Unexpected Loss of portfolio

[BharatSHarma]

Hi David,

Can you please assist me in understanding why the correct answer for below question is $2.83

A portfolio consists of two loans of $1,000. There is no correlation between loans. There are only two outcomes for each loan with equal probability is a loan loss of $8 or $12. The average loss for each position is $10 and the expected loss on the portfolio is $20. What would be the unexpected loss of the portfolio?

A)$8.
B)$10
C)$2.83
D)$0.71

Thanks in advance

 

[Sidharth]

Unexpected loss is nothing but portfolio var when average loss is not counted.

Calculating Std. Dev of Portfolio
Expected Return Average (Return) (Exp-Avg)^2 Probability (Prob*(Exp-Prob)^2)
-0.012 -0.01 0.000004 0.5 0.000002
-0.008 -0.01 0.000004 0.5 0.000002

Single Portfolio Std. Deviation: 0.002 ( Sqrt ( ..00002+.00002))
Combined portfolio Std. Dev : 0.0028284 ( Sqrt { (First port Std. Dev)^2 + (Sec port Std. Dev)^2 ) }

Portfolio value : 2000
Alpha (99%) : 2.57

Unexp Loss = 14.5381154212 (Port Value * Alpha*Std.Dev)

But I am still far away from the answer....

@David: Can you please help..if there is any other way ..??

 

[David Harper CFA FRM]

@Sidharth, your concept is solid, the question seems to want something simpler

@Bharatkant: as Sidharth suggests, the questions seems to just want the standard deviation of the portfolio loss (this is consistent with the Ong's assigned definition UL, where he defines UL as one standard deviation of the loss about the expected loss; however, a better question would clarity that UL = one standard deviation because it is not really a necessary assumption.) But, okay, one you accept that UL here means one standard deviation of the portfolio loss (around the loss, meaning the same as Sidharth says "when the average loss is not counted"), then we simply want:

StdDev [portfolio loss] = SQRT[variance(portfolio loss)]
Variance (portfolio loss) = variance(loan #1 loss) + variance (loan #2 loss) + 2*COV(), but COV()= 0,
So, Var(portfolio loss) =4 + 4 = 8, and SQRT(8) = $2.828
… because VaR(loan loss) = E(deviation^2) = 50%*(8-10)^2 + 50%*(12-10)^2 = 4
... and again, the unstated assumption here is UL = 1 SD of loss about the loss, but technically, UL = f[1 SD of loss about the loss]; e.g., can also be multiple of 1 SD

David