**Solvency II will require an annual profit and loss attribution. Radu Popescu explores whether insurers can learn from investment bankers, who faced a similar requirement**

The profit and loss attribution (P&LA) is a crucial component of the control function in investment banking. It serves as an analysis of the booked profit and loss (P&L) and its variance, giving assurance that the material risk factors have been identified, and hence the exposure to them is known and understood. It has a strong link with the main market risk model, VaR, explained below.

The P&LA is performed daily in investment banking. Let us consider how this is done for a homogenous book of trades that are marked to market on a daily basis. The first step is to identify all material risk factors, these being either variables or parameters.

Changes in the variables are readily available from the market, either directly from stock prices, or implied from other prices, such as volatilities from option prices. Parameters will not change daily, and their values will be subjected to calibrations that will have an element of judgment involved. Examples of such parameters are correlations or parameters of a stochastic volatility model. The risk department will produce the Greeks, or sensitivities (usually first-order and sometimes second-order derivatives) to changes in risk factors, for all variables and for some of the parameters.

We ignore P&L that is not market driven, like new trades or trade lifecycle events such as a change in the notional of the trade or a change in coupons and fees.

There are two ways to do the P&L attribution: full revaluation or risk-based calculation.

Full revaluation attribution is done as follows. The first risk factor is changed from the initial value to the next-day value, and the change in value of each trade is attributed to the first risk factor, with all the other risk factors having the initial value. Then the same is done for the second risk factor and so on. The sum of all these contributions is the explained P&L.

A variant of this full revaluation method is to change the first risk factor, then move to the second one without putting back the value of the first risk factor to the initial value, and so on. The resulting explained P&L will equal the P&L, ie everything is explained. The problem with this method is that the attribution depends on the order in which the risk factors are considered and it is hard to justify the ordering in the first place, ie consider an option on a basket of 50 names.

The risk-based approach is an approximation of the full revaluation. The Greeks are available from risk and the P&L attributed to a risk factor RF is then

?V/?RF * ?RF+ ?2V/?RF2 * ?RF2,

assuming an order two approximation. This ties well with the calculation of VaR. For each scenario - such as a complete set of changes in the risk factors - the change in value of the portfolio is calculated. A one-year historical VaR calculation will involve the calculation of 250 such scenarios.

For the purpose of P&LA, the scenario used for calculation is for the actual change. Comparing the P&L attribution and the distribution of P&L calculated for VaR purposes will give confidence that the material risk factors have been considered and that the approximation used for VaR is correct (within the confidence interval expected). This is part of the backtesting analysis of the VaR model.

Which Greeks are used in this approximation depends on the trades. The basic ones are Delta, Gamma, Vega, Theta and Rho, while second-order cross derivatives account for the cross effect between different risk factors (for instance, Vanna, ?2V/?s?S, which accounts for the cross effect between the spot price S and the implied volatility s). A special Greek is ?, which accounts for the time value decay, and the change in risk factor is known. Note that VaR can also be calculated using a full revaluation method, but computation can be expensive because of the number of scenarios to be considered.

Both methods including the two variants of full revaluation method may be used at the same time to try to isolate some cross effects. The full revaluation method will combine into one risk factor all the contribution from Greeks associated with the same risk factor, but not any cross factor ones.

As a final remark, calculating reliable risk factors for complex trades may be difficult. If this is done in Monte Carlo, the workhorse of numerical calculations, the precision is lower than on the calculation of the valuation itself.** **

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**Link with model risk**

If successful, the unexplained P&L should vary around zero. Because of the numbers involved being both positive and negative, one should be careful about statements such as '90% of the P&L is explained'. Limits can be decided for the unexplained P&L out of which some action to identify more risk factors should be started. If trades are not marked to market but are marked to model, which is the case for complex trades, the unexplained P&L could be used for model risk management purposes. Netting rules should be applied carefully, so that one does not add across model families.

**What about insurers?**

Some differences appear right away with the insurance industry P&LA target. The one-year time horizon in insurance compared with the one-day time horizon in banking is the first big difference, because Taylor approximations are usually reliable over short time intervals.

Also, one has to decide if this should be done with the initial or final market data as the anchor point, ie should you bump up the risk factors from the initial to the final state, or drop them down. This could yield very different results, as Taylor approximations could look very different, especially if market conditions have changed.

The presence of both insurance and market risk factors is also an interesting feature. Structured products combining both could be a challenge, but may deserve special attention if they are designed to implement a specific investment strategy and are hence essential from a risk management point of view. Path- dependent products may require a refinement of the P&LA to the relevant time step and then adding up attributions to obtain the final result. Non-market-driven P&L will play an important role over such a long period of time, when an important number of trade cycle events are bound to happen.

Lastly, if done on simulations, the large number of points required to capture a 99.5% confidence interval will be at odds with the number of risk factors. A daily time step to capture path dependency, and hence some shortcuts, may be required.

Overall, the insurance industry will have to develop solutions that are robust, both in terms of calculations and management.

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*Radu Popescu is a senior manager at Mazars*