Topics 62-64

Question 1

Provide examples of factors that impact asset prices, and explain the theory of factor risk premiums

Answer 1

In the context of factor investing, it is easiest to think of assets as bundles of factor risks, where exposure to the different factor risks earns risk premiums. The underlying factors may include the market (which is a tradable investment factor), interest rates, or investing styles (including value/growth, low volatility, or momentum). Factors may also be classified as fundamental macroeconomic factors, such as inflation and economic growth.

Factor theory is based on an analysis of factor risks. Factor risks represent exposures to bad times, where these exposures are rewarded with risk premiums. Factor theory is based on three primary principles:

1. Factors are important, not assets. It is not exposure to the specific asset that matters, rather the exposure to the underlying risk factors. As a result, investors must look through assets and understand the underlying factor risks.
2. Assets represent bundles of factors. Assets typically represent bundles of risk factors, although some assets, like equities and government bonds, can be thought of as factors themselves. Other assets, including corporate bonds, private equity, and hedge funds, contain many factors, such as equity risk, interest rate risk, volatility risk, and default risk. Assets’ risk premiums reflect these risk factors.
3. Investors have differing optimal risk exposures. Investors each have different optimal exposures to risk factors. One of the important factors is volatility. Higher volatility results in higher asset risks during bad times.

Question 2

Describe the capital asset pricing model (CAPM) including its assumptions, and explain how factor risk is addressed in the CAPM

Answer 2

The capital asset pricing model (CAPM) describes how an asset behaves not in isolation, but in relation to other assets and to the market. The CAPM views not the asset’s own volatility as the relevant measure, but its covariance with the market portfolio, as measured by the assets beta.

The CAPM assumes that the only relevant factor is the market portfolio, and risk premiums are determined solely by beta.

Question 3

Implications of Using the CAPM

Answer 3

The CAPM holds six important lessons.

• Lesson 1: Hold the factor, not the individual asset.
• Lesson 2: Investors have their own optimal factor risk exposures.
• Lesson 3: The average investor is fully invested in the market.
• Lesson 4: Exposure to factor risk must be rewarded.
• Lesson 5: Risk is measured as beta exposure.
• Lesson 6: Valuable assets have low risk premiums.

Question 4

Capital allocation line (CAL)

Answer 4

Mean-variance efficient portfolio. Portfolio diversification and Sharpe ratios can be graphically represented by the mean-variance efficient frontier. When investors hold portfolios that combine the risky asset and the risk-free asset, the various risk-return combinations are represented by the capital allocation line (CAL). The risky asset in this case is the mean-variance efficient (MVE) market portfolio, which is efficient because it represents the maximum Sharpe ratio given investors’ preferences.

Question 5

SML

Answer 5

Question 6

Shortcomings of the CAPM

Answer 6

The assumptions of the CAPM break down especially in illiquid, inefficient markets where information may be costly and not available to all investors. We look at seven of these assumptions:

1. Investors only have financial wealth.
2. Investors have mean-variance utility. Mean-variance utility assumes a symmetric treatment of risk. In reality, investors have an asymmetric view of risk, disliking losses more than they like gains, which deviates from the CAPM assumptions. Therefore, in the real world, stocks exhibit different levels of downside risks. Those with higher downside risks should offer higher returns.
3. Investors have a single period investment horizon.
4. Investors have homogeneous (identical) expectations.
5. Markets are frictionless (no taxes or transaction costs).
6. All investors are price takers. In the real world, investors are often price setters and not price takers. Large (institutional) investors frequently trade on special knowledge, and large trades will often move the market.
7. Information is free and available to everyone.

Question 7

Describe multifactor models, and compare and contrast multifactor models to the CAPM

Answer 7

As mentioned, the CAPM is a single-factor model that looks at the market as the only factor and defines bad times as low returns to the market portfolio. By contrast, multifactor models incorporate other risk factors, including low economic growth, low GDP growth, or low consumption. One of the earliest multifactor models was arbitrage pricing theory (APT), which describes expected returns as a linear function of exposures to common (i.e., macroeconomic) risk factors.

The lessons from multifactor models are similar to the lessons from the CAPM:

1. Diversification is beneficial. In the CAPM, the market removes (diversifies away) idiosyncratic risk. In multifactor models, it is the tradable version of a factor that removes this risk.
2. Investors have optimal exposures. Each investor has an optimal exposure to the market portfolio (in the CAPM) or to factor risks (in multifactor models).
3. The average investor holds the market portfolio. This is true under both the CAPM and multifactor models.
4. Exposure to factor risk must be rewarded. In the CAPM, the market factor is priced in equilibrium. In multifactor models, each factor has a risk premium, assuming no arbitrage or equilibrium.
5. Risk is measured by a beta factor. In the CAPM, an asset’s risk is measured by its beta. In multifactor models, an asset’s risk is measured by its factor exposures (i.e., factor betas).
6. Valuable assets have low risk premiums. Assets that have a positive payoff in bad times are attractive, and, therefore, have low risk premiums. In the CAPM, bad times are explicitly defined as low market returns.

Question 8

Explain how stochastic discount factors are created and apply them in the valuation of assets

Answer 8

Multifactor models define bad times over multiple factors. They use the concept of a pricing kernel, also known as the stochastic discount factor (SDF), which represents a random variable used in pricing an asset. The SDF represents an index of bad times, where the bad times are indexed by a multitude of different factors and states. The SDF is denoted as m in the multifactor model, where m is a single variable that captures all bad times for any given a and b constants:

m = a + b x R_m

The CAPM is a special case of this model, where m moves linearly with the market return. However, modeling returns as linear is a shortcoming of the CAPM, which can be improved upon by using the pricing kernel which allows for the assumption of nonlinearity.

With multifactor pricing kernels, bad times can be defined as periods when an additional $ 1 income becomes very valuable. Looking at bad times this way interprets SDF as a marginal utility. Periods of high marginal utility could arise from the loss of a job (resulting in low income, where the value of an extra dollar is high), low GDP growth, low consumption (resulting in current consumption below past consumption), or generally low economic growth.

Question 9

Describe efficient market theory and explain how markets can be inefficient

Answer 9

• Market efficiency is also described in the efficient market hypothesis (EMH). The EMH implies that speculative trading is costly, and active managers cannot generally beat the market. The average investor, who holds the market portfolio, can beat the market simply by saving on transaction costs.
• The EMH has been refined to improve upon the CAPM’s shortcomings by allowing for imperfect information and various costs, including transaction, financing, and agency costs. Behavioral biases also represent inefficiencies, which have similar effects as frictions.
• Behavioral biases can be described either through a rational or behavioral explanation approach.
• Under the rational explanation approach, losses during bad times are compensated by high returns.
• Under the behavioral explanation approach, it is agents’ reactions (under/overreaction) to news that generates high returns.

Question 10

Explain how different macroeconomic risk factors, including economic growth, inflation, and volatility affect risk premiums and asset returns

Answer 10

Economic growth, inflation, and volatility are the three most important macro factors that affect asset prices.

Economic Growth

• Risky assets like equities generally perform poorly during periods of low economic growth. Less-risky assets like bonds, and especially government bonds, tend to perform well during periods of slow growth.
• During periods of recession, government and investment grade bonds outperform equities and high-yield bonds.
• During expansion periods, equities outperform bonds with large stocks and small stocks.
• High-yield bond returns appear indifferent to changes in economic growth.
• In terms of volatility, both stocks and bonds are more volatile during downturns and periods of low growth.
• Government bonds perform best during recessions but are also more volatile during these periods

Inflation

High inflation is generally bad for both stock and bond prices and returns. Volatilities are also higher in high inflation periods.

Volatility

Volatility is an important risk factor for many asset classes. The CBOE Volatility Index (VIX) represents equity market volatility. The correlation between the VIX and stock returns has historically indicated a negative relationship. This means that stock returns tend to drop when the VIX (equity volatility) increases.

The financial leverage of companies increases during periods of increased volatility because debt stays approximately the same while the market value of equity falls. The negative relationship between stock returns and volatility is called the leverage effect.

Thus, there are two paths to lower stock returns resulting from higher volatility:

1. When market volatility increases, the leverage effect suggests a negative relationship between stock returns and volatility.
2. When market volatility increases, discount rates increase and stock prices decline so that future stock returns can be higher (to compensate for the higher volatility). The capital asset pricing model (CAPM) supports this second path.

Other Macroeconomic Factors

Other macroeconomic factors, including productivity risk, demographic risk, and political risk, also affect asset returns. Productivity shocks affect firm output. In periods of falling productivity, stock prices fall. correlation between productivity shocks and stock returns is relatively high.

New models, called dynamic stochastic general equilibrium (DSGE) macro models, indicate that economic variables change over time due to the actions of agents (i.e., consumers, firms, governments, and central banks), technologies (and their impact on how firms produce goods and services), and the way that agents interact (i.e., markets).

They are: (1) productivity, (2) investment, (3) preferences, (4) inflation, (3) monetary policy, (6) government spending, and (7) labor supply.

Like productivity shocks, demographic risk, which can be interpreted as a shock to labor output, is a shock to firm production. Economic overlapping generation (OLG) models include demographic risk as a factor affecting investor returns. In these models, generations overlap.

Political (or sovereign) risk, once thought only important in emerging markets, increases risk premiums.

Question 11

Assess methods of mitigating volatility risk in a portfolio, and describe challenges that arise when managing volatility risk

Answer 11

There are two basic approaches to mitigating volatility risk. They are:

• Invest in less volatile assets like bonds, understanding that they too can perform poorly during extreme circumstances such as the 2007—2009 financial crisis.
• Buy volatility protection in the derivatives market (e.g., buy out-of-the-money put options).

Volatility Premiums

Typically, an investor buys an asset, like a stock, and the long position produces a positive expected return. In other words, on average, assets have positive premiums. However, volatility has a negative premium. To collect the volatility premium, one must sell volatility protection (e.g., sell out-of-the money put options).

During normal economic periods, selling volatility provides high, stable payoffs. However, when there is a crash, like the 2007—2009 financial crisis, sellers of volatility suffer large, negative returns.

Question 12

Explain how dynamic risk factors can be used in a multifactor model of asset returns, using the Fama-French model as an example

Answer 12

Over the long run, stocks with high betas (i.e., a high market risk factor) should have higher returns than the market return. Returns are higher for high beta stocks to compensate investors for losses during bad periods.

The Fama-French model (called the Fama-French three-factor model) explains asset returns based on three dynamic factors. The model includes:

• The traditional CAPM market risk factor (MKT).
• A factor that captures the size effect (SMB).
• A factor that captures the value/growth effect (HML).

The Fama-French three-factor model is expressed as follows:

E(R_i) = R_F + β_i,MKT x E(R_M - R_F) + β_i,SMB x E(SMB) + β_i,HML x E(HML)

The SMB factor refers to the difference between the returns on small stocks (small market capitalization) versus big stocks (large market capitalization).

The third factor in the model is HML. This factor captures the return differential of high book-to-market stocks versus low-book-to-market stocks. The ratios are calculated as book value divided by market capitalization. Growth stocks have high stock prices relative to book values, and value stocks have low stock prices relative to book values. Historically, value stocks have outperformed growth stocks. Thus, the Fama-French factors are constructed to capture size (SMB) and value (HML) premiums (known as factor-mimicking portfolios).

The CAPM and Fama-French models assume betas are constant, but empirical research indicates they vary and increase during bad times.

Question 13

Performance of small stocks vs. big stocks

Answer 13

The fact that small stocks tend to outperform big stocks, after adjusting for the firm’s beta, was discovered by Banz (1981) and similarly by Reinganum (1981). Following the publication of this finding, the effect disappeared.

The two possible explanations for the disappearing size effect are as follows:

• Data mining.
• Investor actions.

Note that small stocks do tend to have higher returns (i.e., weak size effect), partially because they are less liquid than large-cap stocks. Also, the value and momentum effects, discussed next, are stronger for small stocks. However, the ability to capture small-cap excess returns over the market (on a risk-adj usted basis) is no longer present.

Question 14

Rational Theories of the Value Premium

Answer 14

• Value is risky and, as such, value stocks sometimes perform poorly. The value premium is compensation for these periods of poor performance, for losing money during bad times.
• Consider the difference between growth and value firms. Growth firms are more adaptable and can adjust when times change because the bulk of their capital is human capital. Value firms are more “old school” with capital in the form of fixed assets that cannot be redeployed when times change. Thus, value firms have high and asymmetric adjustment costs. This makes value stocks fundamentally more risky than growth stocks.

Question 15

Behavioral Theories of the Value Premium

Answer 15

Behavioral theories of the value premium revolve around two basic ideas:

1. overextrapolation and overreaction and
2. loss aversion and mental accounting.

Overextrapolation and overreaction. Investors have a tendency to assume that past growth rates will continue in the future. This is called overextrapolation.

Loss aversion and mental accounting. Investors dislike losses more than they like gains (i.e., loss aversion), and they tend to view investment gains and losses on a case-by-case basis rather than on a portfolio basis (known as mental accounting).

The extrapolation/overreaction behavioral explanation of the value premium is different from the rational one in that in the behavioral explanation, value stocks are not riskier, they are just cheap relative to growth stocks. Investors tend to underestimate the growth prospects of value stocks and overestimate the growth prospects of growth stocks. This bids up the prices of growth stocks and bids down the prices of value stocks, allowing value stocks to outperform on average. Investors must determine if they tend to overextrapolate or not. Investors who act like other average, non-over or under-reacting investors should hold the market portfolio. Investors who overextrapolate will lean toward growth stocks, and those who underreact will lean toward value stocks.

Value investing exists in all asset classes. Strategies include:

• Riding the yield curve in fixed income (i.e., capturing the duration premium).
• Roll return in commodities (i.e., an upward or downward sloping futures curve determines the sign of the return).
• Carry in foreign exchange (e.g., long positions in currencies with high interest rates and short positions in currencies with low interest rates). In this case, high yields are akin to low prices in equity value strategies.

Question 16

Momentum Investing

Answer 16

Momentum strategies (also called trend investing) consist of buying stocks that have gone up over a period (e.g., six months or so) and short stocks that have fallen over the same period (i.e., buy past “winners” and sell past “losers”). The momentum factor, WML, stands for “winners minus losers.” It is also sometimes denoted UMD for “up minus down,” buying stocks that have gone up in price and selling stocks that have gone down in price. A momentum premium is observed in fixed income (government and corporate bonds), international equities, commodities, real estate, and specific industries and sectors.

Value and momentum strategies are, however, opposite each other in the following sense. Value investing is inherently stabilizing. It is a negative feedback strategy where stocks that have fallen in value eventually are priced low enough to become value investments, pushing prices back up. Momentum is inherently destabilizing. It is a positive feedback strategy where stocks that have been increasing in value are attractive to investors, so investors buy them, and prices increase even more. Momentum investing can lead to crashes.

Momentum is often added to the Fama-French model as follows:

E(R_i) = R_F + β_i,MKT x E(R_M— R_F) + β_i,SMB x E(SMB) + β_i,HML x E(HML) + β_i,WML x E(WML)

Momentum risk includes:

• Tendency toward crashes.
• Monetary policy and government risk (i.e., the government gets in the way of the natural progression of asset prices).
• Macro factors such as the business cycle, the state of the stock market, and liquidity risk.

Behavioral explanations suggest that investor biases explain momentum. Investors overreact (a delayed overreaction) to good news about firms.

Whether there is momentum that results from overreaction or from underreaction, prices eventually revert to their fundamental values over the long run.

Question 17

Describe and evaluate the low-risk anomaly of asset returns

Answer 17

Higher risk, as measured by beta, should have a higher return. The low-risk anomaly appears to suggest the exact opposite. This anomaly finds that firms with lower betas and lower volatility have higher returns over time.

Question 18

Define and calculate alpha, tracking error, the information ratio, and the Sharpe ratio

Answer 18

Alpha is often interpreted as a measure of investor skill, but it is really just a statement of average performance in excess of a benchmark. Excess return (R_t^ex) can be seen as the difference between the return of an asset (R_t) and the return of the asset’s benchmark (R_t^B).

Excess return is also sometimes called active return. This phrase assumes that the benchmark is passive and can be achieved without investment knowledge or human intervention.

To fully understand the concept of alpha, we also need to understand tracking error and the information ratio. Tracking error is the standard deviation of excess returns. It measures the dispersion of the investors returns relative to their benchmark.

tracking error = <σ> = standard devation(Rt^ex)

One easy way to monitor alpha is to standardize it using tracking error. The ratio of alpha to tracking error is known as the information ratio (IR), and it is a good way to monitor risk-adjusted returns for active asset managers. Active investment choices can be ranked based on their IR scores.

IR = α/<σ>

When the risk-free rate is the appropriate benchmark, the best way to measure risk-adjusted returns is to use the Sharpe ratio. This measure has alpha in the numerator and the standard deviation of the asset in the denominator.

Sharpe ratio = (<r>t&gt; - <r>F&gt;)/ σ</r></r>

Question 19

Explain the impact of benchmark choice on alpha, and describe characteristics of an effective benchmark to measure alpha

Answer 19

An appropriate benchmark can be selected by applying a few different complementary standards.

• First, the benchmark should be well-defined. It should be hosted by an independent index provider, which makes it both verifiable and free of ambiguity.
• Second, an index should be tradeable.
• Third, a benchmark must be replicable

​If it cannot be replicated, then the tracking error will be very high. Fourth, the benchmark must be adjusted for risk. In the previous example, you can see that the alpha and the IR will be calculated too low if the risk level of the benchmark is too high for the investment in question.

Question 20

Describe Grinold’s fundamental law of active management, including its assumptions and limitations, and calculate the information ratio using this law

Answer 20

This fundamental law does not provide a tool for searching for high IR plays, but it does present a good mechanism for systematically evaluating investment strategies. The law states that:

IR = IC x (BR)^0.5

The formula for Grinold’s fundamental law shows that the information ratio (IR) is approximately equal to the product of the information coefficient (IC) and the square root of the breadth (BR) of an investor’s strategy. The information coefficient is essentially the correlation between an investment’s predicted and actual value. This is an explicit evaluation of an investor’s forecasting skill. A higher IC score means that the predictions had a higher correlation (high-quality predictions). Breadth is simply the number of investments deployed.

Grinold’s fundamental law teaches us about a central tradeoff in active management. Investors need to either play smart (a high IC shows high-quality predictions) or play often (a high BR shows a lot of trade activity). Essentially, investors can be very good at making forecasts and place a small number of bets, or they will need to simply place a lot of bets. Grinold’s framework ignores downside risk and makes a critical assumption that all forecasts are independent of one another.

In practice, it has also been noted that as assets under management go up, the IC tends to decline. This is one reason why some mutual funds close to new investors and turn away new assets once they reach an internally set size.

Question 21

Core challenge with using the Fama-French model

Answer 21

One core challenge with using the Fama-French model is replication of indices. Fama and French have created an SMB index and an HML index to increase explanatory power, but there is no way to directly trade an SMB or HML portfolio. These indices are conceptual and not directly tradeable. It is important to include only tradeable factors because the factors chosen will greatly influence the calculated alpha.

Question 22

Style analysis

Answer 22

Style analysis is a form of factor benchmarking where the factor exposures evolve over time.

Question 23

Describe issues that arise when measuring alphas for nonlinear strategies

Answer 23

Alpha is computed using regression, which operates in a linear framework. There are nonlinear strategies, such as uncovered long put options, that can make it appear that alpha exists when it actually does not.

One reason that nonlinear strategies yield a false positive alpha is because the distribution of returns is not a normal distribution. Certain nonlinear strategies will also exhibit negative skewness in their distribution.

Question 24

Compare the volatility anomaly and beta anomaly, and analyze evidence of each anomaly

Answer 24

• Volatility anomaly shows that as standard deviation increased, both the average returns and the Sharpe ratios decreased. Stocks with high betas tend to have lower-risk-adjusted returns.
• Beta anomaly does not suggest that stocks with higher betas have low return because they do not. It means they have lower Sharpe ratios (risk-adjusted performance) because higher betas are paired with higher volatility as measured by standard deviation, which is the denominator in the Sharpe ratio.
• CAPM does not predict that lagged betas (measured over previous periods) should produce higher returns. It does predict that investors should find a contemporaneous relationship between beta and expected returns. This means that stocks with higher betas should also have higher returns during the same time period when the beta was measured.
• The beta anomaly is less a mystery as it is a challenge to find a reliable way of predicting future betas to improve the risk perspective of beta.

Question 25

Describe potential explanations for the risk anomaly

Answer 25

• A comprehensive explanation for the risk anomaly is elusive. It has been speculated that the true explanation is some combination of data mining, investor leverage constraints, institutional manager constraints, and preference theory.
• Since certain investors are leverage constrained, meaning that they cannot borrow funds for investing, they choose to invest in stocks with built-in leverage in the form of high betas. The additional demand for high-beta stocks will bid up their respective prices until the assets are overvalued and they deliver a decreased risk-adjusted return with regard to lower beta stocks. This same theory works to lower the prices of low beta stocks and, therefore, results in higher risk-adjusted returns due to lower entry prices.
• Many institutional investors will have constraints against short selling. Most also have tracking error constraints that only permit a specified deviation from their benchmark. Under either of these constraints, an institutional investor would not be able to capture the alpha that they think exists. One solution for the tracking error constraint is to change the benchmark or the tracking error tolerance bands, but this can be a difficult process requiring formal approval from the investment committee of the fund.
• Sometimes investors simply have a preference for high-volatility and high-beta stocks. This could occur because their capital market expectations are very bullish, so they want to amplify their returns.