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When measuring and managing credit risk, It is important to have a clear understanding of common terms such as expected loss and unexpected loss. As stated in Credit Risk Assessment, credit risk is defined as “the potential that a bank borrower or counterparty will fail to meet its obligations in accordance with agreed terms.” This article will explain the calculations for expected loss (EL) and unexpected loss (UL), for both an individual asset and for a portfolio. Additionally, risk contribution (RC) will be defined.
To begin, let’s review the common terminology used in these calculations:
Terminology
AE = OS + (COM – OS) * (UGD)
Expected Loss (EL)
EL for a single asset is calculated by using the following formula:
EL = AE * LGD * EDF
To calculate EL for a portfolio we must add the expected losses of the individual assets; formula below:
ELP = ∑ELi
Unexpected Loss (UL)
UL for a single asset is calculated by using the following formula:
UL = AE * SQRT [(EDF * σ²LGD) + (LGD² * σ²EDF)]
To calculate the UL for a portfolio, we need to use a more complex formula. Below is the formula for a2 asset portfolio:
ULP = SQRT [ULi² + ULj² +2 (ρi,j) (ULi) (ULj)]
Note: The UL of the portfolio will be less than the sum of the ULs for the individual assets due to diversification benefits.
Risk Contribution
Each asset in a portfolio contributes a portion of its UL to the portfolio UL; known as risk contribution (RC). In other words, it is the incremental risk that the exposure of a single asset contributes to the total risk of the portfolio. The formula for calculating the RC of asset i and j for a two asset portfolio is illustrated below:
RCi = ULi * [ULi + (ρi,j) * (ULj) / ULp]
RCj = ULj * [ULj + (ρi,j) * (ULi) / ULp]
If we take the sum of RCi + RCj we will get the same value as the unexpected loss on the portfolio.
Correlation (ρ) plays an important role in measuring the potential UL in the portfolio. If correlation between the assets increases, the potential UL will increase.
To begin, let’s review the common terminology used in these calculations:
Terminology
AE = OS + (COM – OS) * (UGD)
Expected Loss (EL)
EL for a single asset is calculated by using the following formula:
EL = AE * LGD * EDF
To calculate EL for a portfolio we must add the expected losses of the individual assets; formula below:
ELP = ∑ELi
Unexpected Loss (UL)
UL for a single asset is calculated by using the following formula:
UL = AE * SQRT [(EDF * σ²LGD) + (LGD² * σ²EDF)]
To calculate the UL for a portfolio, we need to use a more complex formula. Below is the formula for a2 asset portfolio:
ULP = SQRT [ULi² + ULj² +2 (ρi,j) (ULi) (ULj)]
Note: The UL of the portfolio will be less than the sum of the ULs for the individual assets due to diversification benefits.
EL = PD * LGD * EAD
EAD is exposure at default, which is generally the exposure given to a corporate
PD is probability of default
LGD =1 - recovery rate