Mango

Question 1

let 2 risks be, X & Y

size of loss distributions assume

Answer 1

losses are from a Poisson process with occurrence rate λ

Question 2

needed surplus

Surplus need before new account is added

Surplus need after new account is added

Answer 2

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

Question 3

2 methods to calculate risk load

Answer 3

Marginal surplus method

Marginal variance method

Question 4

what risk is ignored when using MV or MS method

Answer 4

parameter

Question 5

Marginal Surplus Method, MS method

Answer 5

risk load depends on marginal standard deviation

Question 6

Marginal Variance Method, MV Method

Answer 6

risk load depends on marginal variance

Question 7

reason we set λ is so that

Answer 7

total risk load produced by Marginal Surplus and Marginal Variance methods will be the same

Question 8

Building Up Portfolio of 2 Accounts: in general

Answer 8

assume insurer writes an account X; only account in portfolio until an additional account Y is written

Question 9

building up portfolio: difference between methods

Answer 9

total risk load is the same

distribution of load between X&Y is different

Question 10

Renewing the Portfolio of 2 Accounts: in general

Answer 10

• 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

Question 11

Renewing the Portfolio of 2 Accounts: risk load for Y

Answer 11

-in both scenarios, Y was being added to existing account; therefore there is no difference in risk load between 2 scenarios for Y

Question 12

risk load for X comparison between build up and renewal

Answer 12

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

Question 13

Renewal Additivity

Answer 13

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

Question 14

MS method: renewal additvity

Answer 14

Ʃrenewal risk loads < risk load for portfolio -> accounts will be undercharged

due to sub-additivity of square root

method is sub-additive

Question 15

MV method: renewal additvity

Answer 15

Ʃrenewal risk loads > risk load for portfolio -> accounts will be overcharged

due to double counting the covariance

Question 16

2 methods are alternate methods to MV method

Answer 16

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

Question 17

Shapley Value

Answer 17

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)

Question 18

Covariance Share

Answer 18

-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

Question 19

since Shapley and covariance share methods are based on variance

Answer 19

they use same risk load multiplier as MV method

Question 20

comparison between shapley and covariance share: build up

Answer 20

• 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

Question 21

comparison between shapley and covariance share: renewal

Answer 21

• 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

Question 22

Both the Shapley and Covariance share methods are

Answer 22

renewal additive

• risk load of account X + account Y is = risk load of entire portfolio
• methods do not overstate/understate required premium

Question 23

according to game theory, there are few rules for allocation

Answer 23

1. allocation methods must be renewal additive
2. 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

Question 24

Game theory approach can be applied to risk loads discussed during Mango paper

Answer 24

1. players want to minimize the allocation of total risk load
2. allocation method needs to fairly and objectively assign risk load to each account in proportion to its contribution to the total