Capital Structure in Banks

LOS 1. Identify and describe important factors used to calculate economic capital for credit risk: probability of default, exposure, and loss rate.

The Expected Loss (EL) is defined as the amount a bank expects to lose on average over a certain period of time on it’s loan book. These losses are treated as normal and foreseeable part of a lending business and a prudent bank sets aside reserves to cover these losses. Expected loss is not conditional or based on current state of the economic cycle but rather a long run average loss level over a range of economic conditions.

We need three important ingredients to calculate EL:

1. Probability of Default (PD): This is the borrower specific probability or likelihood that a borrower will default before the end of a predetermined period of time (horizon of 1 year) or prior to the maturity of the loan. It is linked to borrower’s risk rating and it can change with time.
2. Exposure Amount (EA) / Exposure At Default (EAD): Dollar amount of exposure to a customer / credit counterparty at time of default. This includes all outstanding payments (including interest) at that time, which can be very different from the amount at the initiation of the credit (more true for derivatives like swaps). It can also be stated in % of notional amount.
3. Loss Rate (LR): The fraction of the exposure amount that is lost in case of default i.e. the amount that is not recovered after collateral is sold. $LR=1−RR$, where, $RR$ is the fractional recovery rate.

LOS 2. Estimate the variance of default probability assuming a binomial distribution.

We define a random variable $L$ (for loss) that depends on the realization of a Bernoulli variable $D$, which equals 1 (if default) and 0 (otherwise). Note that the expected value of this random variable is $E(D) = PD$ and it’s variance is $\mbox{var}(D) = PD\times (1-PD)$.

LOS 3. Define and calculate expected loss (EL).

The Loss Rate $(LR)$ is a random variable (since we do not know what fraction of $EA$ we will end up losing upon default). We assume it’s average or expected value is ($\overline{LR}$). As an assumption, we assume that occurrence of default ($D$) and severity of losses ($LR$) are independent. The expected loss $E(L)$ will be: $$EL = E(L)=EA \cdot PD \cdot \overline{LR}$$

LOS 4. Define and calculate unexpected loss (UL).

Risk arises when losses vary (and are more than expected), which we call unexpected loss (UL). Unexpected losses occur from unexpected defaults (default risk) and/or unexpected credit migrations (migration risk). These losses require a buffer of loss-absorbing capital.

Assuming probability of default ($PD$) is independent of exposure ($EA$) and loss rate ($LR$), and exposure ($EA$) and loss rate ($LR$) are independent, we have: $$UL=EA\cdot \sqrt{(PD\cdot \sigma_{LR}^2+ (\overline{LR})^2 \cdot \sigma_D^2 )}$$ Note that variance of loss rate ($\sigma_{LR}^2$) is difficult to estimate owing to lack of historical data. We resort to a parametric distribution of $LR$, which can be binomial, uniform or normal.

LOS 5. Calculate UL for a portfolio and the risk contribution of each asset.

For our portfolio, the expected loss ($EL_P$) is given by: $$EL_P=\sum_{i=1}^n EL_i = \sum_{i=1}^n EA_i \cdot PD_i \cdot LR_i$$ The unexpected loss for portfolio or standard deviation of portfolio unexpected losses ($UL_P$) is: $$UL_P=\sqrt{\sum_{i=1}^n \sum_{j=1}^n \rho_{ij} UL_i UL_j }$$ where $\rho_{ij}$ are the pairwise default correlations. The unexpected loss marginal contribution of $i$’th credit ($ULMC_i$) is given by: $$ ULMC_i = \frac{\sum_{j=1}^n UL_j \rho_{ij} }{UL_P} $$ We define $i$’th credit’s contribution to portfolio unexpected loss as: $$ ULC_i=ULMC_i \cdot UL_i $$ These contributions satisfy the property that: $$ UL_P=\sum_{i=1}^n ULC_i = \sum_{i=1}^n ULMC_i\cdot UL_i $$ The formulas for a two credit portfolio are simply: $$ ULMC_1=\frac{UL_1+\rho_{12} UL_2}{UL_P } \\ ULMC_2=\frac{UL_2+\rho_{12} UL_1}{UL_P } \\ ULC_1=ULMC_1 \cdot UL_1; ULC_2=ULMC_2 \cdot UL_2 $$

LOS 6. Explain how the credit loss distribution is modeled.

Credit loss distributions are highly skewed since the upward potential is limited to receiving at most the promised payments and only in rare events to lose a lot of money. Therefore, the normal distribution is not a good choice for modeling these distributions.

Beta distribution is used for modeling credit loss distribution. This distribution can can be characterized by two parameters. It can accommodate a wide variety of shapes (symmetric, positively skewed, negatively skewed) by proper choice of it’s parameters. However, it may become difficult to fit the beta distribution to the tail of a distribution, in which case a Monte Carlo technique can be used.

LOS 7. Evaluate a bank’s economic capital relative to its level of credit risk. Describe how economic capital is derived.

Economic capital is defined as the distance on the loss distribution between expected losses and the unexpected losses (picked at a certain loss confidence level). $$ \begin{align} \mbox{EconomicCapital}&=\mbox{ExtremeUnexpectedLoss} − \mbox{ExpectedLoss}\\ &=L^* – EL_P \end{align} $$ where, $$ L^∗=EL_P+CM \cdot UL_P $$ and $CM$ is picked from the loss distribution such that the standardized loss satisfies: $$ \Pr⁡\left(\frac{L−EL_P}{UL_P }\leq CM\right)=c $$ So, economic capital becomes $L^* – EL_P = EL_P + CM \cdot UL_P – EL_P = CM \cdot UL_P$. This economic capital can also be calculated at the individual transaction level as: $$ \mbox{Economic Capital}_i = ULC_i \cdot CM $$

LOS 8. Describe challenges to quantifying credit risk.

1. [Liquidity of credit assets]: This approach assumes that credits are illiquid assets and hence measures the risk contribution to losses of existing portfolio. If they were more liquid and traded in capital markets, a value approach would be more suitable. We would then model the expected return and value of promised payments and probability distribution of changes in these values.
2. [Extend to multi-period]:In practice, credit risk models use a single period (1-year). They should ideally involve expected and unexpected changes in credit quality of borrowers over several years, which is a difficult task.
3. [Silo-ed risk approach]:We have assumed that other risk components (market, operational) are separated and measured and managed by other bank’s departments.