Enterprise Risk Management
and Strategic Planning
Status of ERM
Started with banking regulation
Banks were not quantifying risk, going broke
Basic risk quantification modeling developed
Expanded to insurance regulation
Modeling more complex, hard to evaluate
Attempts for strategic planning
Manufacturing looking at supply chains, etc.
Financial institutions model financial risk
Guy Carpenter
2
Key Financial Issues
How much capital is needed?
Which business units have best profit
after adjusting for risk?
Building models is an advancing
science with many pitfalls
Difficult to quantify all the risks, even the
largest problems, like competition
Today assume basic models are there
and think about application
Guy Carpenter
3
Risk-Adjusted Profit from ERM Models
ERM quantifies risk of company and
each business unit
Management would like to use that
information to identify units that have
better and worse profitability compared
to risk
Guy Carpenter
4
Uses of Risk Adjusted Profitability
Strategic planning for insurer
Grow business units that have higher
profit in relationship to risk
De-emphasize or restructure business
that does not give enough profit for the
risk
Guy Carpenter
5
Typical Approach
Quantify risk by a percentile of the
distribution of profit = “economic capital”
Maybe start with capital = – 1/3333 quantile
Compute – 1/100 quantile for each business
unit and for company
Allocate capital by ratio of business unit
quantile to company quantile
Divide unit profits by capital so allocated
Guy Carpenter
6
Some Criticisms Historically
Not realistic capital need
Quantile is a very limited risk measure
1/3333 quantile impossible to quantify
accurately
Profit not measured relative to marginal cost
of risk
Arbitrary choices required (1/100, etc.)
Not clear that growing units with higher
returns will actually increase risk adjusted
return or firm value
Guy Carpenter
7
Improvements Round 1 – Realistic Capital
Economic Capital
Economic capital is a non-technical
name for a quantile of a distribution
Not realistic capital need
Useful for regulation as a benchmark
Minimal company information becomes
public – only one point on distribution
Guy Carpenter
9
Realistic Capital Need
Shareholders want to provide as little as
possible
Bondholders and customers want enough
to feel secure with company guarantee
Will demand discounts if capital too low
So shareholders willing to supply enough
to avoid the discounts
But practical perception issue – cannot
calculate as a quantile
Guy Carpenter
10
Relating Capital to Risk
Once actual capital is determined,
return on that capital can be measured
Also riskiness of actual capital can be
quantified with risk measures
Risk measures can also be applied to
business units to get risk-adjusted profit
by unit
Guy Carpenter
11
Improvements Round 2 – Risk Measures
Purposes of Risk Measures
Have a consistent way of comparing
different risks, including asset risk, results
from different businesses
Comparing profit to risk one key application
For strategic planning – which lines to grow,
which to re-organize
Maybe for paying bonuses to managers
Measuring impact of risk-management
All of these work better if risk measures
proportional to economic value of the risk
Guy Carpenter
13
Relating Capital to Risk Measure
Do not have to set capital = risk measure
Useful alternative is capital as a multiple of a
risk measure
Some company standards:
3.5 to 4 times VaR @ 99%; 2.5 times TVaR @ 99%
Capital = 10 times TVaR @ 80%
Average loss in worst 20% of years is 10% of capital
Every risk worth charging for is worth measuring
Models can measure these better than 1/3333
Includes more adverse scenarios
Guy Carpenter
14
Which Risk Measure?
“It has been clearly demonstrated that the
possibility of extreme adverse results is not
the only risk driver of importance.”
Wish I knew who said it, what literature it
refers to, and what other risk is important
But the idea seems sound
Losing part of capital can be a big hit to value
Even profit less than target profit can be also
Guy Carpenter
15
Classification of Risk Measures
Moment based measures
Variance, standard deviation, semi-standard
deviation
Generalized moments, like E[YecY/EY]
Tail based measures
Look only at the tail of the distribution
Transformed distribution measures
Change the probabilities then take mean or other
risk measure with the transformed probabilities
Uses whole distribution but puts more weight in tails
by increasing the probabilities of large losses
Guy Carpenter
16
Variance and Standard Deviation
Do not differentiate between good and poor deviations.
Two distributions with same mean and standard deviation but Risk B has a
much higher loss potential. It will produce losses in excess of 20,000 while
Risk A will not.
1.0%
Semi-variance does
0.9%
.
0.7%
Relative Probability
0.8%
0.6%
Risk A
Risk B
7,500
10,000
0.5%
0.4%
0.3%
0.2%
0.1%
0.0%
5,000
12,500
15,000
17,500
20,000
22,500
25,000
Size of Loss
Guy Carpenter
17
Spectral Measures
  EY  hF Y  for nonnegative function h.
 0, p  1  q gives TVaR .
h p  
q
1 q, 1  q  p
2

1
1  p  1  q   

h p 
exp  
 gives


2 
 
 2

blurred VaR

Co-measure is rj  E X j  h F Y 
Marginal for step function or smooth h.
Guy Carpenter
18
Tail-Based Measures
Probability of default
Value at risk
Tail value at risk
Excess tail value at risk
Expected policyholder deficit
VaR criticized for not being subadditive but not
very important with co-VaR
TVaR criticized for linear treatment of large loss
Guy Carpenter
19
Transformed Probability Measures
Risk measure is the mean (but could be TVaR, etc.)
after transforming the loss probabilities to give
more weight to adverse outcomes
Prices for risky instruments in practice and theory
have been found to be approximated this way
Wang transform for bonds and cat bonds
Esscher transform for compound Poisson process tested
for catastrophe reinsurance
Black-Scholes and CAPM are of this form as well
More potential to be proportional to the market
value of the risk
Guy Carpenter
20
Possible Transforms
G*(x) = Qk[F-1(G(x)) + l] where Qk is the
t-distribution with k dof - Wang transform
l = .0453 and k  [5,6] fit prices of cat bonds
and various grades of commercial bonds
k can be non-integer with beta distribution
Compound Poisson martingale transform
Requires function f(x), with f(x) > – 1 for x>0
l* = l[1+Ef(X)]
g*(x) = g(x)[1+f(x)]/[1+ Ef(X)]
Guy Carpenter
21
Reinsurance Pricing Compared to
Minimum Entropy and Least Squares
g*(y) =
g(y)ecy/EY/EecY/EY
7.4
Loading Factors for Martingale Pricing of FE
6.4
Quadratic
Average
5.4
Loading
l* = lEecY/EY
MMM Loading
MEM Loading
Mixed Loading
4.4
Premium Loading
3.4
2.4
0
Guy Carpenter
0.005
0.01
Expected Loss on Line
0.015
0.02
22
Which Risk Measures?
Useful to be proportional to value of risk
being measured
Favors transformed probability measures
Tail measures are popular but ignore
some of the risk
Guy Carpenter
23
Improvements Round 3 – Co–Measures
Allocation by Co-measures
Goal is additive allocation
Capital allocated separately to lines A and B will equal the
capital allocated to lines A and B on a combined basis.
Start with a risk measure for the company, for example
the average loss in the 1 in 10 and worse years
Then, consider only the cases where the company’s
total losses exceed this threshold. In this example it is
the worst 10% of possible results for the company.
For these scenarios co-measure is how much each line
of business is contributing to the poor results
Guy Carpenter
25
Definition
Denoting loss for the total company as Y,
and for each line of business as Xi let:
R(Y) = E[ Y | F(Y) > a ] . Then
Co-R(Xi) = E[ Xi | F(Y) > a]
More generally:
Risk measure (Y) defined as:
E[h(Y)g(Y)| condition on Y], where h is
additive, i.e., h(U+V) = h(U) + h(V)
Allocate by r(Xj) = E[h(Xj)g(Y)| condition on Y]
VaRa(Y) = E[Y|F(Y) = a], r(Xj) = E[Xj|F(Y) = a]
Guy Carpenter
26
Improvements Round 4 – Marginal
Decompostion
Marginal Decomposition of Risk Measures
Applies when allocation of capital is based on
allocating a risk measure
Marginal impact of a business unit on company risk
measure is decrease in overall risk measure from
ceding a small increment of the line by a quota share
Marginal allocation assigns this marginal risk to every
such increment in the line
Treats every increment as the last one in
If sum of all such allocations over all lines is the
overall company risk measure, this is called a
marginal decomposition of the risk measure
All co-measures are additive but not all are marginal
Guy Carpenter
28
Advantage of Marginal Decomposition
You would like to have it so that:
If you increase business in a unit that has
above average return relative to risk
Then the comparable return for the whole
company goes up
Not all allocation does that; marginal
decomposition does
Thus useful for strategic planning
Guy Carpenter
29
Formal Definition
Marginal r(Xj) = lime0[(Y+eXj) – (Y)]/e .
Take derivative of numerator and denominator wrt e.
L’Hopital’s rule then gives r(Xj) = ’(Y+eXj)|0 .
Consider (Y) = Std(Y)
(Y+eXj) = [Var(Y)+2eCov(Xj,Y)+e2Var(Xj)]½ so ’(Y+eXj)|0 =
[Var(Y)+2eCov(Xj,Y)+e2Var(Xj)]-½ [Cov(Xj,Y) + eVar(Xj)]|0
r(Xj) = Cov(Xj,Y)/Std(Y)
With h(X) = X – EX and g(Y) = (Y – EY)/Std(Y)
(Y) =E[(Y – EY)(Y – EY)/Std(Y)] = Std(Y)
r(Xj) =E[(Xj – EXj)(Y – EY)/Std(Y)] = Cov(Xj,Y)/Std(Y)
So this co-measure gives marginal allocation
Guy Carpenter
30
Example – Tail Value at Risk, etc.
Co-TVaR, co-Var are marginal
decompositions
Increasing Xj by (1+a) increases co-measure and measure by
same amount
EPDa = (1 – a)[TVaRa – VaRa] is expected
insolvency cost if capital = VaRa
Co – EPD is a[co-TVaR – co-VaR] and is
marginal
Guy Carpenter
31
Some Criticisms Historically
Quantile is a very limited risk measure
1/3333 quantile impossible to quantify
accurately
Profit not measured relative to marginal cost
of risk
Arbitrary choices required (1/100, etc.)
Not clear that growing units with higher
returns will actually increase risk adjusted
return or firm value
Guy Carpenter
32
Improvements Round 5 – Capital
Consumption
Risk Adjusted Performance Without Capital
Allocation
Alternative to Capital Allocation
(for measuring risk-adjusted profit)
Charge each business unit for its right to access
the capital of the company
Profit should exceed value of this right
Essentially an economic value added approach
Avoids arbitrary and artificial notions of allocating
capital
Business unit has option to use capital when
premiums plus investment income on premiums run
out (company provides stop-loss reinsurance at
break-even)
Company has option on profits of unit if there are any
Pricing of these options can determine economic
value added
Guy Carpenter
34
Insurance Viewpoint
Company implicitly provides stop-loss reinsurance to
each business unit
Any unit losses above premium and investment income on
premium are covered
Value of this reinsurance is an implicit cost of the
business unit
Higher for higher risk units
Subtracting this value from profit is the value added of
the unit
A form of risk adjusted profitability
Right measure of profit to compare is expected value
of profit if positive times probability it is positive
Company gets the profit if it is positive
Company pays the losses otherwise
Comparing value of these options
Guy Carpenter
35
Some Approaches to Valuing
Units that have big loss when firm overall does
cost more to reinsure, so correlation is an issue
Limits on worth of stop loss
Probably worth more than expected value
Probably worth less than market value
Stop-loss pricing includes moral hazard
Company should be able to control this for unit
Or look at impact of unit loss on firm value
Need to understand relationship of risk and value
Guy Carpenter
36
Capital Consumption Summary
Perhaps more theoretically sound than
allocating capital
Does not provide return on capital by unit
Instead shows economic value of unit
profits after accounting for risk
A few approaches for calculation possible
Really requires market value of risk
Guy Carpenter
37
So …
Marginal decomposition with comeasures improves allocation exercise
Choice of risk measure can make result
more meaningful
Capital consumption removes some
arbitrary choices and artificial notions
Market value of risk is really what is
needed
Guy Carpenter
38
Improvements Round 6
Market Value of Risk
Market Value of Risk Transfer
Needed for right risk measure for
capital allocation
Needed to value options for capital
consumption
If known, could compare directly to
profits, so neither of other approaches
would be needed
Guy Carpenter
40
Two Paradigms
CAPM
Arbitrage-free pricing
And their generalizations
Guy Carpenter
41
CAPM and Insurance Risk
Insurance risk is zero beta so should get
risk-free rate?
But insurance companies lose money
on premiums but make it up with
investment income on float
Really leveraged investment trust, high beta?
Hard to quantify
Cummins-Phillips using full information
betas found required returns around 20%
Guy Carpenter
42
Problems with CAPM
How to interpret Fama-French?
Proxies for higher co-moments?
Could co-moment generating function work?
What about pricing of jump risk?
Earthquakes, hurricanes , …
Two standard approaches to jump risk:
Assume it is priced
Assume it is not priced
Possible compromise: price co-jump risk
Guy Carpenter
43
Arbitrage-Free Pricing
Incomplete market so which transform?
Same transform for all business units?
Guy Carpenter
44
No Good Deals
Generalization of arbitrage-free idea
Rules out arbitrage and good deals
Good deals have some risk but so
much more potential reward that
anyone would take the deal
Defined by some arbitrary standard –
maybe 7 flavors already – but gives
more restricted pricing ranges than no
arbitrage
Guy Carpenter
45
So …
Marginal decomposition with comeasures improves allocation exercise
Choice of risk measure can make result
more meaningful
Capital consumption removes some
arbitrary choices and artificial notions
Market value of risk is really what is
needed
Guy Carpenter
46