## **Learning Outcomes**

This article explains the core building blocks of credit loss and how banks translate them into expected loss, unexpected loss, and capital needs. After reading, you will be able to:

- Define PD, LGD, recovery rate, and EAD, and explain how each enters credit loss measurement
- Calculate expected loss (EL) in percentage and currency terms for a loan or portfolio
- Explain unexpected loss (UL) and connect UL to one-year credit risk capital using a high-percentile loss measure (VaR-style)

## **GARP FRM Part 1 Syllabus**

For the GARP FRM Part 1 exam, you are expected to understand credit risk measurement with a focus on the following syllabus points:

- Define and calculate expected loss (EL)
- Define and explain unexpected loss (UL) and how it relates to credit risk capital
- Explain how economic capital differs from regulatory capital, and how capital is obtained from the credit loss distribution
- Estimate the mean and standard deviation of credit losses using a binomial (default/no default) setting
- Explain why EAD measurement is more difficult for derivatives than for loans

## **Test Your Knowledge**

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision._

1. A bank estimates it will lose 75% of its exposure if a borrower defaults. This 75% is the:
   a) PD  
   b) LGD  
   c) EAD  
   d) Expected loss

2. A \$20,000,000 loan has a one-year PD of 1.2% and a recovery rate of 45%. What is the one-year expected loss in dollars?

3. A portfolio has one-year EL of 1.0% of exposure. Its one-year 99.9% loss quantile is 5.5% of exposure. What is the (VaR-style) capital requirement as a percentage of exposure?

4. True/False (explain briefly): A higher recovery rate implies a higher LGD.

## **What Do You Know Already?**

- Can you quickly convert between recovery rate (RR) and loss given default (LGD)?
- Do you know when to write $EL = PD \times LGD$ versus $EL = EAD \times PD \times LGD$?
- Can you explain why banks “price for EL” but “hold capital for UL”?

Use these prompts to gauge your starting point; then focus your revision where your answers were least confident.

## **Introduction**

Credit risk is the risk of loss when a borrower or counterparty fails to meet contractual obligations. For banks, credit risk is central because loans and counterparty exposures are large, long-dated, and often sensitive to the business cycle.

A practical challenge is that average credit losses are not the same as losses in bad years. Banks therefore separate credit losses into (1) losses expected on average and (2) losses that can exceed the average by a wide margin. This distinction is fundamental to provisioning, loan pricing, and capital adequacy.

The FRM Part 1 treatment is deliberately structured: define the building blocks (PD, LGD, EAD), compute expected loss, then interpret unexpected loss as the variability (and tail) of credit losses that requires a capital buffer.

## **Credit loss building blocks: PD, LGD, and EAD**

Credit losses are often summarized with three inputs:

- **PD**: How likely is default over a horizon (commonly one year)?
- **LGD**: If default occurs, what fraction of exposure is lost (net of recoveries)?
- **EAD**: How much exposure will the bank have at the moment of default?

Together, they provide a compact “actuarial” view of credit loss. In practice, each component is estimated with data, judgment, and models, and each varies with economic conditions.

> **Key Term: Probability of Default (PD)**  
> PD is the probability that an obligor defaults over a specified horizon (often one year). In portfolio credit measurement, PD is the expected default rate.

### PD horizon conventions and common variations

1. **One-year PD (most common in capital frameworks)**  
   Many bank credit risk frameworks, including regulatory capital approaches, use a one-year horizon. This aligns with annual financial reporting and capital planning cycles.

2. **Conditional vs unconditional default probability (multi-period thinking)**  
   When thinking beyond one year, it is useful to distinguish:
   - **Conditional default probability**: probability of default in a future interval given survival up to the start of that interval.
   - **Unconditional default probability**: probability that default occurs by a future time (cumulative default probability).

   In continuous-time credit models, a **hazard rate** (default intensity) links the two. If hazard rate $h$ is constant, the unconditional probability of default by time $t$ is:
   $$PD(0,t) = 1 - e^{-ht}$$
   For FRM Part 1, the main operational message is: always match PD to the same horizon used for EL (and capital).

3. **Through-the-cycle vs point-in-time PD**
   - **Through-the-cycle (TTC)** PDs smooth business cycle effects and move slowly. Rating agencies often prefer stability, which pushes them toward TTC-style thinking.
   - **Point-in-time (PIT)** PDs are the current best estimate of default likelihood and respond more quickly to changing conditions. Banks often require PIT estimates for valuation, provisioning, and active risk management.

### Interpreting PD in a portfolio context

Even when individual PDs appear small, portfolio losses can spike because defaults are not independent. During weak economic conditions, both default rates and default correlations tend to rise (often described as credit contagion). This is one reason unexpected loss can be large relative to expected loss.

> **Key Term: Loss Given Default (LGD) and Recovery Rate (RR)**  
> Recovery rate (RR) is the value just after default as a percentage of face value (or exposure). LGD is the fraction lost if default occurs, typically:
>
> $$
> LGD = 1 - RR
> $$

### LGD drivers and cyclical behavior

LGD depends on seniority, collateral, legal regime, and workout outcomes (restructuring, liquidation, or going concern recovery). Empirically, recovery rates and default rates tend to be **negatively correlated**: in downturns, more firms default and collateral values can be depressed, leading to lower recoveries (higher LGD).

Simple reference points often used in practice (illustrative, not rules):

- Senior secured bonds: recovery rates around 50% (so LGD around 50%)
- Junior bonds: recovery rates around 25% (so LGD around 75%)

For exam questions, the main mechanics are converting RR to LGD correctly and keeping units consistent (percent vs decimal).

> **Key Term: Exposure at Default (EAD)**  
> EAD is the amount of exposure outstanding at the time of default. For loans, it is often close to the current outstanding balance; for derivatives, it is harder because exposure changes with market moves and netting.

### EAD for loans versus derivatives

1. **Loans (typically simpler)**  
   For a standard term loan, a common approximation is:
   - $EAD \approx$ outstanding principal (plus, in some setups, accrued interest/fees)

2. **Revolving credit (conceptually simple, operationally harder)**  
   Borrowers can draw down lines before default. Banks therefore use credit conversion concepts to translate undrawn commitments into an expected EAD at default.

3. **Derivatives (hardest)**  
   Derivative exposure varies daily with market prices. In addition:
   - **Netting** can reduce exposure when multiple trades with the same counterparty offset each other.
   - Rules often set $EAD$ as **current exposure plus an add-on** for potential future exposure.
   - Capital must generally be assessed **by counterparty**, reflecting netting sets.
   - **Wrong-way risk** can arise when exposure increases precisely when the counterparty’s credit quality deteriorates.

For FRM Part 1, be able to explain _why_ EAD is easier for loans than derivatives and how netting changes the measurement unit (trade vs counterparty).

## **Expected loss: definition, calculation, and interpretation**

Expected loss is the average credit loss a bank anticipates over the horizon. It is a “best estimate” of the value decline from credit exposure, but actual losses will vary around it.

> **Key Term: Expected Loss (EL)**  
> Expected loss is the expected decline in value from credit exposure over a horizon. In its standard form:  
> $$EL = PD \times LGD$$  
> and in currency terms:  
> $$EL = EAD \times PD \times LGD.$$

### EL formula and unit discipline

There are two common ways EL appears:

1. **As a rate (percentage of exposure)**  
   $$EL\% = PD \times LGD$$

2. **As a currency amount**  
   $$EL_{\$} = EAD \times PD \times LGD$$

A frequent exam mistake is mixing these: a percentage EL must be multiplied by exposure to produce dollars.

### Why EL matters operationally

Banks usually plan for EL through some combination of:

- **Loan pricing**: charging a spread that, at minimum, covers expected credit losses and operating costs
- **Provisions/reserves**: accounting recognition of expected credit losses (framework depends on accounting regime)
- **Portfolio steering**: adjusting origination, limits, and hedging to manage expected and stressed loss outcomes

EL is not “capital” in the economic sense. Capital is primarily held for adverse outcomes beyond the mean (unexpected loss), while EL is something banks expect to incur on average.

### Worked Example 1.1

A bank expects a 1.8% default rate on its loans over one year. The recovery rate in the event of default is 60%. Calculate the expected loss as a percentage of exposure.

> **Answer:**
>
> 1. Convert recovery rate to LGD:
>    $$
>    LGD = 1 - RR = 1 - 0.60 = 0.40
>    $$
> 2. Multiply by PD:  
>    $$EL = PD \times LGD = 0.018 \times 0.40 = 0.0072$$
> 3. Express as a percentage:
>    $$
>    EL = 0.72\%
>    $$
>    of exposure.

### Worked Example 1.2

A \$100,000,000 loan portfolio has an average one-year PD of 2.0% and an average LGD of 35%. Calculate the one-year expected loss in dollars.

> **Answer:**
>
> 1. Use the currency EL formula:  
>    $$EL = EAD \times PD \times LGD$$
> 2. Substitute values:  
>    $$EL = 100{,}000{,}000 \times 0.020 \times 0.35 = 700{,}000$$
> 3. Interpretation: the portfolio’s one-year expected credit loss is \$700,000.

### Portfolio additivity of EL

Expected loss is additive across exposures under the standard setup:

$$EL_{\text{portfolio}} = \sum_{i=1}^{n} EAD_i \times PD_i \times LGD_i$$

This additivity is one reason EL is widely used in portfolio reporting: it gives a clean decomposition by borrower, sector, geography, or product.

## **Unexpected loss: volatility, tail risk, and capital**

Expected loss is an average. Actual credit losses can be higher or lower, depending on how many obligors default, how correlated defaults are, and what recoveries look like in that particular year. The deviation of realized loss from expected loss is the essence of unexpected loss.

> **Key Term: Unexpected Loss (UL)**  
> Unexpected loss is the difference between actual credit loss and expected loss. In capital terms, UL is often measured as the loss at a high confidence level minus EL (a one-year VaR-style concept).

### Two closely related meanings of UL

You will see “unexpected loss” used in two related ways:

1. **Statistical deviation from the mean**  
   Realized loss can be above or below EL. In this sense, UL is the random variation around EL.

2. **Adverse deviation that capital must absorb**  
   In capital discussions, UL is usually the “buffer” amount above EL needed to cover very bad outcomes. Using VaR language:
   $$\text{Required capital} = \text{VaR}_{X\%, 1\text{y}} - EL$$
   Basel sets $X = 99.9\%$ for regulatory capital in the IRB approach (one-year horizon).

Be clear about context. Capital-related UL is non-negative by construction because it is based on a high loss quantile.

### Economic capital vs regulatory capital

- **Economic capital** is a bank’s internal estimate of capital needed to remain solvent (or to protect a target credit rating) given its risk profile.
- **Regulatory capital** is the amount required by supervisors, commonly based on Basel standards.

Both start from the same conceptual object: a one-year loss distribution. They can differ in confidence level, modeling choices, and constraints imposed by regulation.

### UL from a simple default/no-default model (binomial logic)

A standard FRM Part 1 starting point is a single loan with:

- loss of $0$ if no default
- loss of $EAD \times LGD$ if default occurs

Let loss random variable be $L$ where:

- $L = EAD \times LGD$ with probability $PD$
- $L = 0$ with probability $1 - PD$

Then:

1. **Mean (expected loss):**
   $$E[L] = PD \times EAD \times LGD$$

2. **Standard deviation (one-loan loss volatility):**
   $$\sigma_L = \sqrt{PD - PD^2} \times (EAD \times LGD) = \sqrt{PD(1-PD)} \times (EAD \times LGD)$$

This standard deviation is not “capital” by itself, but it is a common building block for approximations to tail loss.

### Worked Example 2.1

A \$1,000,000 loan has a one-year PD of 0.5%. The recovery rate is 40%. Calculate (a) expected loss and (b) the standard deviation of loss.

> **Answer:**
>
> 1. Convert inputs:
>    $$
>    PD = 0.005
>    $$
>    $$
>    RR = 0.40 \Rightarrow LGD = 1 - 0.40 = 0.60
>    $$
>    $$
>    EAD = 1{,}000{,}000
>    $$
> 2. Expected loss:  
>    $$EL = EAD \times PD \times LGD = 1{,}000{,}000 \times 0.005 \times 0.60 = 3{,}000$$  
>    So:
>    $$
>    EL = \$3{,}000
>    $$
> 3. Loss if default occurs:  
>    $$\text{Default loss} = EAD \times LGD = 1{,}000{,}000 \times 0.60 = 600{,}000$$
> 4. Standard deviation:  
>    $$\sigma_L = \sqrt{PD(1-PD)} \times (EAD \times LGD)$$  
>    $$= \sqrt{0.005 \times 0.995} \times 600{,}000$$  
>    $$\approx 0.07053 \times 600{,}000 \approx 42{,}318$$  
>    So:
>    $$
>    \sigma_L \approx \$42{,}318
>    $$

### From UL to capital using a high-percentile loss measure

Credit risk capital is commonly framed as a one-year high-percentile loss minus EL:

$$K = \text{VaR}_{X\%, 1\text{y}} - EL$$

- Basel IRB uses $X = 99.9\%$
- Banks may use different levels for economic capital, sometimes more conservative to support target credit ratings

In more advanced credit portfolio settings, models (for example, Vasicek-style portfolio models) are used to map an average PD into a stressed or worst-case default rate at the chosen percentile. A common regulatory-capital expression is:

$$UL = (WCDR - PD) \times LGD \times EAD$$

where $WCDR$ is the worst-case default rate (e.g., the 99.9th percentile default rate). Intuition:

- $PD \times LGD \times EAD$ is the mean loss (EL)
- $WCDR \times LGD \times EAD$ is a severe-loss scenario loss level
- the difference is the “excess” loss above the mean that capital must cover

### Credit migration as a source of unexpected loss

Unexpected loss is not only about outright defaults. It can also arise from **credit migration**, meaning deterioration in credit quality that changes the value of credit-sensitive instruments (bond prices, loan marks, CDS spreads), even before default. Models that incorporate rating transitions explicitly treat migration as a driver of UL, not just default.

### **Exam Warning**

> Common exam errors include:
>
> 1. Confusing recovery rate and LGD: always use:
>    $$
>    LGD = 1 - RR
>    $$
> 2. Forgetting EAD when the question asks for dollars (not a percentage).
> 3. Mixing horizons: a four-year PD cannot be used directly in a one-year EL calculation without adjustment.
> 4. Treating UL as “always bigger than EL”: realized losses can be below EL in good years; capital-based UL is a tail measure above EL by construction.

## **Measurement issues: estimation, correlations, and procyclicality**

Even though the formulas look simple, estimating PD, LGD, and EAD is difficult in real portfolios. FRM Part 1 emphasizes several recurring issues:

1. **PIT vs TTC PD needs**  
   Risk management often needs PIT PDs for current conditions, while other processes (including some rating practices) aim for stability.

2. **Downturn co-movement of PD and LGD**  
   In recessions, PD tends to rise and recoveries often fall. This joint movement increases losses more than a “one-factor-at-a-time” view would suggest.

3. **EAD for derivatives and counterparty effects**  
   Exposure changes daily, netting changes the measurement unit, and wrong-way risk can make exposures and credit quality move together.

4. **Default correlation and concentration**  
   Correlations rise in stress, and concentration (single-name, sector, geography) can dominate portfolio tail risk even when EL appears modest.

5. **Model and implementation risk**  
   Credit loss estimates depend on assumptions (data windows, segmentation, downturn calibration, correlation inputs). Errors can come from both model design and operational implementation.

For exam purposes, be ready to (a) compute EL and basic loss volatility cleanly and (b) explain why UL and capital are sensitive to correlation, downturn conditions, and EAD measurement.

## **Summary**

Credit risk measurement begins with three inputs: PD (how often default happens), LGD (how much is lost if it happens), and EAD (how much exposure exists at default). These combine to form expected loss, the average credit loss over the horizon:
$$EL = EAD \times PD \times LGD.$$

Actual losses vary around EL because defaults are random, correlated, and influenced by the business cycle; recoveries also change with conditions. Unexpected loss is the deviation from EL, and in a capital setting it is typically expressed as a high-percentile loss minus EL:
$$K = \text{VaR}_{X\%, 1\text{y}} - EL,$$
with Basel IRB using $X = 99.9\%$.

A strong FRM Part 1 answer is usually built from disciplined units (rate vs dollars), correct conversion between RR and LGD, and a clear statement of how EL relates to pricing/provisions while UL relates to capital and solvency protection.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- PD is the expected default rate over a stated horizon (often one year)
- Recovery rate and LGD are linked by $LGD = 1 - RR$
- EAD is exposure at the time of default; it is straightforward for many loans and harder for derivatives due to market variability and netting
- Expected loss is $EL = EAD \times PD \times LGD$ (or $PD \times LGD$ as a rate)
- Unexpected loss reflects variability around EL; capital is commonly framed as a one-year high-percentile loss minus EL
- Default rates and recovery rates are often negatively correlated, increasing losses in downturns
- Correlation among defaults is a major driver of portfolio UL and capital needs

## **Key Terms and Concepts**

- Probability of default (PD)
- Recovery rate (RR)
- Loss given default (LGD)
- Exposure at default (EAD)
- Expected loss (EL)
- Unexpected loss (UL)
- Economic capital
- Regulatory capital
- Credit migration
- Default correlation and credit contagion
- Value at risk (VaR) as a tail-quantile concept for capital
- Worst-case default rate (WCDR)
- Point-in-time (PIT) PD and through-the-cycle (TTC) PD
- Netting and wrong-way risk (derivatives EAD context)


Credit risk measurement (basics) - PD, LGD, EAD; expected and unexpected loss