Butsic - Solvency Measurement for Risk Based Capital Applications

Question 1

What are the criteria risk-based capital method should satisfy for it to be practical?

Answer 1

1. solvency standard should be the same for all classes of the parties (personal vs. commercial insureds, second- vs. third-party claimants, and primary insurers vs. reinsurers)
2. RBC should be objectively determined. i.e. two insurers with the same risk measures should have identical RBC
3. the method must be able to discriminate between quantifiable items of meaningful value that differ materially in their riskiness. e.g. stocks are significantly riskier than bonds.

Question 2

why probability of ruin is not a adequate measure of insolvency for public policy?

Answer 2

It ignores the severity of ruin.

Question 3

Formula for EPD and EPD ratio

Answer 3

EPD = Expected value of (the amount the insurer is obligated to pay) - the amount actually paid

EPD ratio = EPD/Expected Loss

Question 4

What is the advantage to computing an EPD

Answer 4

it consistently measures insolvency risk in such a way that a standard minimum level of protection is applied to all classes of policyholders and insurers.

Question 5

Asset Risk Under Normal Distribution & Lognormal Distribution

Answer 5

Normal Distribution:

Needs more capital than loss risk because if assets and losses have the same CV, the SD of assets in larger since A>L. d_A = d_L/(1-c).

LogNormal Distribution:

Needs less capital than loss risk because if assets and losses have the same CV, we get C_A = c/(1+c) <c.></c.>

<p>Require less capital since the probability of large losses is higher, and asset values cannot be negative.</p>

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Question 6

c (for a particular EPD ratio) under normal and lognormal distribution.

Answer 6

Under a normal distribution:

is approximately a linear function of k

Under a log normal distribution:

is not as linear a function of k. However, the approximation is OK for modest values of CV.

Question 7

The EPD ratio under a normal distribution

Answer 7

is an increasing function of the CV and the ruin probability. There is no single ruin probability corresponding to a give EPD ratio.

Question 8

Briefly discribe the accounting conventions and bias problem under RBC.

Answer 8

Under accounting conventions:

• bonds and real estate remain book value until sold
• A change in the accounting value per se does not connote risk.

Market value accounting is suitable for solvency assessment.

• Capital is defined as market value of the assets - the market value of liabilities.
• Defining capital as the accounting book value (Statutory Surplus or GAAP Equity) limits the usefulness of an RBC methodology, due to Accounting bias.
• Accounting bias occurs when current recorded value differs from realized value.

Thus, when setting RBC, the financial statement should be adjusted to remove this bias.

Question 9

Insolvency Cost as a Financial Option:

1. Riskless asset paired with Risky liabilities
2. Risky asset paired with Riskless liabilities

Answer 9

1. Riskless asset paired with Risky liabilities

EPD is equivalent to a Call option on the losses (with an exercise price = the value of the assets at the end of the year)

EPD: Max[0, L1 - A1].

2. Risky asset paired with Riskless liabilities

the EPD is equivalent to a Put option on the Ending assets

EPD: Max[0, L1 - A1]. ???double check

Question 10

Correlation and Independence of Risk Element

Correlation under the Normal Distribution

Answer 10

Note that c=C/L, k = σ/L. T

The relationship between c and k is approximately linear for a
fixed EPD ratio, d, and small k. We can further simplify the relationship to c ≈ ak, for some constant a.

Since C = cL, σ=kL, c = ak, then C = akL=aσ

Thus, RBC is proportional to the risk elements standard deviations.

• Risk capital for perfectly correlated items can be added
• Risk capital for independent (and partially correlated) items can be combined according to (the square root rule):
• C=[ΣC_i² + Σp_ijC_iC_j]^0.5

Question 11

The relative error in using the square root rule for 2 independent risks of the same size and standard deviation:

Answer 11

• The relative error (approximated - true capital) / (true capital) decreases as the EPD ratio decreases and as the risk increases for the normal distribution.
• The relative error increases as the risk increases for the lognormal distribution.
• For both distributions, the relative error is positive, meaning that the square root rule overstates required capital.

Question 12

RBC Amounts under differet risk correlation assumptions:

1. fully correlated
2. fully indenpendent
3. has correlations

Answer 12

1. C = sum(Ci)
2. C = sum(Ci^2)^0.5
3. C = sum(Ci^2 + pCiCj)^0.5