CS – 15
Risk Premium
for Insurance Product Pricing
Steve Mildenhall, AON Re
Dave Ingram, Milliman USA
Don Mango, AM Re
Risk Premium for Insurance
Product Pricing
Stephen Mildenhall
CAS/SOA ERM Symposium
Washington DC, July 2003
Why a Risk Premium?
• Need to make a profit
• Need to be reasonably confident of making a
profit
• Risk Premium is an all encompassing term
– Covers frictional costs
– Covers pure risk (toss of fair coin)
– Compensation for bearing risk under uncertainty
• Philosophical distractions should be resisted
Risk Premium: 2000BC-today
State of
the world
w
Policy
Payout
E ( L)   g ( L(w )) Pr(w )
L
w
E ( L)   L(w ) Pr(w )
w
E ( L)   L(w ) Pr* (w )
w


E ( L)  h  L(w ) Pr(w ) 
w

All of the above
Probabilities
Financial
Consequences of policy
Risk Premium
•
•
•
•
•
•
•
•
Standard deviation
Variance
Semi-Variance
Percentile/VaR
Tail-VaR
Wang Transform
Esscher Transform
Utility-based
• Micro-view of single risk
• SD, Variance,… of what?
• Which measure is
appropriate?
Measures of Risk
• Problem: collapse distribution to a number
– All moments may not be enough to determine
distribution!
• No consensus methodology
• Rothschild-Stiglitz offer four possible definitions
of when X is “more risky” than Y
1.
2.
3.
4.
X = Y + noise
Every risk averter prefers Y to X (utility)
X has more weight in the tails
Var(X) > Var(Y)
1, 2, and 3 are equivalent and are different from 4
Parameter Risk: don’t delude
yourself
• Variance of losses in your model is not the same
thing as variance of losses!
– Hayne’s Loss Reserving Example (CLRS)
• Leverage, Excess Policies and Jensen’s inequality
E( f ( X ))  f (E( X ))
– Need to compute the mean correctly
– Risk load should not be used to compensate for
miscellaneous actuarial inadequacies
Don’t believe a risk load formula that says a
new small line is a good thing!
Size: what is a large risk?
• Parameter risk is all that matters…almost
• Process risk matters for large risks
• Large?
–
–
–
–
–
100M households in US
$1M loss = 1¢ per household
$100M loss = $1 per household
$1B loss = $10 per household
$10B loss = $100 per household
Large
Size: what is a large risk?
• Heterogeneous distribution of wealth
• Demographics
– Ultimate risk bearers are individual insureds
– Population concentrations correlated to risk
loads
• Frequency of losses, size of market
Don’t believe a risk load formula that does
not account for population demographics
Big Picture: moving beyond
individual policy risk
All states of the world
States of the world
relevant for one policy
w
Multiple states yielding same
loss L for one policy
Projection
with loss of
information
Policy
Payout
L
Big Picture: moving beyond
individual policy risk
Sim#
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
A
77,490
58,089
78,255
8,934
45,939
37,614
5,379
16,600
36,492
53,382
6,911
42,304
4,114
31,730
15,796
19,180
6,756
3,967
9,401
11,352
B
123,643
44,276
41,085
115,909
66,417
34,151
50,342
19,034
10,658
16,521
43,635
12,079
26,544
16,976
33,017
17,466
18,021
14,584
16,660
3,810
C
12,301
54,757
18,167
1,317
2,677
31,340
24,204
40,084
27,340
3,671
17,632
6,515
29,910
10,725
7,587
13,622
22,012
12,489
3,956
8,277
Total
213,435
157,122
137,507
126,160
115,033
103,105
79,925
75,717
74,490
73,574
68,179
60,898
60,568
59,431
56,401
50,268
46,789
31,040
30,017
23,439
Mean
Loaded
Load
28,484
40,416
42%
36,241
53,667
48%
17,429
19,733
13%
82,155
113,815
39%
SD
CV
23,584
82.8%
31,804
87.8%
13,564
77.8%
46,310
56.4%
Transf Probs
0.160
0.104
0.086
0.074
0.066
0.060
0.054
0.050
0.046
0.042
0.039
0.036
0.033
0.030
0.027
0.025
0.022
0.019
0.016
0.011
Big Picture: moving beyond
individual policy risk
• Rodney Kreps, co-measures
Eg ( X i ) condition on  X i 
• P/C: Catastrophe (re-)insurance
– Cat models explicitly quantify correlation
• Life: Hedging interest rate and investment
risk
Three Points to Remember
• Parameter Risk
• Size
• Think Big-Picture
Pricing for Risk
David Ingram
ERM Symposium
Washington DC, July 2003
Pricing for Risk
1. RMTF Survey of current Practices
2. Methods for Setting Risk Margins
a.
b.
c.
d.
Charge for Risk Capital
Risk Adjusted Hurdle Rates
Adjusted Target Calculation
Replication
How Do you Price for Risk?
Capital
Risk-adjusted
allocation
profit target
ROI
IRR
ROE
CTS
Premium Margin
25%
26%
30%
21%
16%
18%
18%
16%
27%
23%
Stochastic
scenario
analysis
19%
17%
21%
17%
13%
Assumption
PADS
13%
12%
11%
15%
18%
Assumption
stress testing
25%
26%
20%
19%
28%
What is the basis for Risk
Adjustment?
3. If you use capital allocations for reflecting risk, how
are these allocations determined?
Regulatory formula multiple
147
55.26%
Internal formula
69
25.94%
Economic capital
44
16.54%
Other
6
2.26%
Total:
266
What is the basis for Risk
Adjustment?
4. If you use assumption PADS, how are these PADS
determined?
Analysis of recent experience
66
50.00%
Industry standard
36
27.27%
Other
5
3.79%
Stochastic scenario analysis
25
18.94%
Total:
132
What is the basis for Risk
Adjustment?
5. If you risk-adjusted profit objective, how is it
determined?
Judgment
81
Formula
44
Other
6
Total:
131
61.83%
33.59%
4.58%
What is the basis for Risk
Adjustment?
6. If you use assumption stress testing, how are the
parameters determined?
Judgment
137
59.05%
Confidence limits
48
20.69%
Worst case historical experience
42
18.10%
Other
5
2.16%
Total:
232
What is the basis for Risk
Adjustment?
7. If you use stochastic scenario analysis, how is the
distribution of results analyzed?
Percentiles
83
30.51%
Mean-variance analysis
44
16.18%
Conditional tail expectation (CTE)
44
16.18%
Problem scenario analysis
38
13.97%
Value at risk
24
8.82%
Efficient frontier
23
8.46%
Earnings at risk
14
5.15%
Other
2
0.74%
Total:
272
Methods for
Setting Risk Charge
• Judgment Methods
• Quantitative Methods
Judgment Methods
• Risk Premium based on
–
–
–
–
Prior products
Market prices
Comfort with particular risks
Relative perceived risk of company products
Quantitative Methods
1.
2.
3.
4.
Charge for Risk Capital
Risk Adjusted Hurdle Rates
Adjusted Target Calculation
Replication
Charge for Risk Capital
• Most common quantitative risk adjustment
to pricing
• Charge is:
– (HR – is) * RCt
• Where HR is Hurdle Rate
• is is the after tax earnings rate on surplus assets
• RCt is the risk capital in year t
Charge for Risk Capital
• Is it actually a charge for risk?
– Or just a cost of doing business?
• It is a charge that is proportionate to risk
• If there are other risk charges or
adjustments, need to be careful not to
double charge for risk
Risk Adjusted Hurdle Rates
•
Efficient Frontier Analysis
•
Market Analysis
Efficient Frontier Analysis
Process
A.Brainstorming
B.Modeling
C.Display / Identify Frontier
D.Determine Risk/Reward Trade-off Parameters
Efficient Frontier
Premier III
Efficient Frontier
35.00
30.00
Return
25.00
20.00
15.00
10.00
5.00
-
2.00
4.00
6.00
Risk
8.00
10.00
12.00
14.00
Market Analysis
• Study Relationship between Return and
– Product Concentration
– Income/ ROE volatility
For a group of successful companies.
• Develop Target returns
– Based on Products
– Based on volatility
Market Analysis
Product Concentration
ROE Volatility
• Product A – 12%
• Product B – 15%
• Product C – 10%
Target ROE =
• Risk-free rate + 3.7 
• 22.83% +1.83% ln()
• 7.5% + 
Market Analysis
While this is “quantitative”…
Data is so thin that much judgment is needed
to develop targets
Study of Insurance Company
ROE
ROE
Std Dev
Ratio
Group I
13.96%
6.71%
48%
Group II
10.52%
11.32%
107%
Group III 10.12%
16.02%
158%
Group IV
4.86%
25.96%
534%
Group V (3.69%)
21.13%
NM
Adjusted Target
• Instead of concentrating on 50th Percentile
results (or average results)
– In a stochastic pricing model
• Pricing Target adjusted to 60th, 70th or 80th
Percentile
Adjusting Target
Monthly Returns
20%
50th Percentile
0.86%
15%
10%
80th Percentile
(2.35%)
Average
0.72%
5%
0%
100% 95%
-5%
-10%
-15%
-20%
-25%
90%
85%
80%
75%
70%
65%
60%
55%
50%
45%
40%
35%
30%
25%
20%
15%
10%
5%
0%
Replication
• Finance – Law of One Price
– Two sets of securities that have the same
cashflows under all situations will have the
same price
• Replication – if you can replicate the
cashflows of an insurance product with
marketable securities then market price of
securities is the correct price for product
Risk & Return
• Bonds – Volatility of Bond Prices 8.6%
– Average Return on Bonds – 5.8% compound
Average, 6.1% Arithmetic Average
– Risk/Reward = 139% to 148%
• Stocks – Volatility of Stock Returns 20.5%
– Average Return – 10.5%, 12.2%
– Risk Reward = 168% to 194%
Insurance Products
• Cannot easily hedge with 100% efficiency
• But can compare…
VA Product vs.
Common Stocks
• Insurance Product – VA
– $10 B AV
– Std Dev = 200, CTE 90=429
Compare to
• Common Stock Fund A
– $300 M Fund
– Std Dev= 200, CTE 90= 390
• Common Stock Fund B
– $330 M Fund
– Std Dev= 220, CTE 90= 429
Returns
• Insurance Product – VA
– 75 Expected Return
• Common Stock Fund A
– 100 Expected Return
• Common Stock Fund B
– 110 Expected Return
Recommendations
1. Work on evolving from Judgment to
Quantitative
2. Quantitative methods need to be based on
Pricing Risk Metric
3. Ultimately should tie to market pricing for
risks
Risk Premiums
Don Mango
AM Re
Where Are We Going?
•
•
•
•
•
Commonalities
Simulation Modeling
Explicit Valuation
Aggregate Risk Modeling
Interaction Effects
Commonalities
• Valuation of Contingent Obligations
(“VALCON”)
• Levered investment trusts
• Strong dependencies on economic and
capital market conditions
Commonalities
• Long time horizons and held-to-maturity
(“HTM”) portfolios
• We sell “long-dated, illiquid, OTC
derivatives”
• We have an incomplete, inefficient
secondary market
• We retain magnitudes of risk that bankers
would never dream of
Commonalities
• IMPLICATIONS:
• This seminar should be the norm, not the
exception.
• There may be hybrid products in our future.
• We may not be able to simply borrow
capital market techniques.
Simulation Modeling
• Aka “Monte Carlo valuation”
• Financial engineers use it to price longdated, illiquid, OTC derivatives
• Devil is in the parameters and dependence
structure
Simulation Modeling
• IMPLICATIONS:
• We are heading the same direction.
• We need transparency or at least
explicitness of assumptions.
Explicit Valuation
• Complete, efficient market affords
participants the luxury of not having to
think or care or have any opinion of the
fundamental value of a product
• Counting on the continued presence of
counterparties to limit downside
• Bloomberg gives you “the price”
Explicit Valuation
• Incomplete, inefficient market requires
some explicit valuation by its participants
• True, you could be a “delta” off a content
provider
– 10% below Swiss Re or Met Life
Explicit Valuation
• IMPLICATION: If you want to be a leader,
formulate a risk appetite and apply it.
– Read Karl Borch, 1961
• What are your desired payoff profiles, and
please be specific and use quantities!
Aggregate Risk Modeling
• Valuation  develop indicated price based
on the impact of the product on your
portfolio – a “MARKET OF ONE”
• “One Price” does not mean One Value
• Value is idiosyncratic and in the eye, mind,
interpretive filter, and model of the beholder
Aggregate Risk Modeling
•
•
•
•
Requires aggregate portfolio risk modeling
Integration of disparate risks
A critical goal of our ERM efforts
Sounds like it might require actuaries of all
kinds …
Aggregate Risk Modeling
• IMPLICATIONS:
• Get information content into the indicated
prices and (hopefully) the quotes.
• Risk Management is that Content Provider.
Interaction Effects
• Indicated price meets market strategy,
premium goals, expense ratios,
relationships, history, culture, decision
process, …
• Multiple participants selling promises with
indistinguishably small probabilities of nonperformance
Interaction Effects
• Throw in some “momentum sellers” going
delta off the content providers
• Result is an unstable system dynamic = “the
insurance market”
• Mutually reinforcing behaviors, for good or
bad
Interaction Effects
• IMPLICATIONS:
• Theory aside, the attainable risk premium
will rarely be where it “should be.”
• Market Price represents somebody’s quote
(usually the LCD – winner’s curse) – no
“exogenous” source
• No more isolated strategy development –
we have seen the enemy, and it is us.