Mango

Question 1

CAT model outputs (2)

Mango

Answer 1

1. modeled loss for each event

2. probabilities of each event

Question 2

Cov(portfolio, new account)

Mango

Answer 2

Cov(portfolio, new account) = sum over all events of modeled loss(portfolio) * modeled loss(new account) * probability of event * (1 - probability of event)

Question 3

Combined portfolio variance, Var(portfolio + new account)

Mango

Answer 3

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

Question 4

Required surplus (V) for the marginal surplus (MS) method

Mango

Answer 4

V = z * S - R

where S = std. dev(loss)
and R = return
and z = # of std. deviations from the normal distribution

Question 5

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

Mango

Answer 5

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

where S = std. dev(loss)

Question 6

Marginal surplus (MS) method description

Mango

Answer 6

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

Question 7

Required return (y) depends on (3)

Mango

Answer 7

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

Question 8

Marginal variance (MV) method description

Mango

Answer 8

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

Question 9

Risk load (r) for the marginal variance (MV) method

Mango

Answer 9

r = MV multiplier * marginal variance

where marginal variance = Var(new account) + 2Cov(portfolio, new account)

Question 10

Multiplier for the marginal variance (MV) method risk load

Mango

Answer 10

uses MS multiplier converted to an MV basis

multiplier = MS multiplier / std. dev(portfolio + new account)

Question 11

Multiplier for the marginal surplus (MS) method risk load

Mango

Answer 11

multiplier = [(y * z) / (1 + y)]

where y = required return
and z = # of std. deviations from the normal distribution

Question 12

Relationship between combined and account level risk loads under the MS and MV method (general)

(Mango)

Answer 12

total portfolio risk loads are = under both methods but account level risk loads differ

Question 13

Build-up vs. renewal scenario

Mango

Answer 13

build-up = initial adding of new accounts

renewal scenario = steady state portfolio where accounts renew w/no new entrants

Question 14

Account renewal assumption

Mango

Answer 14

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

Question 15

Marginal surplus (MS) method results under the renewal scenario & impact

(Mango)

Answer 15

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

Question 16

Marginal variance (MV) method results under the renewal scenario & impact

(Mango)

Answer 16

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

Question 17

Additivity of marginal surplus (MS) and marginal variance (MV) results under the build-up scenario

(Mango)

Answer 17

sum of individual risk loads = total portfolio risk load

Question 18

Additivity

Mango

Answer 18

when sum of individual risk loads = total portfolio risk load

**specifically, Mango is searching for renewal additivity

Question 19

Order dependency problem

Mango

Answer 19

renewal additivity depends on the entry order of accounts

Question 20

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

(Mango)

Answer 20

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

Question 21

Coalition characteristic function in game theory

Mango

Answer 21

determines the total amount to be allocated

Question 22

Sub-additivity and super-additivity of the coalition characteristic function, v(S) and interpretation of each

(Mango)

Answer 22

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

Question 23

Real life example of a sub-additive coalition characteristic function

(Mango)

Answer 23

insurance premium for a risk purchasing group (each members wants to minimize individual premium)

Question 24

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

(Mango)

Answer 24

1. allocation methods must be additive

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

Question 25

Conditions of fairness under game theory allocation rules (2)

(Mango)

Answer 25

1. individual rationality

2. collective rationality

Question 26

Individual rationality

Mango

Answer 26

players are no worse off for having joined the coalition

Question 27

Collective rationality

Mango

Answer 27

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

Question 28

Core of the game

Mango

Answer 28

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

Question 29

Benefits of the Shapley value allocation method (3)

Mango

Answer 29

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

Question 30

Shapley value

Mango

Answer 30

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

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

Question 31

Shapley value modification with more than 2 accounts

Mango

Answer 31

add additional covariance terms with the remaining combinations

Question 32

Risk load (r) using the Shapley value

Mango

Answer 32

r = MV multiplier * Shapley value(new account)

Question 33

Renewal additivity and the Shapley value

Mango

Answer 33

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

Question 34

Problem with the using the Shapley value to determine risk loads

(Mango)

Answer 34

each account receives an equal share of the mutual covariance, which may be unfair if one account has significantly higher losses

Question 35

Covariance share (CS) method

Mango

Answer 35

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

Question 36

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

(Mango)

Answer 36

1. Shapley value

2. covariance share (CS) method

Question 37

Covariance share for an event with 2 accounts, X & Y

Mango

Answer 37

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

Question 38

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

(Mango)

Answer 38

Shapley method = special case where weight = .5

Question 39

Deferred risk load

Mango

Answer 39

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

Question 40

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

Mango

Answer 40

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

Question 41

Deferred risk load under the covariance share (CS) method

Mango

Answer 41

r(defer) = MV multiplier * CovShare(initial account)

also = MV combined risk load - sum of individual build up risk loads

Question 42

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

(Mango)

Answer 42

identical risk loads

Question 43

Recommended use for MS, MV, Shapley, and CS methods

Mango

Answer 43

use MS/MV for pricing new accounts (b/c additive) and Shapley/CS methods for pricing renewal accounts (b/c of renewal additivity)

Question 44

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

(Mango)

Answer 44

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