Portfolio Risk and Diversification

Interactive exploration of how portfolio weights, individual volatilities, and correlations determine portfolio risk and diversification benefits

When managing a portfolio of multiple assets, risk depends not only on each asset’s volatility but critically on the correlations between them. The portfolio variance for \(n\) assets is (Christoffersen 2012, chap. 7)

\[ \sigma_{PF,t+1}^{2} = w_t' \, \Sigma_{t+1} \, w_t = \sum_{i=1}^{n} \sum_{j=1}^{n} w_{i,t} \, w_{j,t} \, \sigma_{i,t+1} \, \sigma_{j,t+1} \, \rho_{ij,t+1} \]

where \(w_t\) is the vector of portfolio weights, \(\Sigma_{t+1}\) is the covariance matrix, \(\sigma_{i,t+1}\) are individual volatilities, and \(\rho_{ij,t+1}\) are pairwise correlations.

Under the assumption of multivariate normality, portfolio VaR and ES at tail probability \(p\) follow directly:

\[ \text{VaR}_{t+1}^{p} = -\sigma_{PF,t+1} \, \Phi_p^{-1} \qquad \text{ES}_{t+1}^{p} = \sigma_{PF,t+1} \frac{\phi(\Phi_p^{-1})}{p} \]

where \(\Phi_p^{-1}\) is the \(p\)-quantile of the standard normal distribution (i.e., the inverse CDF) and \(\phi(\cdot)\) is the standard normal PDF.

Note

VaR and subadditivity. Under multivariate normality, VaR is subadditive: the portfolio VaR never exceeds the weighted sum of individual VaRs. This means diversification always reduces VaR in this setting. However, VaR is not a coherent risk measure in general (Artzner et al. 1999), and subadditivity can fail for non-normal distributions. ES, being a coherent risk measure, is always subadditive regardless of the distribution. This is one reason we display both measures side by side below.

The undiversified VaR is the weighted sum of individual asset VaRs, \(\text{VaR}_{\text{undiv}} = w_1 \cdot \text{VaR}_1 + w_2 \cdot \text{VaR}_2\). This corresponds to the portfolio risk when assets are perfectly correlated (\(\rho = 1\)), i.e., when there is no diversification at all. The diversification benefit is the difference between this undiversified value and the actual portfolio risk:

\[ \text{Diversification benefit} = \text{VaR}_{\text{undiv}} - \text{VaR}_{PF} \]

It is always non-negative under normality (since \(\sigma_{PF} \leq w_1 \sigma_1 + w_2 \sigma_2\) for \(\rho \leq 1\)) and equals zero only when \(\rho = 1\). The same logic applies to ES.

// ============================================================
// MATH UTILITIES
// ============================================================

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

legend = (items) => {
  const el = document.createElement("div")
  el.style.cssText = "display:flex; flex-wrap:wrap; margin-top:-4px; margin-bottom:6px;"
  const hidden = new Set()

  for (const d of items) {
    const key = d.key || d.label
    const span = document.createElement("span")
    span.style.cssText = "display:inline-flex; align-items:center; gap:4px; margin-right:14px; cursor:pointer; user-select:none; transition:opacity 0.15s;"

    let swatchHTML
    if (d.type === "dot") {
      swatchHTML = `<svg width="12" height="12"><circle cx="6" cy="6" r="5" fill="${d.color}" opacity="0.8"/></svg>`
    } else if (d.type === "dashed") {
      swatchHTML = `<svg width="22" height="12"><line x1="0" y1="6" x2="22" y2="6" stroke="${d.color}" stroke-width="2" stroke-dasharray="4 2"/></svg>`
    } else if (d.type === "rect") {
      swatchHTML = `<svg width="14" height="14"><rect width="14" height="14" fill="${d.color}"/></svg>`
    } else {
      swatchHTML = `<svg width="22" height="12"><line x1="0" y1="6" x2="22" y2="6" stroke="${d.color}" stroke-width="2"/></svg>`
    }

    span.innerHTML = `${swatchHTML}<span style="font-size:0.82rem;">${d.label}</span>`
    span.addEventListener("click", () => {
      const nowHidden = !hidden.has(key)
      if (nowHidden) hidden.add(key); else hidden.delete(key)
      span.style.opacity = nowHidden ? "0.35" : "1"
      span.querySelector("span").style.textDecoration = nowHidden ? "line-through" : "none"
      el.value = new Set(hidden)
      el.dispatchEvent(new Event("input", {bubbles: true}))
    })
    el.appendChild(span)
  }

  el.value = new Set(hidden)
  return el
}

Portfolio risk as a function of weights and correlation

Explore how portfolio risk depends on the allocation between two assets, their individual volatilities, and the correlation between them. The gap between the diversified portfolio risk and the undiversified (weighted-sum) risk is the diversification benefit.

Tip

How to experiment

Start with the default settings and observe the gap between the portfolio \(\sigma\) curve and the straight undiversified line. Then drag the correlation slider toward +1 and watch the gap vanish. Set it to \(-1\) and notice the portfolio volatility can reach zero at a specific weight. Try increasing one asset’s volatility while keeping the other low to see how the optimal diversification weight shifts.

viewof prSig1 = Inputs.range([0.5, 5.0], {
  label: "σ₁ daily volatility (%)",
  step: 0.1,
  value: 2.0
})

viewof prSig2 = Inputs.range([0.5, 5.0], {
  label: "σ₂ daily volatility (%)",
  step: 0.1,
  value: 1.5
})

viewof prRho = Inputs.range([-1.0, 1.0], {
  label: "ρ₁₂ correlation",
  step: 0.01,
  value: 0.40
})

viewof prW1 = Inputs.range([0.0, 1.0], {
  label: "w₁ (weight of asset 1)",
  step: 0.01,
  value: 0.50
})

viewof prP = Inputs.range([0.5, 5.0], {
  label: "Tail probability p (%)",
  step: 0.1,
  value: 1.0
})

// Core calculations
prCalc = {
  const s1 = prSig1 / 100, s2 = prSig2 / 100
  const w1 = prW1, w2 = 1 - prW1
  const rho = prRho, p = prP / 100

  const cov12 = s1 * s2 * rho
  const pfVar = w1*w1*s1*s1 + w2*w2*s2*s2 + 2*w1*w2*cov12
  const pfSig = Math.sqrt(Math.max(0, pfVar))

  const zp = Math.abs(qnorm(p))
  const pfVaR = pfSig * zp
  const pfES = pfSig * normalPDF(qnorm(p)) / p

  const var1 = s1 * zp, var2 = s2 * zp
  const es1 = s1 * normalPDF(qnorm(p)) / p
  const es2 = s2 * normalPDF(qnorm(p)) / p

  const undivVaR = w1 * var1 + w2 * var2
  const undivES = w1 * es1 + w2 * es2
  const undivSig = w1 * s1 + w2 * s2

  const divBenVaR = undivVaR - pfVaR
  const divBenES = undivES - pfES
  const divBenSig = undivSig - pfSig

  const divBenVaRPct = undivVaR > 0 ? divBenVaR / undivVaR * 100 : 0
  const divBenESPct = undivES > 0 ? divBenES / undivES * 100 : 0

  return {
    s1, s2, w1, w2, rho, p, cov12,
    pfVar, pfSig, pfVaR, pfES,
    var1, var2, es1, es2,
    undivVaR, undivES, undivSig,
    divBenVaR, divBenES, divBenSig,
    divBenVaRPct, divBenESPct
  }
}

// Sweep data: portfolio sigma as function of w1 for current rho and reference rhos
prSweepW = {
  const s1 = prSig1 / 100, s2 = prSig2 / 100
  const rhos = [prRho, -1, 0, 0.5, 1]
  const data = []
  for (let w = 0; w <= 1.005; w += 0.01) {
    const w1 = Math.min(w, 1), w2 = 1 - w1
    for (const rho of rhos) {
      const pv = w1*w1*s1*s1 + w2*w2*s2*s2 + 2*w1*w2*s1*s2*rho
      const ps = Math.sqrt(Math.max(0, pv))
      const undiv = w1 * s1 + w2 * s2
      data.push({ w1, sigma: ps * 100, undiv: undiv * 100, rho, isCurrent: rho === prRho })
    }
  }
  return data
}

// Sweep data: portfolio sigma as function of rho for current weights
prSweepRho = {
  const s1 = prSig1 / 100, s2 = prSig2 / 100
  const w1 = prW1, w2 = 1 - prW1
  const data = []
  for (let rho = -1; rho <= 1.005; rho += 0.01) {
    const r = Math.min(rho, 1)
    const pv = w1*w1*s1*s1 + w2*w2*s2*s2 + 2*w1*w2*s1*s2*r
    const ps = Math.sqrt(Math.max(0, pv)) * 100
    data.push({ rho: r, sigma: ps })
  }
  return data
}

Portfolio risk vs. weight

viewof prWLegend = legend([
  { key: "current", label: `ρ = ${fmt(prRho, 2)} (current)`, color: "#2f71d5", type: "line" },
  { key: "undiv", label: "Undiversified (ρ = 1)", color: "#d62728", type: "dashed" },
  { key: "rho0", label: "ρ = 0", color: "#999", type: "dashed" },
  { key: "rhom1", label: "ρ = −1", color: "#2e7d32", type: "dashed" }
])

{
  const h = prWLegend
  const current = prSweepW.filter(d => d.isCurrent)
  const rhom1 = prSweepW.filter(d => d.rho === -1 && !d.isCurrent)
  const rho0 = prSweepW.filter(d => d.rho === 0 && !d.isCurrent)
  const rho1 = prSweepW.filter(d => d.rho === 1 && !d.isCurrent)

  const marks = [
    Plot.ruleY([0], { stroke: "#eee" })
  ]

  if (!h.has("rhom1") && prRho !== -1)
    marks.push(Plot.line(rhom1, { x: "w1", y: "sigma", stroke: "#2e7d32", strokeWidth: 1, strokeDasharray: "4 3" }))
  if (!h.has("rho0") && prRho !== 0)
    marks.push(Plot.line(rho0, { x: "w1", y: "sigma", stroke: "#999", strokeWidth: 1, strokeDasharray: "4 3" }))
  if (!h.has("undiv") && prRho !== 1)
    marks.push(Plot.line(rho1, { x: "w1", y: "sigma", stroke: "#d62728", strokeWidth: 1.5, strokeDasharray: "4 3" }))
  if (!h.has("current"))
    marks.push(Plot.line(current, { x: "w1", y: "sigma", stroke: "#2f71d5", strokeWidth: 2.5 }))

  // Current point
  marks.push(Plot.dot([{ x: prW1, y: prCalc.pfSig * 100 }], { x: "x", y: "y", r: 6, fill: "#2f71d5", stroke: "white", strokeWidth: 2 }))
  // Vertical line at current w1
  marks.push(Plot.ruleX([prW1], { stroke: "#2f71d5", strokeWidth: 1, strokeDasharray: "2 3", strokeOpacity: 0.5 }))

  return Plot.plot({
    height: 400, marginLeft: 60, marginRight: 20,
    x: { label: "w₁ (weight of asset 1)", domain: [0, 1], grid: false },
    y: { label: "Portfolio σ (%)", grid: true },
    marks
  })
}

html`<p style="color:#666;font-size:0.85rem;">Portfolio standard deviation as a function of the weight in asset 1. The <span style="color:#d62728;font-weight:700;">dashed red line</span> is the undiversified case (ρ = 1), where portfolio σ is the weighted average of individual σ values. The <span style="color:#2f71d5;font-weight:700;">blue curve</span> shows the actual portfolio σ at the current correlation. The gap between them is the diversification benefit.</p>`

Portfolio risk vs. correlation

{
  return Plot.plot({
    height: 380, marginLeft: 60, marginRight: 20,
    x: { label: "Correlation ρ₁₂", domain: [-1, 1], grid: false },
    y: { label: "Portfolio σ (%)", grid: true },
    marks: [
      Plot.line(prSweepRho, { x: "rho", y: "sigma", stroke: "#2f71d5", strokeWidth: 2.5 }),
      Plot.dot([{ x: prRho, y: prCalc.pfSig * 100 }], { x: "x", y: "y", r: 6, fill: "#2f71d5", stroke: "white", strokeWidth: 2 }),
      Plot.ruleX([prRho], { stroke: "#2f71d5", strokeWidth: 1, strokeDasharray: "2 3", strokeOpacity: 0.5 })
    ]
  })
}

html`<p style="color:#666;font-size:0.85rem;">Portfolio standard deviation at the current weights (w₁ = ${fmt(prW1, 2)}) as a function of correlation. Risk increases monotonically with correlation. At ρ = +1 there is no diversification benefit; at ρ = −1 perfect hedging is possible.</p>`

VaR and ES comparison

{
  const r = prCalc
  const types = ["Asset 1", "Asset 2", "Undiversified", "Portfolio"]
  const colors = { "Asset 1": "#aaa", "Asset 2": "#ccc", "Undiversified": "#e67e22", "Portfolio": "#2f71d5" }
  const barData = [
    { measure: "VaR", type: "Asset 1", value: r.w1 * r.var1 * 100 },
    { measure: "VaR", type: "Asset 2", value: r.w2 * r.var2 * 100 },
    { measure: "VaR", type: "Undiversified", value: r.undivVaR * 100 },
    { measure: "VaR", type: "Portfolio", value: r.pfVaR * 100 },
    { measure: "ES", type: "Asset 1", value: r.w1 * r.es1 * 100 },
    { measure: "ES", type: "Asset 2", value: r.w2 * r.es2 * 100 },
    { measure: "ES", type: "Undiversified", value: r.undivES * 100 },
    { measure: "ES", type: "Portfolio", value: r.pfES * 100 }
  ]

  // Compute bar positions manually: grouped bars within each measure
  const measures = ["VaR", "ES"]
  const barWidth = 0.18
  const bars = []
  for (const d of barData) {
    const mx = measures.indexOf(d.measure)
    const tx = types.indexOf(d.type)
    const x = mx + (tx - 1.5) * barWidth
    bars.push({ x, x1: x - barWidth/2, x2: x + barWidth/2, y: d.value, type: d.type, measure: d.measure })
  }

  return Plot.plot({
    height: 380, marginLeft: 60, marginBottom: 40,
    x: { label: null, domain: [-0.5, 1.5], ticks: [0, 1], tickFormat: d => measures[d] },
    y: { label: "% of portfolio value", grid: true },
    marks: [
      Plot.ruleY([0]),
      Plot.rectY(bars, {
        x1: "x1", x2: "x2", y: "y", fill: d => colors[d.type], fillOpacity: 0.85
      }),
      Plot.text(bars, {
        x: "x", y: "y", text: d => d.type, dy: -8, fontSize: 9, fill: d => colors[d.type],
        fontWeight: d => d.type === "Portfolio" ? 700 : 500
      })
    ]
  })
}

{
  const r = prCalc
  return html`<div style="display:flex;gap:20px;flex-wrap:wrap;justify-content:center;margin-top:8px;">
    <div style="padding:12px 20px;border-radius:8px;background:#fff3e0;text-align:center;">
      <div style="font-size:0.8rem;color:#666;">VaR diversification benefit</div>
      <div style="font-size:1.3em;font-weight:700;color:#e67e22;">${fmt(r.divBenVaR * 100, 3)}%</div>
      <div style="font-size:0.8rem;color:#666;">(${fmt(r.divBenVaRPct, 1)}% reduction)</div>
    </div>
    <div style="padding:12px 20px;border-radius:8px;background:#e3f2fd;text-align:center;">
      <div style="font-size:0.8rem;color:#666;">ES diversification benefit</div>
      <div style="font-size:1.3em;font-weight:700;color:#2f71d5;">${fmt(r.divBenES * 100, 3)}%</div>
      <div style="font-size:0.8rem;color:#666;">(${fmt(r.divBenESPct, 1)}% reduction)</div>
    </div>
  </div>`
}

html`<p style="color:#666;font-size:0.85rem;">Weighted individual contributions, undiversified (sum of weighted individual measures), and actual portfolio VaR and ES. The diversification benefit is the difference between the undiversified and portfolio values. Under normality, both VaR and ES show positive diversification benefits whenever ρ < 1.</p>`

Summary

{
  const r = prCalc
  return html`<table class="table" style="width:100%;">
<thead><tr>
  <th>Measure</th>
  <th>Asset 1 (w₁ = ${fmt(r.w1, 2)})</th>
  <th>Asset 2 (w₂ = ${fmt(r.w2, 2)})</th>
  <th>Undiversified</th>
  <th style="color:#2f71d5;">Portfolio</th>
  <th>Diversification benefit</th>
</tr></thead>
<tbody>
<tr><td style="font-weight:500;">σ (%)</td>
    <td>${fmt(r.s1 * 100, 3)}%</td>
    <td>${fmt(r.s2 * 100, 3)}%</td>
    <td>${fmt(r.undivSig * 100, 3)}%</td>
    <td style="font-weight:700;">${fmt(r.pfSig * 100, 3)}%</td>
    <td>${fmt(r.divBenSig * 100, 3)}%</td></tr>
<tr><td style="font-weight:500;">VaR (${fmt(r.p * 100, 1)}%)</td>
    <td>${fmt(r.var1 * 100, 3)}%</td>
    <td>${fmt(r.var2 * 100, 3)}%</td>
    <td>${fmt(r.undivVaR * 100, 3)}%</td>
    <td style="font-weight:700;">${fmt(r.pfVaR * 100, 3)}%</td>
    <td>${fmt(r.divBenVaR * 100, 3)}% (${fmt(r.divBenVaRPct, 1)}%)</td></tr>
<tr><td style="font-weight:500;">ES (${fmt(r.p * 100, 1)}%)</td>
    <td>${fmt(r.es1 * 100, 3)}%</td>
    <td>${fmt(r.es2 * 100, 3)}%</td>
    <td>${fmt(r.undivES * 100, 3)}%</td>
    <td style="font-weight:700;">${fmt(r.pfES * 100, 3)}%</td>
    <td>${fmt(r.divBenES * 100, 3)}% (${fmt(r.divBenESPct, 1)}%)</td></tr>
<tr><td style="font-weight:500;">Covariance</td>
    <td colspan="5">${fmt(r.cov12 * 10000, 6)} (σ₁₂ = σ₁ × σ₂ × ρ₁₂ = ${fmt(r.s1*100, 1)}% × ${fmt(r.s2*100, 1)}% × ${fmt(r.rho, 2)})</td></tr>
<tr><td style="font-weight:500;">Portfolio variance</td>
    <td colspan="5">${fmt(r.pfVar * 10000, 6)}</td></tr>
</tbody></table>
<p style="color:#666;font-size:0.85rem;">All risk measures assume multivariate normality and a ${fmt(r.p*100, 1)}% tail probability. Under this assumption, both VaR and ES are proportional to portfolio σ and therefore always exhibit positive diversification benefits when ρ < 1.</p>`
}

References

Artzner, Philippe, Freddy Delbaen, Jean-Marc Eber, and David Heath. 1999. “Coherent Measures of Risk.” Mathematical Finance 9 (3): 203–28.

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