Transcript Download Handout 1

RMK and Covariance
Seminar on Risk and Return in Reinsurance
September 26, 2005
Dave Clark
American Re-Insurance Company
This material is being provided to you for information only, and is not permitted to
be further distributed without the express written permission of American Re. This
material is not intended to be legal, underwriting, financial or any other type of
professional advice. Examples given are for illustrative purposes only.
© Copyright 2005 American Re-Insurance Company. All rights reserved.
1234
RMK Framework
Introduction: A Confusing List of Terms






Capital Allocation
Cost of Capital
Capital Consumption
Risk Load
Risk / Reward trade-offs
RMK ???
1234
RMK Framework
Introduction: The RMK Framework sources
RMK = Ruhm, Mango, Kreps
Mango:
Capital Consumption: An Alternative
Methodology for Pricing Reinsurance
CAS Forum, Winter 2003
Kreps:
Riskiness Leverage Models
PCAS 2005 – Originally circulated in bootleg version as
“A Risk Class with Additive Co-Measures”
1234
RMK Framework
Agenda:
 Examples from Reinsurance Pricing



Example of “Capital Consumption”



Allocation of stop-loss premium
Multi-year profit commission
Properties
The Mathematics
Problems & Challenges
1234
RMK Framework
Resource divided between individuals
Shared Resource:
1234
RMK Framework
Instead of the “capital allocation” problem we will
start with two other examples in which dollar
amounts need to be allocated in reinsurance
applications.
Example #1: A ceding company purchases an
aggregate “stop-loss” cover that applies to all
lines of business combined. How should the
cost of this reinsurance be allocated to the
individual lines of business?
1234
RMK Framework
Example #2: A reinsurer has sold an Excess WC
treaty with a profit commission that applies on a
3-year block. It is now the end of the second
year and we want to evaluate the “expected”
profit commission on the prospective third year.
How do we estimate the expected profit
commission by year?
1234
RMK Framework
Example #3: Risk Measures and “Capital
Consumption”
Given an overall profit target, what is the fairest
method for setting corresponding profit targets
for individual products?
1234
RMK Framework
Review Excel Examples
1234
RMK Framework
Strengths of the RMK Framework:



Results are additive: business segments can be defined
any way you want and it will not affect the answer.
Risk measures by business segment are logically
connected to the risk measure for the company in total.*
Theory works for any correlation or dependence
structure in the losses: If you can simulate it, RMK will
work!
*This is why Kreps calls RMK “additive co-measures”
RMK Framework
1234
Mathematics
We will follow Kreps’ notation for describing formulas for
allocating the risk measure.
Assume that we have a
portfolio “Y”, made up of the
sum of three business
segments: X1, X2, and X3.
Y  X1  X 2  X 3
These three segments do
not have to be independent
or identically distributed.
f ( x1 , x2 , x3 )
RMK Framework
1234
Mathematics
Kreps’ notation continued:
Define a risk measure R, which is based on the total losses
to the portfolio.
R    y   y  L( y ) f ( y ) dy
Amount by which an
actual loss exceeds
the average.
The “Leverage” or
pain associated with
the loss amount.
RMK Framework
1234
Mathematics
Kreps’ notation continued:
We then introduce a simplified notation.
R    y   y  L( y ) dF
where
dF  f ( x1 , x2 , x3 ) dx1 dx2 dx3
RMK Framework
1234
Mathematics
Kreps’ notation continued:
The overall risk-load is allocated as follows.
R    y   y  L( y ) dF
Changes to…
Rk   xk   k   L( y ) dF
RMK Framework
1234
Mathematics
Kreps’ notation continued:
Note that ADDITIVITY is preserved regardless of the
dependence between business segments or the way in
which we define “segment” (line of business, SBU, etc).
R 
R1  R2   R3
 R1  R2  R3 
RMK Framework
1234
Mathematics
The Leverage Ratio L(y) is a “pain” function:




If (y-μy) is negative:
no capital used, L(y)=0; there is no “pain”
If (y- μy) is small but positive:
some capital is consumed, L(y)>0
If (y- μy) is large:
capital is consumed, solvency is imperiled
If (y- μy) is huge (a multiple of capital):
L(y) stops growing, since we no longer care if we are
dead many times over
RMK Framework
1234
Mathematics
Kreps shows that virtually all of the common
proposals for risk measures can be formulated in
terms of a leverage ratio L(y) :
 Variance
 Standard Deviation
 Value at Risk (VaR)
 Conditional Downside (TVaR or “tail value at risk”)
RMK Framework
1234
Mathematics
Kreps’ notation and covariance:
Alternative formulation (from Ruhm & Mango):
Rk  Covxk , L( y) 
This implies that if
L( y )     y   y 
Then a covariance
allocation results:
Rk    Covxk , y 
RMK Framework
1234
Mathematics
Conditions for RMK to reproduce covariance:
(1) If the portfolio risk measure is variance
Or…
R   Var y 
Covxk , y 
Rk  R 
Var  y 
(2) If there is a linear relationship between expected
losses, then for any leverage ratio:
E[ X k | y]  b  m  y
Covxk , y 
Rk  R 
Var  y 
RMK Framework
1234
Mathematics
Conditions for RMK to reproduce covariance:
This second condition is EXACTLY met when
1. All business segments write identical policies (even if
they have different commission percents)
2. All business segments take different shares of a pool
3. All business segments are drawn from certain
multivariate distributions
We can also find cases when it is APPROXIMATELY met…
RMK Framework
1234
Mathematics
Loss Distributions for which RMK = covariance:
Examples:

Elliptical Distributions


Additive Form of Exponential Family



Normal, Student-t, Logistic, etc
Normal, Poisson, Gamma, Inverse Gaussian
Note “Additive” = closed-under-convolution
Other…
RMK Framework
1234
Mathematics
Generalized Pareto (a.k.a. Beta of Second Kind)
f ( x) 
 x    ( x  d x )  

x 
 x     ( x  d x   )
 x 1
x  dx
Bivariate Generalized Pareto (common  and )
f ( x, y ) 
 x   y   

( x  d x ) x 1  ( y  d y )
 y 1
 
 x    y     ( x  d x  y  d y   ) x  y  
RMK Framework
1234
Mathematics
General Principle, informally stated:
Covariance allocation is a linear
approximation to any arbitrary risk
co-measure.
Note: subject to conditions such as all variances existing, and
the risk measure being on a “central” basis (x-μ).
RMK Framework
1234
Mathematics
Condition for RMK to approximate covariance:
Line 1
2,000
1,500
1,000
500
0
0
1,000
2,000
3,000
All Lines Combined
4,000
5,000
RMK Framework
1234
Mathematics
Line subject to Stop
Loss
Condition when RMK does NOT approximate covariance:
1,500
1,000
500
0
0
1,000
2,000
3,000
All Lines Combined
4,000
5,000
RMK Framework
1234
Mathematics
Catastrophe Loss
Condition when RMK does NOT approximate covariance:
2,000
1,500
1,000
500
0
0
1,000
2,000
3,000
All Lines Combined
4,000
5,000
1234
RMK Framework
Problems & Challenges:
Two hurdles:
 Calibration down to individual contract
level will probably always be an
approximation.
 More work needed on theory for including
the time value of money.
1234
RMK Framework
Questions & Discussion
David R. Clark
“Reinsurance Applications of the RMK Framework”;
Spring 2005 CAS Forum
www.casact.org/pubs/forum/05spforum