Stress testing and correlation, liquidity, gap and counterparty risk.

What VaR does not ask

The VaR of lesson 4.2.3 is estimated with recent history and says how much is lost on a bad day of a distribution assumed to be known. A stress test asks another question: what happens to the portfolio if something specific and severe occurs, whether or not it has occurred in the data window. The Basel Committee's stress testing principles (2018) sum it up in one condition: scenarios must be “sufficiently severe but plausible” and cover the material risks of the institution, and they must be reviewed so that they remain relevant (principle 4). There are four families:

flowchart TB
  E[Stress tests] --> H[Historical]
  E --> P[Hypothetical]
  E --> I[Reverse]
  E --> S[Sensitivity]
  H --> R[Revalue portfolio]
  P --> R
  I --> C[Shock that causes L]
  S --> R

• Historical: the moves of a real crisis are replayed (2008, 2020, 2022). They are plausible by definition, but the next crisis will not be the same.
• Hypothetical: a coherent shock is invented (rates rise 200 bp, the dollar 10%) and carried over to each asset with a model, here the factor loadings.
• Reverse: you start from the loss that cannot be tolerated and look for the scenario that produces it.
• Sensitivity: a single factor is moved (one volatility point, 1 bp in rates) to know the exposure. The Greeks of lesson 4.2.6 are exactly that for derivatives.

The level's scenarios

riesgo4_escenarios.csv contains eight scenarios as shock vectors (simple total return) on the 38 assets of the level's synthetic universe. Four are historical and come from the prices themselves: the 2008 crisis from the Lehman bankruptcy to the low, covid in February and March 2020, the 2022 inflation and the worst day of RV_EEUU in the sample. Four are hypothetical: a Black Monday like that of October 19, 1987 (the universe starts in 2006), a parallel rate rise of 200 bp, a 10% appreciation of the dollar and a credit crisis.

import numpy as np
import pandas as pd

esc = pd.read_csv("riesgo4_escenarios.csv")
choques = esc.pivot(index="activo", columns="escenario",
                    values="choque")
orden = ["crisis_2008", "covid_2020", "inflacion_2022", "peor_dia",
         "lunes_negro_1987", "tipos_mas_200pb", "dolar_mas_10",
         "crisis_credito"]
choques = choques[orden]
print(esc.groupby("escenario")["tipo"].first().loc[orden].to_string())
ver = ["RV_EEUU", "RV_EMG", "ALT_PRI", "RF_T710", "RF_T20", "RF_CHY",
       "MP_ORO"]
(100 * choques.loc[ver]).T.round(1)

escenario
crisis_2008          historico
covid_2020           historico
inflacion_2022       historico
peor_dia             historico
lunes_negro_1987    hipotetico
tipos_mas_200pb     hipotetico
dolar_mas_10        hipotetico
crisis_credito      hipotetico
activo            RV_EEUU  RV_EMG  ALT_PRI  RF_T710  RF_T20  RF_CHY  MP_ORO
escenario
crisis_2008         -47.3   -59.8    -63.5     12.5    29.5   -24.6     8.5
covid_2020          -34.3   -41.0    -46.8      5.3    14.0   -20.9    -1.7
inflacion_2022      -25.9   -41.1    -29.9    -14.5   -33.5   -13.6    -1.5
peor_dia            -11.4   -14.0    -15.2     -0.3    -0.6    -2.0    -1.7
lunes_negro_1987    -19.8   -22.8    -28.3      2.0     4.6   -10.3     1.5
tipos_mas_200pb      -5.1    -4.5     -6.5    -14.0   -32.2    -4.2    -5.6
dolar_mas_10          0.0    -8.6      0.0      0.0     0.0     0.0    -6.4
crisis_credito      -15.0   -13.5    -25.5      3.0     6.9   -32.7     1.2

The rows say what matters in each scenario. In 2008 Treasury bonds rise (12.5% for the 7-10 year bond, 29.5% for the 20+ year bond) while equities fall 47%; in 2022 both fall (−14.5% and −33.5%). Listed private equity (ALT_PRI) falls more than the index in every equity scenario. Gold rises in 2008 and barely moves in the others. The hypothetical Black Monday is a −19.8% day for RV_EEUU; the real one in 1987, according to the Federal Reserve, was a 22.6% drop in the Dow Jones in a single session.

The second cell checks where both types come from. The historical ones are the price ratio between the two dates of the scenario; the hypothetical ones, the true loadings of the factor model (universo4_verdad.csv) multiplied by the factor shock, without an idiosyncratic component:

precios = pd.read_parquet("universo4_precios.parquet")
precios = precios.set_index("fecha")
verdad = pd.read_csv("universo4_verdad.csv", index_col="activo")
B = verdad.filter(like="b_").rename(columns=lambda c: c[2:])

# historical: price at the close of `fin` / at the close of `inicio` − 1
ini, fin = "2008-09-12", "2009-03-09"
hist = precios.loc[fin] / precios.loc[ini] - 1
print("Historical 2008, maximum difference:",
      f"{(hist - choques['crisis_2008']).abs().max():.1e}")

# hypothetical: true loadings × factor shock
lunes = pd.Series({"mercado": -0.20, "tamano": -0.02, "tipos": 0.02,
                   "credito": -0.03, "materias": -0.05,
                   "dolar": -0.02, "emergentes": -0.05,
                   "europa": -0.03, "japon": -0.03}
                  ).reindex(B.columns, fill_value=0.0)
print("Black Monday, maximum difference:",
      f"{(B @ lunes - choques['lunes_negro_1987']).abs().max():.1e}")

Historical 2008, maximum difference: 5.2e-06
Black Monday, maximum difference: 5.6e-17

A hypothetical scenario is only as good as its model: if the loadings of an asset change in a crisis, or if the asset has its own risk that the model does not capture, the revaluation is wrong. In the data, the model is the true one; in a real portfolio, the loadings are estimated (lesson 4.1.6).

A multi-asset fund under stress

The lesson's portfolio is a 500 million USD fund spread across fourteen assets: 45% equities, 38% fixed income, 10% alternatives (listed private equity, infrastructure and real estate) and 7% commodities. Its loss in each scenario, by asset class, in percentage points of the fund:

pesos = pd.Series({
    "RV_EEUU": .25, "RV_EUR": .08, "RV_EMG": .07, "RV_EEUU_P": .05,
    "ALT_PRI": .03, "INFRA": .03, "INM_EEUU": .04,
    "RF_T710": .15, "RF_T20": .05, "RF_CIG": .08, "RF_CHY": .07,
    "RF_EMG": .03, "MP_ORO": .04, "MP_AMP": .03})
capital = 500e6
activos = pd.read_csv("universo4_activos.csv", index_col="activo")
clase = activos["clase"].replace({"inmobiliario": "alternativo"})

aporte = choques.loc[pesos.index].mul(pesos, axis=0)   # as a fraction
por_clase = 100 * aporte.groupby(clase.loc[pesos.index]).sum()
por_clase.loc["total"] = por_clase.sum()
print("Weights by class:",
      pesos.groupby(clase.loc[pesos.index]).sum().round(2).to_dict())
por_clase.T.round(2)

Weights by class: {'alternativo': 0.1, 'materias_primas': 0.07, 'renta_fija': 0.38, 'renta_variable': 0.45}
clase             alternativo  materias_primas  renta_fija  renta_variable  total
escenario
crisis_2008             -5.52            -0.78        1.62          -23.09 -27.78
covid_2020              -3.81            -0.68       -0.41          -16.42 -21.31
inflacion_2022          -2.88             0.32       -6.56          -12.22 -21.35
peor_dia                -1.22            -0.12       -0.28           -5.75  -7.37
lunes_negro_1987        -2.06            -0.06       -0.41           -9.26 -11.79
tipos_mas_200pb         -1.06            -0.22       -5.30           -2.18  -8.76
dolar_mas_10            -0.13            -0.45       -0.02           -1.40  -2.01
crisis_credito          -1.90             0.05       -2.68           -6.60 -11.14

The header figure ranks the result. The worst loss is that of 2008 (−27.8%, 139 million), with fixed income adding 1.6 points. In covid (−21.3%) fixed income already subtracts 0.4: high-yield credit falls more than Treasury bonds rise. And in the 2022 inflation (−21.4%) fixed income subtracts 6.6 points: the fund loses almost the same as in covid with a much smaller equity drop, because the leg that should have protected it falls with them. The hypothetical rate rise (−8.8%) confirms it: fixed income is here the largest source of loss.

What would VaR have said? With the 20 years of data, the portfolio's historical VaR:

rend = precios / precios.shift(1).fillna(100) - 1
r_cartera = rend[pesos.index] @ pesos         # simple returns
mes = (1 + r_cartera).rolling(21).apply(np.prod, raw=True) - 1
print(f"1-day 99% VaR (20-year historical): "
      f"{-r_cartera.quantile(0.01):.2%}")
print(f"21-day 99% VaR (20-year historical): "
      f"{-mes.quantile(0.01):.2%}")
print(f"Worst day: {r_cartera.min():.2%} ({r_cartera.idxmin().date()})")
print(f"Worst rolling month: {mes.min():.2%} ({mes.idxmin().date()})")

1-day 99% VaR (20-year historical): 2.27%
21-day 99% VaR (20-year historical): 15.62%
Worst day: -7.44% (2008-09-04)
Worst rolling month: -21.62% (2011-10-24)

The one-day 99% VaR is 2.27% and the worst day of the sample, −7.44%; the hypothetical Black Monday would cost 11.8%. At one month, the VaR is 15.6% and the worst rolling month in history, −21.6% (in 2011). The 2008 and 2022 scenarios last months and lose more than any monthly VaR. This does not make VaR useless: it describes the distribution of normal and bad days; scenarios describe the worst ones. A serious risk limit needs both, as was seen with premium selling in lesson 3.6.1.

The reverse stress test

A reverse stress test starts from the unacceptable loss, for example 20% of the fund in one month, and looks for what produces it. There are infinitely many combinations; the most likely one, with a covariance $\Sigma$ of the factors, is the one with the smallest Mahalanobis distance. If $b = B^\top w$ is the portfolio's exposure to each factor (the loadings weighted by the weights), the shock that loses $L$ with the smallest distance is

The cell uses the true covariance of the factors in the stress regime, scaled to one month (21 sessions), and compares the result with the cumulative move of the factors in covid (February 20 to March 23, 2020):

fv = pd.read_csv("universo4_factores_verdad.csv")


def cov_factores(regimen):
    t = fv[fv.regimen == regimen].set_index("factor")
    corr = t.filter(like="corr_").rename(columns=lambda c: c[5:])
    vol = t["vol_anual"]
    return corr.loc[B.columns, B.columns] * np.outer(vol, vol)


S = cov_factores("estres") * 21 / 252        # one month of stress
b = B.loc[pesos.index].T @ pesos             # exposure to factors
perdida = 0.20
f_inv = -perdida * (S @ b) / (b @ S @ b)     # minimum distance
d = perdida / np.sqrt(b @ S @ b)             # in standard deviations

fd = pd.read_parquet("universo4_factores_diarios.parquet")
fd = fd.set_index("fecha")
f20 = fd.loc["2020-02-20":"2020-03-23"].filter(like="f_").sum()
f20.index = [c[2:] for c in f20.index]
print(f"Exposure to market {b['mercado']:.2f}, "
      f"rates {b['tipos']:.2f}, credit {b['credito']:.2f}; "
      f"distance {d:.2f} standard deviations")
pd.DataFrame({"inverso_%": 100 * f_inv,
              "covid_2020_%": 100 * f20}).round(1)

Exposure to market 0.55, rates 0.43, credit 0.35; distance 2.54 standard deviations
            inverso_%  covid_2020_%
mercado         -32.6         -42.0
tamano           -2.3          -7.3
valor            -1.2          -9.1
tipos             2.2           5.5
credito          -3.9          -6.8
materias        -12.4         -22.8
dolar             3.5           4.6
emergentes       -5.6          -9.1
europa           -0.2           1.7
japon            -0.0          -4.5

The fund has an exposure of 0.55 to the market, 0.43 to rates (because of the bonds) and 0.35 to credit. The most likely shock that costs it 20% in a month of stress is a 32.6% drop in the market factor accompanied by commodities (−12.4%), emerging markets (−5.6%) and credit (−3.9%), with the dollar rising 3.5% and bonds 2.2%: it lies 2.54 standard deviations from a month of stress. It is a crisis like covid, which was worse in the market factor (−42.0%). The reverse stress test does not say that this will happen; it says that the unacceptable loss is one covid away and that the bond hedge barely changes the figure.

Correlations that change in crises

Correlations estimated with the full sample are, above all, those of the calm: 4,367 of the 5,031 sessions. By regime, in the sample and in the model's truth:

exceso = rend.sub(fd["rf"], axis=0)
cov = pd.read_csv("universo4_cov_verdad.csv")


def cov_verdad(regimen, nombres):
    c = cov[cov.regimen == regimen].set_index("activo")
    return c.loc[nombres, nombres]


pares = [("RV_EEUU", "RF_T710"), ("RV_EEUU", "RF_CHY"),
         ("RV_EEUU", "MP_ORO"), ("RV_EEUU", "RV_EMG")]
filas = {}
for reg in ("calma", "estres", "inflacion"):
    dias = fd.index[fd["regimen"] == reg]
    m = exceso.loc[dias].corr()
    for a, b_ in pares:
        c = cov_verdad(reg, [a, b_])
        filas[(f"{a}-{b_}", reg)] = {
            "muestra": m.at[a, b_],
            "verdad": c.iat[0, 1] / np.sqrt(c.iat[0, 0] * c.iat[1, 1])}
print("Sessions by regime:", fd["regimen"].value_counts().to_dict())
pd.DataFrame(filas).T.unstack().round(2)

Sessions by regime: {'calma': 4367, 'estres': 469, 'inflacion': 195}
                muestra                  verdad
                  calma estres inflacion  calma estres inflacion
RV_EEUU-MP_ORO     0.08   0.14      0.26   0.06   0.15      0.19
RV_EEUU-RF_CHY     0.62   0.71      0.75   0.62   0.73      0.75
RV_EEUU-RF_T710   -0.23  -0.41      0.53  -0.24  -0.46      0.51
RV_EEUU-RV_EMG     0.84   0.96      0.85   0.84   0.96      0.88

The correlation between RV_EEUU and the 7-10 year bond is −0.23 in calm, −0.41 under stress (flight to quality) and +0.53 in the inflation regime. That of equities with high yield rises from 0.62 to 0.75; that of the US with emerging markets, from 0.84 to 0.96 under stress: in a crisis, the geographic diversification of equities almost disappears. The figure shows the rolling stock-bond correlation from 2006 to 2025: almost always negative, more so in crises, and positive only in 2022. The sample figures look like the true ones because the regime is known; in reality, the regime is known afterwards.

How much risk does the change in correlations add? The cell computes the fund's true volatility in each regime and what it would have with the volatilities of that regime and the correlations of the calm:

nombres = list(pesos.index)
w = pesos.values
filas = {}
Cc = cov_verdad("calma", nombres).values
corr_calma = Cc / np.outer(np.sqrt(np.diag(Cc)), np.sqrt(np.diag(Cc)))
for reg in ("calma", "estres", "inflacion"):
    C = cov_verdad(reg, nombres).values
    vol = np.sqrt(np.diag(C))
    mixta = np.outer(vol, vol) * corr_calma   # regime vols,
    filas[reg] = {                            # calm correlation
        "vol_cartera": np.sqrt(w @ C @ w),
        "vol_con_corr_calma": np.sqrt(w @ mixta @ w),
        "suma_vols": w @ vol,
        "ratio_diversif": (w @ vol) / np.sqrt(w @ C @ w)}
pd.DataFrame(filas).T.round(3)

           vol_cartera  vol_con_corr_calma  suma_vols  ratio_diversif
calma            0.093               0.093      0.131           1.398
estres           0.273               0.259      0.327           1.198
inflacion        0.158               0.136      0.192           1.218

In calm, the fund has a volatility of 9.3% and the weighted sum of its assets' volatilities is 13.1%: a diversification ratio of 1.40. Under stress, 27.3% (ratio 1.20): with the correlations of the calm it would be 25.9%, so almost all the increase comes from volatilities. In inflation, 15.8% versus 13.6% with the correlations of the calm: here correlations explain a third of the increase (2.2 of the 6.5 points), because the correlation between the two largest legs changes sign. A risk model estimated in calm underestimates both effects; the stress test must move volatilities and correlations at the same time.

Liquidity: how much it costs to get out

The losses of a scenario assume that the portfolio can be valued and sold at those prices. With a large portfolio or illiquid assets that is not the case: selling it takes days and each sale moves the price. riesgo4_liquidez.csv gives, for each asset (as a fictitious ETF), the average daily volume (ADV), the normal and stress bid-ask spread, how much the ADV falls in a crisis and the coefficient of the square-root law, which lesson 4.3.1 studies with the orders of a trading desk: