Review what is gained from pricing financial services according to risk.

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

The idea

Risk-based pricing sets each borrower's rate to cover funding cost, operating cost, expected loss and a return on the capital the loan uses, so higher-risk borrowers pay more. Let r be the loan rate, cost the funding plus operating cost per unit lent, and PD and LGD the default probability and loss given default. The break-even rate for a one-year loan is the one at which expected repayment, including recoveries, equals the amount lent plus its cost. Solving that condition for r gives the following line. r = (1 + cost − PD × (1 − LGD)) ÷ (1 − PD) − 1. The gains are a clearer view of profitability by segment and better capital allocation. Applicants rejected at a single price can be accepted at a higher one, widening access. There is also less cross-subsidy from low-risk to high-risk borrowers, and better protection against adverse selection by competitors that price on risk. The limits are that very high prices can raise default and discourage the best of a risky group, and fair-lending rules constrain the risk factors used. A single price makes good borrowers subsidise bad ones.

Formulas

Break-even rate on a one-year loan

r = [1 + cost − PD × (1 − LGD)] ÷ (1 − PD) − 1

cost: funding and operating cost, as a share of the amount lent
PD: the probability of default
LGD: loss given default

Solves (1 − PD) × (1 + r) + PD × (1 − LGD) = 1 + cost. It exceeds cost + PD × LGD because defaulted loans pay no interest.

Risk-based loan rate covering a hurdle return on capital

rate = funding cost + operating cost + expected loss + hurdle × capital share

funding cost: the cost of funding, as a share of the loan
operating cost: operating costs, as a share of the loan
expected loss: expected loss, as a share of the loan
hurdle: the required return on economic capital
capital share: economic capital as a share of the loan

Apply the hurdle to the capital, not to the whole loan. Ignores any funding benefit from the capital.

A practice question

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

Which is a benefit to a lender of pricing financial services according to risk?

Applicants declined at a single price can be accepted at a higher one, widening access and cutting cross-subsidy. - correct
It eliminates credit losses on high-risk borrowers.Why not: Pricing compensates for expected losses; it does not remove them.
It completely removes adverse selection, because all borrowers accept the quoted price.Why not: Higher prices can still cause the best of a risk group to leave.
It removes the need for scoring models, since the price itself reveals each borrower's risk.Why not: Pricing depends on scoring models to estimate risk first.

With a single price, good borrowers subsidise bad ones and marginal applicants are rejected. Risk-based pricing removes that subsidy and allows more lending.

A harder one

A one-year loan has funding and operating cost of 3% of the amount lent, PD of 8% and LGD of 60%. If a defaulted loan pays nothing but the recovery, the break-even loan rate that solves (1 - PD) x (1 + r) + PD x (1 - LGD) = 1 + cost is closest to:

7.80%Why not: Adds cost and expected loss as a spread; this ignores that defaulted loans pay no interest, so the rate is too low.
8.48% - correct
11.00%Why not: Adds the PD itself instead of the expected loss (PD x LGD).
3.00%Why not: Covers only the funding and operating cost and ignores expected loss.

(1 - 0.08)(1 + r) + 0.08 x 0.40 = 1.03 so 1 + r = 0.998 / 0.92 = 1.0848 and r = 8.48%. The exact solution exceeds the simple addition of cost and EL because interest is not earned on defaulted loans.

The hardest tier

A lender has two equal-sized segments. Good borrowers have PD 1% and bad borrowers 9%. LGD is 50% and funding and operating cost is 3%. Using the break-even rate (1 - PD) x (1 + r) + PD x (1 - LGD) = 1 + cost, a single price is set at the break-even rate for the blended PD of 5%. By how much does the good segment overpay relative to its own break-even rate?

2.45 percentage pointsWhy not: Gives the bad segment's underpayment (its break-even less the single price), not the good segment's overpayment.
4.71 percentage pointsWhy not: Gives the spread between the two segments' break-even rates, not the overpayment against the blended price.
2.00 percentage pointsWhy not: Uses the blended expected loss (2.5%) less the good segment's expected loss (0.5%) and ignores the effect of defaults on interest income.
2.25 percentage points - correct

Break-even: good 3.54%, bad 8.24%, blended 5.79%. Good overpay 5.79% - 3.54% = 2.25 points, while bad underpay 2.45 points: the cross-subsidy.

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