Ch6 Portfolio Effects: Risk Contribution & Unexpected Loss

1. Expected & Unexpected Loss
1-1. EL (Expected Loss): an average loss in value from a risky asset over a specified time horizon.
EL = V1 x LGD x EDF
where:
LGD = loss given default
EDF = expected default frequency
V1 = terminal value of the asset at the end of the period = AE (adjusted exposure)

- EL = sum of the expected losses of the individual assets
- only risky portion of V1 is subject to LGD



1-2. UL (Unexpected Loss): the volatility or standard deviation of loss, whereas expected loss represents the average loss over the same fixed horizon.

- UL of the portfolio = less than a simple sum of the individual asset ULs => diversification benefits

1-3. Recovery Rate, Credit Quality & EDF
- the higher the recovery rate, the lower is the percentage loss for both EL and UL
- EL increases linearly with decreasing credit quality
- UL increases much faster than EL with increasing EDF

**simplified by the binary assumption of the default process

2. Maturity Effects
i. The credit migration can contribute to the volatility of EL & UL
ii. The longer the time to maturity, the greater the variation in asset value due to changes in credit quality
iii. Two-state default process paradigm used by most banks:
- Adjusting a risky asset's internal credit class rating in accordance with its term to maturity
- Use of several analysis horizons: many banks have opted to use several sequential horizons longer than one year. Depending on the confidence they place in their measurements, the most conservative results are chosen for planning

3. Portfolio Expected & Unexpected Loss

3-1. Video Clip: http://www.youtube.com/watch?v=7ykulBzKqFw&feature=related

e.g.) Exposure #1: AE = $1,000, EDF = 1.00%, Std. Dev. of EDF = 9.95%, LGD = 20%, Std. Dev. of LGD = 10%
Exposure #2: AE = $500, EDF = 2.00%, Std. Dev. of EDF = 14.00%, LGD = 50%, Std. Dev. of LGD = 30%
Default correlation = 5.00%
=> Expected Loss #1 = 1,000 x 20% x 1.00% = $2.00
Expected Loss #2 = 500 x 50% x 2.00% = $5.00
Unexpected Loss #1 = 1,000 x (1.00% x 10% ^ 2 + 20% ^ 2 x 9.95% ^ 2) ^ 0.5= $22.27
Unexpected Loss #2 = 500 x (2.00% x 30% ^ 2 + 50% ^ 2 x 14.00% ^ 2) ^ 0.5 = $40.93
Risk Contribution, Exposure #1 = 22.27 x (22.27 x 40.93 x 0.05) / 47.60 = $36.20
Risk Contribution, Exposure #2 = 40.93 x (40.93 x 22.27 x 0.05) /47.60 = $11.40
Sum of Risk Contribution = 36.2 + 11.4 = $47.60
Sum of UL + 22.27 + 40.93 = $63.20

3-2. Diversifiable & Undiversifiable Risk
ULP = Sum of Risk Contributions

***AE = OS + (COM - OS) x UGD
where:
AE = adjusted exposure on default
COM = commitment
OS = outstanding
UGD = unused draw-down on default