Credit risk and expected loss: default, loss given default and pricing
Every loan carries the chance that it will not be repaid. Lenders handle that chance by estimating it, pricing it into the rate and holding reserves and capital against it. This lesson explains probability of default, loss given default and expected loss, shows how they set the rate a lender must charge, tests a loan book against a downturn, and compares the credit-loss figures of DBS and Citigroup. It also covers what credit scores and debt-to-income ratios are for.
- Level
- Beginner
- Reading time
- About 26 minutes
- Know this first
- The true cost of borrowing
By the end you will be able to:
- define probability of default, loss given default and exposure, and calculate expected loss
- work out the rate a lender must charge to cover its cost and expected losses
- explain how security on a loan changes the loss, and why a downturn is tested separately
- describe what credit scores and debt-to-income ratios measure
- explain why banks hold loss reserves before loans actually default
Lending is a bet that most people pay
Every loan has a cost that does not appear in the interest rate: the chance that the borrower will not repay. A lender cannot know which borrowers will fail, but it can estimate how many of a group will, how much it will lose when they do, and charge enough to cover it. That estimate, and the buffers a lender keeps against it, are what credit risk management is about.
The first lesson in this module showed how a thin layer of equity absorbs loan losses, and the second showed what a borrower pays. This lesson looks at the lender's side of the same loan. It uses a fictional book of loans for the mechanics, then compares the credit-loss figures that DBS of Singapore and Citigroup of the United States reported for 2025.
Probability, severity and exposure
Three numbers describe the credit risk of a loan. The probability of default (PD) is the chance that the borrower fails to pay as agreed within a period, usually a year. Under the Basel framework, a borrower already in default has a PD of 100%. The exposure at default (EAD) is the amount owed when that happens. The loss given default (LGD) is the part of that exposure that is lost after anything recovered, expressed as a percentage of the exposure. The Basel framework's supervisory figure for senior unsecured corporate claims is an LGD of 45%, and it is higher for claims that rank below them.
Multiplying the three gives the expected loss: the average loss per year on a group of similar loans. A single loan either loses nothing or loses a lot, so the average is a statement about the group.
The second line says that if the borrower pays, the lender collects the loan and the interest, and if the borrower defaults, it collects only what it recovers. The third solves for the rate that leaves the lender whole on its funding and running cost. This is risk-based pricing: riskier borrowers pay more, and borrowers who provide good security, which lowers the loss given default, pay less.
Price a loan and test a book
Set the chance of default, the share lost if default happens, the lender's cost of funds and running costs, and how much worse defaults get in a downturn. The left page prices a $1,000 loan held for a year. The right page applies the same figures to a book of $1,000 million of such loans, backed by $80 million of equity, which is 8% of the loans and the cushion of the bank in the first lesson.
On a $1,000 loan with a 2.00% chance of default and 45% lost if it happens, the expected loss is $9.00. To cover its 4.0% cost and that loss, the lender must charge 5.00%, which is 1.00 points above its cost. Across a $1,000 million book the expected loss is $9.0 million a year (0.90% of loans). If defaults run at 2.0 times that rate the loss is $18.0 million, 1.80% of loans and 22.5% of an $80 million equity cushion.
Four experiments. First, at the starting settings the expected loss on a $1,000 loan is $9.00, so the lender must charge 5.00% to cover its 4.0% cost. The whole book is expected to lose $9.0 million a year. In a downturn with twice the defaults, the loss doubles to $18.0 million, which is 22.5% of the equity: painful but survivable. Second, raise the chance of default to 5%. The rate needed climbs to 6.58%, and the downturn would use up 56.2% of the equity. Third, return the chance of default to 2% and raise the loss given default to 80%, as for an unsecured loan with little recovered. The rate needed rises to 5.71% even though the chance of default has not changed. Fourth, keep the chance of default at 5% and set the downturn multiple to 4 times. Defaults of 20% would cost 112.5% of the equity, which means the bank would be insolvent. The same book is safe or fatal depending on how bad the downturn is, and that is why lenders and regulators run stress tests.
The rate needed for different combinations, at a 4.0% cost:
| Rate needed to cover a 4.0% cost | 25% lost if default (well secured) | 45% lost if default | 70% lost if default (weak security) |
|---|---|---|---|
| 0.5% chance of default | 4.15% | 4.25% | 4.37% |
| 1% chance of default | 4.29% | 4.49% | 4.75% |
| 2% chance of default | 4.59% | 5.00% | 5.51% |
| 5% chance of default | 5.53% | 6.58% | 7.89% |
| 10% chance of default | 7.22% | 9.44% | 12.22% |
Rates run from 4.15% for a very safe, well-secured loan to 12.22% for a risky loan with weak security. Both factors matter: halving the loss given default does as much for a risky borrower as a large cut in the chance of default. This is the reason lenders ask for collateral, such as a car or a house, and why secured loans cost less than unsecured ones.
Judging a borrower: scores and ratios
Where does a lender get the probability of default for a person? From data about the borrower. A credit report records a person's borrowing and repayment history. A credit score is, in the words of the US Consumer Financial Protection Bureau, a prediction of credit behaviour: how likely the person is to repay on time, calculated by a scoring model from the information in the credit report. Factors include bill-payment history, current unpaid debt, credit utilisation and new applications for credit. Most scores in the United States run from 300 to 850, a person has more than one score, and the number differs by model, data source and date. Other countries use different systems, and some have no central scoring.
Lenders also look at the debt-to-income ratio: all monthly debt payments divided by gross monthly income. The bureau's example is $2,000 of debt payments on $6,000 of income, which is 33%. A higher ratio leaves less room if income falls, and different lenders and loan types set different limits. A score looks back at behaviour, and the ratio looks at present capacity. A lender usually wants both, along with the collateral, which affects the loss given default.
Reserves: recognising losses before they happen
Banks keep an allowance for credit losses, a reserve deducted from the value of their loans for the losses they expect. The accounting rule used in much of the world, IFRS 9, replaced the earlier incurred loss approach, which waited for evidence that a loss had happened, with a forward-looking one. Under it, a loan is placed in one of three stages. In stage 1, the loan has been made and its credit risk has not risen much, and the bank records the expected losses from defaults possible in the next 12 months. In stage 2, credit risk has risen significantly since the loan began, and the bank records the expected losses over the whole life of the loan. In stage 3, the loan is credit-impaired, and lifetime losses are recorded with interest worked out on the amount net of the allowance. In the United States, a similar principle called the current expected credit loss model applies.
The consequence is that a bank's reported credit-loss charge in the income statement rises when forecasts worsen, often before the defaults arrive. A bank can show a large charge in a year when few loans actually failed, and a small charge when it is releasing reserves it no longer needs. The charge is the bank's estimate and not a count of failures.
DBS and Citigroup
DBS is a Singapore bank with business across Asia, and Citigroup is a US bank with operations in many countries. They report in different currencies, Singapore dollars and US dollars, so the table compares only ratios, worked from their published figures for 2025. The credit-loss charge is the amount taken to the income statement in the year.
| Full year 2025, ratios only | DBS | Citigroup |
|---|---|---|
| Credit-loss charge for the year as a share of year-end loans | 0.18% | 1.26% |
| Change in the credit-loss charge from 2024 | +27.2% | −2.4% |
| Growth in year-end loans from 2024 | +3.3% | +8.3% |
| Reserve held against loan losses | 130% of non-performing loans (rate 1.0%) | 2.56% of loans |
Citigroup's charge on loans was about seven times DBS's as a share of year-end loans. The mix of lending usually explains a gap like this, since unsecured loans such as credit cards lose more than secured ones, but the figures here do not show either bank's mix. DBS's charge rose by more than a quarter during the year while Citigroup's edged down, which shows how a small base moves in percentage terms. DBS describes its reserve through coverage of non-performing loans and Citigroup through reserve as a share of loans, so the last row is a reminder that two banks can describe the same thing differently. The reserves are not directly comparable, and the figures say nothing about which bank is better run.
What these measures cannot tell you
- PD and LGD are estimates. They come from history and judgement, and a lender can be wrong about both, especially in a downturn that is unlike the past.
- Defaults come together. The calculator treats a book as a group with average behaviour. In a crisis many borrowers fail at once, which is why the downturn test matters more than the average.
- One-year pricing is simple. Real loans last for years, recoveries take time, and the lender receives interest along the way.
- A score is not a verdict. It predicts behaviour from a record, and a thin or incomplete record can understate a borrower.
- Reported charges mix estimates and events. The credit-loss charge in a single year includes changes in forecasts, so it cannot be read as actual losses.
Common mistakes
Treating the probability of default as the loss
A 2% chance of default does not mean a 2% loss. Multiply by the share lost if default happens, which is often well under 100% when there is collateral.
Pricing only for the average
The expected loss is a long-run average. Equity must cover the bad years too.
Assuming a credit-loss charge equals write-offs
The charge includes the expected future losses. Write-offs happen when a loan is declared unrecoverable, often later.
Comparing banks' reserves without checking the basis
Reserve as a share of loans and coverage of non-performing loans are different measures.
Reading a debt-to-income ratio without the income behind it
The same ratio is more risky for a borrower with unstable income than for one with stable income.
Try it yourself
1. A lender has $50,000 outstanding to a borrower with a 3% chance of default and a 40% loss if default happens. What is the expected loss?
Show the working
Expected loss = 3% × 40% × 50,000 = $600 a year. Over a large group of similar loans, the lender would expect to lose about $600 per $50,000 lent each year.
2. A lender's cost is 5%. A loan has a 4% chance of default and a 60% loss given default. What rate must it charge to break even on a one-year loan?
Show the working
Rate = (1 + 0.05 − 0.04 × 0.40) ÷ (1 − 0.04) − 1 = (1.05 − 0.016) ÷ 0.96 − 1 = 1.034 ÷ 0.96 − 1 = 7.71%. That is 2.71 points above the lender's cost.
3. A borrower earns $5,000 a month before tax and pays $1,500 a month on existing debts. She asks for a new loan with a payment of $400. What are her debt-to-income ratios before and after?
Show the working
Before: 1,500 ÷ 5,000 = 30%. After: (1,500 + 400) ÷ 5,000 = 38%. The new loan raises the ratio by eight points. Whether that is acceptable depends on the lender and the type of loan.
4. A bank has a $2,000 million loan book and equity of 8% of loans. Its probability of default is 3% and its loss given default is 50%. If defaults in a downturn are three times normal, what share of equity is lost?
Show the working
Equity = 8% × 2,000 = $160 million. Downturn default rate = 3% × 3 = 9%. Loss = 9% × 50% × 2,000 = $90 million. Share of equity = 90 ÷ 160 = 56.25%. The bank survives, but with less than half its cushion, before any other losses or profits.
5. Why can a bank's credit-loss charge jump in a year when hardly any borrowers default?
Show the working
Under expected-loss accounting, such as IFRS 9, the charge reflects forecasts as well as events. If the outlook worsens, more loans move to stage 2, where lifetime losses are recorded, and the bank raises its reserve before the defaults arrive.
Where this is used
- A bank's credit team sets the interest rate and the credit limit for a borrower according to the borrower's risk.
- A finance team estimates the credit-loss allowance to put in the accounts each quarter.
- A regulator tests whether a bank's capital can survive a severe recession.
- An investor in a bank looks at the credit-loss charge, the reserve and the quality of the loans before judging its profits.
Key terms
- Credit risk
- The risk that a borrower will not repay as agreed.
- Probability of default (PD)
- The chance that a borrower defaults within a period, usually a year.
- Loss given default (LGD)
- The share of the exposure that is lost after recoveries, if the borrower defaults.
- Exposure at default (EAD)
- The amount owed to the lender when the borrower defaults.
- Expected loss
- The average loss per year, equal to PD × LGD × EAD.
- Collateral
- An asset pledged as security for a loan, which lowers the loss given default.
- Credit score
- A number that predicts how likely a person is to repay debts, built from the credit report.
- Debt-to-income ratio
- Monthly debt payments divided by gross monthly income.
- Allowance for credit losses
- A reserve against the loans on the balance sheet for the losses expected.
- Stress test
- An exercise that tests whether a bank's capital can absorb the losses of a severe downturn.
Questions people ask
Is a default the same as a missed payment?
No. A missed payment is a delinquency, and many are cured. A default is declared under a definition set by the lender or the regulator, commonly when a payment is long overdue or the lender judges that the borrower is unlikely to pay.
Do lenders lose money on every defaulted loan?
Not all of it. Collateral can be sold and some debts are partly recovered, which is why the loss given default is usually below 100%.
Why is my credit score different on different sites?
There are many scoring models and several credit reporting companies, and the score also changes as your report changes.
Why do banks hold equity and reserves?
Reserves cover the losses that are expected. Equity covers the losses that turn out worse than expected.
Sources and review
- DBS Group Holdings, Annual report 2025: five-year summary, for customer loans, the non-performing loan rate, loss allowance coverage and allowances for credit and other losses. DBS reports in Singapore dollars, so only ratios are used here.
- Citigroup Inc., Fourth quarter 2025 financial supplement and results (preliminary), for loans net of unearned income, the allowance for credit losses on loans and the provision for credit losses on loans.
- Bank for International Settlements, Basel Framework, CRE32: IRB approach, risk components, for probability of default, loss given default as a percentage of exposure, and the supervisory loss given default of 45% for senior unsecured corporate claims.
- Financial Stability Institute, Bank for International Settlements, IFRS 9 and expected loss provisioning: executive summary, for the three stages, 12-month and lifetime expected credit losses, and the contrast with the incurred loss approach.
- Consumer Financial Protection Bureau, What is a credit score? and What is a debt-to-income ratio?, for what a credit score predicts and how it is used, and for the definition of the debt-to-income ratio.
How the numbers were checked: the expected losses and the rates needed were recomputed in plain code, and each rate was checked by confirming that the lender's expected receipts equal the loan plus its cost. The calculator, which runs separately, was compared with that code across a grid of inputs. The ratios for DBS and Citigroup were recomputed from the published figures cited above; Citigroup's fourth-quarter figures were preliminary when published.