Industry betas, the market risk premium and country risk

When should you use an industry beta?

When the company’s own regression beta is unreliable or unavailable. The industry beta is the beta of a group of similar companies, adjusted for the leverage difference. Three cases call for it:

• The single-stock regression is noisy (low R-squared (R²), high standard error), because much of one stock’s movement is diversifiable company-specific risk.
• The company has no usable price history: a private company, or a recent initial public offering (IPO).
• The company recently changed its leverage, so past returns reflect an equity risk it no longer has.

Why it works: a portfolio of companies with similar business risk diversifies away much of the company-specific noise. Its regression has a higher correlation with the market and a lower standard error than any one member. Take five comparable logistics companies with single-stock betas of 0.94, 1.03, 1.10, 1.14 and 1.29 and standard errors from 0.13 to 0.17. An equal-weighted portfolio of the five has a beta of 1.10, their average, with a standard error of 0.10 and the highest correlation with the index of any of them.

The precondition. The comparables must have the same asset risk (business risk) as the company. If they do not, the industry beta is simply the wrong beta. Different leverage is fine: that is what unlevering and relevering fix. The peer group lesson covers how to choose comparables.

How do you unlever and relever beta?

Debt makes equity riskier, so an observed beta, the levered beta , mixes business risk with financial risk. Strip the comparables’ leverage out to get the unlevered beta , which holds business risk only. Then put the company’s own leverage back to get its relevered beta . The leverage measure is the debt to market value of equity (D/MVE) : debt divided by the market value of equity, not by debt plus equity.

$$\beta_u = \dfrac{\beta_{L,Comps}}{1 + (1 - t)\,\dfrac{D}{MVE}}$$

Use the comparables’ average levered beta, average tax rate and average D/MVE.

$$\beta_{L,firm} = \beta_u\left[1 + (1 - t)\,\dfrac{D_{firm}}{MVE_{firm}}\right]$$

Here the company’s own tax rate and D/MVE go in.

Two plain rules sum it up. A levered beta is never below its unlevered beta. And the company’s beta ends up above the comparables’ average only if its leverage factor, 1 + (1 − t) × D/MVE, is larger than theirs, and below it only if smaller. The adjustment matters most when the leverage differs a lot; with little debt on both sides it barely moves the beta.

What is the market risk premium?

Memorize

The market risk premium is the extra return investors expect for holding risky equities instead of a risk-free asset. In practice it is the expected return on a broad, diversified stock index minus the return on a government security of the matching country. Together with the government rate, it sets the required return on an average-risk (beta of 1) stock.

It differs across countries and over time, it is an estimate with error, and analysts disagree about its size and about methods. There are two families of methods: historical and forward-looking.

How do you estimate the MRP from history?

The historical market risk premium is the average past total return on a broad stock index minus the average past return on a government security from the same country, over a period long enough for the average to be a fair guide to the future.

$$MRP = \bar{r}_{index} - \bar{r}_{government}$$

Two choices change the answer.

• Arithmetic or geometric mean? The geometric (compound) average is always smaller than the arithmetic average of the same data, because volatility drags on compounding. See the mean returns lesson.
• Bills or bonds? Long-term government bonds earn more than bills on average (a maturity risk premium), so an MRP measured against bonds is lower. Equity cash flows run for decades, so valuation should generally use an MRP measured against long-term government bonds. (Regression betas, by contrast, subtract a short-term rate from each period’s return.)

What is an implied market risk premium?

The implied market risk premium is a forward-looking MRP backed out of today’s index level. Treat the index price as its value, solve the constant growth model for the expected return, and subtract today’s government yield. It uses only current prices plus a growth assumption.

$$E[r_m] = \dfrac{Div_{1,m}}{P_{0,m}} + g_{l,m}, \qquad \text{Implied MRP} = E[r_m] - r_f$$

$Div_{1,m}$ is next year’s expected index dividends, $P_{0,m}$ is today’s index level, and $g_{l,m}$ is the long-run dividend growth rate. The model is the constant growth model from the implied return lesson and the dividend discount models.

The same idea works with free cash flow to equity (FCFE) in place of dividends, or with a two-stage growth model.

Which MRP should you use?

Rising markets pull the two methods apart. After strong years, historical average stock returns rise, so the historical MRP goes up, implying a higher discount rate and lower values just as prices climb. The implied MRP goes down: if the 4,800 index rose 10% to 5,280 with the same dividends, 125.40 ÷ 5,280 + 4.5% − 4.0% = 2.88% (2.875%). Present value models using a historical MRP therefore tend to call a rising market overvalued.

There is a trade-off. Forward-looking estimates fit the purpose better (a cost of equity is about the future) but rest on growth assumptions. Historical estimates are concrete evidence but may not reflect what lies ahead.

Developed markets are typically in the low to middle single digits (about 3%–6%). Developing markets can be in the high single digits or above 10%. Because any point estimate has error, run the valuation over a range of MRPs. That is sensitivity analysis .

How do you add country risk?

For a company in a less developed market, add a country risk premium (CRP) to a developed market’s MRP before multiplying by beta in the capital asset pricing model (CAPM).

$$E[r_i] = r_f + \beta_i\left(MRP_{developed} + CRP\right)$$

The premium itself comes from the country’s government bonds issued in a reserve currency such as USD or EUR. The sovereign bond yield spread is the extra yield those bonds pay over that currency’s benchmark. The equity-to-bond volatility ratio scales the bond-market risk up to equity-market risk.

$$CRP = \text{Sovereign bond yield spread} \times \dfrac{\text{Annualized equity market standard deviation}}{\text{Annualized bond market standard deviation}}$$

Data providers and some academics publish CRP estimates every year. Use a risk-free rate and a developed-market MRP in the same currency as the spread.

Work through all three parts in the interactive: unlever and relever, the two kinds of premium, and the country add-on.

Cost of equity, CAPMr_f + β × (premium)

• Beta
  • Own regression if it is precise
  • Industry beta Memorize unlever the comparables, relever at the company's own D/MVE
• Market risk premium
  • Historical Memorize geometric below arithmetic; against bonds below against bills
  • Implied Memorize Div₁ ÷ P₀ + g − r_f
• Country risk premium Memorize spread × equity σ ÷ bond σ; added to the developed premium before beta

Go deeper Why leverage raises beta

Debt holders are paid first, so shareholders absorb all the swings in the company’s value. The more debt on the balance sheet, the more the equity return moves for a given change in the business, and the higher the equity beta. Tax matters because interest is deductible, which is why the formula uses $(1 - t)$ on the debt term. The Modigliani and Miller lesson explains the logic in full.

Go deeper Why compare against bonds, not bills

A bill matures in a few months, while shares are claims on cash flows that run for decades. The matching risk-free comparison is a long-term government bond. Bonds pay a maturity risk premium over bills, so the stock premium measured against bonds is the smaller one and the more consistent with a long-lived equity cash flow.