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The Actuary The magazine of the Institute & Faculty of Actuaries
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Protection money in Euroland

The level of funding given to liabilities can critically affect solvency, as well as the price to be charged for accepting those liabilities. If a portfolio expects to incur losses of £1m, and only £1m is available to meet those liabilities, then the next time a large storm/air accident occurs, the company could well find itself filing for bankruptcy. An extra margin, which often comes in the form of shareholder funds, is usually held to cover worse-than-expected experience. The dividend profit from insurance business is compensation for bearing this risk.

How much capital is required?
There is no definite answer to this question here. Several approaches are employed in practice, from a practical consideration of capital to highly technical probability simulations. One common method is to allocate capital which will exceed the losses with a certain probability, say 95%. An alternative method for calculating capital requirements above expected losses is to allocate a proportion of the standard deviation of losses.
When transacting overseas business there is the additional exposure to the risk that an adverse currency movement might cause losses to grow in size. For example, a year can see a 20% change in the value of two currencies against each other, and hence in value of currency A liabilities denominated in currency B, eg sterling and the euro in 1999.
The effect can be minimised by currency matching, although some portfolio managers might seek to mismatch in order to try to generate higher investment returns. But the risk cannot be removed entirely a simultaneous movement in the ultimate expected losses and the exchange rate can make cross-border funding inevitable.

Risk-based capital for European business
The debates in the UK over the last decade (indeed over the last 40 years) show that UK entry into the euro is subject to great uncertainty. Some of the possibilities are:
– No attempt at entry.
– Certain to enter at a defined time.
– Dependent on the result of a referendum held at some point in future, subject to great uncertainty.
– Probably enter at a defined time.
– No entry, but an attempt is made over an initial period to stabilise the euro against sterling.
Each gives rise to different probability distributions of the two currencies relative to each other. For example, the first possibility might give rise to fluctuations around a base rate, where the standard deviation increases in time.
The sterling denominated losses of a contract incurring euro losses are given by the expression:
Losses in sterling = losses in euro x exchange rate £/euro.
We can put distributions for each of the above possibilities into the expression to convert losses from one currency to another. The loss distribution itself can be modelled to give a probability distribution for the sterling denominated losses, which then allows the sterling capital requirements to be calculated. The hedging strategy used by the company is to set aside a sterling sum equal to the expected value of losses at the start of the contract. No attempt is made at ongoing hedging.

Example
Since capital requirements differ for every contract, the effect of currency movements on the sterling converted losses will also differ. So I look at a single contract with a certain loss structure: they are assumed to be uniformly spread between e1,000 and e100,000. The loss settlement date is one of the variables, and in the tables is set at the start of 2003, the end of 2006, the start of 2007, and the start of 2010.
As for the currencies, the euro fluctuates around a central value of £0.61, which it is assumed would be the entry value for sterling into the euro. The fluctuations occur under a normal distribution, which introduces an asymmetry between the euro/£ exchange rate and the £/euro rate the reciprocal of a normal distribution is not normal itself. The normal distribution serves only as a model with reasonable theoretical properties (like not giving a negative exchange rate when moderate variance is used), so other distributions could equally be used. For each of the possibilities for exchange rate paths listed above, an appropriate variance is assumed for the distribution:
1 Variance is proportional to time since start of the contract on 1 January 2000.
2 Variance is a quadratic function of time, equal to zero at the start of 2000 and 2010.
3 Variance is equal to a combination of 1 and 2 until the date of a referendum 2007. Subsequent to the referendum, the uncertainty is greater.
4 Variance is equal to that in 2 plus a small proportion of that in 1.
5 Variance is equal to that in 2 until 2007, and is subsequently a multiple (greater than 1) of that in 1.
With these assumptions, we get results shown in figure 1 across (all figures are in sterling).
For the standard deviation table, the risk-based capital actually held depends on the constant multiplier used, which varies according to the risk appetite of the capital provider; the famous market price of risk theory can act as a guide here. The RBC at 95% table indicates the actual level of funding which could be advanced. For this capital allocation scheme, the differences between the various possibilities are more marked, and demonstrate the effect of the volatility which could follow from a failed attempt at entry.
For the reasons outlined, there is merit in looking at the ratios between the different approaches. Table 2 shows how capital charges compare as percentages of the ‘no entry’ value, at any given time.
So, looking down the standard deviation table, in the column ‘end 2006’ we can see that certain entry would entail a 3% lower capital charge than not entering the euro, for insurance with loss settlement date at the end of 2006. Similarly, the RBC at 95% table in the column ‘start 2007’ shows that a failed entry can necessitate a capital charge 15% higher than when no attempt at entry is made, where the contract has a settlement date at the beginning of the year.
The corresponding charges for coverage are shown in table 3. The prices are subject to a number of further assumptions concerning the discount rate, the investment rate, and so on, which are detailed in the footnote.
So, here we can see that, for a contract with loss settlement at the start of 2010, a probable sterling entry into the euro would reduce prices by between 1% and 3% compared with no attempted entry. For a contract with loss settlement at the start of 2007, an entry contingent on an uncertain electoral result would entail about 1% to 2% lower prices than an entry attempt certain to be rejected in the election.
The approach outlined above is single-mindedly microeconomic, and ignores the significant political and macroeconomic effects. The results are subject to the assumptions made in the spreadsheet model which supports the calculations. Other contracts would have different loss profiles, leading to alternative cost reductions. Nevertheless, the methodology remains robust in such circumstances.

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