# Beta & Relative Risk

We can add the risk-free rate and the equity risk premium to determine how much the market, or an average risk stock, will return. But what if our stock is more or less risky than the market? For that, we need a measure of relative risk.

Most analysts use beta as a measure of relative risk. Sometimes you’ll see the shorthand for beta in the form of the lowercase second letter of the Greek alphabet: $\beta$

## Beta

Beta measures how volatile an equity is compared to the market, which for practical purposes, is usually the S&P500 index for US stocks. Beta is convenient because the market’s beta is centered around 1.0. A stock with a beta of 2.0 should move up twice as much as the market in good times and down twice as much in bad times. The expected returns are the same, but they’re twice as volatile. A stock with a beta of 0.5 would move up and down half as much as the market. The former is considered twice as risky as the market while the latter is regarded as half as risky.

Beta can be classified depending upon how it’s calculated:

1. Regression beta
2. Bottom-up beta

## Regression Beta

Most analysts and services calculate beta using a regression. Regression betas are computed using a linear regression comparing the returns of the market with that of a stock. This is what a Bloomberg terminal or Yahoo Finance displays as beta, and it’s a terrible way to think about risk both technically and conceptually.

### Regression Beta Technical Issues

Regression-based betas have three problems inherent in how they are calculated:

1. High standard error
2. Backwards looking
3. Subject to bias

#### High Standard Error

Regression betas are very noisy. Most have a high standard error. For instance, if the beta of a stock is 1.25 with a standard error of 0.5, the actual beta could be .25 or 2.25 using two standard deviations. Check the error next time you’re using a regression beta and be prepared to be shocked.

Additionally, if your stock makes up a significant portion of the comparison market, the beta can look great but it is fundamentally flawed.

#### Backwards Looking

Regression betas look at a single slice of history. If a stock drops significantly while the market is going up, it will be less correlated with the market and its beta measure will go down. If a stock is plummeting, is it really less risky?

#### Subject to Bias

With regression betas, you can pretty much come up with any number you want if you cherry-pick the index and time period.

### Regression Beta Conceptual Issues

A beta is a statistical measure that tries to estimate risk. Risk comes from the choices a company makes regarding:

1. Nature of revenue sources
2. Operating leverage
3. Financial leverage

#### Nature of Revenue Sources

Companies that sell discretionary products will have less predictable earnings as customer purchases will fluctuate more due to market cycles.

#### Operating Leverage

Companies that have higher cost structures will have earnings that vary more in good and bad times than companies that have a less rigid cost structure. While accountants don’t break out financials into variable and fixed costs, you can use the following to get an idea of operating leverage over a period of time.

$EBITVariabilityMeasure = \frac{\%ChangeInEbit}{\%ChangeInRevenues}$

#### Financial Leverage

When you borrow money, you create a fixed expense that you have to pay in good times and in bad. This will magnify earnings in good times and subtract from them in not so good times.

$LeveredBeta = UnleveredBeta * (1 + (1 - TaxRate) * \frac{Debt}{Equity})$

## Bottom-up Beta

A bottom-up beta, sometimes called a fundamental beta, is both technically and conceptually more sound than a regression beta. It inherently incorporates the revenue and leverage risks causing increased earnings volatility. You can then adjust the financial leverage, and potentially the operational leverage, to match that of the firm you’re valuing.

Bottom-up betas have several benefits:

1. They are more precise: $$\frac{AvgStdErrorAcrossRegressionBetas}{\sqrt{NumberOfFirmsInSample}}$$
2. They’re not shackled to the past. You can select the businesses, weights, and financial leverage that a business has today or may have tomorrow

If you’re not interested in calculating the bottom-up beta for yourself or you don’t have access to the resources needed, Aswath Damodaran makes our lives easier once again by calculating the bottom-up betas by sector.

### How to Calculate a Bottom-up Beta

2. Find the publicly traded firms in each of these businesses and obtain their regression betas. Take median of the betas for these companies
3. Estimate how much value your firm derives from each different business it is in
4. Compute the weighted average of the unlevered betas
5. Compute a levered beta (equity beta) for your firm using the market debt to equity ratio of your firm

#### An example for a steel and chemical company:

2. Find the median unlevered regression betas
1. Find the median regression beta for as many steel companies as you can
2. Find the median regression beta for as many chemical companies as you can
3. Remove the effects of debt (unlever): $$UnleveredSteelBeta = BetaAcrossPublicSteelFirms / (1 + (1 - TaxRate) * \frac{AvgDebtRatioPublicSteelFirms}{AvgEquityRatioPublicSteelFirms})$$
4. Remove the effects of cash: $$UnleveredSteelBeta = UnleveredSteelBeta / (1-CashPercentage)$$
3. Estimate how much value the business derives from the steel and chemical business using the weights that make the most sense, generally revenue weights.
4. Compute the weighted average from the unlevered betas: $$WeightedBeta = SteelUnleveredBeta * \frac{SteelRevenue%}{TotalRevenue%} + ChemicalUnleveredBeta * \frac{ChemicalRevenue%}{TotalRevenue%}$$
5. Compute the levered beta (equity beta) for the firm: $$LeveredBeta = (1 + (1 - TaxRate) * \frac{Debt}{Equity})$$

### Adjusting The Market Risk for a Private Company

For a private company, we need to add back the risk that the model assumes to be diversified away. $r^2$ is a statistical measure of how close the data fits the regression line. By dividing by $r^2$, we are adding back that risk.

$NonDiversifiedPrivateCostOfEquity = \frac{MarketBeta}{\sqrt{r^2}}$ 