Risk¶

Tail measures, aggregate-loss models, severity fitting, credibility, and Solvency II aggregation — numpy / base R throughout, with scipy adding polish when it is installed and never being required.

Tail measures¶

PythonR

sc.var(losses, 0.995)        # type-7 quantile of the sample
sc.tvar(losses, 0.995)       # mean beyond it (sc.es is the same function)

sc_var(losses, 0.995)
sc_tvar(losses, 0.995)       # sc_es() is the same function

Aggregate losses¶

PythonR

sc.aggregate_loss("poisson", "lognormal", lam=5, mu=8, sigma=1)
sc.aggregate_loss("negbin", "gamma", r=4, beta=1.2, alpha=2, theta=500, method="fft")
sc.aggregate_loss("poisson", losses_sample, lam=3, method="mc", n_sims=100_000)
sc.simulate_losses("poisson", "lognormal", lam=5, mu=8, sigma=1, n_sims=10_000)
sc.panjer("poisson", severity_pmf, lam=5)     # the raw recursion, if you want the pmf

sc_aggregate_loss("poisson", "lognormal", lam = 5, mu = 8, sigma = 1)
sc_aggregate_loss("negbin", "gamma", r = 4, beta = 1.2, alpha = 2, theta = 500, method = "fft")
sc_aggregate_loss("poisson", losses, lam = 3, method = "mc", n_sims = 100000)
sc_simulate_losses("poisson", "lognormal", lam = 5, mu = 8, sigma = 1)
sc_panjer("poisson", severity_pmf, lam = 5)

       p           VaR           TVaR
0  0.500  20638.189466   37468.522951
3  0.950  58194.894715   75982.678420
5  0.995  99351.284174  123539.901999
— Aggregate loss · poisson(lam=5) × lognormal
  basis: panjer · lattice h = 119.989 × 4096
  · Mean 24,573.18 · sd 18,108.75 · CV 0.737.

Frequencies: poisson(lam), negbin(r, beta), binomial(m, q). Severities: lognormal, gamma, pareto (Lomax), exponential, weibull — Loss-Models parameterisation — or an empirical sample. Methods: panjer (exact on a lattice), fft, mc (seeded). If the lattice is too short to reach the 99.9th percentile, the basis line says WARNING lattice too short rather than quietly truncating the tail.

Fitting severities¶

PythonR

sc.fit(losses)   # lognormal · gamma · pareto · weibull · exponential, ranked by AIC

sc_fit(losses)

Closed-form where a closed form exists, maximum likelihood where not (scipy when available, a numpy Newton fallback when not), with Kolmogorov–Smirnov distances alongside the AICs. A family that fails to fit becomes a failed: … row instead of sinking the whole call.

Credibility¶

PythonR

sc.credibility(df, "group", "loss_ratio")            # Bühlmann
sc.credibility(df, "group", "loss_ratio", weight="exposure")   # Bühlmann–Straub
sc.limited_fluctuation(n=800, p=0.9, k=0.05)          # classical partial credibility
sc.full_credibility(p=0.9, k=0.05)                    # 1082.2 claims

sc_credibility(df, "group", "loss_ratio")
sc_credibility(df, "group", "loss_ratio", weight = "exposure")
sc_limited_fluctuation(n = 800, p = 0.9, k = 0.05)
sc_full_credibility(p = 0.9, k = 0.05)

The Bühlmann table reports each group's Z and credibility premium, with μ, EPV, VHM and K in the basis line. When the variance of hypothetical means comes out non-positive, Z is 0 across the board — the honest reading that the groups are not distinguishable from noise — rather than a negative credibility.

Solvency II aggregation¶

PythonR

sc.aggregate_scr({"mortality": 100, "lapse": 200, "expense": 80})
sc.aggregate_scr(charges, corr=sc.SII_LIFE_CORR)     # the Annex IV matrix, exported
sc.risk_margin([120, 100, 80, 55, 30], rate=0.04)    # cost-of-capital, 6 %

sc_aggregate_scr(c(mortality = 100, lapse = 200, expense = 80))
sc_aggregate_scr(charges, corr = SC_SII_LIFE_CORR)
sc_risk_margin(c(120, 100, 80, 55, 30), rate = 0.04)

aggregate_scr computes SCR = √(vᵀρv), shows each module's Euler marginal (they sum to the SCR) and the diversification credit. Modules missing from the correlation matrix are treated as uncorrelated. The standard-formula matrices ship in both languages: life (7×7, Annex IV), non-life, and BSCR.

Function list¶

var tvar (es) aggregate_loss panjer simulate_losses lognormal_params fit credibility (buhlmann) limited_fluctuation full_credibility aggregate_scr risk_margin · matrices SII_LIFE_CORR SII_NONLIFE_CORR SII_BSCR_CORR — each with the sc_ twin (SC_SII_*) in R.