Climate BMG Factor Lab

Institutional investors face a structural question that balance-sheet emissions data alone cannot fully answer: how do markets price climate transition risk in daily equity returns? When policy tightens, green technologies scale, or fossil-fuel cash flows re-rate, which stocks move—and do they move together for reasons not explained by market beta, size, value, or momentum?

This project applies a return-based climate factor in the spirit of the Brown Minus Green (BMG) literature. Rather than scoring firms on reported carbon footprints, we ask whether a stock's excess returns co-move with a tradable spread between brown and green energy performance. The object of interest is the climate beta ${\beta }_{i,\mathrm{BMG}}$: the sensitivity of stock $i$ to brown-minus-green factor realizations, estimated after standard Fama–French and Carhart controls.

The practical uses are threefold. Risk measurement: quantify implied exposure to a climate-policy or energy-transition shock proxied by ETF spreads. Portfolio diagnostics: rank holdings by ${\beta }_{\mathrm{BMG}}$ and test whether a mandate's risk is concentrated in energy-linked names. Model validation: test whether BMG is a priced dimension orthogonal to $MKT$, $SMB$, $HML$, and $MOM$ in a fixed linear factor framework.

We stress that this is market-implied risk—what prices say, not what companies disclose. It complements—but does not replace—fundamental carbon accounting, scenario analysis, or physical-risk models.

We augment CAPM, Fama–French three-factor, and Carhart four-factor models with a daily BMG return series:

$$BM{G}_t=r(XOP{)}_t-r(SMOG{)}_t$$

where $XOP$ proxies oil & gas exploration & production and $SMOG$ proxies low-carbon energy. For each stock $i$ in a Goncharov (2023) GICS-balanced US panel (2018–2021), we estimate six nested ordinary least squares (OLS) regressions on daily log excess returns and report ${\hat{\beta }}_{i,\mathrm{BMG}}$, $t$-statistics, adjusted $R^2$, and residual standard errors.

The focal specification is Carhart+BMG, isolating marginal climate exposure after market, size, value, and momentum. Positive ${\beta }_{\mathrm{BMG}}$ indicates implied brown exposure: returns tend to rise when brown energy outperforms green. Cross-sectional and sector aggregates summarize where transition risk appears priced.

Definition. Let $P_{b,t}$ and $P_{g,t}$ denote adjusted closes for the brown and green ETFs. Daily log returns are

$$r_{b,t}=\ln P_{b,t}-\ln P_{b,t-1},r_{g,t}=\ln P_{g,t}-\ln P_{g,t-1}.$$

The BMG factor return is the long-short spread:

$$BM{G}_t=r_{b,t}-r_{g,t}.$$

A positive $BM{G}_t$ means brown outperformed green on day $t$. Cumulative ${\sum }_{s\le t}BM{G}_s$ tracks the long-run tilt of that relative performance.

Interpretation. BMG is a reduced-form transition-risk mimicking portfolio in return space. It does not measure tonnes of CO${}_2$; it measures whether markets reward brown over green on a given day. Stocks with high ${\beta }_{\mathrm{BMG}}$ behave like they are long that transition spread.

Volatility. BMG is typically high-variance (energy-policy shocks, oil crashes, green rallies). Rolling annualized volatility of $BM{G}_t$ helps identify regimes where climate betas are estimated against especially informative factor realizations—e.g. the March 2020 oil collapse and subsequent recovery in our sample.

The charts below plot cumulative $r_{XOP}$ vs $r_{SMOG}$, cumulative BMG, and rolling BMG volatility.

For each stock $i$, let $r_{i,t}^e=r_{i,t}-r_{f,t}$ denote the daily log excess return (risk-free $r_{f,t}$ from Ken French). All models include intercept ${\alpha }_i$ and idiosyncratic noise ${\epsilon }_{i,t}$.

CAPM:

$$r_{i,t}^e={\alpha }_i+{\beta }_{i,MKT}MK{T}_t+{\epsilon }_{i,t}$$

Fama–French three-factor (FF3):

$$r_{i,t}^e={\alpha }_i+{\beta }_{i,MKT}MK{T}_t+{\beta }_{i,SMB}SM{B}_t+{\beta }_{i,HML}HM{L}_t+{\epsilon }_{i,t}$$

Carhart four-factor:

$$r_{i,t}^e={\alpha }_i+{\beta }_{i,MKT}MK{T}_t+{\beta }_{i,SMB}SM{B}_t+{\beta }_{i,HML}HM{L}_t+{\beta }_{i,MOM}MO{M}_t+{\epsilon }_{i,t}$$

Climate-augmented models append $BM{G}_t$ to CAPM, FF3, and Carhart. The headline specification is:

$$r_{i,t}^e={\alpha }_i+\sum _{f\in \{MKT,SMB,HML,MOM\}}{\beta }_{i,f}{f}_t+{\beta }_{i,\mathrm{BMG}}BM{G}_t+{\epsilon }_{i,t}.$$

OLS estimation. Stacking $T$ observations, each model is ${\mathbf{y}}_i=\mathbf{X}{\mathbf{\beta }}_i+{\mathbf{\epsilon }}_i$ with ${\hat{\mathbf{\beta }}}_i=({\mathbf{X}}^{\prime }\mathbf{X}{)}^{-1}{\mathbf{X}}^{\prime }{\mathbf{y}}_i$. Standard errors underpin $t$-statistics; $|t_{\mathrm{BMG}}|>1.96$ is reported as significant at 5% (two-sided).

Goodness of fit. Coefficient of determination $R^2$ and adjusted $R^2$:

$$R_{\mathrm{adj}}^2=1-(1-R^2)\frac{T-1}{T-k-1},$$

where $k$ is the number of slope coefficients. Nested comparisons use mean cross-sectional $R_{\mathrm{adj}}^2$ across stocks. Residual standard error:

$$\mathrm{RSE}=\sqrt{\frac{{\sum }_t{\hat{\epsilon }}_{i,t}^2}{T-k-1}}.$$

A climate factor is only useful if it is not fully redundant with $MKT$, $SMB$, $HML$, and $MOM$. Let ${\mathbf{f}}_t=(MK{T}_t,SM{B}_t,HM{L}_t,MO{M}_t,BM{G}_t{)}^{\prime }$. We report the sample correlation matrix $\mathrm{Corr}(\mathbf{f})$.

If $\mathrm{Corr}(BMG,MKT)$ is material, univariate regressions confound energy beta with climate beta. In the multivariate Carhart+BMG regression, ${\beta }_{i,\mathrm{BMG}}$ is the partial effect of BMG holding other factors fixed—analogous to how FF3 isolates value exposure after controlling for market and size.

Near-multicollinearity inflates $\mathrm{Var}({\hat{\beta }}_{i,\mathrm{BMG}})$; the heatmap below is the first diagnostic before interpreting sector or stock rankings.

For each GICS sector $s$, we compute the cross-sectional mean ${\bar{\beta }}_{\mathrm{BMG}}^{(s)}=\frac{1}{N_s}{\sum }_{i\in s}{\hat{\beta }}_{i,\mathrm{BMG}}$ and the fraction of names with $p_{\mathrm{BMG}}<0.05$.

Economic priors: Energy and Materials often load positively on BMG (brown-factor exposure); Utilities with renewable tilt may load negatively; Technology and Healthcare may show weaker direct linkage but can still exhibit significant loadings through supply-chain or discount-rate channels.

The histogram bins ${\hat{\beta }}_{i,\mathrm{BMG}}$ across all estimated stocks—revealing whether climate sensitivity is a few extreme names or dispersed across the panel.

Each row reports the Carhart+BMG fit for one ticker. The climate beta ${\hat{\beta }}_{i,\mathrm{BMG}}$ answers: if BMG rises one unit (one day of brown-minus-green outperformance), how much does the stock's excess return move, holding other factors constant?

Compare Carhart vs Carhart+BMG adjusted $R^2$ to see whether climate explains idiosyncratic variation beyond style factors. Large gains in $R_{\mathrm{adj}}^2$ with insignificant ${\beta }_{\mathrm{BMG}}$ suggest overfitting or correlated noise; significant ${\beta }_{\mathrm{BMG}}$ with modest $R^2$ gains can still matter for portfolio hedging.

Ranking by ${\hat{\beta }}_{\mathrm{BMG}}$ produces an implied brown-to-green spectrum in return space—not a ESG score, but a market-based ordering useful for stress tests.

We follow Goncharov (2023) and report **paired one-tailed $t$-tests** on vectors of stock-level RSEs when comparing nested models (e.g. Carhart vs Carhart+BMG). Null: mean RSE does not fall when adding factors; alternative: nested model tightens residuals.

For individual coefficients, under classical OLS assumptions:

$$t_{\mathrm{BMG}}=\frac{{\hat{\beta }}_{i,\mathrm{BMG}}}{\mathrm{SE}({\hat{\beta }}_{i,\mathrm{BMG}})}\sim t_{T-k-1}.$$

Mean cross-sectional $R_{\mathrm{adj}}^2$ often rises when BMG enters even when paired RSE tests fail to reject at 5%—mirroring the FF3-over-CAPM pattern in multifactor replication studies: explanatory power can improve without statistically lower residual volatility in short panels.

Sample: Thirty thesis stocks (Goncharov 2023); delisted names drop out. Results are not index-representative and suffer survivorship bias.

BMG proxy: Two ETFs summarize a high-dimensional transition; alternative brown/green pairs (e.g. XLE vs ICLN) yield different spreads.

Linear fixed betas: 2018–2021 includes trade-war volatility, the COVID crash, and an oil-price collapse—regimes where a constant ${\beta }_{\mathrm{BMG}}$ may misstate time-varying climate exposure.

Not emissions data: This lab does not report Scope 1–3 emissions, carbon intensity, or net-zero alignment. It infers priced transition risk from returns.

Research and education only—not investment advice.