Mango Flashcards
CAT model outputs (2)
Mango
- modeled loss for each event
2. probabilities of each event
Cov(portfolio, new account)
Mango
Cov(portfolio, new account) = sum over all events of modeled loss(portfolio) * modeled loss(new account) * probability of event * (1 - probability of event)
Combined portfolio variance, Var(portfolio + new account)
Mango
Var(portfolio + new account) = Var(portfolio) + Var(new account) + 2Cov(portfolio, new account)
Required surplus (V) for the marginal surplus (MS) method
Mango
V = z * S - R
where S = std. dev(loss)
and R = return
and z = # of std. deviations from the normal distribution
Risk load (r) for the marginal surplus (MS) method
Mango
r = multiplier * (S(1) - S(0))
where S = std. dev(loss)
Marginal surplus (MS) method description
Mango
uses change in portfolio standard deviation to calculate the risk load for an account
Required return (y) depends on (3)
Mango
- mgmt goals
- market forces
- risk appetite
Marginal variance (MV) method description
Mango
uses change in portfolio variance to calculate the risk load for an account
Risk load (r) for the marginal variance (MV) method
Mango
r = MV multiplier * marginal variance
where marginal variance = Var(new account) + 2Cov(portfolio, new account)
Multiplier for the marginal variance (MV) method risk load
Mango
uses MS multiplier converted to an MV basis
multiplier = MS multiplier / std. dev(portfolio + new account)
Multiplier for the marginal surplus (MS) method risk load
Mango
multiplier = [(y * z) / (1 + y)]
where y = required return
and z = # of std. deviations from the normal distribution
Relationship between combined and account level risk loads under the MS and MV method (general)
(Mango)
total portfolio risk loads are = under both methods but account level risk loads differ
Build-up vs. renewal scenario
Mango
build-up = initial adding of new accounts
renewal scenario = steady state portfolio where accounts renew w/no new entrants
Account renewal assumption
Mango
renewing account X into portfolio Y = adding a new account X to an existing portfolio Y
Marginal surplus (MS) method results under the renewal scenario & impact
(Mango)
sum of individual risk loads < total portfolio risk load
> > b/c of sub-additivity of the square root operator in the std. dev.
impact: undercharge every account
Marginal variance (MV) method results under the renewal scenario & impact
(Mango)
sum of individual risk loads > total portfolio risk load
> > b/c the covariance term is double-counted (MV renewal scenario is super-additive)
impact: overcharge every account
Additivity of marginal surplus (MS) and marginal variance (MV) results under the build-up scenario
(Mango)
sum of individual risk loads = total portfolio risk load
Additivity
Mango
when sum of individual risk loads = total portfolio risk load
**specifically, Mango is searching for renewal additivity
Order dependency problem
Mango
renewal additivity depends on the entry order of accounts
Features of cooperative games with transferrable utilities under game theory (4)
(Mango)
- participants have benefits/costs to share
- opportunity to share benefits/costs from cooperation of all or a sub-group of participants
- freedom for players to negotiate, bargain, & form coalitions
- conflicting player objectives - each wants to maximize benefits/minimize costs
Coalition characteristic function in game theory
Mango
determines the total amount to be allocated
Sub-additivity and super-additivity of the coalition characteristic function, v(S) and interpretation of each
(Mango)
sub-additive: v(S union T) < v(S) + v(T)
each member wants to minimize individual allocation
super-additive: v(S union T) > v(S) + v(T)
each member wants to maximize individual allocation
Real life example of a sub-additive coalition characteristic function
(Mango)
insurance premium for a risk purchasing group (each members wants to minimize individual premium)
Game theory allocation rules to determine the optimal allocation (2)
(Mango)
- allocation methods must be additive
2. coalition must be stable/fair so there is no incentive to leave the group
Conditions of fairness under game theory allocation rules (2)
(Mango)
- individual rationality
2. collective rationality
Individual rationality
Mango
players are no worse off for having joined the coalition
Collective rationality
Mango
no sub-group of players would be better off on its own
Core of the game
Mango
set of all acceptable allocations for each player satisfying fairness and stability rules
Benefits of the Shapley value allocation method (3)
Mango
- additive
- centroid of the core
- order independent
Shapley value
Mango
Shapley value = avg marginal impact taken over all possible entrance permutations
= Var(new account) + Cov(portfolio, new account)
Shapley value modification with more than 2 accounts
Mango
add additional covariance terms with the remaining combinations
Risk load (r) using the Shapley value
Mango
r = MV multiplier * Shapley value(new account)
Renewal additivity and the Shapley value
Mango
is renewal additive b/c each account receives Cov(portfolio, new account)
Problem with the using the Shapley value to determine risk loads
(Mango)
each account receives an equal share of the mutual covariance, which may be unfair if one account has significantly higher losses
Covariance share (CS) method
Mango
allocation method that shares the mutual covariance based on each account’s relative contributions to determine risk loads
Applications of game theory applied to property CAT risk loads (2)
(Mango)
- Shapley value
2. covariance share (CS) method
Covariance share for an event with 2 accounts, X & Y
Mango
CovShare(X-i) = w(X-i) * 2 * x(i) * y(i) * probability of event(i) * (1 - probability of event i)
where x(i), y(i) are modeled losses for accounts X & Y respectively for event i and w = weight of modeled losses
total CovShare = sum of CovShare across all events for an account
Shapley method as a special case of the covariance share (CS) method
(Mango)
Shapley method = special case where weight = .5
Deferred risk load
Mango
remaining risk load when sum of account risk loads < total portfolio risk loads during the build-up phase of the covariance share (CS) & Shapley value methods
Risk load (r) using the covariance share (CS) method
Mango
r = MV multiplier * (var(new account) + CovShare(new account))
Deferred risk load under the covariance share (CS) method
Mango
r(defer) = MV multiplier * CovShare(initial account)
also = MV combined risk load - sum of individual build up risk loads
Risk loads for the MV, Shapley, and CS methods during build-up
(Mango)
identical risk loads
Recommended use for MS, MV, Shapley, and CS methods
Mango
use MS/MV for pricing new accounts (b/c additive) and Shapley/CS methods for pricing renewal accounts (b/c of renewal additivity)
Excel formula for z = # of standard deviations from the normal distribution
(Mango)
norm.inv((1-probability of ruin), 0, 1)
