_{P}^{\star}-r_{M}^{av}$$

When $\beta_{M}=1$ :

$$T_{P}^2=T_{P}-T_{M}$$

Interpretation:

• Higher Treynor and T² indicate better performance per unit of systematic risk (for positive betas).

• $T^2>0$ : Outperforms the market after adjusting for systematic risk.

• w_P>1: Requires borrowing; w_P<1 : Partly invested in the risk-free asset.

• Appropriate when the portfolio is part of a well-diversified overall portfolio.

Why can M^2 and T^2 disagree?

$M^2$ evaluates total risk ($\sigma_P$), whereas $T^2$ evaluates systematic risk ($\beta_P$). Consequently, portfolios with differing levels of idiosyncratic risk can be ranked differently by the two measures.

Jensen's alpha

$\alpha_P = \bar{r}_P - [r_f + \beta_P(\bar{r}_M - r_f)]$. Measures the portfolio return earned above or below the return predicted by the CAPM for its level of market beta.

Information ratio

$IR = \frac{\alpha_P}{\sigma(e_P)}$. Measures abnormal return per unit of residual (active) risk. Used when evaluating an actively managed portfolio that will be combined with a passive market benchmark.

Market timing and timing ability

The strategy of adjusting market beta dynamically ($\beta$ increased before bull markets, decreased before bear markets). Tested via the regression $r_P - r_f = \alpha + b(r_M - r_f) + c(r_M - r_f)^2 + e_P$

• b: Linear market exposure.

• c: Market-timing coefficient.

• c>0: Convexity consistent with successful timing.

• A significantly positive c provides statistical evidence of timing ability.

Performance evaluation and manager compensation

Managers are often evaluated and compensated based on performance relative to peers or a benchmark rather than raw returns.

• Benchmark-relative performance measures whether the manager outperforms the reference portfolio.

• Deviating from the benchmark increases tracking risk, potentially leading to significant underperformance.

• Managers concerned about benchmark underperformance may avoid large active positions.

• Consequently, benchmark-based incentives can encourage benchmark-hugging rather than active risk-taking.

Key insight: Stronger performance incentives do not necessarily lead to greater active portfolio risk.

Treynor-Black: model and required inputs

Treynor–Black combines a passive market portfolio (M) with an actively managed portfolio (A) of potentially mispriced stocks to improve the overall Sharpe ratio.

Required inputs:

• Each stock: alpha $\alpha_{i}$, market beta $\beta_{i}$ , residual variance $\sigma^2\left(e_{i}\right)$.

• Market: expected risk premium $E\left(r_{M}\right)-r_{f}$ and variance $\sigma_{M}^2$

If alpha is not provided:

$$\alpha_{i}=E\left(r_{i}\right)-\left\lbrack r_{f}+\beta_{i}\left(E\left(r_{M}\right)-r_{f}\right)\right\rbrack$$

If residual variance is not provided:

$$\sigma^2\left(\epsilon_{i}\right)=\sigma_{i}^2-\beta_{i}^2\sigma_{M}^2$$

Intuition: Choose stocks with attractive abnormal returns, while accounting for their idiosyncratic risk.

TB Calculate relative weight of shares in active part

Step 1: Calculate the initial position of each stock:

$$w_{i}^0=\frac{\alpha_{i}}{\sigma^2\left(\epsilon_{i}\right)}$$

Positive alpha generates a positive position; negative alpha generates a short position.

Step 2: Normalize the positions so they sum to 1:

$$w_{i}=\frac{w_{i}^0}{\sum_{j=1}^{n}w_{^{}j}^0}$$

TB Calculate active portfolio’s alpha, beta and residual variance

Step 4: Calculate alpha of the active part

$$\alpha_A = \sum_{i=1}^n w_i \alpha_i$$

Step 5: Calculate beta of the active part

$$\beta_A = \sum_{i=1}^n w_i \beta_i$$

Step 6: Calculate the active portfolio’s residual variance

$$\sigma^2(e_{A})=\sum_{i=1}^{n}w_{i}^2\sigma^2(e_{i})$$

(assuming stock residuals are uncorrelated).

TB determine allocation between active portfolio and market

Step 7

Calculate the initial active allocation:

$$w_{A}^0=\frac{\alpha_{A}/\sigma^2(\epsilon_{A})}{(E(r_{M})-r_{f})/\sigma_{M}^2}$$

This compares the active portfolio's alpha per unit of residual variance with the market's risk premium per unit of market variance.

Adjust for the active portfolio's beta:

$$w_A^* = \frac{w_A^0}{1 + (1 - \beta_A)w_A^0}$$

Calculate the passive market weight:

$$w_M^* = 1 - w_A^*$$

These are the weights of the active and passive components in the overall risky portfolio

TB calculate stock’s final weight in the overall portfolio

Multiply its weight within the active portfolio by the overall active allocation:

$$w_{i,P}=w_{A}^{\star}w_{i}$$

TB Calculate overall portfolio’s alpha and beta

The overall portfolio combines the active portfolio and the passive market portfolio

Step 8: Alpha:

$$\alpha_P = w_A^* \alpha_A$$

The market has zero alpha, so only the active component contributes abnormal returns.

Step 9: Beta:

$$\beta_P = w_M^* + w_A^* \beta_A$$

TB Calculate overall portfolio’s expected return and risk

Step 10: Calculate expected excess return:

$$E(r_P) - r_f = \beta_P [E(r_M) - r_f] + w_A^* \alpha_A$$

Step 11: Calculate portfolio variance:

$$\sigma_{P}^2=\beta_{P}^2\sigma_{M}^2+\left(w_{A}^{*}\right)^2\sigma^2\left(e_{A}\right)$$

Step 12: Calculate the portfolio Sharpe ratio:

$$SP=\frac{E\left(r_{P}\right)-r_{f}}{\sigma_{P}}$$

The Treynor–Black maximum squared Sharpe-ratio improvement is:

$$S_P^2 = S_M^2 + \left[\frac{\alpha_A}{\sigma(e_A)}\right]^2$$

Interpretation: Active management can improve the Sharpe ratio when it generates positive abnormal returns relative to residual risk.

TB Calculate Sharpe ratio

Step 12: Calculate the portfolio Sharpe ratio:

$$SP=\frac{E\left(r_{P}\right)-r_{f}}{\sigma_{P}}$$

The Treynor–Black maximum squared Sharpe-ratio improvement is:

$$S_P^2 = S_M^2 + \left[\frac{\alpha_A}{\sigma(e_A)}\right]^2$$

Interpretation: Adding an active portfolio with nonzero alpha can improve the maximum Sharpe ratio. The higher its alpha relative to residual risk, the greater the potential improvement

Tracking error and tracking risk

Tracking error: The difference between the portfolio return and its benchmark return.

$$TE = R_P - R_M$$

Tracking risk: The volatility of those return differences.

$$\sigma(TE)=SD\left(R_{P}-R_{M}\right)$$

Higher tracking risk means the portfolio can deviate more strongly from its benchmark.

Investors may limit tracking risk to avoid excessively large active positions.

TB limitations and alpha correction

1. Alpha estimation errors: Expected abnormal returns are difficult to predict accurately. Incorrect estimates can produce extreme portfolio weights.

2. Uncorrelated residual assumption: The basic model assumes stock-specific residual returns are uncorrelated, which may be unrealistic.

3. Forecast uncertainty: Estimated alphas may need to be reduced toward zero (shrinkage) when forecasts are unreliable.

Forecast R²: Measures forecast reliability in the alpha-adjustment approach. Lower forecast R² implies less reliable predictions and generally stronger shrinkage toward zero

Zero-alpha stock in Treynor-Black

A security with $\alpha = 0$ receives an initial active position of: $w_i^0 = \frac{\alpha_i}{\sigma^2(e_i)} = 0$

Therefore:

• Its normalized active weight is zero.

• The existing stocks' active weights remain unchanged.

• Active portfolio alpha, beta and residual variance remain unchanged.

• The allocation between the active and passive portfolios remains unchanged.

Conclusion: Adding a zero-alpha stock does not change the optimal Treynor–Black portfolio under the model's assumptions