Single-Exposure Credit Loss Calculation Workshop

Why Single-Exposure Calculations Need Consistent Inputs

A single credit position can have several amounts that are easy to confuse: the balance currently drawn, the amount expected to be outstanding at default, and the amount ultimately recovered. A sound expected-loss calculation keeps these quantities separate, aligns them to the same time horizon, and applies recovery only to the exposure that is actually at risk.

This workshop combines default probability, exposure at default, and loss given default for one hypothetical position at a time. It does not combine separate positions or estimate portfolio capital. The calculations use a stated horizon and, unless noted otherwise, an undiscounted monetary loss basis.

Definition: For a specified credit position and time horizon, probability of default (PD) is the probability that the obligor defaults within that horizon. Exposure at default (EAD) is the amount owed or otherwise exposed when default occurs. Recovery is the net amount recovered after default, and loss given default (LGD) is the loss conditional on default. When recovery is a fraction of EAD, the LGD rate is one minus the recovery rate. Expected loss is the probability-weighted average credit loss over the stated horizon.

For a term loan, EAD may include principal expected to remain outstanding and accrued amounts included under the model’s exposure definition. For a revolving facility, EAD may also include expected future draws on the unused commitment. A credit conversion factor (CCF) expresses the modeled proportion of undrawn commitment expected to be drawn by default. The CCF is applied to the undrawn amount, not to the amount already drawn.

Recovery and LGD must use compatible bases. If the recovery rate is a fraction of EAD, then the LGD rate is $1-\text{recovery rate}$, and the loss amount is EAD multiplied by that LGD rate. If recovery is instead supplied as a dollar amount, subtract net recovery dollars from EAD. Net recovery should reflect any recovery costs included in the model. Do not subtract those costs again if they have already been deducted from the stated recovery.

Expected Loss for a Single Position

Start with a one-horizon model in which default either occurs or does not occur. Suppose EAD is fixed, and the loss conditional on default is a fixed amount $L_D$. Credit loss $X$ is zero if there is no default and $L_D$ if default occurs. Thus, $X$ has a two-outcome distribution: $P(X=L_D)=\text{PD}$ and $P(X=0)=1-\text{PD}$.

E[X]=\text{PD}\times L_D =\text{PD}\times\text{EAD}\times\text{LGD rate}.

The formula follows directly from the definition of an expectation. With probability $1-\text{PD}$, the modeled credit loss is zero; with probability PD, the loss is $L_D$. Weighting each outcome by its probability gives $E[X]=(1-\text{PD})\times 0+\text{PD}\times L_D$. If the LGD rate is fixed, $L_D=\text{EAD}\times\text{LGD rate}$, which gives the second expression.

This calculation assumes the specified EAD and LGD rate are fixed for the position’s default outcome and that PD refers to the same horizon. It also assumes the monetary amounts share the same valuation basis. If EAD or severity varies by default outcome, use the conditional expected loss given default rather than multiplying incompatible averages: $E[X]=\text{PD}\times E[X\mid\text{default}]$. If present value is required, discount expected losses at the stated rate and timing; the undiscounted formula alone does not perform that adjustment.

Worked Example: A Term Loan With a Proportional Recovery

A hypothetical lender evaluates a one-year, non-amortizing loan. The principal expected to remain outstanding at default is $1,200,000, and $18,000 of accrued interest is included in exposure. The one-year PD is 1.8%. If default occurs, net recovery is expected to equal 35% of EAD. Calculate EAD, LGD, and expected loss. Use no discounting, so the effective annual interest rate for this calculation is $i=0$. Mortality is not modeled. Assume the default/no-default loss distribution has fixed EAD and recovery rate.

Method and assumptions. The exposure includes the stated principal and accrued interest. The stated recovery rate is a fraction of that total EAD, after any recovery costs. The model’s credit loss is zero without default and EAD less recovery upon default.

Step 1: Calculate EAD. Add the principal and included accrued interest:

\text{EAD}=\$1{,}200{,}000+\$18{,}000=\$1{,}218{,}000.

Step 2: Calculate LGD. The recovery rate is 35%, so the LGD rate is $1-0.35=0.65$, or 65%. The dollar recovery and dollar loss conditional on default are:

\text{Recovery}=0.35\times\$1{,}218{,}000=\$426{,}300, \qquad L_D=0.65\times\$1{,}218{,}000=\$791{,}700.

Step 3: Calculate expected loss. The one-year default probability is 0.018:

E[X]=0.018\times\$791{,}700=\$14{,}250.60.

Check. The loss outcomes are $0 with probability 0.982 and $791,700 with probability 0.018. The probabilities sum to 1, and their weighted loss is $14,250.60. The result is reported to the cent.

Common mistake avoided: Applying the 35% recovery rate directly to the exposure as if it were LGD would produce a $426,300 loss conditional on default. That amount is the recovery, not the loss. The correct LGD rate is 65%.

Revolving Exposure and Net Recovery

For a revolving facility, the drawn balance at the measurement date may understate the amount exposed at default. If the model applies a CCF to undrawn commitment, calculate the modeled additional draw first, then add it to the drawn balance. After calculating EAD, apply the recovery assumption to that EAD. This order keeps the exposure and severity calculations distinct.

Worked Example: A Revolving Facility With a Credit Conversion Factor

A hypothetical business borrower has drawn $420,000 on a revolving facility with a total commitment of $600,000. The one-year CCF on the undrawn commitment is 40%. The one-year PD is 3.2%, and net recovery after default is 28% of EAD. Calculate the modeled EAD, LGD, and one-year expected loss. Assume fixed CCF and recovery rate, no discounting ($i=0$), and no mortality modeling.

Method and assumptions. The undrawn commitment is the total commitment less the drawn balance. The CCF estimates the fraction of that undrawn amount that is drawn by default. Recovery is a percentage of the resulting EAD.

Step 1: Find the undrawn amount and additional draw.

\text{Undrawn}=\$600{,}000-\$420{,}000=\$180{,}000, \qquad \text{Additional draw}=0.40\times\$180{,}000=\$72{,}000.

Step 2: Calculate EAD.

\text{EAD}=\$420{,}000+\$72{,}000=\$492{,}000.

Step 3: Calculate LGD and the loss conditional on default. The LGD rate is $1-0.28=0.72$, or 72%. Net recovery is $137,760, and loss conditional on default is:

\text{Recovery}=0.28\times\$492{,}000=\$137{,}760, \qquad L_D=0.72\times\$492{,}000=\$354{,}240.

Step 4: Calculate expected loss.

E[X]=0.032\times\$354{,}240=\$11{,}335.68.

Check. The implied two-outcome distribution assigns probability 0.032 to a loss of $354,240 and probability 0.968 to zero. The probabilities sum to 1, and the probability-weighted loss is $11,335.68. The result is reported to the cent.

Common mistake avoided: Applying 40% to the drawn balance would give an incorrect EAD of $\$420{,}000+0.40\times\$420{,}000=\$588{,}000$. The CCF applies to the $180,000 undrawn amount. Using the incorrect EAD would also overstate expected loss.

One Position With Several Possible Default Years

When a position is evaluated over several years, an annual default probability may be conditional on the obligor having survived to the start of that year. Such a conditional probability is not, by itself, the probability of default in that particular year measured from the original evaluation date. To calculate that unconditional year-specific probability, multiply the annual conditional default probability by the probability of surviving all preceding years.

P(\text{default in year }k) =P(\text{survive through year }k-1) \times P(\text{default in year }k\mid\text{survive through year }k-1).

If EAD or recovery changes by year, calculate the loss conditional on default separately for each year. The expected cumulative undiscounted loss is the sum, across possible default years, of each year’s unconditional default probability multiplied by its conditional loss. This sum is for one position over the specified horizon; it is not a method for combining separate obligors.

Worked Example: Three-Year Loss With Changing Exposure

A hypothetical three-year loan is assessed at the start of year 1. The conditional default probabilities, given survival to the start of each year, are 1% in year 1, 2% in year 2, and 3% in year 3. EAD at default is $800,000, $600,000, and $300,000 in years 1, 2, and 3, respectively. The corresponding recovery rates are 40%, 35%, and 30%. Calculate cumulative three-year PD and expected cumulative undiscounted loss. Assume these conditional probabilities are fixed, loss occurs in the default year, and $i=0$, so no present-value discounting is applied. Mortality is not modeled.

Method and assumptions. The possible outcomes are default in one of the three years or no default during the horizon. Each year’s loss is EAD multiplied by that year’s LGD rate. Annual probabilities are conditional on survival to the start of the year; expected loss uses the corresponding unconditional probability of default in that year.

Step 1: Calculate survival and default probability by year. Year 1 default probability is 0.01. Survival through year 1 is $1-0.01=0.99$, so year 2 default probability measured from the start of the horizon is $0.99\times0.02=0.0198$. Survival through year 2 is $0.99\times(1-0.02)=0.9702$, so year 3 default probability is $0.9702\times0.03=0.029106$.

P(D_1)=0.010000,\qquad P(D_2)=0.019800,\qquad P(D_3)=0.029106.

The probability of surviving all three years is:

P(\text{no default})=0.99\times0.98\times0.97=0.941094.

Therefore cumulative three-year PD is:

P(D\text{ by year 3})=1-0.941094=0.058906=5.8906\%.

Step 2: Calculate loss conditional on default in each year. The LGD rates are 60%, 65%, and 70%, respectively:

L_{D_1}=0.60\times\$800{,}000=\$480{,}000, $$ $$ L_{D_2}=0.65\times\$600{,}000=\$390{,}000, $$ $$ L_{D_3}=0.70\times\$300{,}000=\$210{,}000.

Step 3: Weight each loss by its unconditional default probability.

E[X_1]=0.010000\times\$480{,}000=\$4{,}800.00, $$ $$ E[X_2]=0.019800\times\$390{,}000=\$7{,}722.00, $$ $$ E[X_3]=0.029106\times\$210{,}000=\$6{,}112.26.

Add the mutually exclusive default-year contributions:

E[X_{\text{cumulative}}] =\$4{,}800.00+\$7{,}722.00+\$6{,}112.26 =\$18{,}634.26.

Check. The outcome probabilities sum to $0.010000+0.019800+0.029106+0.941094=1.000000$. The three-year cumulative PD is also the sum of the three mutually exclusive default-year probabilities, $0.058906$. The expected cumulative undiscounted loss is $18,634.26, reported to the cent.

Common mistake avoided: Adding the conditional annual probabilities gives 6%, which is not the cumulative PD. The year 2 and year 3 probabilities apply only to survivors. The correct cumulative PD is 5.8906%.

Calculation Discipline and Exam Traps

A reliable calculation follows a fixed sequence: specify the horizon and probability basis, determine EAD at default, determine recovery on that exposure, calculate LGD, and only then apply the relevant default probability. In a multi-year model, first convert conditional annual probabilities into probabilities of default in each year from the original evaluation date.

Do not confuse recovery with LGD. A 35% recovery rate implies a 65% LGD rate, not a 35% LGD rate.
Do not apply a CCF to the drawn balance. First find undrawn commitment; apply the CCF to that amount.
Do not mix horizons. A one-year PD cannot be paired with a three-year loss exposure without a model that makes the time bases consistent.
Do not add conditional annual PDs as if they were unconditional. Account for survival through earlier years.
Do not double-count costs or recovery adjustments. Confirm whether the recovery figure is gross or already net of costs.

Key takeaway: For a single position with fixed EAD and LGD over a horizon, expected loss is PD multiplied by EAD and the LGD rate. When exposure, recovery, or default probability varies by possible default year, calculate and probability-weight each outcome separately before adding the results.

Check Your Understanding

For each problem, use the stated horizon and recovery basis; treat losses as undiscounted unless the problem says otherwise.

A hypothetical one-year loan has EAD of $275,000, PD of 2.4%, and a recovery rate of 45%. Find the LGD amount and expected loss. (Answer: LGD = $151,250; expected loss = $3,630.00)
A hypothetical revolving facility has a $600,000 drawn balance and $250,000 undrawn. Its CCF is 30%, one-year PD is 1.6%, and recovery is 32% of EAD. Find EAD and expected loss. (Answer: EAD = $675,000; expected loss = $7,344.00)
For a two-year exposure, conditional default probabilities are 2% in year 1 and 3% in year 2, given survival to that year. EAD is $500,000 at year 1 default and $300,000 at year 2 default; the respective recovery rates are 40% and 30%. Find cumulative PD and expected cumulative undiscounted loss. (Answer: cumulative PD = 4.9400%; expected loss = $12,174.00)
A hypothetical one-year position has EAD of $1,400,000, PD of 4%, gross recovery of $420,000, and recovery costs of $20,000. Find net recovery, LGD amount, and expected loss. (Answer: net recovery = $400,000; LGD = $1,000,000; expected loss = $40,000.00)
A two-year position has conditional PDs of 1.5% in year 1 and 2.5% in year 2, given survival to that year. EAD at default is $1,000,000 in year 1 and $700,000 in year 2; recovery rates are 50% and 40%, respectively. Find cumulative PD and expected cumulative undiscounted loss. (Answer: cumulative PD = 3.9625%; expected loss = $17,842.50)

What You'll Learn