Risk

Misc

• Also see
  • Decision Intelligence >> Decision Curves
    • Not sure if it can be applied to other types of risk analysis outside of medicine, but it might be worth looking at.
• Packages
  • {PerformanceAnalytics} - Econometric Tools for Performance and Risk Analysis
    • Lots of vignettes, but there’s also a substantial amount of material in the introduction section of the Reference Manual

Value at Risk (VaR)

• Notes from GPT-4o

Copulas

• Also see Association, Copulas

• Copula Workflow

  • Data Collection and Preprocessing:
    • Collect historical data for the two variables of interest.
    • Clean the data by handling missing values, outliers, and ensuring consistency.
  • Marginal Distribution Estimation:
    • Estimate the marginal distributions of each variable independently.
    • Fit appropriate probability distributions to the marginal data.
  • Copula Selection and Fitting:
    • Choose a copula family (e.g., Gaussian, t, Clayton, Gumbel, etc.).
    • Fit the copula model to the data, estimating the copula parameters.
  • Simulation and Analysis:
    • Generate joint samples from the fitted copula model.
    • Use the joint distribution to compute risk measures such as VaR and CVaR.

• Example: Value at Risk and Conditional Value at Risk Using Copulas

  set.seed(123)
  n <- 1000
  asset1 <- 
    rnorm(n, 
          mean = 0.001, 
          sd = 0.02)
  asset2 <- 
    rnorm(n, 
          mean = 0.001, 
          sd = 0.03)
  data <- data.frame(asset1, asset2)
  
  1margins <- list(m1 = fitdistrplus::fitdist(asset1, "norm"),
                  m2 = fitdistrplus::fitdist(asset2, "norm"))
  
  2normal.cop <- copula::normalCopula(param = 0.5, dim = 2)
  
  u1 <- 
    pnorm(asset1, 
          mean = margins$m1$estimate["mean"], 
  3        sd = margins$m1$estimate["sd"])
  u2 <- 
    pnorm(asset2, 
          mean = margins$m2$estimate["mean"], 
          sd = margins$m2$estimate["sd"])
  fit.cop <- 
    copula::fitCopula(normal.cop, 
                      cbind(u1, u2), 
                      method = "ml")
  
  copula.samples <- 
  4  copula::rCopula(n, fit.cop@copula)
  
  5simulated.returns <- data.frame(
    asset1 = qnorm(copula.samples[, 1], 
                   mean = margins$m1$estimate["mean"], 
                   sd = margins$m1$estimate["sd"]),
    asset2 = qnorm(copula.samples[, 2], 
                   mean = margins$m2$estimate["mean"], 
                   sd = margins$m2$estimate["sd"])
  )
  simulated.portfolio.returns <- rowMeans(simulated.returns)
  
  6VaR_95 <- quantile(simulated.portfolio.returns, probs = 0.05)
  CVaR_95 <- mean(simulated.portfolio.returns[simulated.portfolio.returns <= VaR_95])
  

  1

  Estimate marginal distributions (assuming normal for simplicity)

  2

  normalCopula doesn’t have it’s own listing in the reference manual. It’s under ellipCopula, or you can just use ?normalCopula.

  3

  Fit the copula to the data

  4

  Generate samples from the fitted copula. The @ operator is used to access copula object since it’s a S4 object. Also, rCopula doesn’t have it’s own listing in the reference manual. It’s under Copula, or you can just use ?rCopula.

  5

  Simulate portfolio returns by ransforming copula samples back to original scale and averaging (i.e. equal weighting)

  6

  Calculate VaR and CVaR at 95% confidence level

  • {PerformanceAnalytics} also has functions for calculation the VaR and CVaR values

    VaR_95 <- VaR(simulated.portfolio.returns, 
                  p = 0.95, 
                  method = "historical")
    CVaR_95 <- ES(simulated.portfolio.returns, 
                  p = 0.95, 
                  method = "historical")