FINC6023 · Financial Risk Management
Historical Simulation, Mapping & the Linear/Quadratic Model
Historical Simulation, Mapping & the Linear/Quadratic Model is the toolkit for portfolios that do not fit a clean delta-normal mould. Historical simulation replays past factor moves on today's book and reads VaR off the ranked P&L; extreme-value theory models the tail beyond a threshold. For interest-rate books you choose a mapping — principal, duration, cash-flow or PCA — with cash-flow mapping (split a cash flow onto adjacent vertices by matching variance) being the core Part-B question. Non-linear option books need the delta–gamma quadratic model or Monte Carlo.
What this chapter covers
- 01Historical simulation: replay factor moves, rank P&L, read the quantile
- 02Weighted historical sim and volatility-updating
- 03Extreme-value theory: GPD tail and the power law
- 04Interest-rate mappings: principal, duration, cash-flow, PCA
- 05Duration VaR: ΔP = −D·P·Δy, VaR = D·P·z·σ_y
- 06Cash-flow mapping by variance-matching (CORE Part-B)
- 07Linear (delta) vs quadratic (delta–gamma) model for options
- 08Monte Carlo simulation and the bootstrap for VaR confidence intervals
Duration-based VaR on a bond position
- 2 marksRelate 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 = $65,000,000 per unit of yield.
- 2 marksDollar volatility of the position: σ_P($) = D · P · σ_y = 65,000,000 × 0.0007 = $45,500 per day.
- 2 marksApply the VaR multiplier: VaR = z · σ_P($) = 2.326 × 45,500.
- 1 markCompute: 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.
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.
Exam move
Memorise the four interest-rate mappings and which scenario each fits, and drill the cash-flow-mapping recipe end-to-end because it is a guaranteed Part-B candidate. For duration VaR, double-check yield-volatility units every time.
