Expected loss

Graybeck Plastics owes RiskAtoms Bank £12m: about 2% of the corporate book. Yet it accounts for 36% of the book's expected loss. Expected loss is the cost of lending that the bank knows about in advance. It has to be priced into every loan and set aside in provisions, and it shows at once where the risk in a book sits.

The three parameters of the last three lessons meet here. Expected lossExpected loss: the average credit loss over a horizon (usually one year), PD × LGD × EAD for one exposure, summed over a portfolio. (EL) is $\text{PD} \times \text{LGD} \times \text{EAD}$, a one-year figure unless stated (L2.2.03: Credit and counterparty risk). For the twelve corporate borrowers, with PDs from the master scale (L3.3.04: Internal ratings and master scales), LGDs after collateral (L3.3.08: Loss given default and recovery) and EADs after CCFs (L3.3.09: Exposure at default):

The book loses 0.48% of its exposure in an average year: its EL rateExpected loss divided by EAD: the share of the exposure lost on average each year, PD × LGD.. Two weak borrowers, Graybeck and Hastemoor, hold 6% of the EAD and 58% of the EL. The large, well-rated, secured loans barely register. Retail is a much bigger book with a similar EL rate:

Retail loses 0.45% a year on average, and the bank as a whole expects to lose £22.8m a year. Mortgages are most of the retail exposure, but cards and personal loans, with high PDs and LGDs, carry most of its expected loss.

An expectation, not a forecast

Expected loss is an expectation in the sense of M-C02: Expectation: a probability-weighted average over everything that can happen. Take Bramwick Stores plc, and suppose that if it defaults, the bank loses either 30% (a quick sale of the business) or 60% (a long administration), equally likely:

The average LGD is 45%, so EL is the same £386,460 as before. But in 98.9% of years the loss is zero, and otherwise it is £22.8m or more. The standard deviation of the loss is £3.81m, about 10 times the EL. Expected loss is the cost the bank charges for and provides for; the spread around it is unexpected loss, which needs capital (L3.3.11: Unexpected loss and default correlation).

Risk-based pricing

A loan's interest must pay for four things. Sort each cost first:

With funding at 4.50%, operating cost 0.40% a year and a pre-tax hurdle rateThe minimum return shareholders require on the capital they put up. of 15% on capitalCapital the bank holds against a loan so that losses above the expected level, up to a high quantile, do not make it insolvent., the required rate is $r = f + \text{PD}\times\text{LGD} + c + K(h - f)$. The capital per £, $K$, comes from the one-factor model of the next lesson: the extra loss, above EL, in a 1-in-1,000 year, with an asset correlation of 0.20. For now, take it as given. It is a simplified version of the regulatory IRB formula, which also adds a maturity adjustment and lets the asset correlation fall as PD rises (M4.2: Credit risk capital). By grade, for an unsecured loan with LGD 45%:

The price rises slowly through the strong grades and steeply through the weak ones, because both EL and capital grow with PD. This is risk-based pricingSetting each loan's rate from its own risk: funding, expected loss, costs and a return on the capital it needs.. Suppose instead the bank charged every corporate borrower the same rate, the book's EAD-weighted average required rate of 5.82%. The 7 strong borrowers would be overcharged and could refinance elsewhere; the 5 weak ones, Graybeck at a required 17.03% among them, would be glad to stay. That is adverse selectionWhen one price is charged to borrowers of different risk, the safer ones leave for cheaper lenders and the riskier ones stay..

Expected loss. Let $\mathbf 1_D$ be 1 if the borrower defaults within the year (probability PD) and 0 otherwise. The loss is $L = \text{EAD} \times \text{LGD} \times \mathbf 1_D$. If EAD and LGD are taken as their averages given default:

The LGD and EAD that belong in the formula are those given default. If recoveries are worse and lines fuller in the years when defaults are high, those averages are higher than in a normal year, which is why banks use downturn LGDs and CCFs measured on defaulted facilities.

Price and return on capital. Per £1 of EAD for a year, with capital $K$ and the rest funded at $f$, the profit after expected loss and costs must give the hurdle return on $K$:

A loan priced at $r$ earns exactly $h$; any lower rate earns less. (Tax, fees and the loan's term are left out to keep the structure clear.)

Move the grade from 1 to 9 and watch the capital charge overtake the expected loss. Then propose a rate and see whether it meets the hurdle.

Regulatory capital for credit risk is set for unexpected loss; expected loss is meant to be covered by pricing and provisions. Banks on the IRB approach compare their expected loss with their provisions, and a shortfall reduces capital (M4.2: Credit risk capital). Accounting provisions use expected credit losses too, over twelve months or the loan's lifetime under IFRS 9 (L3.3.18: Provisioning basics, M4.7: Accounting for risk). Risk-adjusted performance measures and capital allocation come back in M6.5: Portfolio credit models.

• EL = PD × LGD × EAD; the corporate book's is £2.81m on £587m (0.48%), the bank's £22.8m.
• Expected loss is an average over outcomes; a single loan almost never loses exactly its EL.
• Portfolio EL is the sum of the loans' ELs: diversification does not reduce it.
• A risk-based price is funding + EL + costs + capital × (hurdle − funding).
• RAROC tests a proposed rate; one price for all borrowers invites adverse selection.

Expected loss (EL)

Expected loss: the average credit loss over a horizon (usually one year), PD × LGD × EAD for one exposure, summed over a portfolio.

EL rate

Expected loss divided by EAD: the share of the exposure lost on average each year, PD × LGD.

Risk-based pricing

Setting each loan's rate from its own risk: funding, expected loss, costs and a return on the capital it needs.

Economic capital

Capital the bank holds against a loan so that losses above the expected level, up to a high quantile, do not make it insolvent.

Hurdle rate

The minimum return shareholders require on the capital they put up.

RAROC

Risk-adjusted return on capital: income after funding, expected loss and costs, divided by the capital the loan needs.

Adverse selection

When one price is charged to borrowers of different risk, the safer ones leave for cheaper lenders and the riskier ones stay.