1. Beware of the Tail
Phelim P. Boyle1
1School of Business and Economics
Wilfrid Laurier University
29 July 2016
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2. Thanks to Kevin Fan and Charlene Liu for research assistance
Phelim P. Boyle (Wilfrid Laurier University) Beware of the Tail 29 July 2016 2 / 24

3. Outline
1 Investment Guarantees
2 Optimal Contract Design
3 Risk Managment of Guarantees
4 Put Option Returns
5 Buffett’s Put Options
6 Actuarial Approaches in UK and Canada
7 Summary
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4. Investment Guarantees
1960’s and 1970’s rise of equity linked products
Strong demand for guarantees
Simplest guarantee money back at maturity
Long position in portfolio plus a put option
Risk management has proved a challenge
Actuarial thinking has evolved
Nowadays some guarantees are very complicated
Phelim P. Boyle (Wilfrid Laurier University) Beware of the Tail 29 July 2016 4 / 24

5. Optimal Contract Design
Assume investors maximize their expected utility
Two assets: risk free asset and market index
Risky asset has lognormal returns
Assume CRRA utility
Can compute optimal payoff in closed form
Payoff profile depends on investors risk aversion
Plot payoff against market index
It does not correspond to long stock plus put profile
Hence maturity guarantee is not optimal in this framework
Phelim P. Boyle (Wilfrid Laurier University) Beware of the Tail 29 July 2016 5 / 24

6. Notation
CPRA utility function: u(w) = 1
1−ρw1−ρ, ρ is risk aversion
parameter; w denotes wealth
Index returns follow lognormal i.i.d distribution with mean µ and
variance σ2; rf is the risk free rate
dS
S
= µdt + σdz (1)
Merton ration α = µ−rf
ρσ2
Optimal payoff ˆX; optimal return ˆR =
ˆX
w
ˆR = e(1−α)(r+1
2
ασ2)T
Rα
T (2)
where RT = ST /S0
Phelim P. Boyle (Wilfrid Laurier University) Beware of the Tail 29 July 2016 6 / 24

7. Parameters
Parameter Value
µ 0.08
rf 0.035
σ 0.15
T 5
Initial wealth 1
S0 1
K 1
ρ 1, 2, 5, 10
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8. Optimal Payoffs with Different ρ and Maturity Guarantee
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.5
1
1.5
2
2.5
3
3.5
Index Return R
PotfolioPayoffx
Payoff of an Optimal Portfolio and Maturity Guarantees
Put option and Long Index
ρ = 1
ρ = 2
ρ =5
ρ=10
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9. A Simpler Contract
Previous payoff assumed continuous trading
Not feasible for retail investor
Suppose we just make initial investment choice
Optimal buy and hold strategy to maximize expected utility
Agent invests some in the index balance in the risk free asset
This also dominates the standard maturity guarantee
Preference model or return model too simple?
The contracts from insurance companies have high transaction costs
This adds to the puzzle
Phelim P. Boyle (Wilfrid Laurier University) Beware of the Tail 29 July 2016 9 / 24

10. Risk Managment of Guarantees
Hold adequate capital
Hedge the risks using option approach
Transfer the risk to a third party
Reinsurance or investment bank
Or buy the options from Berkshire Hathaway?
Modify the contract design
Combination approach
Ignore. Surely not
Phelim P. Boyle (Wilfrid Laurier University) Beware of the Tail 29 July 2016 10 / 24

11. Distribution of Put Options
Consider distribution of put option on market under P measure
Put option pays when the market is down
From consumption CAPM, we expect put to have low risk premium
Put option expected returns are less than the risk free rate
Noted in the literature by Coval and Shumway (2001), Journal of
Finance
Phelim P. Boyle (Wilfrid Laurier University) Beware of the Tail 29 July 2016 11 / 24

12. Expected Return on Put Options
Table: Expected Return on Put(p.a.);
Assume S0 = K = 100, T = 5, rf = 0.035, σ = 0.15
µ Expected return on put
0.04 0.0136
0.05 -0.0314
0.06 -0.0795
0.07 -0.1307
0.08 -0.1851
Phelim P. Boyle (Wilfrid Laurier University) Beware of the Tail 29 July 2016 12 / 24

13. Moments of Return on Put Option
Assume S0 = K = 100, T = 5, rf = 0.035, σ = 0.15, µ = 0.08
Moments Value
Expected return of put option -0.6037
Standard Deviation 1.2114
Skewness 3.6888
Kurtosis 17.6344
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14. Berkshire Put Options
Entered into equity put transactions between 2004 to 2008, majority
trades were executed between 2004 and 2007;
Sold over-the-counter European put options with maturities of 15 to
20 years
Underlying index: S&P 500, EuroStoxx 50, FTSE 100 and Nikkei 225
Berkshire collected $4.9 billion in premiums
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15. Buffett’s Put Options
2008 shareholder letter:
Notional value of the put options totalled $37.1 billion
Recorded put liability $10 billion or a mark-to-market loss of $5.1 billion
2010 shareholder letter:
8 of 47 equity contracts were unwound, left 39 short equity index put
contracts
Received $647 million for selling those puts and paid $425 million to
get out of those obligations
2011 shareholder letter:
Book value of put liability was $ 8.5 billion
Phelim P. Boyle (Wilfrid Laurier University) Beware of the Tail 29 July 2016 15 / 24

16. Features of Buffett’s Put Options
Many features of contracts to Berkshire’s advantage
Contracts had no margin requirements
During crisis Berkshire did not have to post margin
In 2009, Berkshire’s CDS rates skyrocketed
The perceived default risk of Berkshire increased
Puts were European with long maturities
Written on indices so filtered out ailing companies
Berkshire’s insurance float uncorrelated with market
No capital requirements for writing the puts
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17. Put values for h=.002
0 5 10 15
6
7
8
9
10
11
12
Time to maturity
Optionprices
Black Scholes put
Vulnerable put h=.002
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18. Put values for h=.050
0 5 10 15
5
6
7
8
9
10
11
12
Time to maturity
Optionprices
Black Scholes put
Vulnerable put h=.05
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19. UK Maturity Guarantees
Working Party set up in 1977
To recommend bases of reserving for investment guarantees
Actuarial and non-actuarial literature to be considered
Report recommended setting up reserves based on ruin probabilities
Based on Wilkie Model
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20. UK Maturity Guarantees
WP had access to option pricing papers
WP reached varying degrees of confidence that the mathematics in
these papers was sound
WP concluded there were serious practical disadvantages in the
approach
Worried about model error
Concerned that all hedging trades in same direction
Recommended no reduction in reserves for hedging
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21. Why UK did not adopt option approach?
Some of WP concerns were valid
Papers available to WP were not very accessible
Stochastic calculus difficult
Technology not then available (Uber)
Suitable investments did not exist then
Actuarial conservatism
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22. CIA Segregated Funds Task Force
Set up in 1999 reported in 2001
Strongly advocated stochastic models
Move to stochastic approach via factors
Minimum capital requirements based on the tail risk
Introduced hidden Markov Model (in time for the crisis)
Also introduced CTE, a coherent risk measure
Report had immediate impact on actuarial practice
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23. Comparison of Lognormal and RSLN CTE’s
Table: Comparison of CTE under Lognormal and RSLN Based on Same Data
S0 = K = 100, T = 5, h = 5, µ = 0.08, σ = 0.15, rf = 0.035
CTE(0.90) CTE(0.95) CTE(0.975)
Ratio ( RSLN(CTE)
Lognormal(CTE)) 1.6592 1.5072 1.4397
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24. Summary
Guarantees important in equity linked contracts
Risk management: ongoing challenge for actuaries
Academic research has been helpful in this task
Need to communicate with practitioners
Puzzle of optimality
Analyzed put options returns
Berkshire puts
Other topics
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