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Flashcards in Mango Deck (44)
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1

CAT model outputs (2)

(Mango)

1. modeled loss for each event
2. probabilities of each event

2

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)

3

Combined portfolio variance, Var(portfolio + new account)

(Mango)

Var(portfolio + new account) = Var(portfolio) + Var(new account) + 2Cov(portfolio, new account)

4

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

5

Risk load (r) for the marginal surplus (MS) method

(Mango)

r = multiplier * (S(1) - S(0))

where S = std. dev(loss)

6

Marginal surplus (MS) method description

(Mango)

uses change in portfolio standard deviation to calculate the risk load for an account

7

Required return (y) depends on (3)

(Mango)

1. mgmt goals
2. market forces
3. risk appetite

8

Marginal variance (MV) method description

(Mango)

uses change in portfolio variance to calculate the risk load for an account

9

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)

10

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)

11

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

12

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

13

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

14

Account renewal assumption

(Mango)

renewing account X into portfolio Y = adding a new account X to an existing portfolio Y

15

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

16

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

17

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

18

Additivity

(Mango)

when sum of individual risk loads = total portfolio risk load

**specifically, Mango is searching for renewal additivity

19

Order dependency problem

(Mango)

renewal additivity depends on the entry order of accounts

20

Features of cooperative games with transferrable utilities under game theory (4)

(Mango)

1. participants have benefits/costs to share
2. opportunity to share benefits/costs from cooperation of all or a sub-group of participants
3. freedom for players to negotiate, bargain, & form coalitions
4. conflicting player objectives - each wants to maximize benefits/minimize costs

21

Coalition characteristic function in game theory

(Mango)

determines the total amount to be allocated

22

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

23

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)

24

Game theory allocation rules to determine the optimal allocation (2)

(Mango)

1. allocation methods must be additive
2. coalition must be stable/fair so there is no incentive to leave the group

25

Conditions of fairness under game theory allocation rules (2)

(Mango)

1. individual rationality
2. collective rationality

26

Individual rationality

(Mango)

players are no worse off for having joined the coalition

27

Collective rationality

(Mango)

no sub-group of players would be better off on its own

28

Core of the game

(Mango)

set of all acceptable allocations for each player satisfying fairness and stability rules

29

Benefits of the Shapley value allocation method (3)

(Mango)

1. additive
2. centroid of the core
3. order independent

30

Shapley value

(Mango)

Shapley value = avg marginal impact taken over all possible entrance permutations

= Var(new account) + Cov(portfolio, new account)

31

Shapley value modification with more than 2 accounts

(Mango)

add additional covariance terms with the remaining combinations

32

Risk load (r) using the Shapley value

(Mango)

r = MV multiplier * Shapley value(new account)

33

Renewal additivity and the Shapley value

(Mango)

is renewal additive b/c each account receives Cov(portfolio, new account)

34

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

35

Covariance share (CS) method

(Mango)

allocation method that shares the mutual covariance based on each account's relative contributions to determine risk loads

36

Applications of game theory applied to property CAT risk loads (2)

(Mango)

1. Shapley value
2. covariance share (CS) method

37

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

38

Shapley method as a special case of the covariance share (CS) method

(Mango)

Shapley method = special case where weight = .5

39

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

40

Risk load (r) using the covariance share (CS) method

(Mango)

r = MV multiplier * (var(new account) + CovShare(new account))

41

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

42

Risk loads for the MV, Shapley, and CS methods during build-up

(Mango)

identical risk loads

43

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)

44

Excel formula for z = # of standard deviations from the normal distribution

(Mango)

norm.inv((1-probability of ruin), 0, 1)