UL and EL

[Imad]

Hi David,
Below is an example taken from Qbank. Can you please explain the outcome?
Thanks
Imad
---------------------------------------------------------------------
Assume a portfolio consists of two loans of $1,000 with a correlation between loans of 0. Also, assume the only two outcomes for each loan with equal probability are a loan loss of $8 or $12. Note that the average loss for each position is $10 and the expected loss on the portfolio is $20. Find ULp, the unexpected loss of the portfolio.
A)
$0.71.
B)
$2.83.
C)
$8.00.
D)
$10.00.

The correct answer was B) $2.83.

An interpretation of unexpected loss is that it is the standard deviation of the expected loss. There are two loans so 25% of the time the value will be 8 + 8 = 16, 50% of the time the value will be 8 + 12 = 20, and 25% of the time the value will be 12 + 12 = 24.
E(Li) = $10, E(Lp) = $20,
ULp = ((0.25)(16 − 20)2 + (0.5)(20 − 20)2 + (0.25)(24 − 20)2)0.5 = $2.828.

 

[David Harper CFA FRM]

Hi Imad,

It's a useful question. The answer is just finding the standard deviation of a random variable with three outcomes
... see http://en.wikipedia.org/wiki/Standard_deviation#Discrete_random_variable
what can be confusing is: often we are computing a standard deviation (volatility) of an (historical) sample. But here we are given (ex ante) probabilities, so the (exact) population distribution is specified for us. Put another way, here we are "unrealistically" given the population's distribution versus our more realistic need to observe a sample, so we use the square root of average differences squared, rather than divide by (n-1).

So, it's the standard deviation where mean = 20, and: Loss (L) = 16 with p = 25%, L = 20 with p = 50%, L = 24 with p = 25%.
... UL is the standard deviation under Ong's (credit risk) definition.

fwiw, we get the same answer if we use the recovered values. Where mean portfolio recovery = 980:
SQRT[(984 - 980)^2*25% + (980-980)^2*50% + (976-980)^2*25%] = 2.828

Hope that explains,