BEPP Set 1: CAPM and Climate Factor Models

CAPM expected return

E[ri​]=rf​+βi​(E[rm​]−rf​)

• E[ri​] = Expected return of investment i

• rf​ = Risk-free rate

  • return from risk free (U.S. Treasury securities)

• βi​ = Beta of investment i

• E[rm​] = Expected market return

• E[rm​]−rf​ = Market risk premium

  • The extra return investors expect for investing in the market rather than a risk-free asset

Systematic vs. idiosyncratic risk

Systematic risk moves with the market and cannot be diversified away, so it earns a premium; idiosyncratic risk is firm-specific, diversifiable, and earns no premium.

Beta

Cov(r_i, r_m) / Var(r_m); how much a stock moves with the market. Higher beta means more systematic risk and a higher expected return.

Negative beta

The stock moves opposite the market, so it acts as a hedge and has a low expected return.

Does beta tell you total variance?

No. Total variance = systematic + idiosyncratic; beta only measures the systematic part.

Estimating beta

Regress the stock's excess returns (r_i − r_f) on the market's excess returns (r_m − r_f); the slope is beta, and the intercept (alpha) should be about zero under CAPM.

Efficient frontier

The set of portfolios with the highest expected return for each level of risk; rational investors only pick portfolios on it.

Diversification and correlation

Lower (more negative) correlation between assets means more risk reduction from diversification.

Capital market line

The line from the risk-free rate tangent to the efficient frontier at the market portfolio; all investors hold the market portfolio and adjust risk by lending or borrowing at r_f.

Price and expected return link

Holding expected cash flows fixed, a higher price today means a lower expected return.

When is climate risk systematic?

When it hits the economy in aggregate (on net, costs exceed benefits across firms), so it can't be diversified away and must be priced.

When is climate risk idiosyncratic?

When winners and losers offset randomly across firms, so diversified investors can eliminate it and it earns no premium.

Climate factor model

E[ri​]=rf​+βm​(E[rm​]−rf​)+βc​E[rPMC​]

adds a climate factor to the CAPM.

PMC factor

Polluting minus clean: a self-financing portfolio long high-emission firms and short low-emission firms; its return is the payoff to bearing climate risk.

Climate beta

A stock's sensitivity to the climate factor; positive means brown (hurt by climate action), negative means green (a climate hedge).

Brown vs. green expected returns in a climate factor model

Brown stocks (high climate beta) need higher expected returns to compensate for climate risk; green stocks (low or negative climate beta) have lower expected returns.

Climate factor model practice

E[ri​]=rf​+βm​(E[rm​]−rf​)+βc​E[rPMC​]

Given:

rf=3%, E[rm]−rf=5%, E[rPMC]=2%

Stock:

βm​=1, βc​=1

E[rX​] = 3%+(1)(5%) + (1)(2%) = 10%​

Stock Y:

βm​=1, βc​=−0.5

E[rY​] = 3%+(1)(5%) + (−0.5)(2%) = 7%​

Daily return prediction from climate factor news

Realized return ≈ β_m × market return + β_c × PMC return (ignoring r_f); e.g., with PMC −5% on strict policy news, a β_c = 1 stock loses 5% from climate exposure alone.

How to estimate climate betas

Collect stock returns, market returns, the risk-free rate, and emissions data to build a PMC portfolio; regress stock excess returns on market excess returns and PMC returns.

Why climate beta estimates differ across studies

Different emissions data (scope 1/2/3, estimated vs. reported), different ways of sorting firms (emissions vs. ESG scores), different time periods, and different control factors.

Sector pattern in carbon betas

Highest in energy, materials, and utilities; lowest or negative in health care, financials, and IT.