The Vasicek Portfolio Loss Distribution

Interactive exploration of the limiting distribution of the portfolio loss rate in the single-factor Gaussian copula model, the Vasicek distribution

Let each loan default independently conditional on the common factor \(F\) with conditional probability \(p(F)\). If all loans are equally weighted and their number grows, the portfolio loss rate \(L = \frac{1}{n}\sum L_i\) converges to \(p(F)\), whose distribution inherits the factor randomness. This yields the Vasicek distribution (Vasicek 2002; Christoffersen 2012, chap. 12):

\[ F_L(x;\, PD, \rho) \;=\; \Phi\!\left(\frac{\sqrt{1 - \rho}\,\Phi^{-1}(x) - \Phi^{-1}(PD)}{\sqrt{\rho}}\right) \]

\[ f_L(x;\, PD, \rho) \;=\; \sqrt{\frac{1 - \rho}{\rho}}\, \exp\!\left(-\frac{1}{2\rho}\bigl(\sqrt{1 - \rho}\,\Phi^{-1}(x) - \Phi^{-1}(PD)\bigr)^{2} + \tfrac{1}{2}\bigl(\Phi^{-1}(x)\bigr)^{2}\right) \]

The distribution has mean \(PD\), is supported on \([0, 1]\), and is heavily right-skewed. Higher asset correlation thickens the right tail and pulls mass toward the extremes.

normalCDF = x => {
  const a1 = 0.254829592, a2 = -0.284496736, a3 = 1.421413741
  const a4 = -1.453152027, a5 = 1.061405429, p = 0.3275911
  const sign = x < 0 ? -1 : 1
  const z = Math.abs(x) / Math.sqrt(2)
  const t = 1.0 / (1.0 + p * z)
  const y = 1 - (((((a5 * t + a4) * t) + a3) * t + a2) * t + a1) * t * Math.exp(-z * z)
  return 0.5 * (1 + sign * y)
}

normalPDF = x => Math.exp(-x * x / 2) / Math.sqrt(2 * Math.PI)

qnorm = {
  const a1 = -3.969683028665376e+01, a2 =  2.209460984245205e+02
  const a3 = -2.759285104469687e+02, a4 =  1.383577518672690e+02
  const a5 = -3.066479806614716e+01, a6 =  2.506628277459239e+00
  const b1 = -5.447609879822406e+01, b2 =  1.615858368580409e+02
  const b3 = -1.556989798598866e+02, b4 =  6.680131188771972e+01
  const b5 = -1.328068155288572e+01
  const c1 = -7.784894002430293e-03, c2 = -3.223964580411365e-01
  const c3 = -2.400758277161838e+00, c4 = -2.549732539343734e+00
  const c5 =  4.374664141464968e+00, c6 =  2.938163982698783e+00
  const d1 =  7.784695709041462e-03, d2 =  3.224671290700398e-01
  const d3 =  2.445134137142996e+00, d4 =  3.754408661907416e+00
  const pLow = 0.02425, pHigh = 1 - pLow
  return p => {
    if (p <= 0) return -Infinity
    if (p >= 1) return Infinity
    if (p < pLow) {
      const q = Math.sqrt(-2 * Math.log(p))
      return (((((c1*q+c2)*q+c3)*q+c4)*q+c5)*q+c6) / ((((d1*q+d2)*q+d3)*q+d4)*q+1)
    }
    if (p <= pHigh) {
      const q = p - 0.5, r = q * q
      return (((((a1*r+a2)*r+a3)*r+a4)*r+a5)*r+a6)*q / (((((b1*r+b2)*r+b3)*r+b4)*r+b5)*r+1)
    }
    const q = Math.sqrt(-2 * Math.log(1 - p))
    return -(((((c1*q+c2)*q+c3)*q+c4)*q+c5)*q+c6) / ((((d1*q+d2)*q+d3)*q+d4)*q+1)
  }
}

fmt = (x, d) => x === undefined || isNaN(x) ? "N/A" : x.toFixed(d)

pctFmt = (x, d = 2) => (x * 100).toFixed(d) + "%"

vasicekCDF = (x, PD, rho) => {
  if (x <= 0) return 0
  if (x >= 1) return 1
  if (rho < 1e-6) return x < PD ? 0 : 1
  return normalCDF((Math.sqrt(1 - rho) * qnorm(x) - qnorm(PD)) / Math.sqrt(rho))
}

vasicekPDF = (x, PD, rho) => {
  if (x <= 0 || x >= 1) return 0
  if (rho < 1e-6) return 0
  const q = qnorm(x)
  const qpd = qnorm(PD)
  return Math.sqrt((1 - rho) / rho) *
    Math.exp(-0.5 / rho * Math.pow(Math.sqrt(1 - rho) * q - qpd, 2) + 0.5 * q * q)
}

vasicekVaR = (p, PD, rho) => {
  // P[L ≤ VaR] = 1 - p, so VaR = F⁻¹(1-p)
  return normalCDF((Math.sqrt(rho) * qnorm(1 - p) + qnorm(PD)) / Math.sqrt(1 - rho))
}

Inputs

Tip

How to experiment

Set PD = 2%, ρ = 0.10 to reproduce Figure 12.6 in Christoffersen. Push ρ to 0.30 and notice the density mass sliding to the right tail — at ρ = 0.60 the distribution becomes distinctly bimodal. Flip the normal-approximation toggle on: the Gaussian with the same mean and variance misses the skew entirely, which is why credit VaR based on normal assumptions understates tail risk.

viewof vdPD = Inputs.range([0.001, 0.30], {
  label: "Unconditional PD",
  step: 0.001,
  value: 0.02
})

viewof vdRho = Inputs.range([0.005, 0.95], {
  label: "Asset correlation ρ",
  step: 0.005,
  value: 0.10
})

viewof vdAlpha = Inputs.range([0.90, 0.999], {
  label: "Confidence level 1 − p",
  step: 0.001,
  value: 0.99
})

viewof vdShowNormal = Inputs.toggle({
  label: "Overlay normal approximation",
  value: false
})

vdCurves = {
  const PD = vdPD, rho = vdRho
  const N = 500
  // Adapt x range to where mass lives
  const xMax = Math.min(0.999, Math.max(3 * PD, vasicekVaR(0.001, PD, rho)))
  const xs = []
  for (let i = 1; i < N; i++) {
    xs.push(Math.max(1e-5, Math.min(0.9999, xMax * i / N)))
  }
  const data = xs.map(x => ({
    x,
    density: vasicekPDF(x, PD, rho),
    cdf: vasicekCDF(x, PD, rho)
  }))
  // Moments for the normal overlay: use known mean (PD) and numerical variance by integration
  let mean = PD
  let m2 = 0
  for (let i = 0; i < data.length - 1; i++) {
    const dx = data[i + 1].x - data[i].x
    const xMid = (data[i].x + data[i + 1].x) / 2
    const fMid = (data[i].density + data[i + 1].density) / 2
    m2 += xMid * xMid * fMid * dx
  }
  const variance = Math.max(1e-10, m2 - mean * mean)
  const sd = Math.sqrt(variance)
  return { data, xMax, mean, sd }
}

vdQuantiles = {
  const PD = vdPD, rho = vdRho
  const probs = [0.50, 0.90, 0.95, 0.99, 0.999]
  return probs.map(q => ({ level: q, value: vasicekVaR(1 - q, PD, rho) }))
}

vdVar = vasicekVaR(1 - vdAlpha, vdPD, vdRho)

{
  const data = vdCurves.data
  const tail = data.filter(d => d.x >= vdVar)
  const marks = [
    Plot.ruleY([0]),
    Plot.areaY(tail, { x: "x", y: "density", fill: "#d62728", fillOpacity: 0.35 }),
    Plot.line(data, { x: "x", y: "density", stroke: "#2f71d5", strokeWidth: 2.5 }),
    Plot.ruleX([vdVar], { stroke: "#d62728", strokeWidth: 2 }),
    Plot.ruleX([vdPD], { stroke: "#2e7d32", strokeDasharray: "4 3" })
  ]
  if (vdShowNormal) {
    const { mean, sd } = vdCurves
    const normData = data.map(d => ({ x: d.x, density: normalPDF((d.x - mean) / sd) / sd }))
    marks.push(Plot.line(normData, { x: "x", y: "density", stroke: "#888", strokeDasharray: "5 3", strokeWidth: 1.5 }))
  }
  return Plot.plot({
    height: 360, marginLeft: 60, marginRight: 20,
    x: { label: "Portfolio loss rate L", grid: true, tickFormat: d => (d * 100).toFixed(1) + "%" },
    y: { label: "Density", grid: true },
    marks
  })
}

html`<p style="color:#666;font-size:0.85rem;">Vasicek PDF. The <span style="color:#2e7d32;font-weight:700;">green dashed rule</span> marks the mean (= PD = ${pctFmt(vdPD, 2)}). The <span style="color:#d62728;font-weight:700;">red rule</span> marks the ${pctFmt(vdAlpha, 1)} VaR at ${pctFmt(vdVar, 2)}; the red-shaded mass is the upper tail of probability ${pctFmt(1 - vdAlpha, 2)}. ${vdShowNormal ? "The <span style='color:#888;font-weight:700;'>grey dashed line</span> is a Gaussian with the same mean and variance — at low ρ the two are visually similar, but as ρ grows the Vasicek distribution becomes increasingly right-skewed and the Gaussian misses the tail badly." : "Toggle the normal overlay to compare the Vasicek distribution with a Gaussian of equal mean and variance — the gap widens as ρ increases."}</p>`

{
  return Plot.plot({
    height: 360, marginLeft: 60, marginRight: 20,
    x: { label: "Portfolio loss rate L", grid: true, tickFormat: d => (d * 100).toFixed(1) + "%" },
    y: { label: "F_L(x)", domain: [0, 1], grid: true, tickFormat: d => (d * 100).toFixed(1) + "%" },
    marks: [
      Plot.ruleY([0]),
      Plot.line(vdCurves.data, { x: "x", y: "cdf", stroke: "#2f71d5", strokeWidth: 2.5 }),
      Plot.ruleX([vdVar], { stroke: "#d62728", strokeWidth: 2, strokeDasharray: "3 3" }),
      Plot.ruleY([vdAlpha], { stroke: "#d62728", strokeWidth: 2, strokeDasharray: "3 3" }),
      Plot.dot([{ x: vdVar, y: vdAlpha }], { x: "x", y: "y", r: 5, fill: "#d62728", stroke: "white", strokeWidth: 2 })
    ]
  })
}

html`<p style="color:#666;font-size:0.85rem;">Inverting the CDF at ${pctFmt(vdAlpha, 1)} gives the loss-rate VaR of ${pctFmt(vdVar, 2)}.</p>`

{
  const rows = vdQuantiles.map(q =>
    `<tr>
      <td>${pctFmt(q.level, 1)}</td>
      <td style="font-weight:700;">${pctFmt(q.value, 3)}</td>
      <td>${fmt(q.value / vdPD, 2)}× PD</td>
    </tr>`
  ).join("")
  const { mean, sd } = vdCurves
  return html`<table class="table" style="width:100%;">
<thead><tr>
  <th>Quantile</th>
  <th>Loss rate</th>
  <th>Multiple of PD</th>
</tr></thead>
<tbody>${rows}</tbody>
</table>
<div style="margin-top:10px;padding:10px 16px;border-radius:8px;background:#f5f7fa;">
  <strong>Moments.</strong> Mean = ${pctFmt(mean, 3)} (exactly PD). Numerical SD = ${pctFmt(sd, 3)}. Coefficient of variation = ${fmt(sd / mean, 2)}.
</div>`
}

Note

Reading the shape. At \(\rho = 0\) the distribution collapses to a point mass at \(PD\): with independent defaults and infinitely many loans, the law of large numbers kills all dispersion. At \(\rho \to 1\) the distribution becomes Bernoulli with mass \(1 - PD\) at zero and \(PD\) at one: all firms default together or none do. Intermediate \(\rho\) interpolates between these and generates the heavily right-skewed shapes that drive credit tail risk.

References

Christoffersen, Peter F. 2012. Elements of Financial Risk Management. 2nd ed. Academic Press.

Vasicek, Oldrich. 2002. “The Distribution of Loan Portfolio Value.” Risk 15 (12): 160–62.