Mango Flashcards
let 2 risks be, X & Y
size of loss distributions assume
losses are from a Poisson process with occurrence rate λ
needed surplus
Surplus need before new account is added
Surplus need after new account is added
needed surplus: V = z*SD(loss)-expected return
z = # of std dev associated with percentile that surplus allocated is sufficient to cover actual surplus need
V0 = z*S0-R0
V1 = z*S1-R1
*difference in returns (R1-R0) would be due to risk load charged to new account
2 methods to calculate risk load
Marginal surplus method
Marginal variance method
what risk is ignored when using MV or MS method
parameter
Marginal Surplus Method, MS method
risk load depends on marginal standard deviation
Marginal Variance Method, MV Method
risk load depends on marginal variance
reason we set λ is so that
total risk load produced by Marginal Surplus and Marginal Variance methods will be the same
Building Up Portfolio of 2 Accounts: in general
assume insurer writes an account X; only account in portfolio until an additional account Y is written
building up portfolio: difference between methods
total risk load is the same
distribution of load between X&Y is different
Renewing the Portfolio of 2 Accounts: in general
- each account from build-up scenario is renewed
- when X is renewed, assume Y is already in force
- when Y is renewed, assume X is already in force
Renewing the Portfolio of 2 Accounts: risk load for Y
-in both scenarios, Y was being added to existing account; therefore there is no difference in risk load between 2 scenarios for Y
risk load for X comparison between build up and renewal
MS method: renewal risk load is less than build up
*marginal SD for X is lower
MV method: risk load is higher for renewal scenario than build-up
*marginal variance for X is higher -> receives a risk load for full covariance shared with existing accounts
Renewal Additivity
Risk load method is renewal additive if the sum of renewal risk loads of each risk is = risk load for aggregate portfolio
Neither MS nor MV methods are renewal additive
MS method: renewal additvity
Ʃrenewal risk loads < risk load for portfolio -> accounts will be undercharged
due to sub-additivity of square root
method is sub-additive
MV method: renewal additvity
Ʃrenewal risk loads > risk load for portfolio -> accounts will be overcharged
due to double counting the covariance
2 methods are alternate methods to MV method
Shapley Value
Covariance Share
-mutual covariance is split between accounts instead of being allocated to each new account; purpose is to avoid overstating risk load
Shapley Value
Value = average marginal variance from all different combinations in which a new account can be added to a portfolio
Shapley value method allocates the mutual covariance equally between accounts
- if there are 2 accounts, each receive Cov(L,n)
- under MV, each would receive 2Cov(L,n)
Covariance Share
-Shapley value method is not fairest way to spread the mutual covariance, since it allocates the mutual covariance equally, ignoring factors like different size of accounts
Covariance share method divides mutual covariance according to weights selected by user
since Shapley and covariance share methods are based on variance
they use same risk load multiplier as MV method
comparison between shapley and covariance share: build up
- it can be seen that each of these methods produce a lower risk load for Y, new account, than MV method, as only a portion of mutual covariance is allocated to Y
- covariance share method allocates less covariance to Y than Shapley method as Y makes up smaller portion of losses
comparison between shapley and covariance share: renewal
- risk load for both X&Y is less than under MV method
- in addition, risk load for X is greater than build-up scenario; this is because it is now being allocated a portion of mutual covariance
Both the Shapley and Covariance share methods are
renewal additive
- risk load of account X + account Y is = risk load of entire portfolio
- methods do not overstate/understate required premium
according to game theory, there are few rules for allocation
- allocation methods must be renewal additive
- coalition should be stable (fair); this way, there would be no incentive for a player/group of players to split from the group; essentially there should be no possibilities where a subgroup is better on its own
Game theory approach can be applied to risk loads discussed during Mango paper
- players want to minimize the allocation of total risk load
- allocation method needs to fairly and objectively assign risk load to each account in proportion to its contribution to the total
