### Factor Theory

By the end of this chapter, the learner should be able to give an example of factors that affect asset prices and further provide an explanation of the theory of factor risk premiums. The Capital Asset Pricing Model (CAPM) should also be well described, including an explanation of how CAPM addresses factor risk.

The implications of CAPM in assets valuation will also be explained in this chapter, with an inclusion of the CAPM’s shortcomings, how it treats the benefits of diversification, its exposure to risk factors, and equilibrium and optimal holdings.

Furthermore, some multifactor models will be described and be compared to CAPM. In addition, the chapter will explain the creation of stochastic discount factors and use them in asset valuation. Finally, the chapter will provide the reader with a description of the efficient market theory and the inefficiency of real-world markets.

# The 2008-2009 Financial Crisis

The 2008-2009 financial crisis was accompanied by a plunging of the riskiest assets. For example, The U.S. cap equities returned -37%, with even larger losses suffered in the international and emerging markets equities.

For investments trumpeting their immunity to market disruptions, e.g., hedge funds, there was an estimated decline of 20% for equity hedge funds and their fixed income counterparts. For commodities, their losses exceeded 30%.

However, for cash and safe-haven sovereign bonds like the long-term U.S. Treasuries, their value rose.

# Factor Theory

Just the same way nutrients are to food, so are factors to assets. The driving mechanisms behind the risk premiums of assets are factor risks. The CAPM is a crucial theory of factor risk, which states that there is one factor that drives all asset returns, which happens to be the market returns in excess of T-bills.

The exposure to market factors for all assets is usually different. Greater exposures usually imply higher risk premiums. Examples of investment factors that are tradable include the market, interest rates, low volatility investing, value-growth investing, and momentum portfolios.

During times of inflation (high or low) or in recession or expansion periods, the payoffs of assets usually differ.

One critical factor is volatility. In times of high volatility, most assets and strategies usually lose money. In cases of large volatility increases, most investors would seek to be protected against these increases.

The opposite positions may be taken by a few brave investors who can weather losses in harsh times so that during normal times, they can collect a volatility premium. Moreover, the desired exposure to economic growth usually differs among investors.

For example, periods of shrinking GDP growth may be disliked by one investor since his/her employment is likely to be at stake, whereas for another investor, e.g., a bankruptcy lawyer, low GDP growth can be tolerated due to increase in their labor income during such harsh times.

# CAPM

CAPM was the first theory to discover that the behavior of an asset in isolation has nothing to do with the asset’s risk, but rather the movement of the asset relative to other assets (and generally the market as a whole), thus making it revolutionary.

In most instances, the risk was regarded as the volatility of the asset prior to CAPM. According to CAPM, the covariation of the asset with the market portfolio – the asset’s beta – was the relevant risk measure.

In the 1960’s, CAPM was formulated by various individuals, building on the principle of diversification and mean value utility. However, the predictions that only the beta of assets is relied upon by investors and that the only factor that matters is the market portfolio have been demolished in a lot of empirical studies, therefore making CAPM a failure in practice.

However, the basic intuition still holds true. Asset risk premiums are determined by the factors underlying the assets and, during the harsh times, the losses incurred by the investors are compensated by the said risk premiums. The CAPM still remains the workhorse model of finance despite it being firmly rejected by data.

## CAPM Lesson 1: Instead of Holding the Individual Asset, Hold the Factor

According to the CAPM, the market portfolio is the only existing factor, with each stock being held proportionally to its market portfolio. This corresponds to a market index fund. To optimally develop the factor, many assets have to be held in order to diversify away from the idiosyncratic risk (nonfactor).

As compared to individual stocks, asset owners are better off holding the factor. This can be attributed to the fact that individual stocks are exposed to the market factor carrying the risk premium. Furthermore, they have an idiosyncratic risk and it is not rewarded by a risk premium.

By holding the market factor portfolio, the idiosyncratic part can be diversified away by the investors, thereby increasing their returns. Systematic risk is represented by the market portfolio, which is pervasive: the risk premiums of all risky assets are determined by their exposure to the market portfolio only.

With the exclusion of investors who are infinitely risk-averse and hold assets that are only risk-free, all other investors are affected by market risk.

Diversification is the key to this result. Each investor having a mean-variance utility is the basis of CAPM, with diversification being the most crucial concept in mean-variance. With diversification, losses can be partially offset by gains since, without perfect correlation, when one asset performs well, some other asset could be performing badly.

It has never been the wish of investors to hold assets in isolation since by diversifying and holding many assets, their risk-return trade-off is improved. According to CAPM, every investor holds the market portfolio. However, this fact is outright rejected in data.

## Equilibrium

For equilibrium to occur, there must be exact equality between supply and demand for assets. Due to the fact that the market-variance efficient (MVE) portfolio is held by everyone in CAPM land, the market risk is the factor in equilibrium. Suppose that a certain asset, say AAA stock, is assigned zero weight by everyone’s optimal risky portfolio.

In this case, equilibrium does not exist since for supply to equal demand, AAA must be held by someone. However, in the event that no one wants to hold AAA, it must be overpriced and its expected return is too low. AAA price then falls.

The prices of AAA falls to the point that investors want to hold the exact number of the outstanding AAA shares. The expected return is then such that, at equilibrium, there must be equality between supply and demand. The MVE portfolio becomes the market portfolio due to the fact that the MVE portfolio is held by all investors with the market consisting of each asset in terms of market capitalization weights.

At equilibrium, a risk premium that will not disappear is possessed by the factor, which is the market portfolio. All assets are affected by the systematic market factor. The full setup of all people in the economy is reflected in the risk premium of the asset factor.

## CAPM Lesson 2: Each Investor Has His/Her Own Optimal Exposure of Factor Risk

Although in different proportions, the market portfolio will be held by all investors. A different amount of factor exposure is possessed by each individual investor just in the same manner different nutrient requirements are a necessity for different individuals.

## CAPM Lesson 3: The Market is Held by the Average Investor

The average holdings across investors are represented by the market portfolio. An investor holding 100% in the MEV portfolio is represented by the intersection of the capital asset line (CAL) with the mean-variance frontier.

The average investor is represented by this tangency point. Investors will get exposed to more or less market factor risks depending on their own risk preferences, as each investor differs from the average investor.

## CAPM Lesson 4: The Factor Risk Premium Has an Economic Story

The market risk premium is pinned down by the capital market line (CML) equation:

$$E\left( { r }_{ m } \right) -{ r }_{ f }=\bar { \gamma } { \sigma }_{ m }^{ 2 }\quad \quad \quad \quad \quad \quad \left( a \right)$$

Where the market risk premium is represented by the expression $$E\left( { r }_{ m } \right) -{ r }_{ f }$$; the average investor’s risk aversion is given by $$\bar { \gamma }$$ , and $${ \sigma }_{ m }$$ is the market portfolio’s volatility. The risk premium is derived by CAPM in terms of the underlying agent preferences.

Increased volatility in the market leads to an increase in the market’s expected return and a contemporaneous fall in equity prices. This was experienced in 2008 and 2009 with the skyrocketing of volatility and the nosediving of equity prices.

The portfolio with the lowest volatility, as compared to all portfolios sharing the same mean as the market, is the market portfolio. All idiosyncratic risk is removed by the market. A precise equation for the market risk premium is stated by the above-given equation.

An increase in the market’s risk premium can be witnessed as the average investor becomes more risk averse.

## CAPM Lesson 5: Risk Is Factor Exposure

The asset’s factor exposure measures an individual asset’s risk. For a factor whose risk premium is positive, as the exposure to the factor grows higher, so does the expected return of the asset.

The security market line (SML), which is a traditional beta pricing relationship, is the second pricing relationship in the CAPM model. According to CAPM, the risk premium of any stock is proportional to the market risk premium. The $$i$$ return of the stock is denoted as $${ r }_{ i }$$, and the risk-free return denoted as $${ r }_{ f }$$.

$$E\left( { r }_{ i } \right) -{ r }_{ f }=\frac { Cov\left( { r }_{ i },{ r }_{ m } \right) }{ VaR\left( { r }_{ m } \right) } \left( E\left( { r }_{ m } \right) -{ r }_{ f } \right)$$

$$={ \beta }_{ i }\left( E\left( { r }_{ m } \right) -{ r }_{ f } \right)$$

Where, $$E\left( { r }_{ i } \right) -{ r }_{ f }$$ is an individual stock’s risk premium and is a function of the beta of that stock, $${ Cov\left( { r }_{ i },{ r }_{ m } \right) }/{ VaR\left( { r }_{ m } \right) }$$. The CAPM’s risk measure, beta, measures the lack of diversification potential. Beta can also be expressed as:

$${ \beta }_{ i }={ { \rho }_{ i,m }{ \sigma }_{ i } }/{ { \sigma }_{ m } }$$

Where $${ \rho }_{ i,m }$$ is the correlation between the returns of asset $$i$$ and the market return. The market factor’s volatility is given as $${ \rho }_{ i}$$. Lower correlation with a portfolio implies that the benefits of diversification are greater with respect to that portfolio since the likelihood of the asset to have high returns is higher when the portfolio did badly. Therefore, low benefits of diversification can be attributed to high betas.

For assets with low enough beta, their expected returns are actually negative. These assets are attractive to investors due to the large payoffs they have in case the market is crashing. Examples of low beta assets include gold, government bonds, etc.

## CAPM Lesson 6: Assets that Pay Off in Bad Times Have Low-Risk Premiums

The securities market line (SML) relationship can also be viewed in form of the risk premium in CAPM being a reward for how an asset pays off in bad times. The factor (market portfolio) defines bad times, hence the bad times corresponds to market returns that are low.

An asset incurring losses when the market incurs losses is said to be having a high beta. Should the market have gained, so do the asset beta. On average, for gains experienced during good times not to cancel out during bad times, investors have to be risk-averse.

On the other hand, the asset has a low beta in the event that it pays off when the market has losses. It is also possible for the asset’s expected return to be low, hence not much compensation will be needed by investors to hold these attractive assets at equilibrium.

If an asset’s payoff has a tendency of being high in bad times, then the asset is considered valuable to hold with its risk premium being low. An asset with high-risk premium usually has a low payoff in bad times. In CAPM, a low return of the market portfolio is simply defined as a bad return.

## Multifactor Models

With multifactor models, it is recognized that the definition of bad times can be more broadly in comparison to just bad returns on the market portfolio. The arbitrage pricing theory (APT) was the first multifactor model.

Since it is impossible for the factors to be arbitrated or diversified away, the term arbitrage is used. While each notion of bad times is captured by CAPM only through the market portfolio’s low return, the definition of bad times for each factor is provided by each factor.

## Pricing Kernels

The notion of a pricing kernel is applicable in the new asset pricing approach to capture composite bad times over multiple factors. Another term used to refer to this is the stochastic discount factor (SDF). SDF is denoted as $$m$$. The SDF can be described as an index of bad times which are indexed by many distinct factors and different states of nature.

As this concept is applicable to all the associated recent asset pricing theories, it is critical for the relation between this SDF approach and the traditional CAPM approach to be clearly outlined.

The multiple definitions of bad times with multiple variables can be captured by an extremely powerful notation, by capturing all bad times using a single variable $$m$$. Since $$m$$ is linear in the market return, the CAPM is actually a special case: $$m=a+b\times { r }_{ m }$$.

Where $$a$$ and $$b$$ are constants.

By having the SDF rely on a vector of factors, multiple factors can be very easily specified, with the $$m$$ notation:

$$m=a+{ b }_{ i }{ f }_{ i }+{ b }_{ 2 }{ f }_{ 2 }+\cdots +{ b }_{ k }{ f }_{ k }$$

For all $$F=\left( { f }_{ 1 },{ f }_{ 2 },\dots ,{ f }_{ k } \right)$$, different bad times are defined by each of the $$K$$ factors. While the linearity of $$m$$ is restricted by CAPM, the world is non-linear. The model built should capture this non-linearity.

## Pricing Kernels versus Discount Rate Models

The price asset $$i$$ can be determined, in a traditional discount rate model, by discounting its payoff next period back to today.

$${ P }_{ i }=E\left[ \frac { Payof\quad { f }_{ i }\quad }{ 1+E\left( { r }_{ i } \right) } \right] ,$$

The discount rate is given as:

$$E\left( { r }_{ i } \right) ={ r }_{ f }+{ \beta }_{ i }\left( E\left( { r }_{ m } \right) -{ r }_{ f } \right)$$

Using the $$m$$-notation, the asset price can be written in the following way under the SDF model:

$${ P }_{ i }=E\left[ m\times payof\quad { f }_{ i } \right] ,\quad \quad \quad \quad \quad I$$

This gave rise to the name stochastic discount factor, as the payoffs are discounted using $$m$$.

Both the left- and right-hand sides of equation $$I$$ can be divided by the current price $${ P }_{ i }$$ of the asset, such that:

$$\frac { { P }_{ i } }{ { P }_{ i } } =E\left[ m\times \frac { payof\quad { f }_{ i } }{ { P }_{ i } } \right] \quad \quad \quad \quad \quad \quad II$$

$$1=E\times \left( m\times \left( 1+{ r }_{ i } \right) \right)$$

When the payoffs are constant, a risk-free asset would be given by the equation $$II$$, so that a risk free-bond is simply:

$$\frac { 1 }{ \left( 1+{ r }_{ f } \right) } =E\left[ 1\times m \right]$$

The risk premium of an asset can be written in a relation similar to the SML of the CAPM:

$$E\left( { r }_{ f } \right) -{ r }_{ f }=\frac { cov\left( { r }_{ i },m \right) }{ VaR\left( m \right) } \left( -\frac { VaR\left( m \right) }{ E\left( m \right) } \right) \quad \quad \quad \quad \quad III$$

$$={ \beta }_{ i,m }\times { \lambda }_{ m }$$

Where

$${ \beta }_{ i,m }={ cov\left( { r }_{ j },m \right) }/{ VaR\left( m \right) }$$

In equation $$III$$, the higher beta is multiplied by the price of the bad times risk:

$${ \lambda }_{ m }=-\frac { VaR\left( m \right) }{ E\left( m \right) }$$

This is the inverse of the factor risk, hence the negation. Multiple factors in the SDF in the SDF gives rise to a multi-beta relation for the risk premium of an asset:

$$E\left( { r }_{ i } \right) ={ r }_{ f }+{ \beta }_{ i,1 }E\left( { f }_{ 1 } \right) +{ \beta }_{ i,2 }E\left( { f }_{ 2 } \right) +\cdots +{ \beta }_{ i,K }E\left( { f }_{ K } \right)$$

The beta of asset $$i$$ with respect to factor $$k$$ is given by $${ \beta }_{ 1,k }$$, and $$E\left( { f }_{ K } \right)$$ is the risk premium of factor $$k$$.

# Failures of the CAPM

Some very strong assumptions are used in the derivation of CAPM.

1. Investors have only one financial wealthThe income streams and liabilities of investors are unique, and this has to be been taken into consideration by their optimal portfolio choice. The denomination of liabilities has often been in real terms, as the living standard should be maintained even in rising prices. Human capital or labor income risk is a crucial factor that drives asset returns.
2. Investors have mean-variance utilityIt is often the case that more realistic utility functions have an asymmetric risk treatment since investors are generally more distressed by losses as compared to their pleasure with gains. Deviations should then be expected from the CAPM among stocks with different downside risk measures.
3. Single-period investment horizonA one-period investment horizon is by itself a minor assumption. There is a huge implication when the choice of a portfolio is extended to a dynamic, long-horizon setting, despite the long investment horizon being an inconsequential assumption for the CAPM theory.
4. Investors have homogeneous expectationsWith this assumption, the same MVE portfolio is held by all investors in the CAPM world. Moreover, the market-variance efficient portfolio becomes the market portfolio, in equilibrium. However, in reality, people rarely share similar beliefs, since their expectations are heterogeneous.
5. No taxes or transaction costsExpected returns are affected by taxes, therefore, making it regarded as a systematic factor. There is also a variation in transaction costs across all securities. There may be more deviations from CAPM for every illiquid market with high transaction costs.
6. Individual investors are price takersThe reason why the informed investor is trading and moving prices is due to the fact that he/she may have some knowledge that others may lack.
7. Information is consistent and available to all investorsThere are costs incurred in the collection and processing of information, and particular information may not be availed to all investors.

# The 2008-2009 Financial Crisis Redux

During the financial crisis, there was a simultaneous dismal performance by many risky assets. This was consistent with an underlying multifactor model whereby there was exposure of many assets to similar factors.

To compensate for exposure to the said underlying risk factors, risk premiums are earned by assets. Asset returns are low, during bad times, in the midst of manifestation of these factor risks.

According to some compensators, the 2008 events demonstrate the diversification failure. As demonstrated by the financial crisis, the labels of asset classes can be misleading, hence lulling investors into believing that they are safely diversified when in reality they are not.

Through time, factor exposures can and do vary. This gives rise to correlations that are time-varying, thus giving even more reasons to understand the true factor drivers of risk premiums.

# Practice Questions

Why are low beta assets attractive to investors?

1. They have large payoffs when the market is crashing
2. Low beta assets tend to outperform in the long run
3. Low beta portfolios deliver higher volatility than expected and therefore have shown significantly better results in practice
4. None of the above