Survey Results / Overview of Methods
CAS Limited Attendance Seminar
on Risk and Return in Reinsurance
Stephen Lowe
26 September 2005
Sixteen survey participants
 ACE Tempest Re
 Odyssey Re
 AWAC
 Partner Re
 Chubb Re
 Platinum Re
 GE
 QBE Re
 GMAC Re
 Scor Re
 Hannover Re
 Signet Star
 Max Re
 Toa Re
 Montpelier Re
 Transatlantic Re
2
Traditional approaches to pricing
Approach
Measure
Variations
Return on
Sales
Target
Combined
Ratio
 Nominal versus Discounted
Return on
Capital
Target
Return
 ROE based on NPV of Internal
 Fixed versus Variable Target
Cash Flows versus IRR of Free
Cash Flows
 Fixed Versus Variable Target
 Rating Agency Capital versus
Economic Capital
These methods are usually applied to
deterministic (i.e., expected) cash flows
3
Stochastic pricing methods
Thanks,
Don
Approach
Description
Measure
Standalone
Tail VaR
Required capital a fn of contract
outcome distribution Tail VaR
Target Return
Marginal
Tail VaR
Required capital a fn of marginal
impact on portfolio outcome
distribution Tail VaR
Target Return
R2R
Calculate R2R from contract
outcome distribution
Target R2R
Wang
Transform
Price is expected outcome using
modified probabilities
Adjusted Expected
Value
Capital
Consumption
Price is expected outcome using
modified amounts
Adjusted Expected
Value
4
Typical descriptions of method
 Target ROE, comparing NPV of contract cash flows to equity
based on leverage ratios
 Target underwriting profit by class of business
 Target ROE, using NPV model that balances to capital
requirements
 Target IRR, based on free cash flows (capital and profits in/out)
 Target ROE, reflecting corporate cost of capital, based on NPV
of contract cash flows and internal RBC factors
 Variety of methods that look at downside risk and utility metrics;
game theory considered
 Metrics relating to simulated contract results distribution used to
determine leverage required, then target ROE
5
How are profit margins set in pricing?
Return
on Sales
Return
on Capital
Nominal
NPV


 
 
IRR

 One company responded that they used “a variety of methods”
6
Do pricing methods vary by line?
 Most companies indicated that they use the same general
method for all lines
 Exceptions:
 Property catastrophe, where pricing reflects the marginal
impact of the contract on the portfolio
 Clash covers, where a bank approach is taken
 Property business, where volatility of individual contract and
portfolio concentration is taken into account
 Contracts with loss sensitive features treated differently
 One company responded that they used “a variety of methods”
that vary by line
7
How is risk reflected?
At the class of business level




Fixed ROC, but RBC allocates more capital to volatile
classes
Variable ROC, and RBC allocates more capital to volatile
classes
Profit margins vary with volatility of class
At the individual contract level




Risk loads determined by individual contract simulation
Contracts with unusually high risk have target set higher
than the standard target for the class
Underwriters make judgmental adjustments
Volatility of contract is benchmarked to other contracts in
class
8
How is capital allocated?





Rating agency RBC factors
Leverage ratios
Management allocation
Internal capital model (Economic Capital)
Volatility of class



Individual contract simulation distribution
Individual contract downside risk
Contract characteristics

Not allocated
9
How are pricing targets
reconciled with corporate financial goals?






They are the same; they are consistent
No reconciliation is made
Reconciliation assures that aggregate pricing return
is greater than overall financial target
They are expected to be similar
Differences reflect actual versus rating agency
capital
IRR versus ROE make them different
10
What enhancements are
being developed or considered?
 Allocation of capital to contract is being tested
 Researching RAROC
 Researching greater use of marginal portfolio impact in the
allocation of capital
 Need to understand correlations between lines to implement
marginal impact
 Refinements to marginal capital allocation
 Looking at game theoretical constructs
 Researching internal risk models
 Implementing rating agency capital formula into capital allocation
11
 Additional Materials on Stochastic Pricing
12
Pricing to a Target R2R = 8.00
a)
b)
c)
Probability
Premium Expenses
of Loss
$1,000,000 $175,000
20%
15%
15%
15%
10%
10%
10%
2.5%
2.5%
a)
b)
d)
e)
NPV Amount
(a-b-d)
of Loss
Loss or Profit
$0
$825,000
$100,000
$725,000
$250,000
$575,000
$350,000
$475,000
$500,000
$325,000
$750,000
$75,000
$1,000,000
-$175,000
$1,500,000
-$675,000
$2,500,000
-$1,675,000
Exp Value
$395,000
D/U Ratio
Prob of …
Product
R2R
c)
Probability
Premium Expenses
of Loss
$1,067,660 $175,000
20%
15%
15%
15%
10%
10%
10%
2.5%
2.5%
d)
e)
NPV Amount
(a-b-d)
of Loss
Loss or Profit
$0
$892,660
$100,000
$792,660
$250,000
$642,660
$350,000
$542,660
$500,000
$392,660
$750,000
$142,660
$1,000,000
-$107,340
$1,500,000
-$607,340
$2,500,000
-$1,607,340
Exp Value
$462,660
D/U Ratio
Prob of …
Product
R2R
f)
g)
Downside
-$175,000
-$675,000
-$1,675,000
-$508,333
0.92
15.0%
$
76,250 $
6.18
Upside
$825,000
$725,000
$575,000
$475,000
$325,000
$75,000
$554,412
f)
Downside
-$107,340
-$607,340
-$1,607,340
-$440,674
0.71
15.0%
$
66,101 $
8.00
85.0%
471,250
g)
Upside
$892,660
$792,660
$642,660
$542,660
$392,660
$142,660
$622,071
85.0%
528,761
13
WANG TRANSFORM
PENALTY FUNCTION
Transform the distribution
amounts or probabilities?
Value
X:
U (X ) :
X:
E[ X ]q
x1
x2
…
Either approach uses
SUMPRODUCT of
amounts and
probabilities
xn
Prob
p1
p2
…
pn
Value
U(x1)
U(x2)
…
U(xn)
Prob
p1
p2
…
pn
Value
x1
x2
…
xn
Prob
p1
p2
…
pn
Value
x1
x2
…
xn
Prob
q1
q2
…
qn
Downside
penalty function
modifies the
amounts
Probability
Transform or
“Measure
Change”
modifies the
Probabilities
14
Downside Penalty
a.k.a. Capital Consumption
 Risk Load = E[X*] expected value of adjusted amounts
 Adjustment happens by modifying the amounts using a capital
consumption penalty:
— Zero if positive NPV outcome
— Multiple of outcome if negative NPV outcome
 Expected value = SUMPRODUCT of Amounts and Probabilities
15
Capital Consumption
NPV Distribution
6,000,000
Once NPV Falls Below Zero, Penalties
Assessed to Offset Consumption of
Additional Capital
4,000,000
2,000,000
0
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
-2,000,000
NPV Above Zero – No Penalties
-4,000,000
-6,000,000
-8,000,000
16
Capital Consumption Pricing Example
a)
b)
c)
Probability
Premium Expenses
of Loss
$1,000,000 $175,000
20%
15%
15%
15%
10%
10%
10%
2.5%
2.5%
Downside
(Capital Consumed)
Amounts Increased
d)
e)
NPV Amount
(a-b-d)
of Loss
Loss or Profit
$0
$825,000
$100,000
$725,000
$250,000
$575,000
$350,000
$475,000
$500,000
$325,000
$750,000
$75,000
$1,000,000
-$175,000
$1,500,000
-$675,000
$2,500,000
-$1,675,000
$395,000
Penalty Charge
Mean
Risk-Adjusted Mean
f)
Adjusted
Amounts
-$350,000
-$1,350,000
-$3,350,000
-$1,016,667
200.0%
$395,000
$318,750
17
Wang Transform
Modifies the Probabilities

F * ( x)    ( F ( x))  
1

In Excel:
F* = normsdist( normsinv(F) - lambda )
 Makes severe outcomes appear more likely by reducing their
implied percentile
 For example, if lambda = 0.5, a 3 std deviation outcome
becomes a 2.5 std deviation outcome
18
The Wang Transform shifts the NPV distribution,
giving more weight to the tail of the distribution.
100%
90%
80%
F(X)
70%
60%
50%
40%
Input
30%
20%
10%
0%
500
Transformed
1,000
X
1,500
2,000
Unlike TVaR and VaR, WT considers the entire distribution
19
Wang Pricing Transform
Modifies the Probabilities
Premium Expenses
$1,000,000 $175,000
Probability
of Loss
20%
15%
15%
15%
10%
10%
10%
2.5%
2.5%
NPV Amount
(a-b-d)
of Loss
Loss or Profit
$0
$825,000
$100,000
$725,000
$250,000
$575,000
$350,000
$475,000
$500,000
$325,000
$750,000
$75,000
$1,000,000
-$175,000
$1,500,000
-$675,000
$2,500,000
-$1,675,000
$395,000
Applies
Probabilities
a) a Greater
b) Weight to
c) Downsided)…. By Modifying
e)
f)
Downside
Upside
$825,000
$725,000
$575,000
$475,000
$325,000
$75,000
-$175,000
-$675,000
-$1,675,000
-$508,333
g)
Lambda
Probability Cumulative
0.75
Adjusted
Premium Expenses of Loss
Probability NORMSINV Transform Probability
$1,000,000 $175,000
20.0%
20.0%
(0.84)
(1.59)
5.6%
15.0%
35.0%
(0.39)
(1.14)
12.8%
15.0%
50.0%
(0.75)
22.7%
15.0%
65.0%
0.39
(0.36)
35.8%
10.0%
75.0%
0.67
(0.08)
47.0%
10.0%
85.0%
1.04
0.29
61.3%
10.0%
95.0%
1.64
0.89
81.5%
2.5%
97.5%
1.96
1.21
88.7%
2.5%
100.0%
100.0%
h)
$554,412
i)
j)
Implied
NPV Amount
Prob
of Loss
Loss or Profit
5.6%
$0
$825,000
7.2%
$100,000
$725,000
9.9%
$250,000
$575,000
13.1%
$350,000
$475,000
11.2%
$500,000
$325,000
14.3%
$750,000
$75,000
20.2%
$1,000,000
-$175,000
7.2%
$1,500,000
-$675,000
11.3%
$2,500,000
-$1,675,000
Exp Value -- Unadjusted
$395,000
Exp Value -- Adjusted
-$9,100
Target adjusted ENPV
20