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Discuss which variables matter when assessing a mortgage borrower, and how a credit scoring model uses cutoff scores together with default and loss rates.

FRM® Part II · Credit Risk Measurement and Management · Credit Scoring and Retail Credit Risk Management · FRM2.CR.7.e

The idea

Combined LTV adds any second lien to the loan. Plain loan to value is LTV = loan ÷ lower of price and appraised value. The front-end ratio counts housing costs only, so Front-end = housing payment ÷ gross income. The back-end ratio counts every debt payment, so Back-end = all debt payments ÷ gross income. Other variables are the credit bureau score and history, income stability and documentation, occupancy (owner or investor), loan purpose, term and rate type. A scoring model turns these into a score, and the cutoff score is the minimum accepted. Lowering the cutoff raises volume but also raises the default rate and loss rate of the accepted pool. Let R be revenue per good account, L the loss per bad account and b the bad rate of the marginal band. The profit-maximising cutoff is where the marginal band earns zero, so Profit = R × (1 − b) − L × b = 0. Compare marginal bands, not the portfolio average, which can stay positive while the last band loses money. The pool's expected loss rate is Loss rate = default rate × LGD.

Formulas

Combined loan-to-value ratio

CLTV = (first lien + second lien) ÷ min(price, appraised value)

first lien
the first mortgage loan
second lien
any second mortgage on the same property
price
the purchase price
appraised value
the appraised value of the property

Use the lower of price and appraisal and include every lien. Plain LTV leaves out the second lien.

Back-end debt-to-income ratio

back-end DTI = (housing payment + other debt payments) ÷ gross income

housing payment
the proposed mortgage payment including taxes and insurance
other debt payments
every other recurring debt payment, such as car loans and card minimums
gross income
income before tax

The front-end ratio uses the housing payment alone. Dividing by after-tax income is the usual slip.

Profit per account in a marginal score band

band profit = R × (1 − b) − L × b

R
the profit on a good account
L
the loss on a bad account
b
the bad rate of the band

Accept bands down to where this turns negative; break-even is b = R ÷ (R + L). Judge the marginal band, not the portfolio average.

A practice question

From the question bank, with the reason each wrong option is wrong.

What does the loan-to-value (LTV) ratio measure for a mortgage?

  1. The loan amount divided by the value of the property securing it. - correct
  2. The monthly mortgage payment divided by the borrower's gross monthly income.Why not: That is a payment-to-income or debt-to-income ratio, not LTV.
  3. The property value divided by the borrower's annual income.Why not: Confuses LTV with a house-price-to-income affordability measure.
  4. The loan amount divided by the borrower's liquid assets.Why not: A liquidity-cover measure, not the collateral coverage ratio.

LTV = loan / property value. A higher LTV means a thinner equity cushion and a higher expected loss given default.

A harder one

A borrower buys a house for USD 400,000 which is appraised at USD 380,000. The first-lien loan is USD 320,000 and there is a second lien of USD 20,000. Using the lower of price and appraised value, the combined LTV is closest to:

  1. 85.0%Why not: Uses the purchase price instead of the lower appraised value.
  2. 84.2%Why not: Ignores the second lien and so reports only the first-lien LTV.
  3. 89.5% - correct
  4. 111.8%Why not: Inverts the ratio, dividing property value by total debt.

Combined LTV = (320,000 + 20,000) / min(400,000, 380,000) = 340 / 380 = 89.5%. The lender uses the lower of price and appraisal and includes all liens.

The hardest tier

A lender has equal-sized score bands with bad rates of: 700 and above 1%, 650 to 699 3%, 600 to 649 6%, 550 to 599 11%, 500 to 549 18%. A good account earns a present value profit of USD 400 and a bad account loses USD 3,000. Which cutoff score maximises expected profit?

  1. 550 - correct
  2. 500Why not: Looks at the average profit across all bands, which stays positive, and ignores that the lowest band loses money at the margin.
  3. 600Why not: Double counts by subtracting the forgone USD 400 profit from the bad account loss as well, moving the break-even bad rate below 11%.
  4. 650Why not: Applies an arbitrary low bad rate ceiling rather than comparing marginal revenue and marginal loss.

Band profit per account = 400 x (1 - b) - 3,000 x b gives 366, 298, 196, 26, -212. Break-even bad rate = 400 / 3,400 = 11.8%. The 11% band earns about USD 26 and the 18% band loses USD 212, so accept down to 550.

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