Michael Brenndoerfer | Data & AI, Private Equity, Technology

Michael Brenndoerfer

Credit Risk Fundamentals › Unexpected Loss and Credit VaR

Michael BrenndoerferDecember 21, 20254 min read

Quantitative FinanceAll chapters

Part 10 of 15Risk Management and Financial Risk Controls

Chapter 54Credit Risk Fundamentals › Unexpected Loss and Credit VaR

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While expected loss represents the mean outcome, credit risk management also requires understanding the potential for losses exceeding expectations. Expected loss tells us the average, but risk management is fundamentally concerned with tail outcomes. Unexpected loss captures the volatility around expected loss, and gives a measure of how much actual losses might deviate from the average:

\text{UL} = \sqrt{\text{Var}(\text{Loss})} = \text{EAD} \times \sqrt{\text{PD} \times (1-\text{PD}) \times \text{LGD}^2 + \text{PD} \times \text{Var}(\text{LGD})}

where:

$\text{UL}$ : Unexpected Loss (one standard deviation of the loss distribution)
$\text{Var}(\text{Loss})$ : variance of the credit loss
$\text{EAD}$ : Exposure at Default
$\text{PD}$ : Probability of Default
$\text{LGD}$ : Expected Loss Given Default
$\text{Var}(\text{LGD})$ : variance of the Loss Given Default parameter
$\text{PD} \times (1-\text{PD}) \times \text{LGD}^2$ : component of variance due to uncertainty in the default event itself
$\text{PD} \times \text{Var}(\text{LGD})$ : component of variance due to uncertainty in the recovery rate given default occurs

The formula decomposes variance into two sources. The first term captures uncertainty about whether default will occur: this is the variance of a Bernoulli random variable with probability PD, multiplied by the squared loss severity. The second term captures uncertainty about the severity of loss given that default does occur, which reflects the variability in recovery outcomes. Both sources of uncertainty contribute to unexpected loss and must be managed.

For a portfolio, we must also consider default correlation, which is the tendency for defaults to cluster during economic downturns. When the economy weakens, many borrowers experience stress simultaneously, leading to higher-than-expected default rates precisely when the portfolio is most vulnerable. The unexpected loss of a portfolio is determined by the covariance between individual exposures:

\text{UL}_{\text{portfolio}} = \sqrt{\sum_{i} \text{UL}_i^2 + \sum_{i \neq j} \rho_{ij} \text{UL}_i \text{UL}_j}

where:

$\text{UL}_{\text{portfolio}}$ : Unexpected Loss of the portfolio
$\text{UL}_i$ : Unexpected Loss of individual loan $i$
$\rho_{ij}$ : correlation of losses (or defaults) between loan $i$ and loan $j$

The second term captures the correlation effect, and its importance cannot be overstated. If correlations are zero, the portfolio UL is simply the square root of the sum of squared individual ULs, meaning diversification works perfectly and portfolio risk is much lower than the sum of individual risks. If correlations are positive, portfolio risk increases substantially because losses tend to occur together. In the extreme case of perfect correlation, there is no diversification benefit at all, and portfolio UL equals the sum of individual ULs.

In[27]:

Code

def unexpected_loss(pd, lgd, lgd_volatility, ead):
    """
    Calculate unexpected loss for a single exposure.

    Parameters:
    -----------
    pd : float
        Probability of default
    lgd : float
        Expected loss given default
    lgd_volatility : float
        Standard deviation of LGD
    ead : float
        Exposure at default

    Returns:
    --------
    float : Unexpected loss (standard deviation of loss)
    """
    variance_loss = (pd * (1 - pd) * lgd**2) + (pd * lgd_volatility**2)
    ul = ead * np.sqrt(variance_loss)
    return ul


# Calculate UL for each loan
portfolio["UL"] = portfolio.apply(
    lambda row: unexpected_loss(row["PD"], row["LGD"], 0.25, row["EAD"]), axis=1
)

Out[28]:

Console

Expected vs Unexpected Loss by Borrower
=================================================================
Company A    | EL: $     1,350 | UL: $      63,033 | UL/EL:   46.7x
Company B    | EL: $     8,100 | UL: $     218,253 | UL/EL:   26.9x
Company C    | EL: $    18,975 | UL: $     193,436 | UL/EL:   10.2x
Company D    | EL: $    55,800 | UL: $     274,721 | UL/EL:    4.9x
Company E    | EL: $     1,920 | UL: $      92,414 | UL/EL:   48.1x

The ratio of unexpected to expected loss is typically much higher for investment-grade exposures because, while their expected losses are small, there's still meaningful probability of default. For a very safe borrower with low PD, expected loss is tiny, but unexpected loss remains significant because if that borrower does default (even though unlikely), the loss would be substantial. This characteristic drives the need for economic capital well beyond expected loss provisions. Banks must hold capital against unexpected losses because expected losses are covered by pricing and provisions, while unexpected losses threaten solvency.