- Provide examples of factors that impact asset prices and explain the theory of factor risk premiums.
- Describe the capital asset pricing model (๐ถ๐ด๐๐) including its assumptions and explain how factor risk is addressed in the ๐ถ๐ด๐๐.
- Explain implications of using the ๐ถ๐ด๐๐ to value assets, including equilibrium and optimal holdings, exposure to factor risk, its treatment of diversification benefits, and shortcomings of the ๐ถ๐ด๐๐.
- Describe multifactor models and compare and contrast multifactor models to the ๐ถ๐ด๐๐.
- Explain how stochastic discount factors are created and apply them in the valuation of assets.
- Describe efficient market theory and explain how markets can be inefficient.

- Video Lecture
- |
- PDFs
- |
- List of chapters

- Factors are to assets what nutrients are to food. Each type of food is a bundle of nutrients. Many foods contain more than just one macronutrient: for example, rice contains both carbohydrates and fiber. Different individuals, whether sick or healthy, male or female, or young or old, have different macronutrient requirements. Food is consumed by us for the underlying nutrients; it is the nutrients that give sustenance.
- Factor risks are the driving force behind assetsโ risk premiums. An important theory of factor risk is the ๐ถ๐ด๐๐.
- The ๐ถ๐ด๐๐states that there is only one factor driving all asset returns, which is the market return in excess of T-bills.
- All assets have different exposures to the market factor and the greater the exposure, the higher the risk premium.

- The market is an example of a tradeable, investment factor. Other examples include interest rates, value-growth investing, low volatility investing, and momentum portfolios. Factors can also be fundamental macro factors, like inflation and economic growth. Assets have different payoffs during high or low inflation periods or during economic recessions and expansions.
- There are three principles of factor theory are โ
- Factors matter, not assets โ The factors behind the assets matter, not the assets themselves. Investing right requires looking through asset class labels to understand the factor content, just as eating right requires looking through food labels to understand the nutrient content.
- Assets are bundles of factors โ Assets represent packages of risk factors. But some asset classes can be considered factors themselves โ like equities and government fixed income securities โ while other assets contain many different factors. Corporate bonds, hedge funds, and private equity contain different amounts of equity risk, volatility risk, interest rate risk, and default risk. Factor theory predicts these assets have risk premiums that reflect their underlying factor risks.
- Different investors need different risk factors โ Different investors have different optimal exposures to different sets of risk factors. Each investor has different preferences, or risk aversion coefficients, for each different source of factor risk.

- Volatility is an important factor. Many assets and strategies lose money during times of high volatility, like the 2007-2008 financial crisis. Most investors dislike these times and would prefer to be protected against large increases in volatility. A few brave investors can afford to take the opposite position; these investors can weather losses during bad times to collect a volatility premium during normal times. They are paid risk premiums as compensation for taking losses.
- Another example is that investors have different desired exposure to economic growth. One investor may not like times of shrinking GDP growth because he is likely to become unemployed in such circumstances. Another investor โ a bankruptcy lawyer, perhaps โ can tolerate low GDP growth because his labor income increases during recessions.
- Each different factor defines a different set of bad times. Investors exposed to losses during bad times are compensated by risk premiums in good times.

- The ๐ถ๐ด๐๐ was revolutionary because it was the first cogent theory to recognize that the risk of an asset was not how that asset behaved in isolation but how that asset moved in relation to other assets and to the market as a whole.
- Before the ๐ถ๐ด๐๐, risk was often thought to be an assetโs own volatility. The ๐ถ๐ด๐๐said this was irrelevant and that the relevant measure of risk was how the asset covaried with the market portfolio โ the beta of the asset.
- Although it has been found out that asset volatility itself matters, but for the purpose of describing the ๐ถ๐ด๐๐and its incredible implications, this can be ignored for the time being.
- The ๐ถ๐ด๐๐ is well known to be a spectacular failure. But the basic intuition of the ๐ถ๐ด๐๐still holds true โ that the factors underlying the assets determine asset risk premiums and that these risk premiums are compensation for investorsโ losses during bad times. Risk is a property not of an asset in isolation but how the assets move in relation to each other.

**Lesson1โDonโt Hold an Individual Asset, Hold the Factor**

- The ๐ถ๐ด๐๐ states that one factor exists, and that factor is the market portfolio, where each stock is held in proportion to its market capitalization. This corresponds to a market index fund.
- The factor can be optimally constructed by holding many assets so that nonfactor, or idiosyncratic risk, is diversified away.
- Individual stocks are exposed to the market factor, which carries the risk premium (it is the nutrient), but also have idiosyncratic risk, which is not rewarded by a risk premium (this is the part that carries no nutritional value).
- Investors can diversify away the idiosyncratic part and increase their returns by holding the market factor portfolio, rather than any other combination of individual stocks.
- The market portfolio represents a systematic risk, and it is pervasive: all risky assets have risk premiums determined only by their exposure to the market portfolio.
- Market risk also affects all investors, except those who are infinitely risk-averse and hold only risk-free assets.

- The ๐ถ๐ด๐๐ is based on investors having mean-variance utility and the most important concept in mean-variance investing is diversification. In the absence of perfect correlation, diversification ensures that, when one asset performs badly, some other assets will perform well, and so gains can partly offset losses. This balance across many assets that are not perfectly correlated improves Sharpe ratios.
- The market factor is the best, most-well diversified portfolio investors can hold under the CAPM. The CAPM states that this market portfolio is held by every investor โ a strong implication that is outright rejected in data.
- The mean-variance frontier with the capital Capital Allocation Line allocation line (CAL), is shown in this figure. This is the solution to the mean-variance investing problem. Investors hold different amounts of the risk-free asset and the Mean-Variance Efficient (MVE) portfolio depending on their risk aversion. Now here come the strong assumptions of the CAPM. Assume that the set of means, volatilities, and correlations are the same for all investors. Then all investors ย hold the same MVE portfolio โ just in different quantities depending on their own risk aversion. Since everyone holds the same MVE and this is the best portfolio that can be held by all investors, the MVE portfolio becomes the market factor in equilibrium.
- Equilibrium occurs when investor demand for assets is exactly equal to supply. The market is the factor in equilibrium because in ๐ถ๐ด๐๐land, everyone holds the MVE portfolio (except for those who are infinitely risk averse). If everyoneโs optimal risky portfolio (which is the MVE) assigns zero weight to a certain asset, say ๐ด๐ดstock, then this cannot be an equilibrium. Someone must hold ๐ด๐ดso that supply equals demand. If no one wants to hold ๐ด๐ด, then ๐ด๐ดmust be overpriced, and the expected return of ๐ด๐ดis too low. The price of ๐ด๐ดfalls. The expected payoff of ๐ด๐ดstays constant under ๐ถ๐ด๐๐assumptions, so that as the price of ๐ด๐ดfalls, the expected return of ๐ด๐ดincreases. ๐ด๐ด
*โ*๐ price falls until investors want to hold exactly the number of ๐ด๐ดshares outstanding. Then, the expected return is such that supply is equal to demand in equilibrium. Since all investors hold the MVE portfolio, the MVE portfolio becomes the market portfolio, and the market consists of each asset in terms of market capitalization weights. Equilibrium ensures that the factor โ the market portfolio โ will have a risk premium and that this risk premium will not disappear. The market factor is systematic and affects all assets. The market risk premium is a function of the underlying investorsโ risk aversions and utilities. That is, the risk premium of the market factor reflects the full setup of all people in the economy. Investors cannot arbitrage away the market factor and all other systematic factors.

**Lesson 2- Each Investor Has His Own Optimal Exposure of Factor Risk**

- All investors will hold the market portfolio, just in different proportions. Pictorially, they have different proportions of the risk-free asset and the market portfolio and lie on different positions on the CAL line.

**Lesson 3 โ The Average Investor Holds the Market**

- The market portfolio represents the average holdings across investors. The intersection of the CAL with the mean-variance frontier represents an investor who holds
*100%*in the MVE portfolio. This tangency point represents the average investor. The risk aversion corresponding to this*100%*portfolio position is the risk aversion of the market.

**Lesson 4** **โ The Factor Risk Premium Has an Economic Story**

- The CAL for a single investor is called the capital market line (CML) in equilibrium, since under the strong assumptions of the ๐ถ๐ด๐๐every investor has the same CML. (The MVE portfolio is the market factor portfolio.) The equation for the CML pins down the risk premium of the market โ

where

\( E(r_m) โ r_f = \bar{\gamma} \sigma_m^2 \)

\( E(r_m) โ r_f \) is the market risk premium, or the expected return on the market in excess of the risk-free rate

\(\bar{\gamma}\) is the risk aversion of the โaverageโ investor; and

๐* _{m}* is the volatility of the market portfolio.

- As the market becomes more volatile, the expected return of the market increases and equity prices contemporaneously fall, all else equal. The market risk premium is proportional to market variance The market portfolio is the portfolio that has the lowest volatility among all portfoli that share the same mean as the market, or the market has the highest reward-to-risk ratio (or Sharpe ratio). The market removes all idiosyncratic risk, and the remaining risk has to be rewarded. As the average investor becomes more risk averse to variance (so ๐ฆincreases), the risk premium of the market also increases.

**Lesson 5 โ Risk is Factor Exposure**

- The risk of an individual asset is measured in terms of the factor exposure of that asset. If a factor has a positive risk premium, then the higher the exposure to that factor, the higher the expected return of that asset.
- The second pricing relationship from the ๐ถ๐ด๐๐is the traditional beta pricing relationship, which is formally called the security market line (SML). The SML states that any stockโs risk premium is proportional to the market risk premium:

where r_{i} is stock iโs return,

r_{f} is the risk-free return, and

\( \beta_i = \frac{\text{cov}(r_i, r_m)}{\text{var}(r_m)} = \rho_{i,m} \left( \frac{\sigma_i}{\sigma_m} \right) \) is that stockโs beta

var (rm)

- Beta measures the co-movement of the stock with the market portfolio, and the higher the co- movement (higher ๐๐๐ฃ
*(*๐๐_{i},, higher is the assetโs beta, and overall risk premiums are higher._{m}))- Beta is a measure of the lack of diversification potential. High betas mean low diversification benefits. High beta assets act like the diversified portfolio the investor already holds, and so they require high expected returns to be held by investors.
- In contrast, low beta assets pay off when the market goes down. These low beta assets have tremendous diversification benefits and are very attractive to hold. In fact, if the payoffs of these low beta assets are high enough when the market return is low, investors are willing to pay to hold these assets rather than be paid. That is, assets with low enough betas actually have negative expected returns. These assets are so attractive because they have large payoffs when the market is crashing. Gold, or sometimes government bonds, can act as low (or negative) beta assets which tend to pay off when the stock market crashes.

**Lesson 6 โ Assets Paying Off in Bad Times Have Low Risk Premiums**

- Generally, if the payoff of an asset tends to be high in bad times, this is a valuable asset to hold and its risk premium is low. If the payoff of an asset tends to be low in bad times, this is a risky asset, and its risk premium must be high. In the ๐ถ๐ด๐๐, the bad returns are defined as low returns of the market portfolio.
- Investors are, on average, risk averse so that the gains during good times do not cancel out the losses during bad times.
- When the market has gains, the high beta asset also gains in value. High beta assets are risky and require high expected returns to be held in equilibrium by investors.
- Conversely, if the asset pays off when the market has losses, the asset has a low beta. This asset is attractive, and investors do not need much compensation to hold these attractive assets in equilibrium.

- The ๐ถ๐ด๐๐is derived using some very strong assumptions. These donโt hold good practically and are not applicable in inefficient and illiquid markets. The following 7 main impractical assumptions are the weaknesses of the ๐ถ๐ด๐๐-

**Assumption 1-**Investors have only financial wealth

**Reality **โ Investors have unique income streams and liabilities, and their optimal portfolio choice has to take these into consideration. Income streams are usually risky, and income declines during periods of low economic growth. This makes variables like inflation and growth important factors because many investorsโ income and liabilities change as the macro variables change. One particular important factor that drives asset returns is human capital, or labor income risk. In an influential paper, Jagannathan and Wang (1996) found large improvements in the performance of the ๐ถ๐ด๐๐when labor income risk is taken into account.

**Assumption 2-**Investor have mean-variance utility which treats risk as symmetrical

**Reality **โ More realistic utility functions often have an asymmetric treatment of risk because investors are generally more distressed by losses than pleased by gains. Ang, Chen, and Xing (2006) show that stocks with greater downside risk have higher returns. A large number of ย papers show that other higher moment risks, like skewness and kurtosis, also carry risk premiums.

**Assumption 3-**Single-period investment horizon

**Reality **โ The optimal strategy for long-term investors is to rebalance, unlike the average investor of the ๐ถ๐ด๐๐theory, who holds the market portfolio by definition, and does not rebalance. Practically rebalancing will require more than one period.

**Assumption 4-**Investors have homogeneous expectations

**Reality **โ In the real world, people obviously do not all share the same beliefs โ they have heterogeneous expectations.

**Assumption 5-**No taxes or transactions costs

**Reality **โ Taxes matter. Taxes affect expected returns and can be regarded as a systematic factor. Transactions costs, meanwhile, also vary across securities. For very illiquid markets with high transactions costs, there are high deviations from the ๐ถ๐ด๐๐. There is another effect of

transaction costs when trading frictions are combined with heterogeneous investors. If ย investors cannot short, then investor beliefs matter. Optimists may prevail in pricing because the pessimistsโ beliefs are not impounded into stock prices. Pessimists would like to short but cannot, and so stock prices reflect only the belief of optimists. Thus, investor beliefs become a systematic factor.

**Assumption 6-**Individual investors are price takers

**Reality **โ The informed investor is trading and moving prices because he/she has some knowledge that others do not have. And when these trades are large, which can happen in the case of institutional traders/investors, they move prices by a significant amount, making the investors price setters instead of price takers.

**Assumption 7-**Information is costless and available to all investors

**Reality **โ Processing and collecting information is not costless, and certain information is not available to all investors. Information itself can be considered a factor in some economic settings and additional risk premiums can be collected Several deviations from the ๐ถ๐ด๐๐are strongest in stocks that have small market capitalizations and trade in illiquid markets where

information is not promulgated efficiently.

- Multifactor models recognize that bad times can be defined more broadly than just bad returns on the market portfolio. While the ๐ถ๐ด๐๐captures the notion of bad times solely by means of low returns of the market portfolio, each factor in a multifactor model provides its own definition of bad times.
- The first multifactor model was the arbitrage pricing theory (APT), developed by Stephen Ross (1976). It uses the word โarbitrageโ because the factors cannot be arbitraged or diversified away-just like the single market factor in the ๐ถ๐ด๐๐.
- In equilibrium, investors must be compensated for bearing multiple sources of factor risk.

- To capture the composite bad times over multiple factors, the new asset pricing approach uses the notion of a pricing kernel. This is also called a stochastic discount factor (SDF). The SDF is an index of bad times, and the bad times are indexed by many different factors and different states of nature.

- The SDF is denoted as ๐. Capturing all bad times by a single variable ๐gives an extremely powerful notation to capture multiple definitions of bad times with multiple variables. The

๐ถ๐ด๐๐ is actually a special case where ๐is linear in the market return โ

๐*=*๐*+* ๐*ร* ๐_{m} for some constants ๐and ๐.

With the ๐notation, multiple factors can be specified very easily by having the SDP depend on a vector of factors, ๐น* = *๐* _{1}, *๐

๐*=*๐*+*๐* _{1}*๐

- Another advantage of using the pricing kernel ๐is that while the ๐ถ๐ด๐๐ restricts ๐ to be linear, the world is nonlinear. Kernels help to build models describing equities, fixed incomeย etc. that capture these nonlinearities.

- In the traditional discount rate model of the CAPM, the price of asset i is obtained by discounting its payoff next period back to today:

where the discount rate is given by

\( E(r_i) = r_f + \beta_i (E(r_m) โ r_f) \)according to the CAPM.

- Under the SDP model, equivalently, the price of the asset can be written using m-notation:

P_{i} = E[m x payoff_{i}],

and hence the name โstochastic discount factorโ, because the payoffs are using m as the factor. Dividing both the right- and left-hand sides of the above equation by the assetโs current price, P_{i}

\(\frac{P_i}{P_i} = E\left[m \times \frac{payoff_i}{P_i}\right] \quad \text{or} \quad 1 = E[m \times (1 + r_i)]\)

A special case of this occurs when the payoffs are constant. That would give a risk-free asset, so the price of a risk-free bond is simply its \(\frac{1}{1 + r_f} = E[m \times 1]\)

The SDF is called a pricing kernel, borrowing the use of the word โkernelโ from statistics, because one can estimate m using a kernel estimator. Since it is a kernel that prices all assets, it is a โpricing kernelโ. The price Pi in the previous equation is an expectation taken with respect to the pricing kernel, so this gives rise to the SDF also being called the state price density.

- Also, the risk premium of an asset can be written in a relation very similar to the SML of the CAPM

where

\(\beta_{i,m} = \frac{\text{cov}(r_i, m)}{\text{var}(m)}\) is the beta of the asset with respect to the SDF. This equation also captures the โbad timesโ intuition The higher the payoff of the asset is in bad times (so the higher cov (ri, m) and the higher Bi,m), the lower the expected return of that asset. The higher beta is multiplied by the price of โbad timesโ risk, given by

\(\lambda_{m} = -\frac{\text{var}(m)}{E(m)}\), which is the inverse of factor risk, which is why there is a negative sign. This equation states directly the intuition of Lesson 6 from the CAPM โ higher covariances with bad times lead to lower risk premiums. Assets that pay off in bad times are valuable to hold, so prices for these assets are high and expected returns are low.

- Just as the CAPM gives rise to assets having betas with respect to the market, multiple factors in the SDF gives rise to a multi-beta relation for an assetโs risk premium:

where

๐ฝ_{i,k} is the beta of asset i with respect to factor k and

E (f_{k}) is the risk premium of factor k.

For macro factors, f_{1} could be inflation and f_{2} could be economic growth, for example.

- Bad times are characterized by times of high inflation, low economic growth, or both. For an example for multiple investment factors, ๐
could be the market portfolio and ๐_{1}could be an investing strategy based on going long value stocks and short growth stocks._{2} - Value stocks outperform growth stocks in the long run. Bad times are also characterized by low market returns, value stocks underperforming growth stocks, or both.

- The key lessons in the multifactor world are in fact the same from the ๐ถ๐ด๐๐:

CAPM (Market Factor) | Multifactor Models | |
---|---|---|

Lesson 1 | Diversification works. The market diversifies away idiosyncratic risk. | Diversification works. The tradeable version of a factor diversifies idiosyncratic risk. |

Lesson 2 | Each investor has her own optimal exposure of the market portfolio. | Each investor has her own optimal exposure of each factor risk. |

Lesson 3 | The average investor holds the market. | The average investor holds the market. |

Lesson 4 | The market factor is priced in equilibrium under the CAPM assumptions. |
Risk premiums exist for each factor assuming no arbitrage or equilibrium. |

Lesson 5 | Risk of an asset is measured by the CAPM beta. |
Risk of an asset is measured in terms of the factor exposures (factor betas) of that asset. |

Lesson 6 | Assets paying off in bad times when the market returns is low are attractive, and these assets have low risk premiums. | Assets paying off in bad times are attractive, and these assets have low risk premiums. |

- Today, economists do not believe in perfectly efficient markets, In fact, markets cannot be efficient in their pure form. The modern notion of market near-efficiency is developed by Sanford Grossman and Joseph Stiglitz. They address a conundrum that arises from the costless information assumption of the ๐ถ๐ด๐๐. Suppose that it is costly to collect information and to trade on that information, as it is in the real world.
- Then, if all information is in the price already, why would anyone ever invest in gathering the information?
- But if no one invests in gathering the information, how can information be reflected in security prices so that markets are efficient?

- It is then impossible that markets be efficient in their pure form.
- Active managers search for pockets of inefficiency, and in the process, cause the market to be almost efficient. In these pockets, active managers earn excess returns as a reward for gathering and acting on costly information. These pockets of inefficiency are expected to lie in market segments that are illiquid, with poor information dissemination and where outsized profits may be hard to collect because trading on these anomalies will likely move prices.
- The near-efficient market theory fits closely with the multiple factor risk framework of the APT developed by Ross (1976). In this multifactor model, active managers and arbitrageurs drive the expected return of assets toward a value consistent with an equilibrium trade-off between risk and return. The factors in the APT model are systematic ones, or those that affect the whole economy, that agents wish to hedge against. In their purest form the factors represent risk that cannot be arbitraged away, and investors need to be compensated for bearing this risk.
- Despite the modern notion that markets are not perfectly efficient, a large literature continues to test the Efficient Market Hypothesis (EMH). The implication of the EMH is that, to the extent that speculative trading is costly, active management is a loserโs game and investors cannot beat the market. The average investor holds the market portfolio and come out ahead simply by saving on transactions costs. Even if it is known that the market cannot be perfectly efficient, tests of the EMH are still important because they allow talented investors can identify the pockets of inefficiency where active management efforts are best directed.
- The EMH has been refined over the past several decades to rectify many of the original ย shortcomings of the ๐ถ๐ด๐๐including imperfect information and the costs associated with transactions, financing, and agency. Many behavioral biases have the same effect, and some frictions are actually modeled as behavioral biases.
- These behavioral deviations from efficiency have two forms of explanations โ a) rational and b) behavioral.
- In a rational explanation, high returns compensate for losses during bad times. This is the pricing kernel approach to asset pricing. The key is defining those bad times and deciding whether these are actually bad times for an individual investor. Certain investors, for example, benefit from low economic growth even while the majority of investors find these to be bad periods. In a rational explanation, these risks premiums will not go away-unless there is a total regime change of the entire economy. (These are very rare, and the financial crisis in 2008 and 2009 was certainly not a regime change.) In addition, these risk premiums are scalable and suitable for very large asset owners.
- In a behavioral explanation, high expected returns result from agentsโ under- or overreaction to news or events. Behavioral biases can also result from the inefficient updating of beliefs or ย ignoring some information. Perfectly rational investors, who are immune from these biases, should be able to come in with sufficient capital to take advantage of such mispricing. But there are barriers to the entry of capital. Some of these barriers maybe
- structural barriers โ like the inability of certain investors to take advantage of this investment opportunity.
- regulatory requirements โ for example, holding bonds only above a certain credit rating or stocks with market capitalizations above a certain threshold.

- If there is a structural barrier, then the behavioral bias can be exploited for a long time.
- For some risk premiums, the most compelling explanations are rational (as with the volatility risk premium), for some behavioral(e.g., momentum), and for some others, a combination of rational and behavioral stories prevails (like value/growth investing).
- Overall, the investor should not care if the source is rational or behavioral; the more appropriate question is whether she is different from the average investor who is subject to the rational or behavioral constraints and whether the source of returns is expected to persist in the future (at least in the short term).