Causal Insight into CAPM Class Models (Part II)

This is the second of a series of posts into CAPM and asset pricing models that include a “market factor”. In the first, I discussed how CAPM divides risk into systematic (paid) and unsystematic (non-paid) risks, and that systematic risk is essentially the risk you experience by being exposed to the entirety of world asset growth. Strictly speaking, CAPM isn’t the only model that divides risks into paid and unpaid, but it is the simplest case of a set of models that does so.

I also argued that the expected market risk premium over time does not necessarily exist “because no one would invest if it didn’t,” but in fact has a basis in the fact that world productivity tends to increase on average through the application of knowledge, technology, and a better understanding of how to meet human and business needs. People who are not invested in the world market do not share directly in those productivity gains, although they might benefit by lower prices for consumables and an associated consumer surplus.

In this post, I will briefly go through the CAPM equation and then explain what factors affect the key components of the equation. A later post will show how it (and its relatives) fit into causal approaches to asset modeling. Most texts that I have seen use CAPM as a calculation tool rather than an interpretive one. This is lamentable because one can miss out on important economic intuitions that may help to develop better pricing models.

The CAPM equation

The CAPM equation that is introduced in financial theory courses and the CFA curriculum is so straightforward and simple that many do not take time to interpret its meaning carefully. The equation in its most traditional form is:

        Ra = Rfr + Ba * (Rm - Rfr)

or, alternately

        Ra = Rfr + Ba * MRP

Where:

• Ra = expected (predicted) return on an asset
• Rfr = Risk Free Rate, the time value of money
• Rm = Return on the world market portfolio
• MRP = Market Risk Premium, which equals (Rm - Rfr)
• Ba = The asset’s “Beta to the market”, a regression coefficient

The Ra term is simple enough: it is simply the expected percentage return on a particular asset, whether it be a stock, bond, piece of real estate, portfolio, or whatever. It is the best guess for what the return should be, and should be “on average” correct.

The Rfr term is called the “risk free rate,” which is essentially the interest rate on loan that has a 100% chance of being paid, and represents the time value of having money now rather than later. Normally, people use US Treasuries to calculate this rate, because the US Government can print any dollars necessary to pay back its debt, so it’s about as “guaranteed” as you can get. That may not continue forever, of course, and there is some debate about how to consider the inflationary aspects of printing money to pay debt.

Rm is the return on the world market portfolio, a hypothetical portfolio that holds everything in the world (or x% of every single asset in the world).

The MRP or market risk premium is the return “over and above“ the risk-free rate that the world market obtains. On average, the MRP should be positive, because risk takers will get a larger share of world growth than those who protect themselves by lending at guaranteed rates, but in any specific time period, MRP can be positive or negative.

Not mentioned here is that one could define an ”Asset Risk Premium,“ ARP = (Ra - Rfr), which is a specific asset’s return over and above the Rfr. If we define an ARP like this and rearrange the equation, we get the essence of CAPM:

        ARP = MRP * Ba

Or that the risk premium we can expect by owning an asset is proportional to its Beta to the market (defined momentarily) and the premium for owning the entire market. This formulation boils CAPM down to its real meaning for asset pricing. In practice, the CAPM formula generally uses the Rfr formulation rather than the ARP formulation, because its users typically want to know what total return to anticipate, and not simply how much return to expect over and above a risk-free loan. This risk premium formulation captures one of key insights of CAPM.

Ba, or an asset’s ”beta to the market“ is an empirically determined indicator of how much an asset’s return changes compared to the change in the market as a whole and is usually measured through linear regression methods. If an asset has a Beta of 2.5, it means that for each 1% of excess return for the market as a whole, the specific asset is expected to return (Beta)x(1%) or 2.5% excess return. It is called Beta because it is an estimate of a bivariate regression model’s Beta parameter when using the market return to predict the asset’s return.

Beta is sometimes called the covariance with the market, or even the correlation to the market. Neither of these is technically accurate, although both values are influence beta. In fact Beta = COV(A,M) / VAR(M) where COV(A,M) is the covariance with the market and VAR(M) is the variance of the market. A little more statistical manipulation, and you find that:

        Beta = Corr(A,M) * [ SD(A) / SD(M) ]

where SD(x) signifies the relative volatilities (standard deviations) of the Asset and the Market, respectively.

Therefore Beta captures BOTH the correlation of an asset to the market AND its volatility compared to the market as a whole. A low Beta could represent low correlation to the market, OR it could represent low volatility in the asset, or both. However, since correlation coefficients cannot be greater than one, high Beta assets are necessarily more volatile than the market as a while. Indeed, Beta is an interaction between correlation and asset volatility. To be low (in magnitude), either correlation or relative volatility must also be low; to be high, both correlation and relative volatility are probably high.

Thinking about Beta and the Asset Risk Premium

The asset risk premium and the above reformulation of Beta are not commonly part of presenting the CAPM model of expected returns, but they are worth thinking about. If we combine the ARP approach and this reformulated beta, we get:

        ARP = MRP * Corr(A,M), * [ SD(A) / SD(M) ]

or

        ARP = MRP * [Corr of Asset and Market] * [Volatility of Asset relative to Market]

and, to turn it back into observed returns:

        Ra = Rfr + ARP

This formulation focuses thinking about an asset returns in terms of 1) its correlation with the market, and 2) total volatility relative to market volatility. Both are often presented as an interpretation for Beta, but since Beta is essentially an interaction of the two, it is often difficult to know whether Beta represents correlation with the market or its relative volatility to a greater extent. By reformulating Beta in this way, it helps to think through what market correlation represents and what affects the relative volatility of assets.

Correlation: Indicates the degree to which changes in market returns appear to affect changes in the asset return. For example, a correlation of 0.6 suggests (with normality assumptions) that a market return 1 SD higher than its average return will tend to result in an asset returning 0.6 SD more than its average. The actual movement and volatilities of the asset and the market portfolio can be vastly different; correlation only captures how consistently the market and asset move in the same direction at the same time (or move in opposite directions, if the correlation is negative). A 10% increase in the market portfolio return may only bring about a 1% increase in the asset return (or vice versa), but if that proportion is consistent over return after return, the correlation will still be high (even if Beta is low).

Another way to look at correlation is to think about it as describing the proportion of average movement that can be attributed to the market versus idiosyncratic sources. Mathematically, statisticians talk about the square of the correlation indicating the proportion of variance that the correlation can explain, but since many analysts find variance to be non-intuitive, it can help to think of a correlation coefficient as the expected number of standard deviation changes in one variable when the other changes by one SD. So, in the case of a correlation coefficient of 0.6, one can say that, on average, a 1 SD increase in the market return leads, on average, to a 0.6 SD increase in the asset return. And, although it’s not literally true in a mathematical sense, it is not too outrageous to think of approximately 0.4 SD [ = 1 - 0.6] worth of variability in the asset’s return coming from ”other,“ idiosyncratic (i.e. non-paid) sources.

Correlation, then, tells you how well / how consistently market movements work to predict/explain your asset returns: are they synchronized (or anti-synchronized), or not.

Factors affecting correlation: correlation with the market is primarily determined by two factors: 1) the nature of the business (i.e. degree to which the external economic environment affects business revenues and costs), and 2) the degree to which the business is exposed to idiosyncratic events, some of which can be controlled or influenced by management.

For example, revenues from consumer staple businesses like food and milk are likely to stay relatively constant no matter what the economic environment, since people need to eat whether things are going well or badly; on the other hand, luxury goods may sell substantially better in boom times and sell substantially worse in bad, and therefore be more highly correlated with the growth of assets in general. Note however, that if there are consistently small increases or decreases in food staples that track the market as a whole, one can still have a moderate or even strong correlation. However, if the things that most influence the performance consumer staples from period to period tend to be random or idiosyncratic, the correlation with the market portfolio will be low.

The degree that businesses are exposed to idiosyncratic events also can vary by business line. For example, agriculture may be affected by weather and labor issues more easily than something like an internet business, which by its nature is less exposed to these particular of random events. Management expertise and business sense can influence exposure to idiosyncratic events to some extent; for example, a business in an area with frequent power outages can reduce its exposure to idiosyncratic power outages by installing backup systems and business continuity plans, and a company with highly aware human resources departments may be able to reduce the impact of labor disputes on productivity. However, some idiosyncratic risk is unavoidable, and the correlation between an asset and the broader market essentially reflects the degree to which market returns vs. idiosyncratic events influence asset performance over time.

Relative Volatilities: Regardless of their correlation, some assets go up and down considerably more than the market as a whole; others considerably less. By expressing the relative volatility of an asset as a multiple of the market’s volatility, one can get an idea of the relative riskiness of the asset taken all by itself. In general, the riskier the asset, the greater will be its required (expected) return to entice large numbers of investors to purchase it. Linear models like CAPM assume that, in order to compensate investors for taking on twice as much risk, they will need at least twice as much excess return.

Now, if investors require twice as much return for taking on twice as much risk (measured as standard deviation), the ratio of asset volatility to market volatility [SD(a)/SD(m)] simply tells us the scaling factor that is multiplied to the market risk premium to tell us what the asset return must be to attract investors.

Not all of the asset’s risk is systematically rewarded, however. The idiosyncratic portion - the part that has no relation to average world productivity and is therefore effectively random - doesn’t earn any return because (in the absence of any other information about what affects that number) it averages out to zero over all assets in the investible universe. That assumption, about the absence of other information about what affects the idiosyncratic portion, turns out to be important for believers in active management. But in the CAPM world, we will assume for the moment that returns unsystematically related to the market are completely unpredictable, and that assuming this portion of an asset’s risk will, on average, provide zero excess return.

As mentioned earlier, this is why holding non-diversified portfolios are typically considered inefficient - there is extra risk that the investor is taking that has no corresponding excess return. An investor can easily eliminate that risk by holding a diversified portfolio, and this portfolio has less risk because for every asset that underperforms the world average return, there is another asset that outperforms it. Diversification forces the idiosyncratic risks to balance out to zero; the parts that don’t balance go into the world average, which is the systematic risk.

Given that the volatility of an asset can tell an investor how much risk they take by buying it, how much of that risk is the ”systematic“ or ”paid“ portion. This is what the correlation coefficient tells us (at least in a model with one factor).

        Corr(A,M) * SD(A) = systematic portion of Asset Risk

This quantity is essentially the ”reflection“ of the market’s volatility that can be obtained by holding the asset. The remaining portion of volatility is idiosyncratic. And...

        MRP / SD(M) = Sharpe Ratio of Market

Which tells you how much return the market produced for the amount of volatility risk taken over a specific period. It tells you how well the market converts risk taking into excess return.

If one multiplies these two quantities together, one gets:

        [Systemic part of Asset Vol] * [Market Sharpe Ratio] =

        [ Corr(A,M) * SD(A) ] * [ MRP / SD(M) ] =

        [ Corr(A,M) * SD(A)/SD(M) ] * MRP =

        Beta * MRP = Asset Risk Premium.

Factors affecting relative volatility: are similar to factors affecting correlation, with one major addition: financial leverage.

As with correlation, the nature of the business may affect volatility. Companies delivering products with highly elastic demand may experience more booms and busts than staple products that are not easily substituted, and this creates differences in asset volatility.

Also, the effectiveness of company management at addressing everyday threats and annoyances can affect the portion of volatility that comes from idiosyncratic risk. For example, poor strategy communication and management operating procedures may mean that one company will be devastated if the CEO is hit by a truck, while another may be able to weather the crisis more effectively.

A major factor affecting asset volatility is a company’s capital structure, specifically its financial leverage, captured in the debt-equity ratio or the assets-to-equity ratio. If one compares two otherwise identical businesses, one with an all-equity structure, and the other which has 50% debt, the latter should have 2x the volatility in its stock price, because the total value of the equity is only one half as much, and therefore any change in future expected income will represent twice as much equity rate of growth for the indebted company.

Summary of Part II

This section points out the features of a company that determine its expected return under CAPM. Instead of simply citing Beta as a regression result, it breaks down Beta into a component reflecting the correlation of the asset to the market and the relative volatility of the asset to the market. It then pointed out that the factors affecting correlation and relative volatility are 1) the nature of the business itself (elastic or not, countercyclical or not), 2) whether management has good procedures for reducing idiosyncratic risks, and 3) for relative volatility, the degree of financial leverage. Taking this approach allows one to get a more intuitive feel for what determines a company’s Beta to the market and how changes in these features will affect the required return on an asset.