Risk Loads
A Historical Perspective
Glenn Meyers
ISO Innovative Analytics
CAS Ratemaking Seminar
March 8 and 9, 2007
General Idea of Risk Loads
• Less risk is better
• For greater risk
– Greater demand for transfer risk
– Greater reluctance to accept risk
– Higher price to transfer
• The problem — Quantify the price of
risk
An Attempt to Quantify risk
Buhlmann - 1970
Premium Calculation Principles
• Standard deviation principle
– Risk Load = a  Std. Dev[Loss]
• Variance principle
– Risk Load =   Var[Loss]
• Expected utililty principle
– U(Equity) = E[U(Equity + Premium - Loss)]
An Early (Late 70’s) Use of a
Mathematical Formula
• ISO Increased Limits Ratemaking
• Used the Variance Principle
– Reference — Miccolis (PCAS 1977)
– Replaced judgmental risk loads.
Problem
Answers were too high for high limits.
Policy
ILF W/O ILF With
Limit
Risk Load Risk Load
100,000
1.000
1.000
1,000,000
2.436
2.792
5,000,000
3.428
5.240
10,000,000
3.818
7.276
Response (Mid 80’s)
Variance  Standard Deviation
Policy
ILF W/O ILF With
Limit
Risk Load Risk Load
100,000
1.000
1.000
1,000,000
2.436
2.792
5,000,000
3.428
4.736
10,000,000
3.818
5.892
(Mid-Late 80’s)
Tension between
Risk Loads and CAPM
• Interpretation of CAPM
– No risk loads
– Insurance risk is diversifiable
– Profit loading depends upon the
covariance of the line of insurance with
the stock market.
Another Interpretation of CAPM
• H.H. Müller (ASTIN — Nov. 1987)
• If a security is independent of all other
securities in the market, then its price is
given by the variance principle!
• This apparently contradicts prior
interpretation.
????
The Math Behind of
Müller’s Assertion
• Let:
Ri  Re turn on ith sec urity
RM  Re turn on market
n
  w i  Ri
i1
The Math Behind of
Müller’s Assertion
Assume Cov Rk ,Ri  0 for all i  k
Cov Rk ,RM
Then k 
Var RM
n

 w i  Cov Rk ,Ri
i1
Var RM
wk

 Var Rk
Var RM
Late 80’s
Standard Deviation Risk Load Fell Apart
• Variance principle
– Good to spread risk among (re)insurers
 X2 Y   X2  2 X Y   Y2   X2   Y2
• Standard deviation principle
– Good to have a diversified portfolio
 X Y    2 X Y  
2
x
2
x
   2 X Y     X  Y
2
x
2
x
Hans Buhlmann
• Use variance principle for individual risks
• Use standard deviation principle for
insurers
Early 90’s
• ISO Competitive Market Equilibrium (CME) Risk Load
– Derived from constrained optimization
• Maximize return subject to constraint on insurer standard
deviation
• Risk load proportional to marginal variance
• Constant of proportionality determined by insurer’s
total return objectives
• Recognized correlation generated by parameter
uncertainty
• Motivated by CAPM constrained optimization
• Still in effect today
Earlier 90’s - Independently
• Rodney Kreps
– “Risk Load as the Marginal Cost of Capital”
• C = Capital, V = Insurer Variance, T = Constant
C T V
dC
T
T2


 C  V
dV 2 V 2C
• Marginal Capital  Marginal Variance
• Equivalent to ISO CME
Similar Derivations of
CAPM and Risk Loads
• Both derived by constrained optimization
• CAPM (and successors)
– Directed toward pricing securities
– e.g. the expected return for an insurer
– Requires capital allocation to apply to lines of insurance
• Risk Load
– Used to carve up insurer’s return to individual policies
– Provides no guidance on total return to insurer
– Equivalent to allocating capital in proportion to marginal
capital
Tying Risk Load to Capital
• Long tailed lines need capital to support
uncertain reserves.
• Cost of holding (allocated) capital over
time needs to be reflected in risk loads.
• Long-tailed lines are not necessarily more
risky, but if the distribution of ultimate
losses are the same, the longer-tailed line
should have higher risk load.
Recent Developments
• Solvency II and risk-based capital
– British FSA and S&P internal models
• Tail-Value at Risk TVaR is a leading
candidate to replace Standard Deviation
as a measure of insurer risk.
– Regulatory Capital = TVaR – Expected Loss
• Should we use marginal regulatory capital
to calculate risk loads?