The following table shows the value-at-risk of our trading units in 2003 and 2002. The minimum and maximum value-at-risk amounts show the bands within which the values fluctuated during the periods specified. We calculate the value-at-risk with a holding period of one day and a confidence level of 99%. “Diversification effect” refers to the effect that the total value-at-risk on a given day is lower than the sum of the values-at-risk relating to the individual risk factors. Simply adding the value-at-risk figures of the individual risk classes to arrive at an aggregate value-at-risk would imply the assumption that the losses in all risk categories occur simultaneously.
| Total | Diversification | Interest Rate | Equity Price | Foreign | Commodity | |||||||
| Effect | Risk | Risk | Exchange Risk | Price Risk | ||||||||
| in € m. | 2003 | 2002 | 2003 | 2002 | 2003 | 2002 | 2003 | 2002 | 2003 | 2002 | 2003 | 2002 |
| Average | 48.44 | 42.38 | 33.48 | 30.9 | 45.86 | 35.63 | 21.94 | 24.28 | 7.71 | 8.02 | 6.41 | 5.35 |
| Maximum | 72.13 | 88.86 | 57.35 | 47.54 | 64.07 | 58.48 | 37.01 | 89.26 | 17.48 | 29.25 | 16.7 | 8.66 |
| Minimum | 32.27 | 29.36 | 21.89 | 21.17 | 27.62 | 24.67 | 12.97 | 13.43 | 3.17 | 2.64 | 3.33 | 2.28 |
| Year-end | 60.01 | 32.94 | 33.84 | 22.5 | 52.64 | 29.12 | 27.28 | 13.75 | 6.82 | 6.84 | 7.11 | 5.73 |
The following graphs show the daily aggregate value-at-risk of our trading units in 2003 and 2002, including diversification effects.
Our trading value-at-risk increased over the year from an average of € 37.3 million in the first quarter of 2003 to € 62.6 million in the fourth quarter of 2003. Having remained below € 55.0 million in the first half of the year, our trading value-at-risk rose above this level in the third quarter of 2003 as correlations changed between equity markets and interest rate markets and both Global Markets and Global Equities Business Divisions increased positions.
The following histograms show the distribution of actual daily income of our trading units in 2003 and 2002. The histograms indicate, for each year, the number of trading days on which we reached each level of trading income shown on the horizontal axis in millions of euro. The trading units achieved a positive income for over 96% of the trading days in both 2003 and 2002. On no trading day in either year did they incur an actual loss that exceeded the value-at-risk estimate for that day.
The comparison of the distribution of our trading units' actual daily income with the average value-at-risk enables us to ascertain the reasonableness of our value-at-risk estimate. The histogram for 2003 shows that the actual distribution of our trading units' income produces a 99th percentile of only € 45.7 million below the average daily income level of € 36.6 million, which is less than the average value-at-risk estimate of € 48.4 million.
The value-at-risk and actual incomes of the trading units throughout the year are shown in the following graph.
There was no hypothetical buy-and-hold loss that exceeded our value-at-risk estimate for the trading units as a whole in 2003 and one hypothetical loss exceeding the value-at-risk in 2002. This is below the expected two to three outliers a year that a 99% confidence level value-at-risk model ought to predict, showing that our risk estimates are slightly conservative.