I agree with sherbeer. I wanted to try and point to a specific formula in the reading that covers this, because I feel like there was something about total loss = expected loss + unexpected loss, but I can’t seem to find it.
What you could try doing is thinking of the problem as a frequency x severity problem, where the frequency is a Bernoulli random variable with parameter p = 0.10 and 0.20 for the rogue trader and the model risk, respectively. Under that consideration, you might be able to calculate the unexpected loss via the (EDF * Var(LGD) + LGD^2 * Var(EDF)) ^ (1/2), but I don’t know how the 95% level gets in there, so it might not work.
Here’s maybe another way to think about it:
For the rogue trader, 90% of the time, there will be no loss. So VAR(X) for X < 90% would be 0. For the other 10% of the time, the losses range from $0M to $100M. The 95% level would put you halfway between $0m and $100m (b/c of uniform distribution), which would be $50m for TOTAL losses. Since Total Loss = EL + UL, we get that 50 = (.10 * 50) + UL –> UL = $45m. [assuming frequency and severity are independent]
For the model risk, we look at it the same way: 80% of the time, there is no loss. The other 20% of the time, losses are normally distributed with mean $25M and sd $5M. Thus, to get to the 95% level of total losses, we’re looking at the value of the loss (x_hat) such that 75% (95% = 80% + (75%) * (20%)) of the losses fall below x_hat. So, we’re looking for the z-score that gives us F(z) = 0.75. The normal distribution table tells me it’s between 0.67 and 0.68 (will assume 0.675). So, the total loss level we’re looking at is $25M + (0.675 * $5M) = $28.375M. Again, subtract the expected loss from this total losses to get the unexpected loss: UL = $28.375M - (.20 * $25M) = $23.375M.
This is less than the UL for the rogue trader ($23.375m < $45m), so the max UL at the 95% level is grater for the rogue trader than for the model risk.