Separate risk capital, economic capital and regulatory capital from one another, and explain why and by which techniques economic capital is used to apportion risk capital.

FRM® Part II · Operational Risk and Resilience · Risk Capital Attribution and Risk-Adjusted Performance Measurement · FRM2.ORR.18.a

The idea

Three capital concepts are easy to confuse. Regulatory capital is the minimum a supervisor requires, set by rules such as Basel. Economic capital is the bank's own estimate of the capital needed to absorb unexpected losses at a chosen confidence level over a horizon, usually one year, across credit, market and operational risk. The confidence level might be 99.9 percent, or 99.97 percent for an AA target. Let Q be the loss at the confidence level and EL the expected loss, which is covered by pricing and provisions. Risk capital is that buffer measured for a unit, product or position, and it is the difference, RC = Q − EL. Banks apportion economic capital to units so performance can be compared on a risk-adjusted basis, prices set and capital budgeted. The techniques are stand-alone capital, incremental capital (firm capital with the unit minus without it) and contribution or covariance-based allocation, which splits the diversified total so the pieces add up. Remember that stand-alone figures usually sum to more than the firm total.

Formulas

Risk capital

risk capital = loss quantile − EL

loss quantile: the loss at the chosen confidence level over the horizon
EL: the expected loss

Expected loss is covered by pricing and provisions, so capital covers only the unexpected part. Holding capital for the whole quantile double counts EL.

Incremental capital of a unit

incremental capital = EC with unit − EC without unit

EC with unit: firm economic capital including the unit
EC without unit: firm economic capital excluding it

Lower than stand-alone capital because of diversification. Incremental figures do not sum to the firm total.

Pro rata allocation of diversified capital

allocated capital = stand-alone × (diversified total ÷ sum of stand-alone)

stand-alone: the unit's stand-alone economic capital
diversified total: the firm's diversified economic capital
sum of stand-alone: the sum of all units' stand-alone figures

The allocations add up to the firm total, unlike stand-alone figures, which sum to more.

A practice question

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

In a bank's risk management, what is economic capital?

The bank's own estimate of the capital needed to absorb unexpected losses at a chosen confidence level over a stated horizon. - correct
The minimum capital that supervisors require the bank to hold under the Basel rules, set as a share of risk-weighted assets.Why not: Describes regulatory capital, which is set by rules, not by the bank's own internal model.
The book value of shareholders' equity shown on the balance sheet, net of goodwill and intangible assets.Why not: Uses an accounting measure; economic capital is a risk-based measure and bears no fixed link to book equity.
The provisions and reserves set aside to cover the losses the bank expects to suffer in a normal year.Why not: Describes expected-loss reserves; economic capital is held against unexpected losses, not expected ones.

Economic capital is internally modelled capital to cover unexpected loss at a target solvency (confidence) level; regulatory capital is the supervisory minimum; reserves cover expected loss.

A harder one

Three units have stand-alone economic capital of USD 100 million, 60 million and 40 million. The firm's diversified economic capital is USD 160 million. If the diversified total is spread pro rata to stand-alone capital, how much is allocated to the unit with USD 60 million stand-alone capital?

USD 60 millionWhy not: Uses stand-alone capital and gives the unit no share of the diversification benefit.
USD 53.3 millionWhy not: Splits the diversified total equally, ignoring the units' different stand-alone risk.
USD 40 millionWhy not: Reports the total diversification benefit (200 - 160), not the unit's allocation.
USD 48 million - correct

Scale each stand-alone figure by 160/200 = 0.8: 60 x 0.8 = 48. The three allocations (80, 48, 32) sum to the firm total of 160.

The hardest tier

A bank with a target AA rating models one-year economic capital of USD 4.2 billion. Its regulatory minimum is USD 3.6 billion and it holds USD 3.8 billion of capital. Which conclusion is correct?

It meets the regulatory minimum with USD 0.2 billion spare but is USD 0.4 billion short of its own model's need. - correct
It is fully capitalised, because its holdings exceed the regulatory minimum that supervisors set for the bank.Why not: Treats the regulatory minimum as the only standard and ignores the shortfall against economic capital.
It breaches the regulatory minimum by USD 0.4 billion.Why not: Confuses the economic capital figure with the regulatory requirement.
It has a surplus of USD 0.4 billion over regulation and a surplus of USD 0.2 billion over economic capital.Why not: Swaps the benchmarks: the surplus is against regulation and the shortfall against economic capital.

Surplus over regulation = 3.8 - 3.6 = 0.2; shortfall against economic capital = 4.2 - 3.8 = 0.4. Regulatory and economic capital are separate benchmarks and either can be the binding one.

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