Retail credit portfolios · RiskAtoms Academy

By the end you will be able to

1Explain why retail credit is managed by segment, and read the expected loss of each segment of a retail book.
2Use a points scorecard: turn a score into odds and a PD, and set a cut-off from the profit on good and bad loans.
3Read vintage curves and spot a vintage that is going wrong.
4Forecast write-offs from delinquency buckets and roll rates.

Why it matters

RiskAtoms Bank has 152,100 retail accounts and £4.49bn of retail exposure, more than seven times its corporate book. Nobody reads 152,100 files. A personal loan is approved in seconds by a scorecard, and the book is watched month by month as a set of statistics. Retail credit risk is managed like a factory: by measuring the output.

Core explanation

Many small loans, managed by numbers

Illustrative dataRiskAtoms Bank plc's credit book

The module's lending book: twelve invented corporate borrowers (exposure at default £587m, expected loss £2.81m), retail mortgages, personal loans and cards, a ten-grade master scale, personal-loan vintages and twenty past workouts. Every firm and figure is made up for teaching, in pounds, at one date; none is real data.

The retail segments and the personal-loan vintages are the book's. The scorecard, the applicant score distribution, the profit per loan and the card delinquency buckets are extras for this lesson.

Segments

L3.3.03: Analysing a household's credit analysed one household. A retail book is thousands of them, each too small to analyse by hand but similar enough to model as a group. The unit of analysis is the segment: loans with similar risk, measured together.

Segment	Accounts	EAD	PD	LGD	EL	EL / EAD
Mortgages: LTV up to 60%	9,800	£1,470m	0.3%	5%	£0.22m	0.02%
Mortgages: LTV 60–80%	8,200	£1,640m	0.6%	12%	£1.18m	0.07%
Mortgages: LTV 80–95%	3,100	£710m	1.5%	20%	£2.13m	0.30%
Personal loans: Score band A	21,000	£190m	1.2%	65%	£1.48m	0.78%
Personal loans: Score band B	14,000	£130m	3.5%	70%	£3.19m	2.45%
Personal loans: Score band C	6,000	£50m	9.0%	75%	£3.38m	6.75%
Credit cards: Transactors	52,000	£91m	0.8%	80%	£0.58m	0.64%
Credit cards: Revolvers	38,000	£204m	4.5%	85%	£7.80m	3.82%

The total expected loss is £20.0m a year. Mortgages are 85% of the exposure but only 18% of the expected loss: the house protects the lender. Cards are the opposite. Even inside one product the loss rate varies by a factor of ten or more between segments, which is why a bank prices, limits and provisions segment by segment, not product by product.

Scorecards in outline

A scorecard adds up points for the answers on an application and the data in the borrower's credit file. The score is built so that it is a straight-line function of the log of the good:bad odds: here 600 points means odds of 50 to 1, and every 20 points doubles the odds (PDO = 20). A “bad” is usually 90 days past due within a year. The points come from a logistic regression on past loans (M6.4: Credit risk modelling). Try it:

Score an applicant

Illustrative scorecard. Base points 380.

Age of oldest credit account

Missed payments on the credit file, last 12 months

Unsecured debt payments / income

Housing

Years as a customer of the bank

Score

600

Odds

50.0 : 1

PD

2.0%

Above the break-even score of 550.5: approve.

On this scale the book's personal-loan bands sit at scores of about 614 (band A, PD 1.2%), 583 (band B, PD 3.5%), 554 (band C, PD 9.0%). An application score is used once, to decide. A behavioural score is recalculated every month from how the customer runs the account, and drives credit limits, collections and the PDs used for provisions.

Where to set the cut-off is a business decision. Approving one more loan earns about £700 over its life if it is repaid and costs about £6,300 if it goes bad. So it pays to lend while the odds are better than 6,300 / 700 = 9.0 to 1: a score of 550.5. Two cautions. A scorecard learns only from loans that were approved, so banks use reject inference when they rebuild one. And the law limits what may be scored: the UK's Equality Act 2010 and the US Equal Credit Opportunity Act, for example, protect characteristics such as race and sex.

Vintage analysis

Defaults on a personal loan come mostly 6 to 24 months after it is made. If the book grows fast, the overall default rate falls for a while simply because most loans are too young to have defaulted. A vintage curve removes that effect: it follows each quarter's loans by months on book. V1 is the quarter three years ago; V12 the latest, seen for only 3 months.

Personal-loan vintages: cumulative default rate by month on book

Red: the three vintages written under looser lending criteria. Click a vintage to pick it out.

At 12 months on book the normal vintages average 2.99% cumulative defaults; V6 to V8 average 4.46%, about half as much again. They were written when the bank loosened its lending criteria, and the curves showed it within a year, long before those loans reached the end of their term. The oldest vintage reached 5.53% by month 36. Vintage curves are also how a bank checks that a new scorecard or cut-off is doing what it promised.

Delinquency buckets and roll rates

For revolving products such as cards, collections teams track balances by delinquency bucket. Each month some of each bucket cures, and some rolls to the next. The book's card revolvers today:

Bucket	Balance	Rolls to next bucket each month
Current	£141.0m	2.5% → 30–59 days
30–59 days	£4.2m	35.0% → 60–89 days
60–89 days	£2.6m	60.0% → 90–119 days
90–119 days	£2.2m	75.0% → write-off

The product of the roll rates, the flow rate, is 0.394%: the share of a current balance that is written off four months later. Roll rates are early warning in their own right: a rise in the current-to-30 roll is the first sign of trouble, months before any write-off.

The formal version

Scorecard scaling. The score is linear in the log of the odds. With base score $S_0 = 600$ at odds $O_0 = 50$ and $\text{PDO} = 20$ :

\begin{aligned} S &= \text{offset} + \text{factor}\times\ln(\text{odds}) \\ S + \text{PDO} &= \text{offset} + \text{factor}\times\ln(2\times\text{odds}) \;\Rightarrow\; \text{PDO} = \text{factor}\times\ln 2 \;\Rightarrow\; \text{factor} = \frac{20}{\ln 2} = 28.854 \\ \text{offset} &= S_0 - \text{factor}\times\ln O_0 = 600 - 28.854\times\ln 50 = 487.123 \\ \text{odds}(S) &= \exp\!\Big(\frac{S - \text{offset}}{\text{factor}}\Big) = 50\times 2^{(S - 600)/20}, \qquad \text{PD}(S) = \frac{1}{1 + \text{odds}(S)} \end{aligned}

Break-even odds. Approving a loan with PD $p$ adds $(1-p)\,G - p\,L$ to profit, where $G$ is the gain on a good loan and $L$ the loss on a bad one. It is worth approving while this is positive:

\begin{aligned} (1-p)\,G > p\,L \iff \frac{1-p}{p} > \frac{L}{G} = \frac{6{,}300}{700} = 9.00 \iff S > 487.12 + 28.85\ln 9.00 = 550.5 \end{aligned}

Here $L = \text{loan} \times \text{LGD} = 9{,}000 \times 0.70$ .

Vintages and roll rates. For vintage $v$ with $N_v$ loans, of which $D_v(m)$ have defaulted by month $m$ on book, and roll rates $r_k$ from bucket $k$ to $k+1$ :

\begin{aligned} \text{CDR}_v(m) &= \frac{D_v(m)}{N_v}, \qquad \text{flow rate} = \prod_{k=0}^{3} r_k = 0.025\times0.35\times0.60\times0.75 = 0.394\% \end{aligned}

Simplification: each roll rate is a fixed share; balances that do not roll forward are assumed to cure or be paid down. Real roll-rate models are matrices that also let balances move back (L3.3.07: Rating migration uses the same algebra for ratings).

Try it

The bank receives applicants whose scores are spread around 595 (an illustrative, bell-shaped distribution). Move the cut-off from 600 down to the break-even score and watch the profit curve. Then keep going lower: past the break-even, each extra approval loses money on average.

Try itSet the cut-off

Cut-off score (approve at or above)600 (odds 50.0 : 1, PD 2.0%)

Applicants approved45.5%

Bad rate among approved0.90%

Expected profit per 1,000 applicants£289,864

Marginal applicant: profit if approved£563

Break-even score550.5

Approving everyone gives a bad rate of 4.9%; the best cut-off, 551, approves 87% with a bad rate of 2.7%. The best cut-off sits at the break-even score, as the formal version predicts.

Worked example

One applicant, then the card book's next four months

From a score to a decision, and from buckets to write-offs

Step 1 of 5

1

Score the applicant

Characteristic	Answer	Points
Age of oldest credit account	4 to 10 years	+45
Missed payments on the credit file, last 12 months	One	+35
Unsecured debt payments / income	15% to 30%	+40
Housing	Renting	+25
Years as a customer of the bank	1 to 5	+25

Base 380 plus the points gives a score of 550.

Regulatory lens

Under the internal ratings-based approach, retail exposures are usually not rated one by one: banks assign them to pools of similar risk and estimate PD, LGD and EAD for each pool, which is why segmentation matters to capital as well as to management. The definition of default uses the same 90 days past due test as for companies, with a lower absolute materiality threshold for retail in the EU (at most €100, against €500 for other exposures, alongside a relative threshold normally set at 1% of the exposure, under the EBA's technical standards); the PRA set both of the UK's retail thresholds at zero. Conduct rules sit alongside: in the UK, the FCA's consumer credit sourcebook (CONC) requires a creditworthiness assessment before lending, and scorecards must not discriminate unlawfully. Retail IRB capital is in M4.2: Credit risk capital; scorecard building and validation in M6.4: Credit risk modelling.

Spark Atoms

Spark AtomCredit scoring in 1941

In 1941 David Durand published Risk Elements in Consumer Instalment Financing for the US National Bureau of Economic Research. He compared samples of good and bad instalment loans and built “credit-rating formulae” from applicants' characteristics: one of the first statistical scorecards, decades before computers made them routine.

Spark AtomFrom an engineer and a mathematician

Fair, Isaac and Company was founded in 1956 by engineer Bill Fair and mathematician Earl Isaac to sell scoring systems to lenders. Its first general-purpose FICO score came in 1989; base FICO scores run from 300 to 850.

Key takeaways

Retail credit is managed by segment: loss rates differ by a factor of ten or more between segments of one product.
A scorecard's score is linear in ln(odds): odds = 50 × 2^((S − 600)/20) on this scale, and PD = 1/(1 + odds).
The break-even cut-off is where the odds equal loss per bad / gain per good; the profit-maximising cut-off sits there.
Vintage curves compare loans of the same age, so growth cannot hide a bad batch; V6 to V8 showed looser lending within a year.
Roll rates move balances bucket by bucket; their product, the flow rate, forecasts write-offs months ahead.

Reaction

Check what stuck

Question 1 · objective 2

The scorecard gives 600 points at odds of 50:1 and doubles the odds every 20 points. What are the good:bad odds at a score of 640?

: 1

Question 2 · objective 2

Using the same scaling, what PD does a score of 560 imply, in % (two decimals)?

%

Question 3 · objective 2

The bank raises its cut-off from 570 to 600. What happens?

AMore applicants are approved and the bad rate risesBFewer applicants are approved and the bad rate falls, but profit can fall too, because good customers are turned awayCThe PD of each applicant fallsDNothing changes for applicants scoring above 570

Question 4 · objective 3

A personal-loan book has grown fast. Its overall default rate this year is lower than last year. Why might the vintage curves still show a problem?

AThey cannot: a lower default rate means better loansBNew loans have not had time to default, so fast growth dilutes the overall rate; vintages compare loans of the same ageCVintage curves only show mortgagesDVintage curves ignore loans that defaulted

Question 5 · objective 4

Monthly roll rates are 4% (current to 30 days), 40% (30 to 60), 50% (60 to 90) and 80% (90+ to write-off). What share of today's current balance is expected to be written off, in % (two decimals)?

%

Glossary

Segment: A group of retail loans with similar risk (product, LTV band, score band, months on book), managed and measured together.
Scorecard: A table that gives points for each answer on an application and in the credit file. The total score ranks applicants by their chance of going bad.
Good:bad odds: Good:bad odds: the number of borrowers expected to repay for each one expected to go bad. Odds of 50 mean a PD of 1/51.
Points to double the odds (PDO): Points to double the odds: how many points a score must rise for the odds to double. Here, 20.
Cut-off: The lowest score the bank will approve. Applicants below it are declined or referred to an underwriter.
Bad (in scoring): The outcome a scorecard predicts, usually 90 days or more past due (or default) within 12 months of the loan being made.
Behavioural score: A score recalculated every month for existing customers from how they use and repay their accounts. Used for limits, collections and PDs.
Reject inference: Estimating how declined applicants would have behaved, so a new scorecard is not built only on past approvals.
Vintage: All the loans made in one period (here a quarter), followed together as they age.
Months on book: Months on book: how long ago a loan was made. Vintage curves line loans up by it.
Delinquency bucket: A delinquency band: current, 30–59 days past due, 60–89 days and so on.
Roll rate: The share of balances in one delinquency bucket that move to the next bucket a month later.
Flow rate: The product of the roll rates from current to write-off: the share of today's current balance expected to be written off.

Deep Dives & Origins

Deep Dives

N. Siddiqi, Intelligent Credit Scoring: Building and Implementing Better Credit Risk Scorecards, 2nd ed., Wiley, 2017. Intermediate.
L. C. Thomas, J. N. Crook and D. B. Edelman, Credit Scoring and Its Applications, 2nd ed., SIAM, 2017. Advanced.

Origins

D. Durand, Risk Elements in Consumer Instalment Financing, Technical Edition, National Bureau of Economic Research, 1941.
European Banking Authority, Final draft RTS on the materiality threshold for credit obligations past due (EBA/RTS/2016/06), 2016.
Financial Conduct Authority, Consumer Credit sourcebook (CONC), chapter 5.

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