Scenario analysis: portfolio value at risk

Interactive VaR estimation using real-world stock price distributions

In scenario analysis, we work in the real world to answer: “How bad can things get?” (see Hull 2023, chap. 10). The risk-neutral world is an artificial device for valuation — risk managers interested in future outcomes should use real-world parameters.

For a stock portfolio, we compute the stock price that has only a small probability \(q\) of the actual price being lower:

\[ V = S_0 \exp\!\left[\left(\mu - \frac{\sigma^2}{2}\right)T + N^{-1}(q)\,\sigma\sqrt{T}\right] \]

This worst-case stock price gives us the portfolio’s Value at Risk — the loss that will not be exceeded with confidence \(1 - q\).

Tip

How to experiment

1. Increase \(\sigma\) to see the worst-case price drop and losses widen.
2. Increase \(T\) for a similar effect over longer horizons.
3. Switch between confidence levels to see how the tail quantile changes.

viewof varS0 = Inputs.range([10, 200], {
  label: "Current stock price S₀ ($)",
  step: 5,
  value: 80
})

viewof varMu = Inputs.range([0.01, 0.20], {
  label: "Expected return μ (per year)",
  step: 0.01,
  value: 0.08
})

viewof varSigma = Inputs.range([0.05, 0.60], {
  label: "Volatility σ (per year)",
  step: 0.01,
  value: 0.20
})

viewof varT = Inputs.range([0.25, 3], {
  label: "Time horizon T (years)",
  step: 0.25,
  value: 0.5
})

viewof varConf = Inputs.range([0.90, 0.99], {
  label: "Confidence level",
  step: 0.01,
  value: 0.99
})

viewof varShares = Inputs.range([100, 100000], {
  label: "Number of shares",
  step: 100,
  value: 10000
})

// Inverse normal CDF (rational approximation, Beasley-Springer-Moro)
norminv = p => {
  if (p <= 0) return -Infinity
  if (p >= 1) return Infinity
  if (p === 0.5) return 0
  const a = [-3.969683028665376e+01, 2.209460984245205e+02, -2.759285104469687e+02,
    1.383577518672690e+02, -3.066479806614716e+01, 2.506628277459239e+00]
  const b = [-5.447609879822406e+01, 1.615858368580409e+02, -1.556989798598866e+02,
    6.680131188771972e+01, -1.328068155288572e+01]
  const c = [-7.784894002430293e-03, -3.223964580411365e-01, -2.400758277161838e+00,
    -2.549732539343734e+00, 4.374664141464968e+00, 2.938163982698783e+00]
  const d = [7.784695709041462e-03, 3.224671290700398e-01, 2.445134137142996e+00,
    3.754408661907416e+00]
  const pLow = 0.02425, pHigh = 1 - pLow
  let q, r
  if (p < pLow) {
    q = Math.sqrt(-2 * Math.log(p))
    return (((((c[0]*q+c[1])*q+c[2])*q+c[3])*q+c[4])*q+c[5]) /
      ((((d[0]*q+d[1])*q+d[2])*q+d[3])*q+1)
  } else if (p <= pHigh) {
    q = p - 0.5
    r = q * q
    return (((((a[0]*r+a[1])*r+a[2])*r+a[3])*r+a[4])*r+a[5])*q /
      (((((b[0]*r+b[1])*r+b[2])*r+b[3])*r+b[4])*r+1)
  } else {
    q = Math.sqrt(-2 * Math.log(1 - p))
    return -(((((c[0]*q+c[1])*q+c[2])*q+c[3])*q+c[4])*q+c[5]) /
      ((((d[0]*q+d[1])*q+d[2])*q+d[3])*q+1)
  }
}

varQ = 1 - varConf
varZq = norminv(varQ)
varWorstPrice = varS0 * Math.exp((varMu - varSigma ** 2 / 2) * varT + varZq * varSigma * Math.sqrt(varT))
varCurrentPortfolio = varShares * varS0
varWorstPortfolio = varShares * varWorstPrice
varMaxLoss = varCurrentPortfolio - varWorstPortfolio

// PDF for the distribution
varPDF = {
  const meanLog = Math.log(varS0) + (varMu - varSigma ** 2 / 2) * varT
  const sdLog = varSigma * Math.sqrt(varT)
  const meanST = varS0 * Math.exp(varMu * varT)
  const sdST = meanST * Math.sqrt(Math.exp(varSigma ** 2 * varT) - 1)
  const xMax = meanST + 3.5 * sdST
  const xMin = Math.max(0.01, meanST - 4 * sdST)
  const n = 400
  const dx = (xMax - xMin) / n
  const pts = []
  for (let i = 0; i <= n; i++) {
    const x = xMin + i * dx
    if (x <= 0) continue
    const logx = Math.log(x)
    const pdf = (1 / (x * sdLog * Math.sqrt(2 * Math.PI))) *
      Math.exp(-0.5 * ((logx - meanLog) / sdLog) ** 2)
    pts.push({ x, pdf })
  }
  return pts
}

fmtDollar = v => v.toFixed(2)
fmtLargeNum = v => v.toLocaleString("en-US", { minimumFractionDigits: 0, maximumFractionDigits: 0 })

Plot.plot({
  height: 320,
  marginLeft: 50,
  marginRight: 20,
  x: { label: "Stock price at time T ($)", grid: false },
  y: { label: "Density", grid: true },
  marks: [
    Plot.areaY(varPDF.filter(d => d.x <= varWorstPrice), {
      x: "x", y: "pdf",
      fill: "#d62728", fillOpacity: 0.25,
      curve: "natural"
    }),
    Plot.line(varPDF, {
      x: "x", y: "pdf",
      stroke: "#4682b4", strokeWidth: 2.5, curve: "natural"
    }),
    Plot.ruleX([varWorstPrice], { stroke: "#d62728", strokeWidth: 2, strokeDash: [6, 3] }),
    Plot.text([{ x: varWorstPrice, label: `$${fmtDollar(varWorstPrice)}` }], {
      x: "x",
      y: varPDF.reduce((m, d) => Math.max(m, d.pdf), 0) * 0.85,
      text: "label", fill: "#d62728", fontWeight: "bold", fontSize: 11, dx: 5, textAnchor: "start"
    }),
    Plot.ruleX([varS0], { stroke: "#888", strokeDash: [4, 4], strokeOpacity: 0.5 }),
    Plot.text([{ x: varS0, label: `S₀ = $${varS0}` }], {
      x: "x",
      y: varPDF.reduce((m, d) => Math.max(m, d.pdf), 0) * 0.95,
      text: "label", fill: "#888", fontSize: 10, dx: 5, textAnchor: "start"
    })
  ]
})

html`<table class="table" style="width:100%;">
<thead><tr><th colspan="2">Scenario analysis (${(varConf * 100).toFixed(0)}% confidence, T = ${varT} years)</th></tr></thead>
<tbody>
<tr><td style="font-weight:500;">Current portfolio value</td><td>$${fmtLargeNum(varCurrentPortfolio)}</td></tr>
<tr><td style="font-weight:500;">Worst-case stock price</td><td>$${fmtDollar(varWorstPrice)}</td></tr>
<tr><td style="font-weight:500;">Worst-case portfolio value</td><td>$${fmtLargeNum(Math.round(varWorstPortfolio))}</td></tr>
<tr style="border-top: 2px solid #888;"><td style="font-weight:700;">Maximum loss (VaR)</td><td style="font-weight:700;">$${fmtLargeNum(Math.round(varMaxLoss))}</td></tr>
</tbody></table>
<p style="margin-top:0.5rem; color:#666; font-size:0.85rem;">There is only a ${(varQ * 100).toFixed(0)}% chance the actual loss will exceed this amount. Note that this calculation uses the <strong>real-world</strong> expected return μ = ${(varMu * 100).toFixed(0)}%, not the risk-free rate --- because scenario analysis is about what might actually happen.</p>`

Note

Real world, not risk-neutral

A common mistake in scenario analysis is to use risk-neutral parameters. The risk-neutral world is an artificial device for valuation — it does not describe what is likely to happen in reality. For scenario analysis and VaR estimation, we must use real-world expected returns, which include the risk premium investors demand for bearing systematic risk.

References

Hull, John. 2023. Risk Management and Financial Institutions. 6th ed. New Jersey: John Wiley & Sons.