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FRM Exam Part II · Range of Practices and Issues in Economic Capital Frameworks

Risk Measures and Confidence Levels in Economic Capital

Updated 11 October 2026 · Fact-checked

Economic capital is the loss buffer a bank holds at a chosen risk measure (VaR or expected shortfall), confidence level and time horizon. The confidence level is usually set from the bank's target credit rating: higher rating means lower default probability, so a higher percentile. The horizon is commonly one year.

Understand Risk Measures and Confidence Levels in Economic Capital

Economic capital is the capital a bank thinks it needs to absorb unexpected losses. You measure it from a loss distribution. Three choices define the number: the risk measure, the confidence level and the time horizon.

The most common risk measure is VaR. Economic capital is often VaR at the chosen confidence level minus expected loss, because expected loss is covered by pricing and provisions. The alternative is expected shortfall (ES), the average loss in the tail beyond the VaR point. ES is coherent and captures tail size. VaR ignores losses beyond the cut-off. Practice is mixed: many banks still use VaR because it is simple and easy to link to ratings, and some use ES or a tail-based measure.

The confidence level reflects the bank's solvency standard. If a bank wants to hold a target rating such as AA, it sets the confidence level so that the chance of the loss exceeding capital over the horizon roughly equals the default probability that rating historically shows. A 99.97% level implies a 0.03% one-year failure probability, which is in the range banks often associate with AA or higher. A lower target rating means a lower confidence level and less capital.

The time horizon is the period over which losses are measured. One year is the usual choice for credit and operational risk, because it matches the planning, reporting and capital-raising cycle. Market risk VaR often uses shorter horizons for trading, but economic capital for trading books may use a longer one to reflect how long it takes to liquidate or hedge positions.

The link to remember: a higher confidence level, a longer horizon or a tail-sensitive measure all raise capital. These choices are judgement, not fact. Confidence levels that are very high are hard to validate, because there is little data in the far tail.

Key formulas to remember

Economic capital (VaR basis)
EC = VaR(α) − Expected Loss
Unexpected loss at confidence level α over the horizon. Some banks hold the full VaR instead, so read the question.
Expected shortfall
ES(α) = E[Loss | Loss ≥ VaR(α)]
Average tail loss. ES(α) is always at least VaR(α).
Link of confidence level to rating
Confidence level α = 1 − target one-year default probability
Example: target default probability 0.03% gives α = 99.97%.
Normal VaR
VaR(α) = μ + z(α) × σ
For normal losses over the horizon. z(99%) ≈ 2.33, z(99.9%) ≈ 3.09, z(95%) ≈ 1.645.
Normal ES
ES(α) = μ + σ × φ(z) ÷ (1 − α)
φ is the standard normal density. At 99%, ES ≈ μ + 2.665σ.
Square-root-of-time scaling
VaR(T days) ≈ VaR(1 day) × √T
Valid only for independent, identically distributed returns with zero mean. Weak for credit and illiquid positions.

How to solve Risk Measures and Confidence Levels in Economic Capital questions

Use this order for any question on measures, confidence levels and horizons.

  1. 1Identify what is asked: the choice of measure, the confidence level, the horizon, or a capital number.
  2. 2If a target rating is given, convert it to a default probability and set α = 1 − that probability.
  3. 3Fix the horizon. Default to one year unless the question gives a liquidation or holding period.
  4. 4If numbers are given, compute VaR from the loss distribution, using z-values if losses are normal.
  5. 5Subtract expected loss if the question defines economic capital as unexpected loss.
  6. 6If ES is asked, use the tail average, not the cut-off, and check ES ≥ VaR.
  7. 7Scale the horizon only if the independence assumption is stated or reasonable.
  8. 8Interpret: say what the figure means for solvency, and name the limitation of the measure.

Quickest way: Rating-to-capital shortcut

When to use it: Use when a question gives a target rating or default probability and a normal loss distribution.

  1. Write α = 1 − default probability.
  2. Pick the z-value you know: 2.33 for 99%, 3.09 for 99.9%, 1.645 for 95%.
  3. Compute capital = z × σ (or μ + z × σ, then subtract expected loss).
  4. For a horizon change, multiply by √T only if independence holds.
  5. Eliminate options that put ES below VaR or give higher capital for a lower confidence level.

Common mistakes in Risk Measures and Confidence Levels in Economic Capital

  • Treating a higher target rating as needing a lower confidence level

    Confusing the rating with the tail probability.

    Fix: Better rating means lower default probability, so a higher α and more capital.

  • Saying VaR shows how large losses are beyond the cut-off

    VaR is read as a worst case.

    Fix: VaR is only a threshold. ES measures the average size of losses beyond it.

  • Forgetting to subtract expected loss

    Candidates assume capital equals VaR.

    Fix: Check the definition in the question. Economic capital as unexpected loss is VaR minus expected loss.

  • Applying √T scaling to credit losses

    The rule is learned from market risk.

    Fix: Use it only for independent, zero-mean returns. Credit risk is normally measured directly over one year.

  • Claiming ES is always bigger capital for the same α in a way that is comparable

    ES and VaR at the same α are mixed up.

    Fix: ES at α is at least VaR at α. To compare fairly, match the measures, for example ES at 97.5% is close to normal VaR at 99%.

  • Treating the confidence level as a fixed regulatory number for economic capital

    Confusing it with the 99.9% Basel regulatory credit and operational standard.

    Fix: Economic capital is internal. The bank chooses α, usually from its target rating.

Worked examples

Example 1

A bank has a target one-year default probability of 0.1%. Its one-year portfolio loss is normal with mean $40 million and standard deviation $50 million. Using z(99.9%) = 3.09, what is economic capital defined as unexpected loss?

Show the solution
  1. α = 1 − 0.001 = 99.9%.
  2. VaR = 40 + 3.09 × 50 = 40 + 154.5 = $194.5 million.
  3. Economic capital = VaR − expected loss = 194.5 − 40 = $154.5 million.

Answer: $154.5 million

Example 2

One-year losses are normal with mean 0 and standard deviation $20 million. At 99% confidence, VaR = 2.33σ and ES = 2.665σ. Which statement is correct, and what is the extra capital if the bank moves from VaR to ES? Options: (A) ES is $46.6 million, below VaR; (B) ES is $53.3 million, VaR is $46.6 million, difference $6.7 million; (C) ES equals VaR at 99%; (D) VaR is $53.3 million and ES is $46.6 million.

Show the solution
  1. VaR = 2.33 × 20 = $46.6 million.
  2. ES = 2.665 × 20 = $53.3 million.
  3. Difference = 53.3 − 46.6 = $6.7 million.
  4. ES is above VaR, so only option B fits.

Answer: B: VaR $46.6 million, ES $53.3 million, extra capital $6.7 million

Exam tips

  • Expect case-style questions that give a rating and ask for the confidence level. Convert rating to default probability first.
  • Know that ES is coherent and VaR is not, and that VaR ignores tail size. These appear as conceptual options.
  • Check whether the question defines capital as VaR or VaR minus expected loss.
  • Watch for horizon traps: one year is the usual economic capital horizon, and √T scaling needs stated assumptions.
  • Remember that very high confidence levels are hard to validate with data.

Practice questions from Range of Practices and Issues in Economic Capital Frameworks

Risk Measures and Confidence Levels in Economic Capital: frequently asked questions

How is the confidence level linked to a target credit rating?

The bank takes the default probability that its target rating implies and sets α equal to one minus that probability. A higher target rating means a smaller default probability and a higher confidence level. This is the solvency standard the capital is meant to meet.

Should economic capital use VaR or expected shortfall?

Both are used in practice. VaR is simple and ties neatly to a default probability. ES is coherent and reflects how bad tail losses are, so it suits fat-tailed risks. Be ready to argue the trade-off in an answer.

Why is a one-year horizon common for economic capital?

It matches the budgeting, reporting and capital-raising cycle, and it is long enough for credit and operational losses to emerge. Trading books may use shorter or liquidity-adjusted horizons.

Is economic capital the same as regulatory capital?

No. Regulatory capital follows Basel rules set by supervisors. Economic capital is the bank's own estimate, based on its chosen measure, confidence level and horizon.