FINC6023 Chap.8 Historical Simulation, Mapping & the Linear/Quadratic Model
Historical Simulation, Mapping & the Linear/Quadratic Model
Historical Simulation, Mapping and the Linear or Quadratic Model covers portfolios that do not fit a simple delta-normal approximation. Historical simulation applies observed risk-factor changes to today's positions, revalues the current portfolio, ranks the hypothetical profit and loss outcomes, and reads the desired tail measure.
Weighting or volatility updating can make recent or comparably volatile observations more influential, while extreme-value methods focus directly on exceedances beyond a high threshold.
Fixed-income portfolios introduce a dimensionality problem because cash flows occur at many dates while risk systems use a smaller set of standard maturity vertices.
Principal mapping assigns a position to a nearby vertex, duration mapping matches interest-rate sensitivity, and cash-flow mapping splits present value between adjacent vertices while matching variance. Principal-component mapping instead represents yield-curve changes through a few common shapes.
Cash-flow mapping is the main written calculation.
Interpolate the rate at the cash-flow maturity, discount the cash flow, interpolate its volatility, solve the variance-matching equation for the split weight, and allocate present value to the adjacent bonds.
After mapping every cash flow, aggregate the vertex exposures with their covariance and scale the resulting portfolio VaR to the required horizon.
Options and other curved payoffs require more than a linear delta approximation. A delta-gamma expansion adds the curvature term; full Monte Carlo simulation can revalue a highly nonlinear portfolio under many correlated scenarios.
The available final paper assigns a full Part B question to mapping two cash flows and then calculating ten-day portfolio VaR. Clear presentation matters because the answer joins several distinct steps and unit conventions.
What this chapter covers
- 01
Historical simulation: replay factor moves, rank P&L, and apply weighting or volatility updating
- 02
Extreme-value theory: GPD tail and the power law
- 03
Interest-rate mappings: principal, duration, cash-flow, PCA
- 04
Duration VaR: ΔP = −D·P·Δy, VaR = D·P·z·σ_y
- 05
Cash-flow mapping by variance-matching (CORE Part-B)
- 06
Linear and quadratic option models, Monte Carlo simulation, and bootstrap confidence intervals
Worked example · free
Duration-based VaR on a bond position
- 2Relate price change to yield change: ΔP = −D · P · Δy, so the dollar sensitivity to a 1-unit yield move is D · P = 6.5 × 10,000,000 = $65,000,000 per unit of yield.
- 2Dollar volatility of the position: σ_P($) = D · P · σ_y = 65,000,000 × 0.0007 = $45,500 per day.
- 2Apply the VaR multiplier: VaR = z · σ_P($) = 2.326 × 45,500.
- 1Compute: 2.326 × 45,500 = $105,833.
Key terms
- Historical simulation
- Apply each past period's factor moves to today's portfolio to build a distribution of hypothetical P&L, then read VaR as the relevant percentile. It makes no normality assumption and captures observed tail behaviour.
- Duration mapping
- Approximate a bond's price change as ΔP = −D·P·Δy and compute VaR = D·P·z·σ_y, collapsing the whole book onto a single yield factor.
- Cash-flow mapping
- Split a cash flow that falls between two standard maturity vertices onto those vertices by matching both present value and variance (solving σ² = a²σ₁² + (1−a)²σ₂² + 2ρσ₁σ₂a(1−a) for the weight a). This is a core Part-B presentation question.
- Delta–gamma (quadratic) model
- ΔP ≈ Sδ·Δx + ½S²γ(Δx)², adding the gamma (curvature) term that the linear delta model misses for options, improving VaR accuracy for non-linear books.
- Maturity vertex
- A maturity vertex is a standard point on the yield curve to which off-grid cash flows are assigned so a fixed-income portfolio can be measured with a manageable set of risk factors.
- Variance matching
- Variance matching chooses mapping weights so the return variance of the replacement positions equals the interpolated variance of the original cash flow.
Historical Simulation, Mapping & the Linear/Quadratic Model FAQ
When do I use the linear model versus delta–gamma or Monte Carlo?
Use the linear (delta) model when the portfolio's value is roughly linear in the risk factors. For options and other convex payoffs the linear model misses curvature and skew, so switch to the quadratic delta–gamma approximation or, when payoffs are highly non-linear, full Monte Carlo revaluation.
What makes cash-flow mapping a recurring Part-B question?
Because it has a clean four-step recipe — interpolate the rate to the cash flow's maturity, present-value it, interpolate the volatility, then split the PV onto two adjacent vertices by matching variance. The variance-match form is on the sheet, but presenting the steps clearly is what earns the marks.
What is the most common error in duration VaR?
Mismatching the units of the yield volatility and the duration. Modified duration multiplies an ABSOLUTE change in yield, so the yield volatility must be in the same absolute units (e.g. 0.0007, not 0.07). Getting this wrong scales the answer by 100.
How is cash-flow mapping marked in a written answer?
Show each stage separately: interpolated yield, present value, interpolated volatility, variance-matching equation, solved allocation weight, and amounts assigned to adjacent maturity vertices. After mapping all cash flows, aggregate the vertex positions and calculate portfolio VaR. The official solution demonstrates the method sequence but does not publish a sub-part mark split.
When is delta-gamma preferable to a linear approximation?
Use delta-gamma when curvature is material, as with options whose value changes nonlinearly with the underlying risk factor. The gamma term captures the second-order effect ignored by a linear delta model. If the payoff is highly nonlinear or several factors interact, full Monte Carlo revaluation may be more reliable.
Exam move
Learn the mapping methods as answers to different approximation goals rather than as a list. Principal mapping preserves amount, duration mapping preserves first-order rate sensitivity, cash-flow mapping preserves present value and matches variance across adjacent vertices, and principal components preserve dominant yield-curve movements.
For cash-flow mapping, rehearse the complete sequence on fresh numbers: interpolate the yield, discount the cash flow, interpolate volatility, solve the variance match, allocate present value, aggregate all mapped exposures, and calculate horizon-scaled VaR. Keep a running table with one row per original cash flow so the two vertex allocations can be checked against its present value.
In duration VaR, write yield changes in absolute decimal units. For nonlinear portfolios, decide whether delta is adequate, whether gamma must be added, or whether full revaluation is needed. Time a complete mapping solution because the available paper gives this method a full written question and rewards an orderly chain more than an unexplained final number.
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