Net Zero 2050 | -0.712 | -0.692 | -0.181 | -0.051 | -0.362
Divergent Net Zero | -1.187 | -0.978 | -0.310 | -0.099 | -0.554
Table 12: Average annual output growth evolution with respect to the Current
Policies scenario between 2020 and 2030 (in %)

It follows from the Total column in Table 12 that the average annual growth
between 2020 and 2030 is decreasing. The Divergent Net Zero is the economic
worst case (the best one for the climate) where the carbon ton would cost
395.21€ in 2030. The Current Policies is the economic best case (the worst one
for the climate) where the carbon ton would cost 39.05€ in 2030. The
difference of the annual output growth between the worst and the best
scenarios is of about $-0.554\%$.

Figure 4: Mean and 95% confidence interval of the annual output growth

The four scenarios are clearly discriminating. In the Divergent Net Zero
scenario, our model shows, on the last subplot in Figure 4, a drop in output
growth, with respect to the Current Policies scenario, that starts at 0.405%
in 2020 and increases every year until a 0.746% drop is reached in 2030.
Cumulatively, from 2020 to 2030, a drop of 5.539% is witnessed.

We can compare this value to 2.270% which is the GDP drop between the Net Zero
2050 and Current Policies scenarios obtained with the REMIND model in [32].
The difference observed with REMIND can be explained by the fact that our
model does not specify how the revenues generated by charging GHG emissions
and collected by the regulator are reinvested or redistributed. We could, for
example, head the investment towards low-carbon energies, which would have the
effect of reducing the GHG emissions costs on these sectors. Moreover, in our
model, the carbon price is assumed to increase uniformly (which implies that
emissions would increase indefinitely - which is not desirable) from 2021 to
2030, while in REMIND an adjustment of the carbon price growth rate is being
made in 2025. Furthermore, productivity is totally exogenous in our model
while there are exogenous labor productivity and endogenous technological
change for green energies in REMIND, which is expected to have a downward
effect on the evolution of the carbon price. However, we recall that our model
has the benefit to be stochastic and multisectoral.

Now, it follows from both Figure 4 and Table 12 that the introduction of the
carbon price is less adverse for the Very Low Emitting and Low Emitting groups
than for the High Emitting and Very High Emitting ones. The slowdown is
highest for the Very High Emitting group, which was anticipated given that the
emissions cost on firms was the highest. Moreover, the slowdown could be
accelerated by the climate transition, not only because this sector emits GHG,
but also because its intermediary inputs are from the High Emitting and Very
High Emitting sectors. On the other hand, the Very Low Emitting sector
continues its strong growth because it emits less and because France is driven
by the service industry. In addition, the consumption in the two most
polluting sectors suffers from a slowdown higher than the whole consumption
slowdown and lower than in the two least polluting ones.

Figure 5: Mean and 95% confidence interval of the contributions (in %) of each
sector in the total output

Finally, from figure 5 666We ignore years 2020 and 2021 due to COVID-19
jumps., due to deindustrialization and the reduction in agricultural
production, the share of production from sectors Very High Emitting and Low
Emitting is tending to decline in the French economy. This decline accelerates
with the severity of the transition. However, it is evident that the French
economy is shifting toward a service-based economy. This phenomenon becomes
more pronounced as the carbon price increases. It could imply that companies
are increasingly leaving the most polluting (and therefore most taxed)
sectors. To mitigate these effects, we could intelligently reinvest the
collected carbon taxes.

####  6.3.2 Greenhouse gases

We observe in Figure 6 777We ignore years 2020 and 2021 due to COVID-19 jumps.
a slow reduction in GHG emissions concurrent with deindustrialization, the
reduction of agricultural production and probably thanks to the efforts that
are beginning to be made. With the introduction of a carbon price, this
downward trend accelerates. However, for the sector Very High Emitting, GHG
emissions continue to increase for soft transition scenarios.

Figure 6: Mean and 95% confidence interval of the direct GHG emissions (in ton
of CO2-equivalent) of firms in each sector

Finally, it should be noted that these trends in GHG emissions can also be
explained by the structure chosen for our carbon intensities.

####  6.3.3 Firm valuation

Here, we consider a representative firm characterized by its cashflow
$F_{t_{\circ}-1}$ at $t_{\circ}-1$, with standard deviation
$\sigma_{\mathfrak{b}}$ and by the contribution $\mathfrak{a}$ of sectoral
consumption growth to its cash flows growth. We would like to know how the
value of this company evolves during the transition period and with the carbon
price introduced in the economy. Consider
$F_{t_{\circ}-1}=\text{€{1,000,000}}$, $\sigma_{\mathfrak{b}}=5.0\%$,
$\mathfrak{a}=[0.25,0.25,0.25,0.25]$ (each sector has the same contribution to
the growth of the cash flows of the firm), the interest rate $r=5\%$. For
$M=5000$ simulations of the productivity processes
$(\Theta_{t},\mathcal{A}_{t})_{t_{\circ}\leq t\leq t_{\star}}$, we compute the
firm value using (3.8). We can analyze both the average evolution of the firm
value per year and per scenario (Figure 7) and the empirical distribution of
the firm value per scenario (Figure 8).

(a) Annual firm value per scenario in million euros per year

(b) Annual firm value growth per scenario in % per year

Figure 7: Firm value

We see that even if the value of the firm grows each year, this growth is
affected by the severity of the transition scenario (Figure 7(a)). The
presence of a carbon price in the economy clearly reduces the firm value
yearly by -2.440% for NDCs, -5.009% for Net Zero 2050, and -7.412% for
Divergent Net Zero. The faster the transition, the steeper the drop over 10
years. We start thus from a decrease of 1.351% in 2021 to 3.483% in 2030 for
NDCs, from 3.541% to 6.248% for Net Zero 2050, and from 5.307% to 8.238% for
Divergent Net Zero (Figure 7(b)).

Figure 8: Firm value distribution per scenario and per year

The introduction of the transition scenario distorts the density function of
the firm value, and in particular, moves it to the left.

##### Credit risk

Consider a fictitious portfolio of $N=16$ firms described in Table 13 below.
This choice is made to ease the reproducibility of the result since the
default data are proprietary data of BPCE. Note that the firms $1$ to $4$, $5$
to $8$, $9$ to $12$, and $13$ to $16$ respectively belong to the Very High
Emitting, High Emitting, Low Emitting, and Very Low Emitting groups.

n° | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\sigma_{\mathfrak{b}^{n}}$ | 0.05 | 0.05 | 0.06 | 0.06 | 0.06 | 0.07 | 0.07 | 0.07 | 0.08 | 0.08 | 0.08 | 0.09 | 0.09 | 0.09 | 0.10 | 0.10
$F_{0}^{n}$ | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0
$B^{n}$ | 2.95 | 2.94 | 2.93 | 2.92 | 3.06 | 3.02 | 2.98 | 2.94 | 2.94 | 2.93 | 2.92 | 2.90 | 2.99 | 2.96 | 2.94 | 2.92
$\mathfrak{a}^{n}$(Very High) | 1.0 | 0.75 | 0.50 | 0.25 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
$\mathfrak{a}^{n}$(High) | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.75 | 0.50 | 0.25 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
$\mathfrak{a}^{n}$(Low) | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.75 | 0.50 | 0.25 | 0.0 | 0.0 | 0.0 | 0.0
$\mathfrak{a}^{n}$(Very Low) | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.75 | 0.50 | 0.25
Table 13: Characteristics of the portfolio

####  6.3.4 Probabilities of default

We use the parameters of the portfolio and firms as detailed in Table 13 to
compute the annual PDs over ten years using the closed-form formulae (5.7). We
then report, in Figure 9, the average annual PD and its annual evolution.

The remarks raised for the output growth remain valid, only the monotony
changes: we can clearly distinguish the fourth various climate transition
scenario. The probability of default slightly grows each year, this is due to
the fact that $\mathrm{PD}$ (see (4.6)) is driven in particular by the
productivity growth which, in France, tends to decrease slightly toward a
stationary position (see Figure 15). Even in the Current Policies scenario,
the PD goes from 0.132% in 2021 to 0.352% in 2030.

Figure 9: Average annual probability of default and 95% confident interval of
the portfolio, per scenario and year in %

Moreover, the increase is emphasized when the transition scenario gets tougher
from an economic point of view. Between the worst-case (Divergent Net Zero)
scenario and the best-case (Current Policies) one, the difference in the
average PD reaches 1.307% in 2030 and gradually decreases after the
transition. Over the transition period of 10 years, the annual average PD is
0.344% for the Current Policies scenario, 0.471% for the NDCs scenario, 0.856%
for the Net Zero 2050 scenario, and 1.487% for the Divergent Net Zero
scenario. It is no surprise that the introduction of a carbon price increases
the portfolio’s average probability of default.

Figure 10: Average annual probability of default and 95% confident interval,
per scenario and per sub-portfolio

In Figure 10 above, we can also observe that, for each sub-portfolio, the
evolution of the PD depends on the sector that is at the origin of the growth
of its cash flows. As expected, the PD grows throughout the years, and the
growth is even more abrupt when the sub-portfolio is polluting.
Diversification also has a positive effect on the portfolio: the average PD of
the overall portfolio is higher than the average PD of the least polluting
sub-portfolios, and lower than the average PD of the most polluting
portfolios.

Bouchet-Le Guenedal [8] and Bourgey-Gobet-Jiao [9] also worked on the impact
of a carbon price on credit portfolios. They computed the PDs from 2020 up to
2050/2060. More precisely, [8] presents the percentage of companies by sector
and by scenario whose probabilities of default are above 99%, and [9] focuses
on default intensities and probabilities of default. However, they fixed the
time at which the PDs are computed and varied the time horizon (maturity)
while in our case, we are doing the other way around. Moreover, they do not
comment much about the uncertainties on the dynamics of the balance sheet,
productivity, carbon intensities, and the carbon price, while such
uncertainties are expected to significantly increase with the transition time
horizon, and therefore to substantially impact any credit risk metrics. As the
average length of small and medium-sized enterprises loans is of about seven
years, we prefer to focus on short-term risk measures.

####  6.3.5 Expected and unexpected losses

We compute the EL and UL using (5.8) and (5.9), assuming that LGD and EAD are
constant over the years and $\mathrm{LGD}^{n}=45\%$ and
$\mathrm{EAD}^{n}=\text{€{10 million}}$ for each firm $n$ described in Table
13. The annual exposure of the notional portfolio of the $N=16$ firms thus
remains fixed and is equal to €160 millions, while each sub-portfolio exposure
is of €40 millions. We then express losses as a percentage of the firm’s or
portfolio’s exposure. Table 14 and Table 15 show the average annual EL and UL.

Emissions level |  | Very High
---
| High
---
| Low
---
| Very Low
---
Portfolio
Current Policies | 0.329 | 0.034 | 0.097 | 0.160 | 0.119
NDCs | 0.504 | 0.050 | 0.107 | 0.186 | 0.161
Net Zero 2050 | 1.066 | 0.100 | 0.128 | 0.246 | 0.292
Divergent Net Zero | 2.057 | 0.138 | 0.155 | 0.327 | 0.512
Table 14: Average annual EL as a percentage of exposure Figure 11: EL of the
portfolio in % of the exposure per scenario and per year

We observe in Table 14 and Figure 11 that, as expected (notably because the
LGD is assumed to be deterministic and constant), the different scenarios
remain clearly differentiated for the EL. The latter as a percentage of the
portfolio’s exposure increases with the year and the carbon price. For the
portfolio as a whole, we see that the average annual EL increases from 330%
between the two extreme scenarios. Moreover, still focusing on the two extreme
scenarios, the average annual EL increases by 525% for the Very High Emitting
portfolio while it increases by 143% for the Very Low Emitting portfolio. The
EL being covered by the provisions coming from the fees charged to the client,
an increase in the EL implies an increase in credit cost. Therefore companies
from the most polluting sectors should be charged more than those from the
least polluting sectors.

Emissions level |  | Very High
---
| High
---
| Low
---
| Very Low
---
Portfolio
Current Policies | 1.191 | 0.066 | 0.147 | 0.277 | 0.109
NDCs | 1.691 | 0.098 | 0.163 | 0.316 | 0.161
Net Zero 2050 | 2.964 | 0.193 | 0.197 | 0.400 | 0.307
Divergent Net Zero | 4.585 | 0.264 | 0.239 | 0.507 | 0.520
Table 15: Average annual UL as a percentage of exposure Figure 12: UL of the
portfolio in % of the exposure per scenario and per year

Similarly for the UL, we observe the difference between the scenarios from
Table 15 and Figure 12. For the portfolio as a whole, we see that the average
annual UL increases by 377% between the two extreme scenarios. Moreover, still
focusing on the two extreme scenarios, the average annual UL increases by 284%
for the Very High Emitting portfolio while it increases by 83% for the Very
Low Emitting portfolio.

The UL being covered by the economic capital coming from the capital gathered
by the shareholders, an increase in the UL implies a decrease in the bank’s
profitability. Therefore, in some way, granting loans to companies from the
most polluting sectors will affect banks more negatively than doing so to
companies from the least polluting sectors. We therefore observe that the
introduction of a carbon price will not only increase the banking fees charged
to the client (materialized by the provisions via the expected loss) but will
also reduce the bank’s profitability (via the economic capital that is
calculated from the unexpected loss). Finally, for more in-depth analysis,
Figure 19 (respectively Figure 20) shows the distortions of the distribution
of the EL (respectively the UL) per scenario and per year.

####  6.3.6 Losses’ sensitivities to carbon price

Finally, we compute the sensitivity of our portfolio losses to carbon price
using (5.10). Since the scenarios are deterministic, this quantity allows us
to measure some form of model uncertainty. Indeed, for a given scenario, it
allows to capture the level by which the computed loss would vary should that
assumed deterministic scenario deviate by a certain percentage. For each time
$t$, we choose the direction
$\mathfrak{U}=\begin{bmatrix}1&\ldots&1\end{bmatrix}^{\top}\in\mathbb{R}_{+}^{t_{\star}+1}$,
and a step $\vartheta=1\%$. A carbon price change of 1% will cause a change in
the EL of $\widehat{\Gamma}^{N,T}_{t,\delta}(\mathrm{EL})$ and a change in the
UL of $\widehat{\Gamma}^{N,T}_{t,\delta,\alpha}(\mathrm{UL})$. We report the
results in Table 16 and Table 17.

For example, over the next ten years, if the price of carbon varies by 1%
around the scenario NDCs, the portfolio’s EL will vary by $1.402$% while the
portfolio’s UL will change by $1.148$% around this scenario.

Emissions level |  | Very High
---
| High
---
| Low
---
| Very Low
---
Portfolio
Current Policies | 1.561 | 1.581 | 1.191 | 0.827 | 1.280
NDCs | 1.777 | 1.687 | 1.261 | 0.904 | 1.402
Net Zero 2050 | 2.142 | 1.864 | 1.386 | 1.035 | 1.631
Divergent Net Zero | 2.668 | 2.096 | 1.562 | 1.215 | 1.973
Table 16: Average annual EL sensitivity to carbon price in % Emissions level |  | Very High
---
| High
---
| Low
---
| Very Low
---
Portfolio
Current Policies | 1.299 | 1.290 | 1.042 | 0.547 | 1.135
NDCs | 1.463 | 1.365 | 1.102 | 0.583 | 1.148
Net Zero 2050 | 1.726 | 1.485 | 1.206 | 0.634 | 1.197
Divergent Net Zero | 2.070 | 1.632 | 1.352 | 0.681 | 1.472
Table 17: Average annual UL sensitivity to carbon price in %

The greater the sensitivity, the more polluting the sector is. This is to be
expected as carbon prices are higher in these sectors. In addition, the
sensitivity of the portfolio is smaller than that in the most polluting
sectors, and greater than that in the least polluting ones. Finally, we notice
that the variation of the EL is slightly more sensitive than the variation of
the UL. This means that the bank’s provisions will increase a bit more than
the bank’s capital, or that the growth of the carbon price will impact
customers more than shareholders.

## Conclusion

In this work, we study how the introduction of a carbon price would propagate
in a credit portfolio. To this aim, we first build a dynamic stochastic
multisectoral model in which firms (resp. households) are charged for the GHG
they emit when they consume intermediary inputs from other sectors and when
they produce goods/services (resp. for the GHG they emit when they consume
goods/services). We later use the Discounted Cash Flows methodology to compute
the firm value and introduce the latter in a structural credit risk model to
project $\mathrm{PD}$, $\mathrm{EL}$ and $\mathrm{UL}$. We finally introduce
losses’ sensitivities to carbon price to measure the uncertainty of the losses
to the transition scenarios. This work opens the way to numerous extensions.
In the climate-economic model, exogenous and deterministic scenarios as well
as homogeneous agents are assumed while one could consider agent-based or
mean-field games models where a central planner decides on the carbon price
and agents (companies or households) optimize production, prices, and
consumption according to the carbon price/tax level. In the credit risk part,
the LGD is assumed to be deterministic, constant, and independent of the
carbon price. In our forthcoming research, we will analyze how the LGD is
affected by the stranding of assets. We furthermore assume that $\mathrm{EAD}$
and thus bank balance sheets remain static over the years while the transition
will require huge investments. One could thus introduce capital in the model.
Finally, we have adopted a sectoral view, while one could alternatively assess
the credit risk at the counterpart level and thus penalize or reward companies
according to their individual and not sectoral emissions.

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##  Appendix A Vector Autoregressive Model (VAR):

Detailed proofs can be found in Hamilton [25], and Kilian and Lütkepohl [28].
Assume that $(\Theta_{t})_{t\in\mathbb{N}}$ follows a VAR, i.e. for all
$t\in\mathbb{N}^{*}$,

$\Theta_{t}=\mu+\Gamma\Theta_{t-1}+\mathcal{E}_{t},\quad\text{where for}\quad
t\in\mathbb{Z},\quad\mathcal{E}_{t}\sim\mathcal{N}(\mathbf{0},\Sigma)$ (A.1)

with $\mu\in\mathbb{R}^{I}$ and where the matrix $\Gamma\in\mathbb{R}^{I\times
I}$ has eigenvalues all strictly less than $1$ in absolute value. We have the
following result that is shown in the VAR’s literature.

  * 1.

$(\Theta_{t})_{t\in\mathbb{N}}$ is weak-stationary.

  * 2.

If $\Theta_{0}\sim\mathcal{N}(\overline{\mu},\overline{\Sigma})$ with
$\overline{\mu}:=(\mathbf{I}_{I}-\Gamma)^{-1}\mu$, and
$\mathrm{vec}(\overline{\Sigma})=(\mathbf{I}_{I\times
I}-\Gamma\bigotimes\Gamma)^{-1}\mathrm{vec}(\Sigma)$, then for
$t\in\mathbb{Z}$, $\mathcal{E}_{t}\sim\mathcal{N}(\mathbf{0},\Sigma)$ with
$\Sigma\in\mathbb{R}^{I\times I}$.

  * 3.

For $t,T\in\mathbb{N}$, we note $\Upsilon_{t}:=\sum_{v=0}^{t}\Gamma^{v}$, then

$\sum_{u=1}^{T}\sum_{v=1}^{u}\Gamma^{u-v}\mathcal{E}_{t+v}=\sum_{u=1}^{T}\Upsilon_{T-u}\mathcal{E}_{t+u},$

  * 4.

For $t,u\in\mathbb{N}$,

$\Theta_{t}=\overline{\mu}+\sum_{v=1}^{\infty}\Gamma^{v}\mathcal{E}_{t-v}\qquad\text{and}\qquad\Theta_{t+T}=\Gamma^{T}\Theta_{t}+\Upsilon_{T-1}\mu+\sum_{v=1}^{T}\Gamma^{T-v}\mathcal{E}_{t+v}.$
(A.2)

  * 5.

For $t,T\in\mathbb{N}$,

$\left(\sum_{u=1}^{T}\Theta_{t+u}\middle|\Theta_{t}\right)\sim\mathcal{N}\left(\Gamma\Upsilon_{T-1}\Theta_{t}+\left(\sum_{u=1}^{T}\Upsilon_{u-1}\right)\mu,\sum_{u=1}^{T}\Upsilon_{T-u}\Sigma(\Upsilon_{T-u})^{\top}\right),$
(A.3)

and in particular
$\left(\Theta_{t+1}\middle|\Theta_{t}\right)\sim\mathcal{N}\left(\mu+\Gamma\Theta_{t},\Sigma\right)$.

##  Appendix B Proofs

###  B.1 Existence condition of the firm value

###### Proof of Proposition 3.4.

Let $t\in\mathbb{N}$, $n\in\\{1,\ldots,N\\}$. For $s>0$, from Assumption 3.1,
we observe that

$\displaystyle
F^{n}_{t+s}=F^{n}_{t}\exp\left(\sum_{u=1}^{s}w^{n}_{t+u}\right).$ (B.1)

Let $K\in\mathbb{N}^{*}$ and define

$\displaystyle V^{n,K}_{t}$
$\displaystyle:=\mathbb{E}_{t}\left[\sum_{s=0}^{K}e^{-rs}F^{n}_{t+s}\right].$
(B.2)

We now show that $\lim_{K\rightarrow+\infty}V^{n,K}$ exists, in particular
that $\mathbb{E}_{t}\left[e^{-rs}F^{n}_{t+s}\right]$ is summable. To this end,
we first observe that

$V^{n,K}_{t}=F^{n}_{t}\left(1+\sum_{s=1}^{K}e^{-rs}\mathbb{E}_{t}\left[\exp\left(\sum_{u=1}^{s}w^{n}_{t+u}\right)\right]\right).$

We now give an upper bound for
$\|\exp\left(\sum_{u=1}^{s}w^{n}_{t+u}\right)\|_{p}$ for some $p>1$. We
observe that, using (3.2),

$\displaystyle\sum_{u=1}^{s}w^{n}_{t+u}=\mathfrak{a}^{n\cdot}\left(\sum_{u=1}^{s}\Theta_{t+u}+\mathfrak{v}(\mathfrak{d}_{t+s})-\mathfrak{v}(\mathfrak{d}_{t})\right)+\sum_{u=1}^{s}\mathfrak{b}^{n}_{t+u}.$
(B.3)

From Assumption 2.1 and (2.3), it follows

$\Theta_{t+u}=\overline{\mu}+\varepsilon\left(\Gamma^{u}\mathcal{Z}_{t}+\sum_{v=1}^{u}\Gamma^{u-v}\mathcal{E}_{t+v}\right).$

We define $\Upsilon_{k}:=\sum_{v=0}^{k}\Gamma^{v}$ and observe that

$\displaystyle|\Upsilon_{k}|\leq(1-|\Gamma|)^{-1}\;.$ (B.4)

Since

$\displaystyle\sum_{u=1}^{s}\sum_{v=1}^{u}\Gamma^{u-v}\mathcal{E}_{t+v}=\sum_{v=1}^{s}\Upsilon_{s-v}\mathcal{E}_{t+v},$
(B.5)

we compute

$\sum_{u=1}^{s}\Theta_{t+u}=\overline{\mu}s+\varepsilon\Gamma\Upsilon_{s-1}\mathcal{Z}_{t}+\varepsilon\sum_{v=1}^{s}\Upsilon_{s-v}\mathcal{E}_{t+v}.$

Then (B.3) reads

$\sum_{u=1}^{s}w^{n}_{t+u}=\varepsilon\mathfrak{a}^{n\cdot}\Gamma\Upsilon_{s-1}\mathcal{Z}_{t}+s\mathfrak{a}^{n\cdot}\overline{\mu}+\varepsilon\sum_{v=1}^{s}\mathfrak{a}^{n\cdot}\Upsilon_{s-v}\mathcal{E}_{t+v}+\mathfrak{a}^{n\cdot}\left(\mathfrak{v}(\mathfrak{d}_{t+s})-\mathfrak{v}(\mathfrak{d}_{t})\right)+\sum_{u=1}^{s}\mathfrak{b}^{n}_{t+u}.$

Observe that under Assumption 2.3, there exists a constant $\mathfrak{C}>0$
such that

$\displaystyle\sup_{n,s,t}\exp\left(\mathfrak{a}^{n\cdot}\left(\mathfrak{v}(\mathfrak{d}_{t+s})-\mathfrak{v}(\mathfrak{d}_{t})\right)\right)\leq\mathfrak{C}\,.$
(B.6)

Thus, using the independence of $\mathcal{Z}_{t},(\mathcal{E}_{t+v})_{v\geq
1},(\mathfrak{b}^{n}_{t+v})_{v\geq 1}$, we obtain

$\displaystyle\scriptstyle\mathbb{E}_{t}\left[\exp\left(p\sum_{u=1}^{s}w^{n}_{t+u}\right)\right]\leq$
$\displaystyle\mathfrak{C}^{p}\exp\Big{(}p\varepsilon\mathfrak{a}^{n\cdot}\Gamma\Upsilon_{s-1}\mathcal{Z}_{t}+ps\mathfrak{a}^{n\cdot}\overline{\mu}\Big{)}\mathbb{E}\left[\exp\left(p\varepsilon\sum_{v=1}^{s}\mathfrak{a}^{n\cdot}\Upsilon_{s-v}\mathcal{E}_{t+v}+p\sum_{u=1}^{s}\mathfrak{b}^{n}_{t+u}\right)\right].$
(B.7)

Since

$\displaystyle\mathbb{E}\left[\exp\left(p\sum_{u=1}^{s}\mathfrak{b}^{n}_{t+u}\right)\right]=\exp\left(\frac{p^{2}}{2}s\sigma_{\mathfrak{b}^{n}}^{2}\right),$
(B.8)

we compute

$\mathbb{E}\left[\exp\left(p\varepsilon\mathfrak{a}^{n\cdot}\Upsilon_{s-v}\mathcal{E}_{t+v}\right)\right]=\exp\left(\frac{\varepsilon^{2}p^{2}}{2}|\mathfrak{a}^{n\cdot}\Upsilon_{s-v}\sqrt{\Sigma}|^{2}\right)\leq\exp\left(\frac{\varepsilon^{2}p^{2}}{2}|\mathfrak{a}^{n\cdot}|^{2}|\sqrt{\Sigma}|^{2}(1-|\Gamma|)^{-2}\right).$
(B.9)

One could also have found above a finer upper bound. Combining (B.8)-(B.9)
with (B.7), we obtain

$\mathbb{E}_{t}\left[\exp\left(p\sum_{u=1}^{s}w^{n}_{t+u}\right)\right]\leq\mathfrak{C}^{p}\exp\left(p\varepsilon\mathfrak{a}^{n\cdot}\Gamma\Upsilon_{s-1}\mathcal{Z}_{t}+p^{2}\rho
s\right).$

Using similar computations as above, we also get (because $\Upsilon_{s-1}$ is
bounded and $\mathcal{Z}_{t}$ is stationary and Gaussian)

$\mathbb{E}\left[\exp\left(p\varepsilon\mathfrak{a}^{n\cdot}\Gamma\Upsilon_{s-1}\mathcal{Z}_{t}\right)\right]\leq
C_{p},$ (B.10)

and hence

$\left\|\exp\left(\sum_{u=1}^{s}w^{n}_{t+u}\right)\right\|_{p}\leq
C_{p}e^{p\rho s}.$

Under (3.6), we then obtain

$\displaystyle\sum_{s\geq
0}e^{-rs}\left\|\exp\left(\sum_{u=1}^{s}w^{n}_{t+u}\right)\right\|_{p}<+\infty,$
(B.11)

for some $p>1$. Set $1<\tilde{p}:=\frac{p}{1+\epsilon}$, for $\epsilon>0$
small enough. Then, using Hölder’s inequality (with
$\frac{1}{\tilde{p}}=\frac{1}{p}+\frac{1}{p/\epsilon}$),

$\mathbb{E}\left[|V^{n,K}_{t}|^{\tilde{p}}\right]\leq
C_{p}\mathbb{E}\left[|F_{t}^{n}|^{\frac{p}{\epsilon}}\right]<+\infty,$

since $\|F^{n}_{t}\|_{q}<\infty$ for any $q\geq 1$. ∎

###  B.2 Conditional distribution of the firm value

###### Proof of Remark 3.8.

Let $t,T\geq 1$, we have from (3.8),

$\begin{split}\mathcal{V}^{n}_{t+T}&=F^{n}_{0}\mathfrak{R}^{n}_{t+T}(\mathfrak{d})\exp{\left(\mathfrak{a}^{n\cdot}(\mathcal{A}^{\circ}_{t+T}-\mathfrak{v}(\mathfrak{d}_{0}))\right)}\exp\left(\sum_{u=1}^{t+T}\mathfrak{b}^{n}_{u}\right)\\\
&=F^{n}_{0}\mathfrak{R}^{n}_{t+T}(\mathfrak{d})\exp{\left(-\mathfrak{a}^{n\cdot}\mathfrak{v}(\mathfrak{d}_{0})\right)}\exp{\left(\mathfrak{a}^{n\cdot}\mathcal{A}^{\circ}_{t+T}\right)}\exp\left(\mathcal{W}_{t+T}^{n}\right).\end{split}$

But
$\mathcal{A}^{\circ}_{t+T}=\mathcal{A}^{\circ}_{t}+\sum_{u=t+1}^{t+T}\Theta_{u}$,
then

$\begin{split}\mathcal{V}^{n}_{t+T}&=F^{n}_{0}\mathfrak{R}^{n}_{t+T}(\mathfrak{d})\exp{\left(\mathfrak{a}^{n\cdot}(\mathcal{A}^{\circ}_{t}-\mathfrak{v}(\mathfrak{d}_{0}))\right)}\exp{\left(\mathfrak{a}^{n\cdot}\sum_{u=t+1}^{t+T}\Theta_{u}\right)}\exp\left(\mathcal{W}^{n}_{t+T}\right)\\\
&=F^{n}_{0}\mathfrak{R}^{n}_{t+T}(\mathfrak{d})\exp{\left(\mathfrak{a}^{n\cdot}(\mathcal{A}^{\circ}_{t}-\mathfrak{v}(\mathfrak{d}_{0}))\right)}\exp{\left(\mathfrak{a}^{n\cdot}\sum_{u=1}^{T}\Theta_{t+u}+\mathcal{W}^{n}_{t+T}\right)}.\end{split}$

But recall from Remark 2.2 and A, for all $u\in\\{1,\ldots,T\\}$,

$\Theta_{t+u}=\Gamma^{u}\Theta_{t}+\Upsilon_{u-1}\mu+\varepsilon\sum_{v=1}^{u}\Gamma^{u-v}\mathcal{E}_{t+v},$

then

$\scriptstyle\sum_{u=1}^{T}\Theta_{t+u}=\sum_{u=1}^{T}\Gamma^{u}\Theta_{t}+\sum_{u=1}^{T}\Upsilon_{u-1}\mu+\varepsilon\sum_{u=1}^{T}\sum_{v=1}^{u}\Gamma^{u-v}\mathcal{E}_{t+v}=\Gamma\Upsilon_{T-1}\Theta_{t}+\left(\sum_{u=1}^{T}\Upsilon_{u-1}\right)\mu+\varepsilon\sum_{v=1}^{T}\Upsilon_{T-v}\mathcal{E}_{t+v}.$

From Assumptions 2.1 and 3.1, we have

$\scriptstyle\left(\sum_{u=1}^{T}\mathfrak{a}^{n\cdot}\Theta_{t+u}+\mathcal{W}^{n}_{t+T}\middle|\mathcal{G}_{t}\right)\sim\mathcal{N}\left(\mathfrak{a}^{n\cdot}\Gamma\Upsilon_{T-1}\Theta_{t}+\mathfrak{a}^{n\cdot}\left(\sum_{u=1}^{T}\Upsilon_{u-1}\right)\mu,\varepsilon^{2}\sum_{u=1}^{T}(\mathfrak{a}^{n\cdot}\Upsilon_{T-u})\Sigma(\Upsilon_{T-u}\mathfrak{a}^{n\cdot})^{\top}+\sigma_{\mathfrak{b}^{n}}^{2}(t+T)\right),$

and the conclusion follows. ∎

###  B.3 Convergence of $(\mathcal{V}^{n}_{t}-V^{n}_{t})/F^{n}_{t}\text{ to
zero }$

###### Proof of Proposition 3.9.

For $K\in\mathbb{N}^{*}$, recall the expressions of $V^{n,K}_{t}$ in (B.2) and
$\mathcal{V}^{n,K}_{t}$ in (3.11) and note that

$\displaystyle\mathbb{E}\left[\left|\frac{V^{n}_{t}}{F^{n}_{t}}-\frac{\mathcal{V}^{n}_{t}}{F^{n}_{t}}\right|\right]\leq\mathbb{E}\left[\left|\frac{V^{n}_{t}}{F^{n}_{t}}-\frac{V^{n,K}_{t}}{F^{n}_{t}}\right|\right]+\mathbb{E}\left[\left|\frac{V^{n,K}_{t}}{F^{n}_{t}}-\frac{\mathcal{V}^{n,K}_{t}}{F^{n}_{t}}\right|\right]+\mathbb{E}\left[\left|\frac{\mathcal{V}^{n,K}_{t}}{F^{n}_{t}}-\frac{\mathcal{V}^{n}_{t}}{F^{n}_{t}}\right|\right]\,.$
(B.12)

Using Hölder’s inequality and Proposition 3.4, one gets that the first term in
the right hand side of the above inequality goes to zero as $K$ goes to
$+\infty$. Similarly, using Hölder’s inequality and (the beginning of the
proof of) Lemma 3.6, one shows that the last term in the right hand side of
the above inequality goes to zero as $K$ goes to infinity. It remains thus to
study the middle term to obtain the desired result. Observe that

$\displaystyle\frac{V^{n,K}_{t}-\mathcal{V}^{n,K}_{t}}{F^{n}_{t}}=\left(\sum_{s=1}^{K}e^{-rs}\mathbb{E}_{t}\left[\exp\left(s\mathfrak{a}^{n\cdot}\overline{\mu}+\mathfrak{a}^{n\cdot}(\mathfrak{v}(\mathfrak{d}_{t+s})-\mathfrak{v}(\mathfrak{d}_{t}))+\sum_{u=1}^{s}\mathfrak{b}^{n}_{t+u}\right)\Delta_{s}\right]\right),$

with

$\Delta_{s}:=\exp\left\\{\varepsilon\sum_{u=1}^{s}\mathfrak{a}^{n\cdot}\mathcal{Z}_{t+u}\right\\}-1,$
(B.13)

using (B.3) and (2.3). We first compute, by independence,

$\begin{split}&\left|\mathbb{E}_{t}\left[\exp\left(s\mathfrak{a}^{n\cdot}\overline{\mu}+\mathfrak{a}^{n\cdot}(\mathfrak{v}(\mathfrak{d}_{t+s})-\mathfrak{v}(\mathfrak{d}_{t}))+\sum_{u=1}^{s}\mathfrak{b}^{n}_{t+u}\right)\Delta_{s}\right]\right|\\\
&=\mathbb{E}\left[\exp\left(s\mathfrak{a}^{n\cdot}\overline{\mu}+\mathfrak{a}^{n\cdot}(\mathfrak{v}(\mathfrak{d}_{t+s})-\mathfrak{v}(\mathfrak{d}_{t}))+\sum_{u=1}^{s}\mathfrak{b}^{n}_{t+u}\right)\right]\left|\mathbb{E}_{t}\left[\Delta_{s}\right]\right|\\\
&\leq\mathfrak{C}\exp\left(s\mathfrak{a}^{n\cdot}\overline{\mu}+\frac{1}{2}s\sigma_{\mathfrak{b}^{n}}^{2}\right)\mathbb{E}_{t}\left[|\Delta_{s}|\right],\end{split}$

using (B.6). We then obtain

$\|(V^{n,K}_{t}-\mathcal{V}^{n,K}_{t})/F^{n}_{t}\|_{1}\leq\left(\sum_{s=1}^{K}\mathfrak{C}e^{\varrho_{n}s}\mathbb{E}\left[|\Delta_{s}|\right]\right),$

where $\varrho_{n}$ is defined in Lemma 3.6. We can rewrite (B.13) as

$\Delta_{s}=\varepsilon\int_{0}^{1}\exp{\left(\varepsilon\lambda\sum_{u=1}^{s}\mathfrak{a}^{n\cdot}\mathcal{Z}_{t+u}\right)}\sum_{u=1}^{s}\mathfrak{a}^{n\cdot}\mathcal{Z}_{t+u}d\lambda.$

For $p>1$, using Hölder’s inequality, we deduce from the previous expression

$\displaystyle\mathbb{E}\left[|\Delta_{s}|\right]\leq\varepsilon\mathbb{E}\left[\left|\int_{0}^{1}\exp{\left(\varepsilon\lambda\sum_{u=1}^{s}\mathfrak{a}^{n\cdot}\mathcal{Z}_{t+u}\right)}\mathrm{d}\lambda\right|^{p}\right]^{\frac{1}{p}}\mathbb{E}\left[\left|\sum_{u=1}^{s}\mathfrak{a}^{n\cdot}\mathcal{Z}_{t+u}\right|^{q}\right]^{\frac{1}{q}},$
(B.14)

with $q$ the conjugate exponent to $p$.

We first compute by convexity

$\mathbb{E}\left[\left|\sum_{u=1}^{s}\mathfrak{a}^{n\cdot}\mathcal{Z}_{t+u}\right|^{q}\right]\leq
s^{q-1}\sum_{u=1}^{s}\mathbb{E}\left[\left|\mathfrak{a}^{n\cdot}\mathcal{Z}_{t+u}\right|^{q}\right]\leq
C_{q}s^{q},$

where the last inequality follows since
$\mathcal{Z}_{t+u}\sim\mathcal{N}(0,\overline{\Sigma})$.

We now turn to the first term in the right hand side of (B.14),

Using Jensen’s inequality, we have

$\displaystyle\mathbb{E}\left[\left|\int_{0}^{1}\exp{\left(\varepsilon\lambda\sum_{u=1}^{s}\mathfrak{a}^{n\cdot}\mathcal{Z}_{t+u}\right)}\mathrm{d}\lambda\right|^{p}\right]\leq\int_{0}^{1}\mathbb{E}\left[\exp{\left(\varepsilon\lambda
p\sum_{u=1}^{s}\mathfrak{a}^{n\cdot}\mathcal{Z}_{t+u}\right)}\right]\mathrm{d}\lambda.$

Since
$\mathcal{Z}_{t+u}=\Gamma^{u}\mathcal{Z}_{t}+\sum_{v=1}^{u}\Gamma^{u-v}\mathcal{E}_{t+v}$,
we write

$\displaystyle\mathbb{E}_{t}\left[\exp\left(p\varepsilon\lambda\sum_{u=1}^{s}\mathfrak{a}^{n\cdot}\mathcal{Z}_{t+u}\right)\right]$
$\displaystyle=\exp\left(p\varepsilon\lambda\sum_{u=1}^{s}\mathfrak{a}^{n\cdot}\Gamma^{u}\mathcal{Z}_{t}\right)\times\mathbb{E}_{t}\left[\exp\left(p\varepsilon\lambda\sum_{u=1}^{s}\mathfrak{a}^{n\cdot}\sum_{v=1}^{u}\Gamma^{u-v}\mathcal{E}_{t+v}\right)\right].$

By (B.4), $|\Upsilon_{k}|\leq(1-|\Gamma|)^{-1}$ where
$\Upsilon_{k}:=\sum_{v=0}^{k}\Gamma^{v}$. We compute

$\displaystyle\sum_{u=1}^{s}\mathfrak{a}^{n\cdot}\sum_{v=1}^{u}\Gamma^{u-v}\mathcal{E}_{t+v}=\sum_{v=1}^{s}\mathfrak{a}^{n\cdot}\Upsilon_{s-v}\mathcal{E}_{t+v}.$

Using (B.9) and recalling that $\lambda\in[0,1]$, we get

$\displaystyle\mathbb{E}_{t}\left[\exp\left(p\varepsilon\lambda\sum_{u=1}^{s}\mathfrak{a}^{n\cdot}\sum_{v=1}^{u}\Gamma^{u-v}\mathcal{E}_{t+v}\right)\right]\leq\exp\left(s\frac{\varepsilon^{2}p^{2}}{2}|\mathfrak{a}^{n\cdot}|^{2}|\sqrt{\Sigma}|^{2}(1-|\Gamma|)^{-2}\right).$
(B.15)

Thus, appealing to (B.10), we get

$\displaystyle\mathbb{E}\left[\exp(\varepsilon
p\lambda\sum_{u=1}^{s}\mathfrak{a}^{n\cdot}\mathcal{Z}_{t+u})\right]$
$\displaystyle\leq
C_{p,\epsilon}\exp\left(s\frac{\varepsilon^{2}p^{2}}{2}|\mathfrak{a}^{n\cdot}|^{2}|\sqrt{\Sigma}|^{2}(1-|\Gamma|)^{-2}\right).$

Finally, combining the above inequalities, we obtain

$\displaystyle\mathbb{E}\left[|\Delta_{s}|\right]\leq
C_{p,\epsilon}\varepsilon
s\exp\left(s\frac{\varepsilon^{2}p}{2}|\mathfrak{a}^{n\cdot}|^{2}|\sqrt{\Sigma}|^{2}(1-|\Gamma|)^{-2}\right),$

and then

$\sum_{s=1}^{K}\mathfrak{C}e^{\varrho_{n}s}\mathbb{E}\left[|\Delta_{s}|\right]\leq\varepsilon\sum_{s=1}^{K}C_{p,\epsilon}se^{p(\rho-r)s}.$

For $p-1>0$ small enough, we thus get

$\displaystyle\|(V^{n,K}_{t}-\mathcal{V}^{n,K}_{t})/F^{n}_{t}\|_{1}\leq\sum_{s=1}^{K}\mathfrak{C}e^{\varrho_{n}s}\mathbb{E}\left[|\Delta_{s}|\right]\leq
C\varepsilon.$ (B.16)

The proof is thus concluded letting $K$ goes to infinity in (B.12). ∎

##  Appendix C Factor selection by LASSO regression

We perform LASSO regression questioning the relationship between credit risk
(described by the logit of the default rate) and economics conditions
(described by the macroeconomic variables as we assumed in Section 4), we use
S&P ratings for data on the ratings and default, on a yearly basis from 1995
to 2019, of 7046 large US companies belonging to 13 sectors. We can analyze
and use them to compute the historical probability of default (displayed
Figure 16) and the migration matrix by sector. The USA macroeconomic time
series can be found in the World Bank database and in the FRED Saint-Louis
database [17].

| Coef_ | Importance | Percentage
---|---|---|---
Industry value added growth | -0.433 | 0.433 | 73.979
Real GDP per capita growth | -0.073 | 0.073 | 12.485
Unemployment rate | 0.046 | 0.046 | 7.934
Stocks returns | -0.033 | 0.033 | 5.602
Export of goods and services | 0 | 0 | 0
Real GDP growth | 0 | 0 | 0
Inflation rate | 0 | 0 | 0
10-year interest rate | 0 | 0 | 0
Table 18: Factor selection by LASSO

##  Appendix D Sectoral groups

We use the output and GHG emissions by sector to compute the carbon intensity
(which is the tons of GHG emitted per euro of output) per sector. Then we
compute their annual average and we group the sectors together if their annual
average carbon intensities are close.

  1. 1.

Very High Emitting

     *      * (a)

Manufacture of basic metals and fabricated metal products, except machinery
and equipment

     * (b)

Water supply; sewerage, waste management and remediation activities

     * (c)

Manufacture of rubber and plastics products, and other non-metallic mineral
products

     * (d)

Agriculture, forestry and fishing

  2. 2.

High Emitting

     *      * (a)

Electricity, gas, steam and air conditioning supply

     * (b)

Transportation and storage

     * (c)

Manufacture of chemicals and chemical products

     * (d)

Manufacture of coke and refined petroleum products

  3. 3.

Low Emitting

     *      * (a)

Manufacture of food products, beverages and tobacco products

     * (b)

Manufacture of wood and paper products, and printing

     * (c)

Mining and quarrying

  4. 4.

Very Low Emitting

     *      * (a)

Other Service Activities

     * (b)

Arts, Entertainment and Recreation

     * (c)

Social Work Activities

     * (d)

Human Health Activities

     * (e)

Education

     * (f)

Public

     * (g)

Administration and Defence; Compulsory Social Security

     * (h)

Administrative and Support Service Activities

     * (i)

Advertising and Market Research; Other Professional, Scientific and Technical
Activities; Veterinary Activities

     * (j)

Scientific Research and Development Legal and Accounting Activities;
Activities of Head Offices; Management Consultancy Activities; Architecture
and Engineering Activities; Technical Testing and Analysis

     * (k)

Real Estate Activities

     * (l)

Financial and Insurance Activities

     * (m)

Computer Programming, Consultancy and Related Activities; Information Service
Activities

     * (n)

Telecommunications

     * (o)

Publishing, Audiovisual and Broadcasting Activities

     * (p)

Accommodation and Food Service Activities

     * (q)

Wholesale and Retail Trade, Repair of Motor Vehicles and Motorcycles

     * (r)

Construction

     * (s)

Manufacture of Furniture; Other Manufacturing; Repair and Installation of
Machinery and Equipment

     * (t)

Manufacture of Transport Equipment

     * (u)

Manufacture of Machinery and Equipment N.E.C.

     * (v)

Manufacture of Electrical Equipment

     * (w)

Manufacture of Computer, Electronic and Optical Products

     * (x)

Manufacture of Basic Pharmaceutical Products and Pharmaceutical Preparations

     * (y)

Manufacture of Textiles, Wearing Apparel and Leather Products

##  Appendix E Plots of historical data

We plot the data described in Section 6.1.

Figure 13: Nominal consumption, labor, and output (described in item 1) Figure 14: Consumption, labor, and output growth (described in item 1) Figure 15: Annual and 10-year moving average productivity growth Figure 16: Historical data of a chosen portfolio - France - from 1995 to 2018 (described in item C) Emissions Level |  | $\varphi_{0}$
---
| $g_{\varphi,0}$
---
|  $\theta_{\varphi}$(%)
---
$\kappa_{\text{Very High}}$ | 0.003 | -0.026 | 0.001
$\kappa_{\text{High}}$ | 1.123 | -0.040 | 0.001
$\kappa_{\text{Low}}$ | 0.003 | -0.026 | 0.001
$\kappa_{\text{Very Low}}$ | 0.003 | -0.026 | 0.001
$\zeta_{\text{Very High, Very High}}$ | 0.124 | -0.043 | 1.5
$\zeta_{\text{Very High, High}}$ | 0.017 | -0.045 | 0.001
$\zeta_{\text{Very High, Low}}$ | 0.088 | -0.065 | 0.001
$\zeta_{\text{Very High, Very Low}}$ | 0.034 | -0.042 | 3.6
$\zeta_{\text{High, Very High}}$ | 0.051 | -0.049 | 0.001
$\zeta_{\text{High, High}}$ | 0.177 | -0.046 | 1.1
$\zeta_{\text{High, Low}}$ | 0.022 | -0.081 | 0.001
$\zeta_{\text{High, Very Low}}$ | 0.026 | -0.030 | 0.001
$\zeta_{\text{Low, Very High}}$ | 0.037 | -0.055 | 11.1
$\zeta_{\text{Low, High}}$ | 0.117 | -0.079 | 0.001
$\zeta_{\text{Low, Low}}$ | 0.111 | -0.065 | 0.3
$\zeta_{\text{Low, Very Low}}$ | 0.026 | -0.018 | 0.001
$\zeta_{\text{Very Low, Very High}}$ | 0.019 | -0.052 | 0.1
$\zeta_{\text{Very Low, High}}$ | 0.025 | -0.05 | 2.1
$\zeta_{\text{Very Low, Low}}$ | 0.016 | -0.088 | 0.001
$\zeta_{\text{Very Low, Very Low}}$ | 0.059 | -0.034 | 0.001
Table 19: Carbon intensities parameters

##  Appendix F Figures

Figure 17: Average annual EL per scenario for some firms Figure 18: Average
annual UL per scenario for some firms Figure 19: Annual EL distribution per
scenario Figure 20: Annual UL distribution per scenario