Expected loss calculator: PD × LGD × EAD

8,140.00expected loss of 3 positions

The 3 positions are expected to lose 8,140.00 together, or 0.866% of the total exposure of 940,000.00. Position 1 accounts for 76.7% of it.

Expected loss by position
Position	PD × LGD	Expected loss	Share of expected loss	Share of EAD	Loss if it defaults	Loss standard deviation
1	0.96%	6,240.00	76.7%	69.1%	260,000.00	39,792.74
2	0.16%	400.00	4.9%	26.6%	50,000.00	4,454.21
3	3.75%	1,500.00	18.4%	4.3%	30,000.00	6,538.35
Total	0.866%	8,140.00	100%	100%	–	see below

Spread of the portfolio loss (standard deviation)
Defaults independent of each other	40,571.57
Upper limit: the standard deviations added up	50,785.30
Expected loss, whatever the dependence	8,140.00

The upper limit would need the positions to always default together, which is impossible with different PDs; the true spread lies below it. Defaults that tend to happen together push the spread above the independent figure.

How it is calculated

Expected loss of position 10.024 × 0.4 × 650,000.00 = 6,240.00
Expected loss of position 20.008 × 0.2 × 250,000.00 = 400.00
Expected loss of position 30.05 × 0.75 × 40,000.00 = 1,500.00
Expected loss of the portfolio6,240.00 + 400.00 + 1,500.00 = 8,140.00
Expected loss rate8,140.00 ÷ 940,000.00 = 0.866%
Loss standard deviation of position 1260,000.00 × √(0.024 × 0.976) = 39,792.74
Loss standard deviation of the portfolio, independent defaults√(39,792.74² + 4,454.21² + 6,538.35²) = 40,571.57

Limits: LGD and EAD are treated as fixed and the PD as given. Real recoveries and drawings vary and can move with defaults, and the expected loss says nothing about the size of a bad year.

Key facts

Expected loss = PD × LGD × EAD, with PD and LGD as decimals and EAD as an amount; PD and EAD must refer to the same horizon.
LGD = 1 − recovery rate: a 60% recovery means an LGD of 40%.
Expected losses add up across positions whatever the link between their defaults: the default list of three positions is expected to lose £8,140, 0.866% of its £940,000 exposure.
With fixed LGD and EAD, the loss standard deviation of a position is LGD × EAD × √(PD × (1 − PD)): about £39,793 for the £650,000 credit line.
Two such loans have a loss standard deviation of about £56,275 if they default independently and £79,585 if they always default together; the expected loss stays £12,480.

How is the result calculated?

One position. In a default-only model a position loses nothing if the borrower does not default and LGD × EAD if it does. With LGD and EAD fixed, the probability-weighted average of the two outcomes is the expected loss (NIST, binomial moments (secondary source)) (BCBS, CRE35):

EL = PD \times LGD \times EAD,\qquad LGD = 1 - \text{recovery rate}

PD is the probability of default over the horizon, usually one year; LGD the share of the exposure lost on default; EAD the amount exposed at default, which for a credit line includes what the borrower is expected to draw by then (BCBS, CRE36). PD × LGD is the expected loss rate, the expected loss as a share of EAD. The Basel rules for the internal ratings-based (IRB) approach use the same product for corporate, sovereign, bank and retail exposures not in default (BCBS, CRE35).

A list of positions. Expected values add up, so the expected loss of a list is the sum of its positions’ expected losses, however their defaults are linked (Chen, probability moments (secondary source)). The calculator also shows each position’s share of the total and the expected loss rate of the list: total expected loss ÷ total EAD.

EL_{\text{list}} = \sum_i PD_i \times LGD_i \times EAD_i

The spread around the average. Whether a position defaults is a yes-or-no variable with variance $PD(1-PD)$ . With fixed LGD and EAD, the standard deviation of the loss – unexpected loss in the dispersion sense – is therefore (NIST, binomial moments (secondary source))

UL_i = LGD_i \times EAD_i \times \sqrt{PD_i\,(1 - PD_i)}

Standard deviations do not add up. The variance of a sum is the sum of the variances plus the covariances (Chen, probability moments (secondary source)): with independent defaults the covariances are zero, and since no correlation exceeds 1 (NYU Stern, covariance and correlation (secondary source)), the standard deviations added up are the most the spread can be.

UL_{\text{independent}} = \sqrt{\textstyle\sum_i UL_i^2}\qquad UL_{\text{list}} \le \textstyle\sum_i UL_i

Worked example

1. A credit line

A borrower has drawn £500,000 of a line and £300,000 is still available. Assume a one-year PD of 2.4%, a recovery of 60% and, by default, further drawings of half the undrawn amount (our assumption, not a Basel value):

\begin{aligned}EAD &= 500{,}000 + 0.5 \times 300{,}000 = 650{,}000\\ LGD &= 1 - 0.60 = 0.40\\ EL &= 0.024 \times 0.40 \times 650{,}000 = \pounds 6{,}240\end{aligned}

That is 0.96% of the exposure. The loan never actually loses £6,240: it either defaults and loses £260,000, or it does not and loses nothing. The standard deviation shows how far those outcomes lie from the average:

UL = 260{,}000 \times \sqrt{0.024 \times 0.976} \approx \pounds 39{,}793

2. Three positions together

Add a loan of £250,000 with a 0.8% PD and 20% LGD, and a small loan of £40,000 with a 5% PD and 75% LGD – the default list in the calculator. Their expected losses are £6,240, £400, £1,500:

EL = 6{,}240 + 400 + 1{,}500 = \pounds 8{,}140

That is 0.866% of the total exposure of £940,000. The credit line carries 69.1% of the exposure and 76.7% of the expected loss; the small loan only 4.3% of the exposure but 18.4% of the expected loss, because its PD and LGD are both high. With independent defaults the loss standard deviation of the three is about £40,572; the upper limit, the three standard deviations added up, is about £50,785.

3. Two identical loans: dependence changes the spread, not the average

Take two loans exactly like the credit line. The expected loss is £12,480 whatever the link between their defaults. If they default independently of each other, the standard deviation is √2 times that of one loan:

UL = \sqrt{2} \times 39{,}792.74 \approx \pounds 56{,}275

If they always default together, every outcome doubles, and so does the standard deviation: about £79,585. The same figures appear in the expected and unexpected loss guide, which goes on to credit VaR.

What do the results mean for you?

Expected loss is the average cost of lending. The Basel Committee describes it as a cost component of doing business that lenders manage through the pricing of loans and through provisions, while capital is meant for the larger, rarer losses above it (BCBS, IRB explanatory note). For the credit line that cost is £6,240 a year, 0.96% of the exposure.

For one loan the average never happens, and over a few loans it can be far from the result of a given year. Read the expected loss together with the loss if a position defaults and the standard deviation: in the default list, a single default of the credit line would cost £260,000, more than 31 times the expected loss of all three positions.

For exam practice, work a question by hand first: write PD and LGD as decimals, check that PD and EAD refer to the same horizon, then compare your figure and steps with the calculator’s.

Which expected loss mistakes are common?

The recovery rate in place of LGD. A 60% recovery means an LGD of 40%. Using 60% gives £9,360 instead of £6,240 for the credit line.
Only the drawn balance as EAD. Borrowers can draw more before they default, so for a credit line EAD can be above today’s balance (BCBS, CRE36). With £500,000 the expected loss would be £4,800, not £6,240.
Mismatched horizons. A one-year PD gives a one-year expected loss. For a longer period you need the probability of default over that period, not the one-year figure.
Adding standard deviations. That gives the upper limit, which needs every position to default together. With independent defaults, square, add and take the square root.
Reading the standard deviation as a worst case. It measures spread, not a loss that is not exceeded with some probability. A quantile of the loss needs the whole loss distribution; see the value at risk guide.

What does the calculator leave out?

Random LGD and EAD. The calculator treats them as fixed. In practice recoveries and drawings vary and can be worse when defaults are frequent; Basel requires LGD estimates that reflect economic downturns (BCBS, CRE36). When severity is random, expected loss is PD × E[LGD × EAD | default], which equals PD × E[LGD | default] × E[EAD | default] only when LGD and EAD are uncorrelated given default (Chen, probability moments (secondary source)).
A model of joint defaults. Without one there is no loss distribution for the list, so no credit VaR. The calculator shows the two cases that need no model: independent defaults and the upper limit.
Capital requirements. The IRB risk-weight functions add correlation parameters and, for relevant exposures, a maturity adjustment (BCBS, CRE31). The expected loss here is not a capital figure.
Accounting provisions and discounting. Provisions follow their own accounting standards, which this calculator does not model.

Can you use this calculator for the exam?

This calculator supports study for the FRM® Exam Part I, run by the Global Association of Risk Professionals (GARP) (GARP, FRM Exam information). Our tools complement GARP’s own study materials; they do not replace them. It runs in a web page, so it is not an exam calculator: which calculators you may use is set by GARP, and our page on which calculators are allowed in the exams sets out the models with their sources.

What this tool covers

Last updated

30 October 2026

Based on

The default-only model of credit loss and the Basel IRB definitions of PD, LGD and EAD (see the sources)
Your own PD, LGD and EAD; the default list is illustrative

Includes

Expected loss of one to 20 positions and of the list, with each position’s share and the expected loss rate
The loss if a position defaults and its loss standard deviation with fixed LGD and EAD
The loss standard deviation of the list with independent defaults and its upper limit
Worked steps and a link that keeps your numbers

Not included

Estimating PD, LGD or EAD from ratings, data or collateral
Models of joint defaults, credit VaR and regulatory capital
Accounting provisions, discounting and losses over several years

Frequently asked questions

How do you calculate expected loss?

Multiply the probability of default by the loss given default and the exposure at default, with PD and LGD as decimals. A PD of 2.4%, an LGD of 40% and an EAD of £650,000 give 0.024 × 0.40 × 650,000 = £6,240.

What is the difference between expected and unexpected loss?

Expected loss is the average loss; unexpected loss is the variation around it. As a standard deviation, with fixed LGD and EAD, it is LGD × EAD × √(PD × (1 − PD)): about £39,793 for a credit line whose expected loss is £6,240.

How do you get LGD from the recovery rate?

LGD is 1 minus the recovery rate: a 60% recovery gives an LGD of 40%. Using the recovery rate in place of LGD is a common mistake.

How do you calculate the expected loss of a portfolio?

Work out each position’s expected loss and add them up. For the three default positions that is £6,240 + £400 + £1,500 = £8,140, or 0.866% of the total exposure.

Does correlation between defaults change the expected loss?

No: expected losses add up however defaults are linked. Correlation changes the spread. Two loans like the credit line have an expected loss of £12,480 either way, but a standard deviation of about £56,275 if independent and £79,585 if they always default together.

What exposure should I use for a credit line?

The amount expected to be owed at default: the drawn balance plus what the borrower is expected to draw before defaulting. With £500,000 drawn and half of a further £300,000 assumed to be drawn, EAD is £650,000.

Does the calculator save my numbers?

No. The calculation runs in your browser. Your numbers appear only in the page address after the # sign, so you can share a result; they are not sent to us.

Sources

Basel Committee on Banking Supervision, Bank for International Settlements: Basel Framework CRE35.1–35.9 — IRB expected losses, eligible provisions and capital treatment. Accessed 29 October 2026.
Basel Committee on Banking Supervision, Bank for International Settlements: Basel Framework CRE36.63, 36.76–36.93 — PD horizon, economic loss, downturn LGD and additional drawings in EAD. Accessed 29 October 2026.
Basel Committee on Banking Supervision, Bank for International Settlements: An Explanatory Note on the Basel II IRB Risk Weight Functions — §§2 and 4.4, expected loss and capital above the mean (5 July 2005). Accessed 29 October 2026.
Basel Committee on Banking Supervision, Bank for International Settlements: Basel Framework CRE31 — IRB risk-weight functions, input units, correlation and maturity. Accessed 29 October 2026.
National Institute of Standards and Technology (statistical handbook, secondary source): NIST/SEMATECH e-Handbook §1.3.6.6.18 — Binomial distribution: mean and standard deviation. Accessed 29 October 2026.
Yen-Chi Chen, University of Washington (lecture notes, secondary source): STAT 535, Lecture 1: Review on Probability and Statistics — expectation, covariance and total expectation. Accessed 29 October 2026.
Aswath Damodaran, NYU Stern School of Business (secondary source): A Primer on Statistics — covariance and correlation. Accessed 29 October 2026.
Global Association of Risk Professionals (GARP): FRM Exam information: format and exam topics. Accessed 26 October 2026.

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