Charta is on Google Play: practise on the go. Get the app
Charta

Assess what capital a life insurer and a property-casualty insurer must hold.

FRM® Part I · Financial Markets and Products · Insurance Companies and Pension Plans · FRM1.FM.2.h

The idea

Insurers hold capital so that policyholder claims can still be paid after an adverse year. A life insurer holds long-dated liabilities against a large asset portfolio. Its capital is sensitive to interest rates, credit losses and the asset-liability duration gap, as well as to mortality and longevity. A property-casualty insurer has shorter policies but volatile, partly correlated claims, so capital is driven by catastrophe risk, reserve risk and pricing risk. Modern regimes set capital with a confidence level. Solvency II in Europe defines the solvency capital requirement as a one-year 99.5 percent value at risk. Risk modules are combined using correlations, so total capital is below the simple sum. Economic capital = loss quantile − expected loss. Reinsurance reduces underwriting capital but adds counterparty credit risk, which is also charged.

Formulas

Economic capital for a normally distributed loss

EC = z × σ

z
the standard normal quantile for the confidence level, such as 2.576 at 99.5%
σ
the standard deviation of the loss

Economic capital is the loss quantile minus the expected loss, so the mean drops out. Adding the mean back is the trap.

Aggregating two capital charges with a correlation

Total capital = √(C₁² + C₂² + 2 × ρ × C₁ × C₂)

C₁
the stand-alone capital charge for the first risk
C₂
the stand-alone capital charge for the second risk
ρ
the correlation between the two risks

Below the simple sum unless the correlation is 1. Forgetting the factor 2 on the cross term is the usual slip.

A practice question

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

Why must an insurance company hold capital in addition to its reserves?

  1. To absorb losses beyond expected levels so claims can still be paid in adverse years - correct
  2. To fund the dividends promised to shareholders each yearWhy not: Dividends are not the purpose of regulatory capital.
  3. To replace the premium income lost when policyholders cancelWhy not: Lapse of policies is not the main reason for holding capital.
  4. To pay the expected claims on all the policies that the insurer has already writtenWhy not: Expected claims are covered by reserves and premiums; capital covers unexpected losses.

Reserves and premiums are set for expected claims; capital is the buffer against unexpected losses, so that policyholders are still paid.

A harder one

A property-casualty insurer models next year's aggregate loss as normally distributed with mean USD 40 million and standard deviation USD 25 million. Economic capital is defined as the 99.5% loss quantile minus the expected loss. Capital is closest to:

  1. USD 58.2 millionWhy not: Uses the 99% quantile (2.326) instead of the 99.5% quantile (2.576).
  2. USD 64.4 million - correct
  3. USD 41.1 millionWhy not: Uses the 95% quantile (1.645) instead of the 99.5% quantile.
  4. USD 104.4 millionWhy not: Includes the expected loss; the definition subtracts it because expected loss is covered by premiums and reserves.

The 99.5% standard normal quantile is 2.576. Capital = 2.576 x 25 = 64.4 million; the mean is excluded by definition.

The hardest tier

A life insurer has assets of USD 1,000 million with a duration of 7 and liabilities of USD 900 million with a duration of 12, so capital is USD 100 million. Using duration approximations, interest rates fall by 1 percentage point. Capital after the fall is closest to:

  1. USD 107 millionWhy not: Applies the asset duration of 7 to the liabilities as well, ignoring the longer liability duration.
  2. USD 170 millionWhy not: Revalues the assets but leaves the liabilities unchanged.
  3. USD 62 million - correct
  4. USD 138 millionWhy not: Applies a rise in rates instead of a fall; falling rates raise both values, but the liability more.

Assets: 1,000 x (1 + 7 x 0.01) = 1,070. Liabilities: 900 x (1 + 12 x 0.01) = 1,008. Capital = 1,070 - 1,008 = 62 million, down from 100 because the liabilities are longer.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in Insurance Companies and Pension Plans