P1.T4.24. Expected loss (M. Ong)

[David Harper CFA FRM]

AIMs: Define, calculate and interpret the expected loss for an individual credit instrument. Explain how a credit downgrade or loan default affects the return of a loan.

Questions:

24.1. A bank has a $10 million commitment (COM) of which $6 million is outstanding (OS) and the usage given default (UGD) assumption is 50.0%. The probability of default (PD) is 1.0% and the loss conditional on default (LGD) has a beta distribution with a mean of 70.0% and a standard deviation of 25.0%. The PD and LGD are not independent; they are positively correlated. What is the expected loss (EL) of the adjusted exposure (AE)?

a. Less than $56,000
b. $56,000
c. More than $56,000
d. $112,000

24.2. Assume a bank sets the contractually promised gross return on a loan (k) according to the following formula which equates the expected (net) return on the loan equal to the risk-free rate of return, per a risk-neutral assumption: PD*RR + (1-PD)*(1+k) = 1+Rf, where PD=probability of default, RR = recovery rate, k = promised gross return, and Rf = risk-free rate. If the risk-free rate (Rf) is 3.0%, the probability of default (PD) is 4.0%, and the loss given default (LGD) is 75.0%, what is the risk-neutral promised gross return on the loan (k)?

a. 3.00%
b. 3.75%
c. 5.75%
d. 6.25%

24.3. A bank has a portfolio of two lines of credit extended to a corporate customer (commitments, COM). The $12.0 million commitment has a 3.0% probability of default (EDF) and the $18.0 million commitment has a 5.0% EDF. Both commitments are already 20% drawn (outstanding, OS) with 80% unused. For both lines, the bank's usage given default (UGD) assumption is 60.0% and the recovery rate assumption is 35.0%. Finally, since the lines are with the same customer, their default correlation is high at 0.70. What is the expected loss (EL) of the portfolio that contains both exposures?

a. $299,880
b. $397,800
c. $556,920
d. $819,000
Answers:

• Here in forum

 

[ggaoshen]

Hello David,

I have a question regarding 24.1. Can you please explain how the positive correlation between PD and LGD affects the expected loss (56k = expected loss given PD and LGD are independent)?

Thank you!

 

[David Harper CFA FRM]

Hi ggaoshen, expected loss (EL) in this context is not the portfolio EL that is the sum of individual ELs. The sum of ELs is unaffected by correlation, as E(X+Y) = E(X)+(Y).
However, this EL is a product and the product of means is influenced by correlation: EL = PD*LGD, and E(XY) = E(X)*E(Y) only if they are independent.
See http://en.wikipedia.org/wiki/Expected_value#Non-multiplicativity
This is useful: Cov[X,Y] = E[XY] - E[X]*E[Y]. In this case, cov(pd, lgd) = E[pd*lgd] = e[pd]*e[lgd], such that E[pd*lgd] = e[pd]*e[lgd] + cov(.)

I'm a big fan of memorizing Cov[X,Y] = E[XY] - E[X]*E[Y] because it gives you also the highly useful: Cov[X,X] = VaR[X] = E[X^2] - E[X]^2

 

[ggaoshen]

That's very helpful. Thank you, David!