In Reckziegel/CMA: Copula Marginal Algorithm (CMA)

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(cma)
library(dplyr, warn.conflicts = FALSE)

# stationarity - "invariance"
x <- matrix(diff(log(EuStockMarkets)), ncol = 4)
colnames(x) <- colnames(EuStockMarkets)
head(x)

Assume there is a equally-weighted portfolio of stocks from the indexes that appears in object x.

# weights
w <- rep(0.25, 4)

pnl <- tibble::tibble(
  base_pnl = as.double(x %*% w)
)

The main statistics of the P&L can be seen with empirical_stats():

empirical_stats(pnl)

The big question here is: how would the P&L statistics change in response to a massive sudden sell-off?

To address this question we follow A New Breed for Copulas for Risk and Portfolio Management and model the market as a mixture of "calm" vs. "panic" distributions. For details on the full specification of this market, please, see the reference above.

# For the details on how the market is modeled, please, see the paper: 
# "A New Breed for Copulas for Risk and Portfolio Management"
panic <- panic_copula(x, n = 50000, panic_cor = 0.97, panic_prob = 0.02, dist = "normal")
calm  <- panic_copula(x, n = 50000, panic_cor = 0.00, panic_prob = 0.00, dist = "normal")

We simulate 50.000 scenarios that matches the normal distribution exactly, with the sample counterparts of $\hat{\mu}$ and $\hat{\sigma}$.

In the first scenario we assume there is a 2% probability of panic, in which all the cross-correlations are set equal to 0.97:

co <- matrix(0.97, 4, 4)
rownames(co) <- colnames(EuStockMarkets)
colnames(co) <- colnames(EuStockMarkets)
diag(co) <- 1
co

We also model a "calm" scenario without changing the historical correlation structure.

For consistency, we continue with an equal-weight strategy for both simulations:

# Equal-Weight Portfolio Under the Panic Market
pnl_panic <- tibble::tibble(
  pnl_panic = as.matrix(panic$simulation) %*% w
)

# Equal-Weight Portfolio Under the Calm Market
pnl_calm <- tibble::tibble(
  pnl_calm = as.matrix(calm$simulation) %*% w
)

The "panic" P&L can be seen with plot_panic_distribution():

# PnL under the Panic Regime
plot_panic_distribution(pnl_panic, panic$p, breaks = 200)

The new marginal distribution shows a hump shape format around -2%, which is a direct consequence of the panic being instated. Nevertheless, the location and dispersion parameters still matches the sample counterparts. The panic is hidden!

The full picture - with the similarities and differences - among these markets regimes can be seen with help of ggplot2:

stats_panic <- empirical_stats(pnl_panic)
stats_calm  <- empirical_stats(pnl_calm)

bind_rows(stats_panic, stats_calm) |> 
  ggplot2::ggplot(ggplot2::aes(x = stat, y = value, fill = name)) + 
  ggplot2::geom_col(position = "dodge") +
  ggplot2::facet_wrap(~ stat, scales = "free") + 
  ggplot2::labs(x = NULL, y = NULL, fill = "regime") + 
  ggplot2::scale_fill_viridis_d()

While the location and dispersion doesn't change, all the other metrics are twisted for markets that operate under dual market rules. The objects stats_panic and stats_calm imply that a 2% probability of panic increases the kurtosis by 6% and VaR and CVaR by 10%.

Reckziegel/CMA documentation built on July 13, 2022, 10:31 p.m.