Credit Risk Actuarial Science One on One — Part 4: Credit Expected Losses and Credit Unexpected… (2024)

The ‘standard scenario’ which people intuitively expect to happen when they consider the default risk of an obligor is ‘no default’, that is, no loss. This is indeed the most likely scenario if we consider each obligor individually (unless we consider an obligor of extremely low credit quality).

This scenario is also still used quite frequently for accounting purposes: a loan or bond is booked at its notional value (essentially assuming zero loss), and only if it gets into distress is it depreciated. In some institutions return on capital is still (incorrectly) calculated this way.

Unfortunately, this is one of the cases wheren naive intuition can lead us astray: the ‘standard scenario’ is not the mathematical expectation of the loss on the individual obligor. If we assume that exposure Ei and loss given default Li are known and constant, the credit expected loss is

E[Di(T)] =pi(T) ⋅ Ei ⋅ Li􀂘≠ 0

where pi(T) is the default probability of the obligor.

At the level of individual obligors in isolation, the concept of credit expected loss may be counterintuitive at first because we will never observe a realisation of the expected loss: either the obligor survives (then the realised loss will be zero) or the obligor defaults (then we will have a realised loss which is much larger than the expected loss).

There is a related trick question. Next time you go out, offer to buy your friend a drink if the next person entering the bar does not have an above average number of legs.

Of course your friend will have to buy you a drink if the person does indeed have an above average number of legs. You will win the bet if the next person has two legs: the average number of legs per person in the population must be slightly less than two because there are some unfortunate people who have lost one or both legs (but there are no people with more than two legs).

The same idea applies to credit obligors: most obligors will perform better than expected (they will not default), but there are some who perform significantly worse than expected. But nobody will perform exactly as expected.

Typically, the credit expected loss is small (because pi will be small) but it is positive. These small errors will accumulate when we consider a portfolio of many obligors. In a portfolio of 1,000 obligors we may no longer assume that none of these obligors will default.

Even if each of the obligors has a default probability of only 1%, we will have to expect 10 defaults. In the figure below the level of the credit expected loss of the portfolio is shown by the first (leftmost) vertical line; it is at a level of about $35 millions, that is, 3.5% of the portfolio’s notional amount.

Credit expected loss is an important concept when it comes to performance measurement, in particular in connection with risk-adjusted return on capital (RAROC) calculations. When a loan’s expected gain (in terms of excess earnings over funding and administration costs) is not sufficient to cover the credit expected loss on this loan, then the transaction should not be undertaken.

Suffering the credit expected loss (in particular, on a portfolio) is not bad luck: it is what you should expect to happen. Consequently, the credit expected loss should be covered from the portfolio’s earnings.

It should not require capital reserves or the intervention of risk management. The credit expected loss on a portfolio is the sum of the credit expected losses of the individual obligors:

This follows from the property of the expectation operator:

E(X + Y) = E(X) + E(Y)

However, this simple summation property will not hold for the credit unexpected loss! Credit unexpected loss is usually defined with respect to a VaR quantile and the probability distribution of the portfolio’s loss. Let us assume that D99% is the portfolio’s 99% VaR quantile, that is

P[D(T)≤D99%]=99%

Then the credit unexpected loss of the portfolio at a VaR quantile of 99% is defined as the difference between the 99% quantile level and the credit expected loss of the portfolio:

CUEL = D99% - E[D(T)]

If another risk measure such as conditional VaR is used in place of VaR, then is easily extended. We define credit unexpected loss in these situations by replacing D99% with the general risk measure.

The term ‘credit unexpected loss’ may be confusing at first, because it does not concern credit losses that were unexpected, but only something like a worst-case scenario. Intuitively, one might define credit unexpected loss as the amount by which the portfolio’s credit loss turns out, in the end, to exceed the originally credit expected loss:

max{D(T) - E[D(T)];0}

Here, we will use credit unexpected loss in the sense of the CUEL equation and not in the sense of the above equation.

In the above figure, the 99% and 99.9% VaR loss quantiles were shown with two vertical lines intersecting the tail of the loss distribution. The 99% VaR level is at approximately $160 millions, which yields a credir unexpected loss of $160 millions - $35 millions = $125 millions.

The 99.9% VaR level is at approximately $220 millions, with a credit unexpected loss of $185 millions. It is no coincidence that, even at a very high VaR quantile of 99.9%, the credit unexpected loss is still much less than the maximum possible loss of $1 billion that is suffered when the total portfolio defaults with zero recovery.

This effect stems from the partial diversification, which is still present in the portfolio despite an asset correlation of 20%.

As opposed to credit expected loss, the credit unexpected loss is not additive in the exposures. If, for example, we assume that with zero recovery and unit exposure each obligor defaults with a probability of 3%, then each obligor’s individual 99% VaR will be 1, its total exposure.

But the 99% VaR of a large portfolio of such obligors will not be the total exposure of the portfolio (unless we have the extreme case of perfect dependency between all obligors).

The credit unexpected loss is frequently used to determine the capital reserves that have to be held against the credit risk of the portfolio. It is not economically viable to hold full reserves against total loss of the portfolio, but reserving against credit unexpected loss at a sufficiently high quantile is viable and effective if it is done centrally under exploitation of all diversification effects.

Thus, coverage of the (linear) credit expected loss is in the domain of responsibility of the business lines, but the management of the (highly non-linear) unexpected loss is usually a task for a centralised risk management department. The risk management department then makes appropriate risk charges to the business lines.

A stylised capital allocation procedure is as follows:

1. Fix a VaR quantile for credit losses (usually 99% or 99.9%). This quantile should reflect the institution’s desired survival probability due to credit losses (this probability can be derived from its targeted credit rating). This is a management decision that has to be made at the top level.
2. Determine the portfolio’s credit expected loss.
3. Determine the credit unexpected loss of the portfolio.
4. Allocate risk capital to the portfolio to the amount of the credit unexpected loss.
5. Split up the portfolio’s risk capital over the individual components of the portfolio according to their risk capital contributions.

Losses in the portfolio up to the amount of the credit expected loss will have to be borne by the individual business lines (because these losses are economic losses), but any losses that exceed the credit expected loss will hit the risk capital reserves. Should these reserves not suffice to cover all losses, the bank itself will have to default.

But by setting the original VaR level, the probability of this event can be controlled.

Roi Polanitzer, CRA, QFV, FEM, F.IL.A.V.F.A., FRM, CRM, PDS, is a well-known authority in Israel the field of credit risk actuarial science and has written hundreds of papers that articulate many of the concepts used in modern credit risk actuarial science around the world. Mr. Polanitzer is the Owner and Chief Risk Actuary of Intrinsic Value — Independent Business Appraisers, a credit risk actuarial science consulting firm headquartered in Rishon LeZion, Israel. He is also the Owner and Chief Data Scientist of Prediction Consultants, a consulting firm that specializes in advanced analysis and model development.

Over more than 17 years, he has performed credit risk actuarial science consulting engagements in the areas of: subprime mortgages and securitization, counterparty risk (e.g., mitigation techniques, credit exposure profiles, collateralization and netting effects and pricing credit value adjustments — CVA), credit derivatives (e.g., types and uses, mechanics and structure, valuation and spread curves), structured finance and securitization (e.g.. structuring and securitization process, agency problems and moral hazard in the securitization process, tranching, subordination, and support), default risk (e.g., quantitative methodologies, and estimating defaults and recoveries from market prices and spreads), credit expected losses and credit unexpected losses and credit VaR.

Mr. Polanitzer holds an undergraduate degree in economics and a graduate degree in business administration, majoring in finance, both from the Ben-Gurion University of the Negev. He is a Full Actuary (Fellow), a Certified Risk Actuary (CRA), a Quantitative Finance Valuator (QFV) and a Financial and Economic Modeler (FEM) from the Israel Association of Valuators and Financial Actuaries (IAVFA). Mr. Polanitzer is the Founder of the IAVFA and currently serves as its chairman.

Mr. Polanitzer’s professional recognitions include being designated a Financial Risk Manager (FRM) by the Global Association of Risk Professionals (GARP), a Certified Risk Manager (CRM) by the Israel Association of Risk Managers (IARM), as well as being designated a Python Data Analyst (PDA), a Machine Learning Specialist (MLS), an Accredited in Deep Learning (ADL) and a Professional Data Scientist (PDS) by the Professional Data Scientists’ Israel Association (PDSIA). Mr. Polanitzer is the Founder of the PDSIA and currently serves as its CEO.

He is the editor of IAVFA’s weekly newsletter since its inception (primarily for the professional appraisal community in Israel).

Mr. Polanitzer develops and teaches credit risk actuarial science professional trainings and courses for the Israel Association of Valuators and Financial Actuaries, and frequently speaks on credit risk actuarial science at professional meetings and conferences in Israel. He also developed IAVFA’s certification programs in the field of actuarial science and he is responsible for writing the IAVFA’s statement of financial valuation standards.