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Carbon footprint for dynamically

rebalanced portfolios.

Carmine de Franco

December 14th, 2017

Green Finance Research Advances

Challenges for dynamically rebalanced portfolios: the turnover effect

Market practice measures carbon footprints with low-frequency snapshots of portfolio holdings (usually End-of-

Year or averages of End-of-Quarter).

While this approach is somehow valid for standard cap-weighted benchmarks, it is definitely not adapted for

strategies with relatively high turnover (as Smart Beta).

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Historical sector breakdown for the Barclays Shiller CAPE US Sector Value Index (left) and the S&P 500 Index (right). Source S&P, Barclays, Bloomberg.

Challenges for dynamically rebalanced portfolios: the performance effect

When we compare a dynamic strategy with a (static) benchmark, the carbon footprint (CFP) can be heavily

impacted by performance differentials. Assume a market with Stock A and B.

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Stock A Stock B

5E+05 2E+06

Carbon Measure tCO2

and a Strategy that switched from stock A to stock B

At the end of the year, the strategy has a CFP of 5.5 tCO2 while the benchmark (with the same initial 1000€)

realizes 4.2 tCO2. The difference is 1.33 tCO2, or 32% in relative terms.

But is the benchmark really a “greener” strategy?

If we account for the performance effect (+44% for the strategy versus +2% for the benchmark), then the

CFP of this adjusted (Natural) benchmark is 5 tCO2, so that the strategy has a CFP roughly 10% higher.

Date Stock A Stock B Stock A Stock B Market Cap A Value CFP

t0 100 100 1 5 600 16.7% 1000 0.0

t0 + 6M 120 80 1 5 520 23.1% 867 2.1

t0 + 1Y 132 96 1 5 612 21.6% 1020 4.2

Nb Outstanding

Shares (mln)Performances Benchmark

Market

Capitalization

(mln)

Weight A Value CFP Weight in A Value Adj Value CFP

100% 1000 0.0 16.7% 1000 1000 0.0

0% 1200 2.5 23.1% 867 1200 2.1

0% 1440 5.5 21.6% 1020 1440 5.0

Natural BenchmarkStrategy CFP

Strategy

tCO2

Delta over

Benchmark

Delta over

Natural

Benchmark

5.50 1.33 (+32%) 0.53 (+10.7%)

The Framework

Ft is the fund‘s value (AUM). Weights are denoted by

𝑤𝑡𝐹. Bt is a reference cap-weighted index to which the

fund is benchmarked, and stock weights are denoted by

𝑤𝑡𝐵 (ex. Stoxx Europe 600 Index or the S&P 500 Index).

We assume 𝑤𝑡𝐵 = 0 ⇒ 𝑤𝑡

𝐹 = 0

For each stock in the reference index, we denote

𝐶𝐵𝑖,𝑦 = 𝐶𝐵𝑖,𝑦1 , 𝐶𝐵𝑖,𝑦

2 , … , 𝐶𝐵𝑖,𝑦𝑗

We denote by 𝑅𝑖,𝑦 the revenue of company i for year y.

We adjust data on a daily basis:

𝐶𝐵𝑖,𝑡𝑑𝑎𝑖𝑙𝑦=𝐶𝐵𝑖,𝑦𝑛 𝑦

where 𝑛(𝑦) is the number of trading days.

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Absolute and Relative Contributions

We define the proportion of daily emissions

attributable to fund F as the Fund's Daily Contribution

DCF:

𝐷𝐶𝑡𝐹 𝐶𝐵𝑘 =

𝜃𝑖,𝑡𝐹

𝜃𝑖,𝑡𝐵 𝐶𝐵𝑖,𝑡

𝑘

𝑛

𝑖

=𝐹𝑡𝐵𝑡 𝑤𝑖,𝑡𝐹

𝑤𝑖,𝑡𝐵 𝐶𝐵𝑖,𝑡

𝑘

𝑛

𝑖

We define the absolute contribution A of the fund F

relative to the measure 𝐶𝐵𝑘 over the period [T1; T2]

as the sum of daily values of 𝐶𝐵𝑘 attributable to the

fund

𝐴𝐹 𝑇1, 𝑇2, 𝐶𝐵𝑘 = 𝐷𝐶𝑡

𝐹 𝐶𝐵𝑘𝑇2

𝑡=𝑇1

= 𝐹𝑡𝐵𝑡 𝑤𝑖,𝑡𝐹

𝑤𝑖,𝑡𝐵 𝐶𝐵𝑖,𝑡

𝑘

𝑛

𝑖

𝑇2

𝑡=𝑇1

And the relative contribution (amount of emissions for

1 million Revenue)

𝐼𝐹 𝑇1, 𝑇2, 𝐶𝐵𝑘 =𝐴𝐹 𝑇1, 𝑇2, 𝐶𝐵

𝑘

𝐴𝐹 𝑇1, 𝑇2, 𝑅

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It is very common to see CFP calculated as the

weighted average of single stock’s carbon

emissions or carbon intensity.

If we look at instantaneous carbon intensity :

𝐼𝐹 𝐶𝐵𝑘 =

𝑤𝑖,𝑡𝐹

𝑤𝑖,𝑡𝐵 𝐶𝐵𝑖,𝑡

𝑘𝑛𝑖

𝑤𝑖,𝑡𝐹

𝑤𝑖,𝑡𝐵 𝑅𝑖,𝑡

𝑛𝑖

= 𝜆𝑖,𝑡𝐶𝐵𝑖,𝑡𝑘

𝑅𝑖,𝑡, 𝑤ℎ𝑒𝑟𝑒 𝜆𝑖,𝑡 =

𝑤𝑖,𝑡𝐹

𝑤𝑖,𝑡𝐵 𝑅𝑖,𝑡

𝑤𝑖,𝑡𝐹

𝑤𝑖,𝑡𝐵 𝑅𝑖,𝑡

𝑛𝑖

𝑛

𝑖

Here 𝐶𝐵𝑖,𝑡𝑘

𝑅𝑖,𝑡 is the stock's intensity measure while

𝜆𝑖,𝑡 is the fraction of fund's revenue attributable

to stock i.

We highlight that weighted averages of stocks'

intensities, where the weights are the ones in

the fund as opposed to the revenues, will give a

biased measure on the fund's intensity.

The Natural benchmark

• As we show in our example, a strategy can end up with a higher CFP simply because it has significantly

outperformed the benchmark.

• We recognize that the difference in financial performances could imply higher measures AF(X).

• In line with our fair accounting approach, we propose a specific benchmark that neutralizes differences

between the fund and the reference index market values.

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Definition Let 𝑇1 ≤ ℎ ≤ 𝑇2. The Natural Benchmark 𝐵𝐹ℎ 𝑇1≤ℎ≤𝑇2 for the fund F is a family of theoretical

portfolios, such that

∀𝑡 ∈ 𝑇1, 𝑇2 , 𝐵𝐹𝑡 𝑡 = 𝐹𝑡 𝑎𝑛𝑑 𝑤𝑡,𝑖𝐵𝐹𝑡 = 𝑤𝑖,𝑡

𝐵

i.e. at time t the t-th portfolios of the family 𝐵𝐹𝑡 invests the amount 𝐹𝑡 according to the weights in the reference index.

Application 1: the size effect

Consider two portfolios invested in the largest and the smallest terciles of the Stoxx Europe 600

Index at each rebalancing date. Denote these portfolio as Large and Small. We assume that two

investors are buying-and-holding these two portfolios for 1,000,000 EUR at inception (December,

30, 2005).

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Application 1: the size effect

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Application 2: Eurozone vs rest of Europe

Consider two portfolios invested in the stocks of Eurozone and the rest of Europe that belongs

to the Stoxx Europe 600 Index at each rebalancing date. Denote these portfolio as Euro and

Ex-Euro.

9

Application 2: Eurozone vs rest of Europe

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Application 3: Carbon-friendly portfolios

• For benchmarked investors, it is interesting to look at strategies that mimic the financial behaviour of the

standard cap-weighted benchmarks (ex. MSCI World Index) while targeting a given CFP.

• This becomes an optimization problem: Minimize the ex-ante tracking error w.r.t the benchmark under

carbon performance constraints as follows

– 𝐴𝐹 𝑋 ≤ 1 − 𝑥% × 𝐴𝐵𝐹(𝑋) for Carbon Measures X

– 𝐴𝐹 𝑋 ≥ 1 + 𝑥% × 𝐴𝐵𝐹 (𝑋) for “Green” Measures X.

– 𝐼𝐹 𝑋 ≤ 1 − 𝑥% × 𝐼𝐵𝐹 (𝑋) for Carbon Measures X

• In the example we use

– Carbon emission reduction of 50%

– Carbon intensity reduction of 50%

– Carbon emission from reserves reduction of 50%

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Application 3: Carbon-friendly portfolios

12

Application 3: Carbon-friendly portfolios

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References & Disclaimer

De Franco, C. and Monnier, B. (2018) “Carbon footprint for dynamically rebalanced portfolios”. To appear in the Journal of Investing, Vol. 1,

https://www.ossiam.fr/files/research_papers/1487341494_17_Carbon_Footprint_for_dynamically_rebalanced_portfolios.pdf

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