# Allocating capital to insurance products

A friend “volunteered” me to answer an insurance question from Aardvark on allocating economic capital across different insurance products. After writing a short response, I received the frighteningly useful message: “Error”.

Having written a brief summary of the different techniques used in this really important area, I thought I should use it as a blog post. Maybe “Daan d.” from Cape Town will stumble across this answer eventually.

The question:

What is the standard practice to allow for diversification benefits when allocating capital required between different insurance products?

My brief answer (this is a huge topic!):

There is no standard practice. It’s one of the more irritating and subjective aspects of allocating capital between imperfectly correlated product

Economic Capital doesn’t have to be calculated as VaR, but I will use VaR below as a generalisation. Banks are typically slightly more mature in their capital allocation processes so what I’m describing below is often used in the banking world, but applies equally to insurance (life and non-life / P&C).

Splitting the capital in proportion to the sum of the components is frequently used, but is flawed and usually doesn’t give good results unless speed and simplicity are primary objectives.

Incremental VaR is the increase in VaR from the current position when adding the new product. Measuring the added value from the new product using Incremental VaR leads to optimal decisions. However, it doesn’t sum to the total capital requirement and so is less useful for capital allocation for existing business.

Marginal VaR, based on the marginal risk contribution of each product, is quite often used and generally produces sensible, consistent results. The Marginal VaR for all risks sums to the total VaR and allocates diversification benefits in a single, objective measure. (It’s not necessarily “correct” since there is no single correct answer.)

You calculate Marginal VaR or Marginal Economic Capital for each by multiplying the size of the product by the rate of change of the total Economic Capital with respect to size. This requires the calculation of the first derivative of Economic Capital. This can sometimes be done analytically when using delta normal or volatility based methods, but is usually a fair amount of calculation work. It also requires a fair definition of product “size”.

Marginal VaR works well when each product is small in relation to the size of the overall firm since it is based on derivatives. When the product groups are large, the sum of Marginal VaR will differ slightly from total VaR due to the discrete approximation to the continuous case. Here I recommend rescaling the final results to ensure the sum still equals the total economic capital.

In short, there is no standard method, but much to recommend Marginal VaR in many cases, or Incremental VaR in others.

This is an incredibly important aspect to managing an insurer on a risk-reward basis, calculating risk-adjusted performance metrics and allocating capital to best create shareholder wealth. With Solvency II on the way in Europe and equivalent measures already planned by the FSB in South Africa, these questions will be asked over and over again in the coming years.