Hazard Rates and Survival

Interactive exploration of how hazard rates determine cumulative default probability and the link between the two, including the common cumulative vs. conditional pitfall

The hazard rate (or default intensity) \(\lambda(t)\) is the instantaneous conditional probability of default given survival up to time \(t\) (Hull 2023, chap. 17; Christoffersen 2012, chap. 12):

\[ \Pr(t < \tau \leq t + \Delta t \mid \tau > t) \;\approx\; \lambda(t)\,\Delta t \]

Let \(V(t) = \Pr(\tau > t)\) denote the survival probability to time \(t\) and \(Q(t) = 1 - V(t)\) the cumulative default probability. Taking \(\Delta t \to 0\) gives \(dV/dt = -\lambda(t) V(t)\), which integrates to:

\[ V(t) = \exp\!\left(-\int_0^t \lambda(\tau)\,d\tau\right), \qquad Q(t) = 1 - \exp\!\left(-\int_0^t \lambda(\tau)\,d\tau\right) \]

For a constant average hazard rate \(\bar\lambda\) over \([0, t]\):

\[ Q(t) = 1 - e^{-\bar\lambda\,t}, \qquad \bar\lambda = -\frac{1}{t}\ln(1 - Q(t)) \]

Note

Cumulative vs. conditional. The unconditional default probability during year \(k\) is \(Q(k) - Q(k-1)\). The conditional probability of defaulting in year \(k\) given survival to year \(k-1\) is \([Q(k)-Q(k-1)]/[1-Q(k-1)]\). Dividing the cumulative figure \(Q(k)\) by the survival probability is a common error.

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

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

Two views of the same object

The hazard rate \(\bar\lambda\) and the cumulative default probability \(Q(T)\) over a horizon \(T\) are two ways of describing the same survival process. Move either slider to see how they are tied.

Tip

How to experiment

Start with a flat \(\bar\lambda\) in “Specify hazard rate” mode and watch \(Q(T)\) climb. Switch to “Specify Q(T)” mode, move \(Q(10)\) to 28.3% (CCC/C 1-year rate), and notice that as \(T\) varies the implied \(\bar\lambda\) stays high even if you are reading off a 10-year window. Then switch to the two-piece mode and introduce an elevated early hazard followed by a lower mature-firm hazard — the shape of \(Q(t)\) that emerges matches the speculative-grade pattern Hull describes.

viewof hrMode = Inputs.radio(["Specify hazard rate", "Specify Q(T)", "Two-piece hazard"], {
  label: "Input mode",
  value: "Specify hazard rate"
})

viewof hrT = Inputs.range([1, 20], {
  label: "Horizon T (years)",
  step: 1,
  value: 10
})

viewof hrLambda = hrMode === "Specify hazard rate"
  ? Inputs.range([0.001, 0.25], {
      label: "Average hazard rate λ̄ (per year)",
      step: 0.001,
      value: 0.02
    })
  : (() => { const el = html`<span></span>`; el.value = 0.02; return el })()

viewof hrQT = hrMode === "Specify Q(T)"
  ? Inputs.range([0.005, 0.80], {
      label: "Cumulative default Q(T) (fraction)",
      step: 0.005,
      value: 0.10
    })
  : (() => { const el = html`<span></span>`; el.value = 0.10; return el })()

viewof hrLambdaA = hrMode === "Two-piece hazard"
  ? Inputs.range([0.0, 0.4], {
      label: "λ₁ early hazard, 0 to t*",
      step: 0.005,
      value: 0.10
    })
  : (() => { const el = html`<span></span>`; el.value = 0.10; return el })()

viewof hrLambdaB = hrMode === "Two-piece hazard"
  ? Inputs.range([0.0, 0.4], {
      label: "λ₂ late hazard, after t*",
      step: 0.005,
      value: 0.02
    })
  : (() => { const el = html`<span></span>`; el.value = 0.02; return el })()

viewof hrTstar = hrMode === "Two-piece hazard"
  ? Inputs.range([1, 14], {
      label: "t* breakpoint (years)",
      step: 1,
      value: 3
    })
  : (() => { const el = html`<span></span>`; el.value = 3; return el })()

hrBuild = {
  const N = 241
  const tMax = 20
  const xs = Array.from({length: N}, (_, i) => i * tMax / (N - 1))

  function cumHazardConst(lambda) {
    return xs.map(t => lambda * t)
  }
  function cumHazardPiece(lA, lB, tstar) {
    return xs.map(t => t <= tstar ? lA * t : lA * tstar + lB * (t - tstar))
  }

  let Lambda, lambdaBar
  if (hrMode === "Specify hazard rate") {
    Lambda = cumHazardConst(hrLambda)
    lambdaBar = t => hrLambda
  } else if (hrMode === "Specify Q(T)") {
    const lam = -Math.log(1 - hrQT) / hrT
    Lambda = cumHazardConst(lam)
    lambdaBar = t => lam
  } else {
    Lambda = cumHazardPiece(hrLambdaA, hrLambdaB, hrTstar)
    lambdaBar = t => t <= hrTstar ? hrLambdaA : hrLambdaB
  }

  const V = Lambda.map(L => Math.exp(-L))
  const Q = V.map(v => 1 - v)
  const data = xs.map((t, i) => ({t, V: V[i], Q: Q[i], lambda: lambdaBar(t)}))

  const iT = xs.findIndex(t => t >= hrT)
  const QatT = Q[iT]
  const VatT = V[iT]
  const lamAvg = -Math.log(1 - QatT) / hrT

  return { data, xs, Q, V, QatT, VatT, lamAvg, lambdaBar }
}

hrConditional = {
  // Conditional default probability over each year k: [Q(k)-Q(k-1)] / [1-Q(k-1)]
  const years = Array.from({length: hrT}, (_, k) => k + 1)
  const Qyear = k => {
    if (hrMode === "Specify hazard rate") return 1 - Math.exp(-hrLambda * k)
    if (hrMode === "Specify Q(T)") {
      const lam = -Math.log(1 - hrQT) / hrT
      return 1 - Math.exp(-lam * k)
    }
    const L = k <= hrTstar ? hrLambdaA * k : hrLambdaA * hrTstar + hrLambdaB * (k - hrTstar)
    return 1 - Math.exp(-L)
  }
  const rows = []
  let Qprev = 0
  for (const k of years) {
    const Qk = Qyear(k)
    const inc = Qk - Qprev
    const cond = inc / (1 - Qprev)
    rows.push({year: k, Qk, Qprev, inc, cond})
    Qprev = Qk
  }
  return rows
}

Q(t) and V(t)

{
  return Plot.plot({
    height: 360, marginLeft: 60, marginRight: 20,
    x: { label: "Time t (years)", domain: [0, 20], grid: true },
    y: { label: "Probability", domain: [0, 1], grid: true, tickFormat: d => (d * 100).toFixed(0) + "%" },
    marks: [
      Plot.ruleY([0]),
      Plot.line(hrBuild.data, { x: "t", y: "Q", stroke: "#d62728", strokeWidth: 2.5 }),
      Plot.line(hrBuild.data, { x: "t", y: "V", stroke: "#2f71d5", strokeWidth: 2.5 }),
      Plot.ruleX([hrT], { stroke: "#666", strokeWidth: 1, strokeDasharray: "3 3" }),
      Plot.dot([{ x: hrT, y: hrBuild.QatT }], { x: "x", y: "y", r: 5, fill: "#d62728", stroke: "white", strokeWidth: 2 }),
      Plot.dot([{ x: hrT, y: hrBuild.VatT }], { x: "x", y: "y", r: 5, fill: "#2f71d5", stroke: "white", strokeWidth: 2 }),
      Plot.text([
        { x: 19.5, y: hrBuild.QatT, text: "Q(t)", fill: "#d62728" },
        { x: 19.5, y: hrBuild.VatT, text: "V(t)", fill: "#2f71d5" }
      ], { x: "x", y: "y", text: "text", fill: "fill", textAnchor: "end", fontWeight: 700 })
    ]
  })
}

html`<p style="color:#666;font-size:0.85rem;">Cumulative default probability <span style="color:#d62728;font-weight:700;">Q(t)</span> and survival probability <span style="color:#2f71d5;font-weight:700;">V(t) = 1 − Q(t)</span>. The dashed rule marks the horizon T. At T, Q(${hrT}) = ${pctFmt(hrBuild.QatT, 2)} and the implied average hazard rate is λ̄ = ${pctFmt(hrBuild.lamAvg, 3)} per year.</p>`

Hazard rate profile

{
  const lamMax = Math.max(...hrBuild.data.map(d => d.lambda))
  const yMax = Math.max(lamMax * 1.25, 0.005)
  return Plot.plot({
    height: 320, marginLeft: 60, marginRight: 20,
    x: { label: "Time t (years)", domain: [0, 20], grid: true },
    y: { label: "λ(t) per year", domain: [0, yMax], grid: true, tickFormat: d => (d * 100).toFixed(1) + "%" },
    marks: [
      Plot.ruleY([0]),
      Plot.line(hrBuild.data, { x: "t", y: "lambda", stroke: "#2e7d32", strokeWidth: 2.5 }),
      Plot.ruleX([hrT], { stroke: "#666", strokeWidth: 1, strokeDasharray: "3 3" })
    ]
  })
}

html`<p style="color:#666;font-size:0.85rem;">The instantaneous hazard rate profile. In the two-piece mode, a drop at t* generates the downward-sloping cumulative hazard that characterises speculative-grade bonds that survive the critical early period.</p>`

Cumulative vs. conditional by year

{
  const rows = hrConditional
  const barW = 0.35
  const bars = []
  for (const r of rows) {
    bars.push({ x1: r.year - barW, x2: r.year, y: r.inc, series: "unconditional" })
    bars.push({ x1: r.year, x2: r.year + barW, y: r.cond, series: "conditional" })
  }
  return Plot.plot({
    height: 340, marginLeft: 60, marginRight: 20, marginBottom: 40,
    x: { label: "Year k", domain: [0.4, hrT + 0.6], ticks: rows.map(r => r.year) },
    y: { label: "Probability", grid: true, tickFormat: d => (d * 100).toFixed(1) + "%" },
    color: { legend: true, domain: ["unconditional", "conditional"], range: ["#7aa7d9", "#e67e22"] },
    marks: [
      Plot.ruleY([0]),
      Plot.rectY(bars, { x1: "x1", x2: "x2", y: "y", fill: "series", fillOpacity: 0.85 })
    ]
  })
}

{
  const rows = hrConditional
  const rowHtml = rows.map(r =>
    `<tr><td>${r.year}</td><td>${pctFmt(r.Qk, 3)}</td><td>${pctFmt(r.inc, 3)}</td><td>${pctFmt(1 - r.Qprev, 3)}</td><td style="color:#e67e22;font-weight:700;">${pctFmt(r.cond, 3)}</td></tr>`
  ).join("")
  return html`<table class="table" style="width:100%;">
<thead><tr>
  <th>Year k</th>
  <th>Q(k)</th>
  <th>Q(k) − Q(k−1)</th>
  <th>V(k−1)</th>
  <th>Conditional λ<sub>k</sub></th>
</tr></thead>
<tbody>${rowHtml}</tbody>
</table>
<p style="color:#666;font-size:0.85rem;">
The conditional probability of defaulting in year k equals the incremental default probability divided by the survival probability at the start of year k. For a flat hazard, this conditional probability is constant across years.
</p>`
}

{
  const r = hrBuild
  return html`<div style="margin-top:10px;padding:12px 16px;border-radius:8px;background:#f5f7fa;">
<strong>Summary at T = ${hrT} years</strong><br/>
Q(T) = <span style="color:#d62728;font-weight:700;">${pctFmt(r.QatT, 3)}</span>,
V(T) = <span style="color:#2f71d5;font-weight:700;">${pctFmt(r.VatT, 3)}</span>,
average hazard λ̄ = <span style="color:#2e7d32;font-weight:700;">${pctFmt(r.lamAvg, 3)}</span> per year.
</div>`
}

References

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

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