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Show in what ways stress tests can draw on the mapping of risk factors.

FRM® Part II · Market Risk Measurement and Management · VaR Mapping · FRM2.MR.5.e

The idea

Mapping gives stress testing a common language. Once every position is expressed as exposures to a few risk factors, a shock to a factor translates directly into portfolio P&L. Writing Change for the change in portfolio value, with each exposure in money and each shock as a return, Change = Σ exposure × factor shock, summed across factors. A manager can apply a parallel or twisting shift to the yield-curve vertices, a fall in an equity index, a currency devaluation or a replay of a historical episode, and revalue the whole book consistently. Shocks can also target the covariance matrix, such as raising volatilities or moving correlations. The adverse direction depends on position signs. For a long-short pair, correlation falling, not rising, hurts. Limits: linear mapping misstates the result for options because it ignores gamma, and it ignores basis risk, so large shocks call for delta-gamma or full revaluation. Exam point: stress tests reuse the mapping but are not bound by the normal distribution that VaR assumes.

Formulas

Stress P&L of a mapped bond portfolio

ΔV = −Σ xᵢ × Dᵢ × Δy

xᵢ
the exposure at vertex i
Dᵢ
the modified duration at vertex i
Δy
the yield shock, as a decimal

Linear, so it ignores convexity under large shocks. A rise in yields gives a loss; getting the sign wrong is the trap.

Delta-gamma change in an option's value

ΔV = Δ × ΔS + ½ × Γ × ΔS²

Δ
the option delta
ΔS
the change in the underlying price
Γ
the option gamma

For a long option the gamma term is positive, so delta alone overstates the loss in a large fall.

A practice question

From the question bank, with the reason each wrong option is wrong.

How does mapping help in stress testing a large portfolio?

  1. It removes the need to choose scenarios, because mapped factors move according to their historical distribution.Why not: Stress tests still require scenarios chosen by the analyst; mapping only provides the translation.
  2. Shocks to a few risk factors can be turned into portfolio P&L through the mapped exposures, consistently across all positions. - correct
  3. It converts stress losses into a VaR at 99.9 percent confidence.Why not: Stress losses are scenario outcomes, not quantiles of a fitted distribution.
  4. It ensures that non-linear positions are revalued exactly under any shock.Why not: Mapping is typically linear and can misstate non-linear positions under large shocks.

Because every position is expressed as exposures to the same risk factors, a scenario for those factors (for example +200 bp on the curve or -25 percent on equities) can be applied once and revalued across the book. The scenarios themselves are a judgement.

A harder one

A bond portfolio maps to zero-coupon exposures of USD 10 million at a vertex with modified duration 1.9, USD 20 million at a vertex with modified duration 4.6 and USD 5 million at a vertex with modified duration 8.4. In a stress test, all vertex yields rise by 100 basis points. The change in portfolio value is closest to:

  1. a gain of USD 1.53 millionWhy not: Gets the sign wrong; a rise in yields lowers bond values.
  2. a loss of USD 0.15 millionWhy not: Applies a 10 basis point shock instead of 100 basis points.
  3. a loss of USD 1.53 million - correct
  4. a loss of USD 1.74 millionWhy not: Applies the unweighted average of the three durations to the total exposure instead of weighting by exposure.

Change in value = -sum of exposure x modified duration x yield shock = -(10 x 1.9 + 20 x 4.6 + 5 x 8.4) x 0.01 = -(19 + 92 + 42) x 0.01 = -1.53m.

The hardest tier

A stress test shocks an underlying priced at 100 down by 20 percent. The portfolio holds 5,000 long call options, each with delta 0.40 and gamma 0.03 per USD of underlying move. Using a delta-gamma approximation for the change in value of the options, the stressed P&L is closest to:

  1. a loss of USD 40,000Why not: Uses delta alone, as a linear mapping would, and omits the offsetting gamma gain.
  2. a loss of USD 70,000Why not: Subtracts the gamma term instead of adding it, although convexity benefits a long option.
  3. a loss of USD 10,000 - correct
  4. a gain of USD 30,000Why not: Includes only the gamma term, 0.5 x gamma x shock squared, and drops the delta loss.

Per option: delta term = 0.40 x (-20) = -8.0; gamma term = 0.5 x 0.03 x 20² = +6.0, so the change is -2.0. For 5,000 options it is -10,000. A delta-only mapping would show -40,000, overstating the loss for a long-gamma position under a large shock.

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