Suppose that the economic capital estimate for two business units are as follows: Type of risk                  Business Unit1                             Business Unit 2 Market risk                       10                                                        50 Credit risk                        30                                                        30 Operational risk                50                                                       10  The correlation between market risk and credit risk in the same business unit is 0

Answer:

The Total Economic Capital is

=the square root of 10²+50²+30²+30²+50²+10²+2*10*30*0.3+2*50*30*0.3+2*30*30*0.7+2*10*50*0.2

=

=

=97.673

When Business Unit 1 is increased by 1%, i.e the market risk, capital risk and operational risk would be 10.1; 30.3; 50.5 respectively, the total economic capital increases by 0.452 to 98.125. The amount of capital that should be allocated to Business Unit 1 is therefore 0.452/0.01=45.2

When Business Unit 2 is increased by 1% i.e the market risk, capital risk and operational risk would be 50.5; 30.3; 10.1 respectively, the total economic capital increases by 0.526 to 98.199. The amount of capital that should be allocated to Business Unit 2 is therefore 0.526/0.01=52.6

Note that the total of the economic capital allocated is nearly exactly equal to the original economic capital. If the partial derivatives had been calculated exactly we would have exact equality, as shown by Euler's theorem