P&L Attribution: Why Two Models Can Agree on Direction and Still Disagree

A model can produce a plausible total risk number while valuing a simplified version of the actual portfolio. P&L attribution tests this mismatch directly. It compares what the frozen portfolio earns under the full valuation process with what the risk model can explain using its own factors and methods.

REFERENCE · NO CALL

The worked desk in this report is an explicit teaching construction. It holds a ₹300 crore equity basket, short index puts and ₹120 crore of government bonds. Its figures are retained to explain the mechanics, with institutional and product branding removed. They are not confidential position disclosures or forecasts about a real bank.

Three P&L measures, three purposes

Actual, hypothetical and risk-theoretical P&L have different purposes

The unexplained residual is

$$U_t=HPL_t-RTPL_t.$$

Here $t$ identifies the same valuation day on both sides. A negative residual on a loss day means the full valuation lost more than the risk representation explained. A positive residual means the opposite. The residual is not automatically a trading loss, a forecast error or an operational defect: it is the difference between two controlled valuation constructions.

The experiment requires matching positions, valuation dates, currencies and agreed treatment of time effects, fees and valuation adjustments. If the two calculations use different books, the residual cannot be interpreted purely as model fidelity. Correct pairing is therefore a prerequisite, not a housekeeping step after the statistics are calculated.

Correct beta only where it belongs

The equity basket has a source-example beta of 1.28 to its reference index. Its index-equivalent cash exposure is therefore

$$D_{\mathrm{cash}}=₹300\text{ crore}\times1.28=₹384\text{ crore}.$$

The short index puts already have an index delta-equivalent exposure of ₹41.50 crore. The source model incorrectly maps the cash basket one-for-one, producing total index exposure of ₹341.50 crore. Correcting only the cash mapping gives

$$D_{\mathrm{corrected}}=₹384+₹41.50=₹425.50\text{ crore}.$$

Multiplying the entire ₹341.50 crore by 1.28 would also apply the basket’s beta to the index-option exposure. That changes an exposure that did not require the basket-to-index conversion. It is not the intended correction.

For a 5% index decline, the cash-only correction is easy to isolate:

Cash beta correction leaves index-put delta unchanged

The ₹4.20 crore difference equals the ₹84 crore mapping gap multiplied by 5%. This isolates the beta effect. It does not explain every difference between HPL and RTPL because gamma, vega, skew, basis and other valuation effects remain outside this controlled comparison.

Basis remains after the beta correction

A simple basket representation is

$$r_{\mathrm{basket},t}=\beta r_{\mathrm{index},t}+\varepsilon_t,$$

where $r$ denotes a return and $\varepsilon_t$ is the basket-specific residual. Beta captures systematic co-movement with the index. The residual captures composition, idiosyncratic effects and other deviations from that linear relation.

The teaching material uses a basis residual of approximately 0.52% per day as an illustrative assumption. Multiplying by ₹300 crore gives ₹1.56 crore as a scale for the cash basket’s residual movement under that simplification. This is not an independent prediction of total unexplained P&L: its joint behaviour with other missing factors matters.

The lesson is that beta correction and basis modelling are distinct interventions. The former rescales the common factor. The latter adds a source of variation that the common factor cannot explain. Treating beta as a universal multiplier for total desk risk conflates the two.

Optionality adds separate dimensions

The short-put book has positive delta but negative gamma and vega. The teaching inputs include gamma of −₹831 crore per squared return and vega of −₹0.353 crore per volatility point. Its government-bond exposure has DV01 of ₹8.40 lakh per basis point.

A local P&L expansion can be written

$$\Delta V\approx D r+\frac12\Gamma r^2+\nu\Delta\sigma-DV01\,\Delta y,$$

In words: Approximate the value change by adding the effects of the index move, its curvature, changing volatility and changing bond yields.

where $D$ is the relevant index-equivalent exposure, $r$ the index return, $\Gamma$ gamma in the stated return units, $\nu$ vega per volatility point, $\Delta\sigma$ the volatility-point change and $\Delta y$ the yield change in basis points. The displayed DV01 is a positive loss magnitude for rising yields.

This approximation excludes any unmodelled skew dimension, nonparallel curve move, basket basis and higher-order valuation terms. The formula should therefore be used to isolate contributions, not to assert that every large-shock loss has been captured.

For a 7% decline, a ten-point volatility rise and a 15-basis-point yield rise, the source’s original mapping gives approximately −₹30.731 crore. Correcting only the cash beta changes the result to −₹36.611 crore. The incremental ₹5.880 crore again equals ₹84 crore times 7%. Because the other terms are held fixed, the difference has a clear causal interpretation.

Distributional evidence reveals a quieter model

Over the teaching case’s 250 paired observations, the reported statistics are:

Retained paired-P&L distribution diagnostics

The worst P&L day and the worst residual day need not be the same. Combining them into one event would misstate the evidence. Likewise, the residual standard deviation is not the difference between the two component standard deviations; covariance is essential.

The source also reports an 84% explained-variance diagnostic. Its meaning depends on the exact calculation used. A variance ratio based on the residual is not automatically the same as regression $R^2$, so a production report should preserve the definition rather than using those labels interchangeably.

Rank agreement does not establish magnitude agreement

Spearman correlation compares the ranks of paired HPL and RTPL observations. It tests whether the two series broadly order good and bad days similarly. A positive rescaling of every RTPL value leaves its ranks unchanged, so a model that consistently understates magnitudes can still score extremely well.

The Kolmogorov–Smirnov distance compares their empirical cumulative distributions:

$$D_{KS}=\sup_x\left|F_{HPL}(x)-F_{RTPL}(x)\right|.$$

Here $F$ is the fraction of observations no greater than $x$. The maximum gap measures distributional mismatch. It does not itself identify whether beta, skew, basis or a valuation approximation caused the difference.

The teaching case gives Spearman correlation 0.9078 and KS distance 0.0960. Under the historical thresholds used in the material, rank correlation clears the favourable threshold while the distributional distance does not, producing an intermediate overall zone. The combination illustrates why the two diagnostics are complementary.

Quantify uncertainty without turning it into a forecast

The source resamples 250 paired days 2,000 times.

Reported bootstrap intervals and zone frequencies

These are bootstrap results conditional on the observed sample and resampling scheme. They are not probabilities of next quarter’s regulatory outcome. Independent resampling also removes temporal dependence; a block bootstrap or another design may be more appropriate when observations cluster. The numerical exhibit nevertheless shows that a reported metric close to a boundary can carry substantial sampling uncertainty.

Fix the representation, then test it independently

Remediation should follow the diagnosed mechanism. Correct the basket mapping; represent residual basis where material; add the volatility-surface dimensions needed by the products; compare curve representations; and benchmark approximate valuation over the relevant shock range. Each change should predict a reduction in a particular residual signature.

It is legitimate to reconcile data conventions through an approved documented process. It is not legitimate to adjust the benchmark P&L until it agrees with the model. The purpose of the experiment is to expose economically relevant differences, not to eliminate all differences by construction.

A defensible conclusion identifies which parts of HPL are represented, which remain unexplained, how large those residuals are and what controls cover them. The aim is a faithful risk representation with known limits, not simply a more favourable correlation.