Book 5. Risk and Investment Management

FRM Part 2

IM 1. Factor Theory

Presented by: Sudhanshu

Module 1. Factors that Affect Prices and the CAPM

Module 2. Multifactor Models, Pricing Kernels and Efficient Market Theory

Module 1. Factors that Impact Asset Prices and the CAPM

Topic 1. Factors that Impact Asset Prices and Risk Premium

Topic 2. Capital Asset Pricing Model

Topic 3. Six Important Lessons of the CAPM

Topic 4. Shortcomings of the CAPM

Topic 1. Factors that Impact Asset Prices and Risk Premium

Core Idea: It's not the asset itself, but the exposure to the underlying factor risks that earns a risk premium.
What are Factor Risks? Exposures to "bad times" (e.g., low economic growth, high inflation). Investors are compensated with a risk premium for bearing these risks.
Examples of Factors:
- The Market: A fundamental tradable factor.
- Economic Factors: Inflation, economic growth.
- Investing Styles: Value/growth, low volatility, momentum.
Three Principles of Factor Risk:
1. Factors over Assets: What matters is the exposure to the underlying risk factors, not the specific asset.
2. Bundles of Factors: Assets are typically a collection of different risk factors (e.g., a corporate bond has interest rate risk and default risk). Some assets, like equities and government bonds, can be thought of as factors themselves.
3. Optimal Exposures: Each investor has their own optimal level of exposure to various risk factors, including volatility.

Practice Questions: Q1

Q1. Which of the following concepts would least likely meet the definition of a factor?

A. Market.

B. Volatility.

C. Hedge funds.

D. Momentum investing style.

Practice Questions: Q1 Answer

Explanation: C is correct.

Assets, including corporate bonds, private equity, and hedge funds, are not
considered factors themselves, but contain many factors, such as equity risk,
interest rate risk, volatility risk, and default risk.
Some assets, like equities and government bonds, can be thought of as factors
themselves. Factors may also include the market (a tradable investment factor),
interest rates, or investing styles (including value/growth, low volatility, or
momentum).

Topic 2. Capital Asset Pricing Model (CAPM)

What is the CAPM? A single-factor model that describes how an asset's price and behavior relate to the overall market.
Key Assumptions:
- Single Factor: The only relevant factor is the market portfolio.
- Risk Measure: An asset's risk is not its own volatility, but its covariance with the market, measured by beta ( $β$ ).
How it Works: The CAPM posits that an asset's risk premium is determined solely by its beta. Risk is defined by an asset's movement relative to the market, and not in isolation.
Formula: $E (R_{i}) = R_{F} + β_{i} \times (E (R_{M}) - R_{F})$
- $E (R_{i})$ = Expected return of the asset
- $R_{F}$ = Risk-free rate
- $E (R_{M})$ = Expected return of the market
- $β_{i}$ = Beta of the asset

Topic 3. Six Important Lessons of the CAPM

Lesson 1: Hold the factor, not the individual asset
- CAPM assumes that stocks are held proportionally to market capitalization with the market portfolio as the sole factor, diversifying away idiosyncratic risk while retaining systematic market risk.
- CAPM assumes investors prefer diversified portfolios as diversification offsets poor performers with strong performers, improving Sharpe ratios until reaching the optimal market portfolio.
- The mean-variance efficient (MVE) market portfolio represents maximum Sharpe ratio given investor preferences, with risk-return combinations of the risky MVE portfolio and risk-free asset represented by the capital allocation line (CAL).
- In CAPM equilibrium, demand equals supply with all investors holding the MVE market portfolio; an unowned risky asset would be overpriced with too-low expected returns, preventing equilibrium.
- The market factor is determined by investor risk aversions and utilities, with risk premiums persisting since investors cannot use arbitrage to eliminate systematic risk.

Topic 3. Implications of Using the CAPM

Lesson 2: Investors have their own optimal factor risk exposures
- Every investor holds the same risky MVE market portfolio, but the proportion in which they hold it differs.
- Investors hold different combinations of the risk-free asset and the risky portfolio, representing various positions along the CAL.
Lesson 3: The average investor is fully invested in the market
- An investor with average risk aversion holds 100% of the risky MVE market portfolio, representing the tangency point of the MVE frontier and the CAL.
- The average investor's risk aversion equals the risk aversion of the market.
Lesson 4: Exposure to factor risk must be rewarded
- In equilibrium, when all investors hold the same risky MVE portfolio, the capital allocation line (CAL) becomes the capital market line (CML).
- The CML risk premium depends on investor risk aversion and market portfolio volatility:
  - where γ̄ is average risk aversion and is market variance.
- During volatile periods (e.g., 2007-2009 financial crisis), equity prices fall and expected returns increase, with risk premium proportional to market variance.
- Market variance captures only systematic risk after diversifying away idiosyncratic risk, and this systematic risk is rewarded through the risk premium.
- When average investor risk aversion increases, the market risk premium also increases proportionally.

\mathrm{E}\left(\mathrm{R}_{\mathrm{M}}\right)-\mathrm{R}_{\mathrm{F}}=\bar{\gamma} \times \sigma_{\mathrm{M}}^2

\sigma_{\mathrm{M}}^2

Topic 3. Implications of Using the CAPM

Lesson 5: Risk is measured as beta exposure
- An asset’s risk is measured by its factor exposure (beta) — higher beta → higher expected return (if risk premium > 0).
- Under CAPM, the risk premium is defined as:
  $E(Ri)−RF=βi[E(RM)−RF]E(R_i) - R_F = \beta_i [E(R_M) - R_F]$
- Beta measures how much an asset’s returns co-move with the market:
  - $βi=cov(Ri,RM)var(RM)\beta_i = \frac{\text{cov}(R_i, R_M)}{\text{var}(R_M)}$
- High beta → greater market sensitivity → higher risk premium but lower diversification benefit.
- Low beta → performs better in down markets → valuable for diversification.
- Investor behavior: High-beta assets are riskier, requiring higher compensation; low-beta assets are defensive and desirable.
- During the 2007–2009 crisis, safe-haven assets (e.g., gold, government bonds) became so demanded that they had negative expected returns — investors effectively paid for safety.
Lesson 6: Valuable assets have low risk premiums
- CAPM risk premium represents the reward for holding assets during bad times, defined as periods of low market returns when the market portfolio (risk factor) underperforms.
- Assets with losses during low market return periods have high betas, indicating higher risk and requiring higher risk premiums.
- Low beta assets provide positive payoffs when markets perform poorly, making them valuable to investors and requiring lower risk premiums.

E\left(R_i\right)-R_F=\beta_i\left[E\left(R_M\right)-R_F\right]

\beta_i=\frac{\operatorname{cov}\left(R_i, R_M\right)}{\operatorname{var}\left(R_M\right)}

Practice Questions: Q2

Q2. According to the capital asset pricing model (CAPM), in equilibrium, all investors hold the mean-variance efficient portfolio. Which of the following investor types is an exception to this assumption?

A. Infinitely risk-averse investors.

B. Infinitely risk-tolerant investors.

C. Investors who hold some of the risk-free asset.

D. Investors who hold the market portfolio.

Practice Questions: Q2 Answer

Explanation: A is correct.

According to the CAPM, all investors hold a combination of the risky mean- variance efficient market portfolio and the risk-free asset. All investors hold the same market portfolio (therefore the mean-variance efficient portfolio is the market portfolio), and it is only the quantity of holdings that differs among investors. The only exception to this assumption is an infinitely risk-averse investor, who would only hold the risk-free asset.

Practice Questions: Q3

Q3. Assets that have losses during periods of low market returns have:

A. low betas and low risk premiums.

B. high betas and low risk premiums.

C. low betas and high risk premiums.

D. high betas and high risk premiums.

Practice Questions: Q3 Answer

Explanation: D is correct.

Assets that have losses during periods of low market returns have high betas (high sensitivity to market movements), which indicates they are risky and, therefore, should have high risk premiums. Low beta assets have positive payoffs when the market performs poorly, making them valuable to investors. As a result, investors do not require high risk premiums to hold these assets.

Practice Questions: Q4

Q4. Which of the following statements best describes the relationship between asset payoffs and "bad times" events (high inflation, low economic growth, or both)?

A. The higher the expected payoff of an asset in bad times, the higher the asset's expected return.

B. The higher the expected payoff of an asset in bad times, the lower the asset's expected return.

C. The expected payoff of an asset in bad times is unrelated to the asset's expected return, because it depends on investor preferences.

D. The expected payoff of an asset in bad times is unrelated to the asset's expected return, because arbitrageurs eliminate any expected return potential.

Practice Questions: Q4 Answer

Explanation: B is correct.

The higher the expected payoff of an asset in bad times, the lower the asset’s expected return. Assets that have a positive payoff in bad times are valuable to hold, leading to high prices and, therefore, low expected returns.

Topic 4. Shortcomings of the CAPM

The CAPM relies on several simplifying assumptions that are unrealistic in the real world:
1. Investors only have financial wealth: Ignores other sources of wealth and liabilities, such as human capital and inflation-linked debt.
2. Mean-variance utility: Assumes a symmetric view of risk, when in reality investors have asymmetric preferences (disliking losses more than they like gains).
3. Single-period horizon: Assumes all investors have a single-period investment horizon, ignoring the multi-period strategies like rebalancing.
4. Homogeneous (identical) expectations: Assumes all investors have identical expectations about asset returns, which is not realistic.
5. Frictionless markets: Ignores real-world factors like taxes, transaction costs, and trading restrictions (e.g., short selling).
6. Price takers: Assumes all investors are small price takers, when in reality, large institutional investors can act as price setters.
7. Free information: Assumes information is free and available to everyone, ignoring the fact that information can be costly and a factor in itself.

Practice Questions: Q5

Q5. Which of the following statements least likely represents a limitation of the capital asset pricing model (CAPM)?

A. All investors are price takers.

B. Information is costless to obtain.

C. All investors have the same expectations.

D. There are uniform taxes and transaction costs.

Practice Questions: Q5 Answer

Explanation: D is correct.

The CAPM does not assume uniform taxes and transaction costs; it assumes there are no taxes or transaction costs (i.e., frictionless markets). The other limiting assumptions of the CAPM include:

Investors only have financial wealth.
Investors have mean-variance utility.
Investors have a single period investment horizon.
Investors have homogeneous (identical) expectations.
All investors are price takers.

Module 2. Multifactor Models, Pricing Kernels, And Efficient Market Theory

Topic 1. Multifactor Models

Topic 2. Pricing Kernels

Topic 3. Pricing Kernels vs. Discount Rate Models

Topic 4. Efficient Market Theory

Topic 1. Multifactor Models

Definition: Multifactor models expand on the CAPM by incorporating multiple risk factors beyond just the market.
Comparison to CAPM:
- CAPM: A single-factor model where the only factor is the market portfolio. It defines "bad times" as low market returns.
- Multifactor Models: Incorporate multiple risk factors like low economic growth, low GDP growth, and low consumption, providing a more comprehensive view of "bad times."
Arbitrage Pricing Theory (APT): One of the earliest multifactor models, the APT describes expected returns as a linear function of exposures to common macroeconomic risk factors.
Core Lessons (similar to CAPM):
1. Diversification is Key: Diversification helps remove idiosyncratic risk, with factors being the tradable components that remain.
2. Optimal Exposures: Investors have an optimal exposure to each factor risk.
3. Average Investor: The average investor still holds the market portfolio.
4. Factor Risk is Rewarded: Each factor has its own risk premium.
5. Beta Measures Risk: An asset's risk is measured by its exposures to each factor (its factor betas).
6. Valuable Assets: Assets that perform well in "bad times" have low risk premiums.

Topic 2. Pricing Kernels

Definition: A pricing kernel, or stochastic discount factor (SDF), is a random variable that serves as an index of "bad times" for asset valuation.
Purpose: It is a single variable that captures the collective effect of all relevant factors and states that define "bad times."
CAPM as a Special Case: The CAPM is a special case of this model where the SDF moves linearly with the market return:
- $m = a + b \times R_{m}$
  - where
    - $m$ = Stochastic Discount Factor (SDF)
    - $R_{m}$ = Market Return
Multifactor Expansion: The model can be expanded to include multiple factors, where m depends on a vector of these factors:
- $m = a + b_{1} f_{1} + b_{2} f_{2} + ... + b_{k} f_{k}$
SDF as Marginal Utility: The SDF can be interpreted as marginal utility. "Bad times" are periods when an extra dollar of income is highly valuable (i.e., marginal utility is high), such as during periods of low income or low economic growth.

Topic 3. Pricing Kernels vs. Discount Rate Models

Traditional Discount Rate Model: The price of an asset is calculated by discounting future cash flows at an appropriate discount rate.
- The discount rate, $E (R_{i})$ , is often determined by models like the CAPM.
SDF (Pricing Kernel) Model: The asset's price is determined by the expected value of its payoff multiplied by the stochastic discount factor.
- $P_{i} = E [m \times p a yo f f_{i}]$
- The SDF, m, acts as the "stochastic discount factor" because the payoffs are discounted using m.
Risk Premium in the SDF Model:
- where
  - $β_{i, m}$ measures the asset's co-movement with the SDF.
  - $λ_{m}$ is the price of the "bad times" risk.
- The formula shows that assets with a positive payoff in "bad times" (low covariance with m) are valuable, leading to high prices and low expected returns.
The equation for expected return can also be modeled as having exposure to the risk-free rate and multiple betas in the SDF model.
Each beta represents a different macroeconomic factor, such as inflation, economic growth, the market portfolio, or investment strategy:

P_i=E\left[\frac{payoff_i}{1+E\left(R_i\right)}\right]

E\left(R_i\right)-R_F=\frac{\operatorname{cov}\left(R_i, m\right)}{\operatorname{var}(m)} \times\left(-\frac{\operatorname{var}(m)}{E(m)}\right)=\beta_{i, m} \times \lambda_m

E\left(R_i\right)=R_F+\beta_{i, 1} \times E\left(f_1\right)+\beta_{i, 2} \times E\left(f_2\right)+\ldots+\beta_{i, k} \times E\left(f_{i k}\right)

Topic 4. Efficient Market Theory

Efficient Market Hypothesis (EMH): The EMH suggests that speculative trading is costly and active managers cannot consistently beat the market. The average investor can "beat the market" simply by saving on transaction costs.
Grossman and Stiglitz (1980): Proposed that markets are "near-efficient" and that information has a cost. Active managers' search for inefficiencies makes markets more efficient. This suggests that inefficiencies are most likely to be found in illiquid segments where information is not freely available.
Explanations for Inefficiencies (or market mispricings):
- Rational Explanation: High returns compensate for losses during "bad times." The key is accurately defining what constitutes "bad times" for a specific investor.
- Behavioral Explanation: Market inefficiencies are caused by agents' over- or under-reactions to news. These biases can persist because of market barriers (e.g., structural or regulatory) that prevent rational investors from taking advantage of the mispricing.

Practice Questions: Q1

Q1. Market efficiency can be described with the efficient market hypothesis (EMH). Regarding the definition of EMH and the rational and behavioral explanations for this approach, the EMH suggests that:

A. speculative trading is costless.

B. active managers cannot generally beat the market.

C. under the behavioral explanation, losses during bad times are compensated for by high returns.

D. under the rational explanation, it is agents' under- or overreactions to news that generates high returns.

Practice Questions: Q1 Answer

Explanation: B is correct.

The EMH implies that speculative trading is costly, and active managers cannot generally beat the market. Under the rational explanation of behavioral biases, losses during bad times are compensated for by high returns. Under the behavioral explanation, it is agents' under- or overreactions to news that generates high returns. Market barriers may make it difficult to take advantage of mispricings.