We talked extensively about valuing a stream of cash flows in the previous articles. By building on the idea of present value, we can use it to value more complex investments like stocks. We start our section on valuation with CAPM.

*The article covers several basic concepts in statistics. If you have taken a college level stats course, these may be familiar or even trivial to you. If so, feel free to skip the sections.*

### What is CAPM?

CAPM (pronounced Cap-Em) stands for Capital Asset Pricing Model – a simple model used to price capital assets, usually stocks. Specifically, it relates return on a capital asset to its systematic risk (Beta). CAPM is usually used to estimate a required return on equity, which is in turn used to calculate WACC (Weighted Average Cost of Capital) for DCF (Discounted Cash Flow) analysis – we will get to that later. To understand CAPM, we must first review a few concepts: risk and return, systemic and unsystemic risk, and correlation and covariance.

### What is Risk and Return?

Risk and return is a fundamental concept in finance – the riskier the asset is, the higher the expected return. This should be intuitive – if safe strategies paid more than risky strategies, who would take the risk? This trade-off is validated empirically, as the historical returns of riskier assets (equities) is higher than less risky assets (fixed income), which in turn is higher than risk free assets (cash). In finance, there are many types of risk – in this article we are only concerned with the risk associated with the price volatility of an asset (the more the price of an asset swings, the riskier it is).

### What is Systematic Risk and Unsystematic Risk?

There are two broad categories of risk in portfolio theory – systematic (or undiversifiable, sometimes called systemic) risk and unsystematic (or idiosyncratic) risk. Unsystematic risk is typically confined to a single company or industry, and is not correlated with the rest of the market. This means with diversification (holding uncorrelated assets such as Egyptian Debt and Bombardier stock), unsystematic risk can be minimized. Systematic risk however cannot be diversified away – a good example would be market risk. Even if you hold a well diversified portfolio of real estate, stocks and bonds, there are market events where all asset classes crash (see 2008 subprime mortgage crisis). The most common measure of systematic risk is Beta, denoted by the Greek alphabet β.

#### What is Beta?

Beta is another important concept in finance. Market Beta quantifies risk by describing a security’s (usually a stock) correlation with the broader market, usually represented by an index like the S&P 500. A stock that tracks the S&P 500 exactly would have a beta of 1, and a stock that moves opposite to the movements of S&P 500 would have a beta of -1. In other words, beta is a measure of the correlation between the movements of the stock to the movements of the market.

### What is Correlation and Covariance?

#### Variance and Standard Deviation

Before getting into correlation and covariance, we need to first cover variance and standard deviation. Variance is a simple concept – it is a statistical measure of spread – how far a set of numbers deviates away from their average. Specifically, the variance of a set of numbers is the average squared deviation from the average. The formula for variance is:

Var = ∑ (x_{i} – µ)^{2} / (N – 1)

Sigma (∑) is the mathematical notation for summation, and mu (µ) is the notation for the average of a set of numbers. You simply sum the squared difference (so negative differences do not cancel out with positive differences, ∑ (x_{i} – µ) is always 0) between each number and the average of the set, and divide that sum of squared difference by the size of the set minus 1 (N – 1). The standard deviation is the squared root of variance, its formula is:

Standard Deviation (σ) = √(Var) = √(∑ (x_{i} – µ)^{2} / (N – 1))

These formulas should be memorized, especially if you are in school. Excel has nice functions that calculate the Variance and the Standard Deviation of a sample/population for you, but understanding the formula gives context to what the values represent.

Standard deviation, rather than variance, is a more commonly used measure of spread. In the context of finance, standard deviation is one measurement of risk – the less stable the return of an investment is, the “riskier” the investment is. Investors are typically risk averse, which means that given two investments with the same expected return, investors will pick the one with the lowest uncertainty (standard deviation of historical returns).

#### Correlation and Covariance

Covariance is a measure of how 2 sets of random numbers vary with respect to each other (how the 2 sets covary). Positive covariance occurs when the 2 sets of numbers move together (when one rises, the other rises as well), and negative covariance occurs when the 2 sets of numbers move opposite of each other.

The equation for covariance is a bit more involved than the one for variance:

Cov (X, Y) = ∑ (x_{i} – x̄) (y_{i} – ȳ) / N

Where x_{i} is the i-th data point and x̄ (x bar) is the average value of x (same notation goes for y, it can also be denoted by µ_{x} and µ_{y}). The intuition here is that if x_{i} and y_{i} are both larger or smaller than their respective averages, that contributes positively to covariance. If one is larger and one is smaller, that contributes negatively to covariance. The average of each contribution to covariance is then the covariance of the 2 sets of numbers. Covariance is expressed in the same units as x and y, and the value is reflects both the variance of each set of numbers (if var (x) increases, cov (x, y) also increases) and the strength of the relationship.

Correlation is covariance normalized – a coefficient between -1 and 1 that describes covariance. Unlike covariance, correlation is unitless. By removing the standard deviation of x and y from cov (x, y), correlation between x and y solely reflects the strength of the relationship between the two sets of numbers. Correlation is described by the following formula:

Cor (X, Y) = Cov (X, Y) / (σ_{X} σ_{Y})

### Breaking Down CAPM

Now that we have gone though the foundation of CAPM, we can introduce the specifics of the model. CAPM is based on the theory that an investment’s expected return is composed of the expected return from a risk free investment, and the investment’s risk premium. The investment’s risk premium is estimated by the market’s risk premium multiplied by Beta, to adjust for the market sensitivity of the investment. The equations is as follows:

E(r_{i}) = r_{F} + (E(r_{M}) – r_{F}) * β_{i}

Where E(r_{i}) is the expected return on the investment, r_{F} is the risk free rate, E(r_{M}) is the expected return on the market and β_{i} is the beta of the investment. If you think of this equation as an equation of a line (y = mx + b), the risk free rate would be the intercept, E(r_{M}) – r_{F }the slope and β_{i} the x value. Graphically, the CAPM might look something like this.

Statistics 101 | · · · |
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