Risk Sensitivities: The Units, Curvature and Mapping Errors That Matter

A risk system can calculate every derivative correctly and still measure the wrong portfolio. The problem often lies between the pricing model and the risk-factor grid: an exposure is assigned to an unsuitable factor, a volatility cube loses a dimension, or a number expressed per basis point is interpreted as a response to a percentage change. Validation must therefore establish the meaning of a sensitivity before testing its numerical precision.

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This report brings together sensitivity measurement, nonlinear approximation and the translation of internal risk factors into a standardised regulatory representation. Its conclusion is operational: preserve the economic meaning of the shock throughout the calculation, and quantify the information discarded by the mapping. The worked numbers are teaching inputs, not observations about an identified institution. The regulatory mapping research dates to December 2020; the discussion explains that historical methodology without asserting that its implementation details are current requirements.

Start with the variable and its unit

Let a portfolio have value $V$, and let $x$ be a market variable. Its local sensitivity is

$$S_x=\frac{\partial V}{\partial x},\qquad \Delta V\approx S_x\Delta x.$$

In words: Sensitivity is how quickly value changes, and multiplying it by a small market move estimates the value change.

Here $S_x$ measures the change in value per unit of $x$, and $\Delta x$ is the proposed market movement. Neither symbol is meaningful without units. A derivative with respect to a decimal interest rate differs by a factor of 10,000 from a sensitivity per basis point. A volatility sensitivity per unit of decimal volatility differs by a factor of 100 from vega per volatility point. A spot-price delta is different again from delta-equivalent currency exposure to a percentage return.

The control record should state the factor name, quote convention, quotation currency, bump size, output currency, sign convention, valuation date and model version. These fields are part of the calculation. Relegating them to informal knowledge makes a plausible-looking factor-of-100 error difficult to detect.

A long fixed-rate bond usually loses value as yield rises. Some systems report DV01 as a positive loss magnitude for a one-basis-point rise; others report signed value sensitivity. Both conventions can work, but mixing them reverses the hedge. Portfolio aggregation should use a signed common convention before any absolute values are taken for a capital calculation.

Numerical sensitivities need convergence evidence

When an analytical derivative is unavailable, a forward bump estimates

$$S_x^{F}(h)=\frac{V(x+h)-V(x)}{h},$$

where $h$ is the bump size. A central bump uses

$$S_x^{C}(h)=\frac{V(x+h)-V(x-h)}{2h}.$$

For a smooth pricing function, the central construction typically removes the leading forward-difference truncation error. That does not make an arbitrarily small bump desirable. Solver tolerances, rounding, interpolation changes and simulation noise can dominate the numerator when the two valuations become too close.

A useful validation exhibit compares several bump sizes rather than displaying one sensitivity. The reviewer looks for a stable region between large-bump nonlinearity and small-bump numerical noise. For simulation-based valuation, consistent random numbers across bumped and unbumped runs can reduce noise, provided the technique is documented and applied consistently. Discontinuous payoffs, exercise decisions and barrier monitoring require particular care because smooth-function intuition may fail.

Validation tests distinguish numerical instability from mapping defects

Curvature explains why local accuracy is insufficient

For one factor, a second-order approximation is

$$\Delta V\approx S_x\Delta x+\frac12\Gamma_x(\Delta x)^2, \qquad \Gamma_x=\frac{\partial^2V}{\partial x^2}.$$

In words: A small move changes value through both the initial slope and the way that slope changes.

The curvature $\Gamma_x$ measures how the slope itself changes. With several factors, the corresponding expression is

$$\Delta V\approx \mathbf{s}^{\mathsf T}\Delta\mathbf{x} +\frac12\Delta\mathbf{x}^{\mathsf T}H\Delta\mathbf{x},$$

In words: Several market moves affect value through their individual slopes, their bending effects and the interactions between them.

where $\mathbf{s}$ is the sensitivity vector and $H$ is the Hessian matrix of second derivatives. Off-diagonal entries capture interactions between different factors. A delta-gamma calculation that keeps only diagonal terms has made an additional approximation, even if it is called second order.

Zero delta is therefore a local statement, not proof of safety. A delta-neutral option portfolio may retain substantial gamma, vega, skew, basis and liquidity exposure. A hedge that neutralises the first derivative can make curvature more important as a proportion of the remaining risk. The validation question becomes whether the selected approximation remains adequate over the shocks used for its intended purpose.

Regulatory curvature is also not simply a raw second derivative. In sensitivity-based capital frameworks it is constructed from prescribed upward and downward revaluations after the corresponding linear contribution is removed, followed by rule-specific aggregation. A mathematically correct gamma does not, by itself, reproduce that capital measure.

A curve example makes the aggregation tangible

Consider the following signed rate exposures, retained from the teaching material:

Signed curve exposures and their first-order loss contributions — Curve bucket: Two years

Signed curve exposures and their first-order loss contributions — Curve bucket: Five years

Signed curve exposures and their first-order loss contributions — Curve bucket: Ten years

Signed curve exposures and their first-order loss contributions — Curve bucket: Total

The five-year point produces the largest contribution. Reporting only a parallel-shift DV01 would conceal that concentration. Reporting only the net total would conceal that every bucket is exposed in the same adverse direction.

The result remains a local approximation. It excludes convexity, changes in spreads, changes in cash-flow projections and any factor absent from the table. A precise statement is that these three specified yield movements explain approximately ₹41 lakh of loss through the stated first-order mapping.

Internal and regulatory grids need a bridge

A pricing system may use dozens of curve nodes while a regulatory calculation uses a smaller prescribed tenor grid. Let $\mathbf{x}$ be the internal factors and $\mathbf{y}$ the regulatory factors. If a small regulatory shock generates an internal shock through

$$\Delta\mathbf{x}=W\Delta\mathbf{y},$$

then local P&L consistency requires

$$\mathbf{s}_{y}=W^{\mathsf T}\mathbf{s}_{x}.$$

Here $W$ is the shock-mapping matrix, and the subscripts identify which factor space the sensitivities describe. This is the chain rule expressed as an operational mapping. The mapping is defensible when the same proposed regulatory shock produces approximately the same P&L whether calculated in the internal representation or the mapped one.

Interpolation determines the entries of $W$. A node between two regulatory tenors may allocate its exposure between them. Extrapolation at the ends needs an explicit policy. Currency changes, rate/spread decompositions and volatility surfaces require more than one-dimensional interpolation. A system that maps the final output while its stress engine shocks the internal curve differently can produce two individually plausible but mutually inconsistent answers.

The historical research also describes an equivalent directional-bump construction. Instead of first calculating every internal sensitivity, the engine applies the internal shock pattern corresponding to one regulatory factor and revalues directly. Agreement between this route and the matrix route provides a useful independent check.

A volatility cube can lose information legitimately—and dangerously

An internal swaption model can distinguish option expiry, underlying swap tenor and moneyness. A regulatory representation may retain only the first two dimensions. The historical worked example aggregates five moneyness contributions of 0, 942, 629, 255 and 118 into 1,944 before mapping the remaining expiry and tenor axes. These are source-example sensitivity units, not currency amounts inferred here.

That aggregation is meaningful only under its stated shock convention: the moneyness dimension is being moved together for the relevant calculation. It does not establish that volatility skew is economically irrelevant. The same portfolio may need a finer surface for P&L explanation, hedging and stress testing.

This distinction prevents a common mistake. Regulatory factor reduction answers how to express a prescribed calculation. Risk management asks whether the portfolio’s actual sources of loss are represented. The former cannot be used as evidence that the latter has been completed.

What a controlled mapping release should contain

A mapping change should have a dated specification, instrument coverage, interpolation and extrapolation rules, unit checks, representative revaluation comparisons and an assessment of exposures that no longer remain distinct. The change record should quantify its effect on both aggregate outputs and concentrated positions. Netting can make a total look stable while individual buckets move sharply.

The reviewer should also distinguish an implementation defect from a conscious approximation. A missing conversion factor is a defect. Collapsing a fine grid into fewer nodes may be an accepted approximation, but it needs a measurable error bound for the relevant use. New products can invalidate that bound without changing the code.

The useful result of sensitivity validation is therefore a chain of reproducible meaning: position, factor, shock convention, sensitivity, aggregation and explained P&L. Numerical precision supports that chain. It does not replace it.