Kreps Flashcards
Total assets and what is supported by each component (2)
Kreps
total assets = reserves + surplus
reserves support mean of assets & liabilities
surplus supports variability of assets & liabilities
Total capital (C)
Kreps
total capital = mean outcome + risk load
Desirable qualities for an allocatable risk load (3)
Kreps
- ability to be allocated to any level
- allocated risk load for a sum of random variables should = sum of individually allocated risk load amounts
- same additive formula is used to calculate risk loads for any sub-group or grouping
General form of riskiness leverage models
Kreps
R = integral of f(x) * (x - mu) * L(x) dx
f(x)dx can be called dF-bar for joint probability distributions
where f(x) = joint probability distribution and L(x) = riskiness leverage function for total losses
Risk load (R) and capital (C) across multiple LOB
Kreps
R = sum of R(k)'s C = sum of C(k)'s
where k = individual LOB
Advantage of co-measures
Kreps
they are automatically additive
Disadvantage of co-measures
Kreps
can be challenging to find appropriate forms of the riskiness leverage function L(x)
Conditions for negative risk loads (2) and when it is desirable
(Kreps)
- x(k) < mean
- large L(x)
desirable for hedges, occurs when there is a low correlation with total losses
Properties of riskiness leverage models (4)
Kreps
- desirable qualities for allocatable risk loads are satisfied
- no risk load for constant variables - R(c) = 0
- risk load will scale with change in currency - R(lambda * x) = lambda * R(x)
- may not produce a coherent risk measure
Coherent risk measures
Kreps
satisfy sub-additivity requirement
R(x + y) <= R(x) + R(y)
Super-additivity
Kreps
R(x + y) > R(x) + R(y)
not coherent
Types of riskiness leverage functions, L(x) (7)
Kreps
- risk-neutral
- variance
- VaR
- TVaR
- semi-variance, SVaR
- mean downside deviation
- proportional excess
Risk-neutral form of the riskiness leverage function, L(x)
Kreps
L(x) = c
Situations when a risk-neutral form of L(x) might be appropriate (2)
(Kreps)
- risk of ruin if risk of ruin is very small compared to capital OR capital is infinite
- risk of not meeting plan if indifferent about making plan
Variance form of the riskiness leverage function, L(x)
Kreps
L(x) = (beta / surplus) * (x - mu)
Relevant part of the distribution when using the variance form of L(x)
(Kreps)
entire distribution (just as much risk associated with good & bad outcomes)
Surplus (S) when using the variance form of L(x)
Kreps
S = sqrt(beta * var(x))
Forms of the riskiness leverage function, L(x) where the risk load increases quadratically (2)
(Kreps)
- variance
2. semi-variance, SVaR
TVaR form of the riskiness leverage function, L(x)
Kreps
L(x) = theta(x - x(q)) / (1 - q)
where theta is a step function with:
theta(x) = 0 for x <= 0 and
theta(x) = 1 for x > 0
x(q) = value of x so F(x(q)) = q
Relevant part of the distribution when using the TVaR form of L(x)
(Kreps)
only the high end of the distribution is relevant
VaR form of the riskiness leverage function, L(x)
Kreps
L(x) = delta(x - x(q)) / f(x(q))
where delta(x) = 0 everywhere except 0 and integrates to 1
Coherent riskiness leverage function, L(x)
Kreps
TVaR
Capital (C) when using the VaR form of L(x)
Kreps
C = x(q) = VaR
x(q) = value of x so F(x(q)) = q
Relevant part of the distribution when using the VaR form of L(x)
(Kreps)
only the single VaR point is relevant
Semi-variance, SVaR, form of the riskiness leverage function, L(x)
(Kreps)
L(x) = (beta / surplus) * (x - mu) * theta(x - mu)
where theta is a step function with:
theta(x) = 0 for x <= 0 and
theta(x) = 1 for x > 0
Risk load when using the risk-neutral form of L(x)
Kreps
risk load = 0
Risk load when using the semi-variance, SVaR, form of L(x)
Kreps
risk-load = semi-variance (only non-zero for results worse than the mean)
Relevant part of the distribution when using the semi-variance, SVaR, form of L(x)
(Kreps)
only bad results are relevant
Mean downside deviation form of the riskiness leverage function, L(x)
(Kreps)
L(x) = beta * theta(x - mu) / (1 - F(mu))
where theta is a step function with:
theta(x) = 0 for x <= 0 and
theta(x) = 1 for x > 0
Risk load when using the mean downside deviation form of L(x)
(Kreps)
risk load = multiple of mean downside deviation
Condition when the mean downside deviation = TVaR form of the riskiness leverage function, L(x)
(Kreps)
x(q) = mu
Riskiness leverage ratio for the mean downside deviation form of L(x)
(Kreps)
riskiness leverage ratio = 0 below the mean and constant above the mean
Capital allocation using the mean downside deviation form of L(x)
(Kreps)
assigns capital for bad outcomes in proportion to how bad they are
Proportional excess form of the riskiness leverage function, L(x)
(Kreps)
L(x) = h(x) * theta(x - (mu + delta)) / (x - mu)
Capital allocation using the proportional excess form of L(x)
(Kreps)
individual allocation for any given outcome is pro rata to its contribution to the excess over the mean
Sources of risk that could impact riskiness leverage ratio selection (7)
(Kreps)
- not making plan
- serious deviation from plan
- not meeting investor expectations
- ratings downgrade
- triggering regulatory notice
- going into receivership
- not getting a bonus
Management properties of a riskiness leverage ratio (4)
Kreps
must:
1. be a downside measure
2. be approximately constant for excess losses that are small compared to capital
3. become much larger for excess losses significantly impacting capital
4. go to 0 for excess losses significantly exceeding capital
Regulator properties of a riskiness leverage ratio (2)
Kreps
must:
1. be 0 until capital is seriously impacted
2. not decrease (due to risk to state guaranty fund)
Reinsurance premium
Kreps
reinsurance premium = E[ceded loss] + load % * std. dev(ceded losses)
Reinsurance net cost
Kreps
reinsurance net cost = reinsurance premium - E[ceded loss]
Impact of reinsurance on total return and capital
Kreps
positive net cost of reinsurance will reduce total return, but allow firm to release capital
Total return
Kreps
total return = income / surplus
Surplus released from purchasing reinsurance
Kreps
surplus release = current surplus - target surplus
Reduction in cost of capital from purchasing reinsurance
Kreps
reduction in cost of capital = cost of capital * surplus release
Determining whether reinsurance treaties are worth pursuing
Kreps
if reduction in cost of capital > net avg cost of reinsurance, then reinsurance is worth pursuing
Potential actions to take if return on surplus < target return on surplus (3)
(Kreps)
- write less business (shift volume to more profitable LOB)
- purchase reinsurance to reduce required surplus
- raise premium by increasing the profit load
Potential problems with changing LOB volume when return on surplus < target (3)
(Kreps)
- LOBs may be two parts of an indivisible policy
- regulatory requirements
- may not be cost effective to undergo major UW effort
Risk load, R(k), for a line of business using riskiness leverage models
(Kreps)
R(k) = average of all L(x) * (x(k) - mu(k)) for individual loss amounts
**L(x) is always for total losses, NOT individual LOB
