Butsic Flashcards
Relationship between capital, assets, and liabilities
Butsic
capital = assets - liabilities
Criteria for risk-based capital (RBC) methods (3)
Butsic
- the solvency standard is the same across all classes
- RBC is objectively determined
- ability to differentiate relative riskiness b/w quantifiable measures of risk
Reason that EPD is a better measure of insolvency risk compared to probability of ruin (advantage of EPD method)
(Butsic)
EPD also contemplates severity of ruin
Expected policyholder deficit (EPD)
Butsic
expected value of the difference b/w insurer’s full obligation and actual amount paid
EPD ratio
Butsic
EPD ratio = EPD / expected losses
*always ratio to losses even when assets are risky
EPD when losses are risky (discrete & continuous)
Butsic
discrete: sum where losses > assets of pr(L(i)) * (L(i) - A)
continuous: integral from A to infinity of pr(L) * (L - A) dL
EPD when assets are risky (discrete & continuous)
Butsic
discrete: sum where losses > assets of pr(A(i)) * (L - A(i))
continuous: integral from 0 to L of pr(A) * (L - A) dA
EPD method for capital allocation
Butsic
if the EPD ratio > the target EPD ratio, then increase capital
cannot solve for capital directly, so need to use an iterative process starting with the largest loss or smallest asset producing a deficit & working backwards to solve for A
Modification to the EPD method for capital allocation when assets are risky
(Butsic)
use asset relativities since asset values can change throughout the year
relativity = A(i) / E[A]
EPD ratio when losses are risky, d(L), under a normal distribution
(Butsic)
EPD ratio = d(L) = k * ϕ( -c / k) - c * Φ( -c / k)
where c = capital / E[L]
k = coefficient of variation (L)
EPD ratio when assets are risky, d(A), under a normal distribution
(Butsic)
EPD ratio = d(A) = (1 / (1 - c(A))) * [ k(A) * ϕ( - c(A) / k(A)) - c(A) * Φ( -c(A) / k(A))
where c(A) = capital / E[A] k(A) = coefficient of variation (A)
Standard normal density function and negative values ϕ(-x)
Butsic
ϕ(-x) = ϕ(x) because it is symmetric around 0
Cumulative standard normal distribution and negative values Φ(-x)
(Butsic)
Φ(-x) = 1 - Φ(x)
Capital needed for asset risk relative to loss risk with the same beginning balance sheet & EPD ratio under a normal distribution
(Butsic)
more capital needed for asset risk compared to loss risk
standard deviation of capital will be larger under the risky asset scenario because assets > losses
Most appropriate time to use the normal distribution for EPD ratios
(Butsic)
population with known mean where individual losses are independent
EPD ratio when losses are risky, d(L), under a lognormal distribution
(Butsic)
EPD ratio = d(L) = Φ(a) - (1 + c) * Φ(a - k)
where a = k / 2 - ln(1 + c) / k
EPD ratio when assets are risky, d(A), under a lognormal distribution
(Butsic)
EPD ratio = d(A) = Φ(b) - Φ(b - k(A)) / (1 - c(A))
where b = k(A) / 2 + ln(1 - c(A)) / k(A)
Required capital when assets are risky under the lognormal distribution compared to the normal distribution & rationale
(Butsic)
lower required capital to achieve the same EPD ratio
b/c the lognormal distribution does not allow negative values
Required capital when losses are risky under the lognormal distribution compared to the normal distribution & rationale
(Butsic)
higher required capital to achieve the same EPD ratio
b/c the lognormal distribution is skewed and produces a higher probability of large losses
Capital needed for asset risk relative to loss risk under the same beginning balance sheet & EPD ratio under a lognormal distribution
(Butsic)
more capital needed for loss risk compared to asset risk
Relationship between the difference in required capital under the normal vs. lognormal distribution and coefficient of variation when losses are risky
(Butsic)
difference in required capital increases as CV(L) increases
Factors influencing required capital (2)
Butsic
- accounting conventions
2. time horizons
Most appropriate valuation for solvency risk measurement and rationale
(Butsic)
market value
rationale: firm insolvency would lead to liquidation, which would be evaluated based on realizable market value vs. accounting book value which is subject to accounting bias in recorded values
Accounting practices causing bias for solvency risk measurement (2)
(Butsic)
- statutory accounting allows insurers to consistently under- or over-value certain items (ex: undiscounted loss reserves)
- statutory & GAAP accounting allow inconsistent measurement for identical items across companies (ex: risk margin in loss reserves)
Reason time horizons impact required capital
Butsic
the market value of assets & liabilities change over time which impacts the market value of realizable capital
Determinants of the degree of insolvency risk (2)
Butsic
- time horizon - longer time intervals lead to more risk
2. volatility of financial statement values - more volatility leads to more risk
Goal of the EPD method considering time horizons
Butsic
meet the target EPD ratio criterion over each time interval by adjusting capital requirements
Factors determining the market value of reserves (2)
Butsic
- market interest rates
2. risk of adverse development
Difference in evaluating RBC requirements for a going concern business vs. one in run-off
(Butsic)
need to consider losses & premiums from future business as well as the return on that premium
EPD as a financial option when losses are risky
Butsic
PV(EPD ratio) = call option on ending losses with strike price = ending asset value
insurer has the option to abandon it’s promise of full claim payment
EPD as a financial option when assets are risky
Butsic
PV(EPD ratio) = put option on ending assets with strike price = ending liabilities
Required capital for multiple LOB vs. required capital for individual LOB
(Butsic)
required capital for all LOB < sum of required capital for individual LOB
*unless there is perfect correlation (rho = 1), in which case, the required capital for all LOB = sum of required capital for individual LOB
Relationship between the diversification benefit on required capital and correlation
(Butsic)
diversification benefit decreases as correlation approaches 1
Total risk-based capital (formula)
Butsic
total risk-based capital = sqrt( sum of capital(i)^2 across all risks + double sum of correlation(i,j) * capital(i) * capital(j))
Modification required when correlated risk elements are on opposite sides of the balance sheet
(Butsic)
reverse the sign of the correlation coefficient (ex: bonds & loss reserves)
Key results of the Butsic paper (5)
Butsic
- PV(EPD ratio) = relevant measure of solvency
- RBC should use market value of risk elements to remove accounting bias
- common time horizons should be used to compare risk b/c insurance risk increases as time elapses
- EPD ratio is based on expected market value at the end of each valuation interval
- total RBC < sum RBC amounts for each risk element unless the risk elements are fully correlated
Risk elements with positive correlation (rho = 1) and which one needs modification (3)
(Butsic)
- stocks & bonds
- loss reserves & LAE reserves
- bonds & loss reserves (modification)
Risk elements with no correlation (rho = 0) and which one needs modification (3)
(Butsic)
- cash & real estate
- loss reserve & property UPR
- stocks & UPR (modification)
Risk elements with negative correlation (rho = -1) and which one needs modification (5)
(Butsic)
- stocks & put options
- loss reserves & income tax liabilities
- loss reserves & dividend reserves
- property/liability stocks & loss reserves (modification)
- reinsurance recoveries & loss reserves (modification)
Disadvantage of the EPD method for allocating capital
Butsic
does not reflect diversification benefits
