Modeling and Managing Basis Risk

1. Modeling and Managing
Basis Risk
Society of Actuaries
Equity-
Equity-Based Insurance Guarantees Conference
Boston MA, Oct 12th 2009
Dr. Pin Chung, Vice President, Allianz Investment Management
Dr. Thiemo Krink, Director, Allianz Investment Management
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2. Agenda
§ Overview
§ What is Basis Risk
§ Sources of Basis Risk
§ Fund Mapping
§ Example
§ Basis Risk Management
§ Conclusions
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7. What is Basis Risk?
Basis risk is the deviation between
expected vs. actual funds performance
Under-
Under- or over-performance
over-
relative to benchmark portfolio
Correlation risk
7

8. Tracking Error
n
Tracking Error = ∑(R
i =1
p.i − RB,i )2 /(n − 1)
Measure the severity of deviation
Annualized volatility of the difference
between return of fund and benchmark
5% tracking error commonly allowed
for actively managed funds
8

9. Possible sources of Basis Risk
Possible Sources
Changes of fund manager’style
s
Deviation of benchmark & surrogate portfolio
Unanticipated expenses (fund, tax, etc.)
Changes of policyholder’investment strategy
s
Changes of funds’return profiles
Changes of indices correlation relationships
9

10. Fund Mapping
Fund mapping goal
To reflect the systematic components
(beta coefficients) of the underlying funds
across selected hedging indices
Considerations
How many indices to use?
How long period of data to use?
How frequent to update model?
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11. Fund Mapping Methods
Fund mapping methods
Seriatim on seriatim: Fund Level
Forced asset allocation: Investment Style
Others
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12. Seriatim on seriatim method
Collect historical daily or weekly returns of each
fund and candidate hedging indices return
profile for 3 to 7 years
Perform OLS regression on each fund to
indices to obtain beta coefficients for each
fund
One could apply different weighting
scheme to return data; e.g., equal
weighted or exponential decay
12

13. Forced asset allocation method
Collect historical daily or weekly returns profile
of each asset allocation and candidate hedging
indices return profile for 3 to 7 years
Perform OLS regression on each asset
allocation to indices to obtain beta
coefficients for each asset allocation
One could apply different weighting
scheme to return data; e.g., equal
weighted or exponential decay
13

14. Seriatim on seriatim
Mutual Fund 1 Replicating Portfolio 1
Name 1, 1 Index 1
Name 1, 2
Index 2
Name 1, 20
Index 3
Index 1
Mutual Fund 2 Replicating Portfolio 2
Name 2, 1 Index 1
Name Index 2
Index 2
2, 2
Name 2, 15 Index 3
Index 3
Mutual Fund 3
Replicating Portfolio 3
Name 3, 1 Index 1
Name 3, 2 Index 2
Name Index 3
3, 25 14

15. Forced asset allocation
Aggressive Replicating Portfolio 1
Mutual Fund A, 1 Index 1
Mutual Fund A, 2
Index 2
Mutual Fund A, 30
Index 3
Index 1
Moderate Replicating Portfolio 2
Mutual Fund M, 1 Index 1
Mutual Fund Index 2
Index 2
M, 2
Mutual Fund M, 20 Index 3
Index 3
Conservative
Replicating Portfolio 3
Mutual Fund C, 1 Index 1
Mutual Fund C, 2 Index 2
Mutual Fund Index 3
C, 10 15

16. Example (forced allocation)
Investment style Underlying fund
Equity fund
Aggressive
Balance fund
Moderate
Bond fund
Conservative
Real Estate fund
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17. Example (continued)
§ In matrix form:
 Equity Fund 
 Aggressive Allocation   80 10 7 3  
    Balance Fund 
 Moderate Allocation  =  15 70 10 5  
 Conservative Allocation   5 15 70 10  Bond Fund
    

 RealEstate Fund 
Investment Styles Policy Weights Underlying Funds
n
Note that: ExpReturn(Equ ityFund) = ∑ wi ⋅ ExpReturn( Namei )
i =1
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18. Example (continued)
§ Regress return of each investment style to return
of three liquid and tradable indices, e.g., SPTR,
AGG, and EAFE to obtain the following:
 Aggressive AssetAllocation   0.72 0.13 0.15  SPTR 
    
 ModerateAssetAllocation  =  0.55 0.12 0.33 AGG 
 ConservativeassetAllocation   0.14 0.75 0.11  EAFE 
    
0.83
2  
With R =0.76 For given
Σ3×3
0.82
 
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19. Example (continued)
§ Map each investment style to four
hedging indices, SPTR, AGG, EAFE, RUT
 SPTR 
 AggressiveAssetAllocation   0.63 0.08 0.15 0.14  
    AGG 
 ModerateAssetAllocation  =  0.45 0.22 0.13 0.20 
 ConservativeassetAllocation   0.08 0.72 0.05 0.15  EAFE
    RUT 
 
0.87

2  
With R =0.79
 For given
Σ4×4
0.84

 
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20. Deficiencies of procedure
What went wrong?
Less quick access to fund composition
Fund manager deviates from policy mandate
Policyholders exercise leeway too frequently
Linear relationship breaks down
Correlation relationship breaks down
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21. Basis Risk Management (continued)
Fund Managers
Limit mutual funds choice
Establish direct relationship
Provide “
real time”allocation
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22. Basis Risk Management
hedge develop price in
more indices, contingency basis risk
currencies plan
set review fund
aside capital performance
VA writers
overlay qualitative lower number
considerations of funds
frequent
direct contact use index, ETFs,
monitor fund
with managers passive funds
mapping
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23. Conclusions
§ Basis risk is a risk with great potential to cause
severe financial loss.
§ A few ways to reduce basis risk:
Ø Moving away from actively managed funds into
indices, ETFs or passive funds.
Ø Simplify product design and price in the basis risk
component.
Ø Increase frequency of monitoring fund mapping
procedure.
Ø Have a back-up plan to react appropriately.
back-
23

24. Q&A
§ A journey of a thousand miles begins
with a single step. Lao-tzu
Lao-
§ Thanks for your attention. Questions?
Dr. Pin Chung, Vice President, Allianz Investment Management
pin.chung@allianzlife.com,
pin.chung@allianzlife.com, (763) 765-7647
765-
Dr. Thiemo Krink, Director, Allianz Investment Management
thiemo.krink@allianzlife.com,
thiemo.krink@allianzlife.com, (763) 765-7979
765-
24

25. Appendix
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26. Ordinary Least Squares (OLS)
General linear model: yi=β0+β1xi,1+β2xi,2+…+βp-1xi,p-1+εI
+…+β i,p-
yi is the ith value of the dependent variable
[Fund returns]
β0,β1,…,βp-1 are the regression parameters
,…,β
[Beta coefficients]
xi,1,xi,2,…,xi,p-1 are known values of independent variables
i,p-
[Indices returns]
εi is the independent random error, with N(0,σ2)
N(0,σ
[Residuals]
26

27. Review of OLS (continued)
To estimate β , we minimize the total sum of squares S,
where:
S=Σεi2=Σ(yi-β0-β1xi,1-β2xi,2-…-βp-1xi,p-1-εi)2, where i=1,2,…,n
S=Σε i,p-
Next, simultaneously solving the normal equations:
∂S/∂β0=0, ∂S/∂β1=0,…,∂S/∂βp-1=0
∂S/∂β ∂S/∂β =0,…,∂S/∂β
In matrix form: minimizing S=(Y-Xβ)’ -Xβ)
S=(Y (Y
By solving: ∂[(Y-Xβ)’ -Xβ)]/∂β=0
∂[(Y (Y )]/∂β
With the resulting estimator for β expressed as:
b=(X’ -1X’
X) Y
27

28. Review of OLS (continued)
Let mean(Y) be the mean of the observed values;
mean(Y
fit(Y
fit(Y) be the vector of the predicted values;
mean[fit(Y
mean[fit(Y)] be the mean of the predicted values;
e denote the vector of residuals from the model fit:
e(nx1) =Y-Xb=Y-fit(Y)
Xb= fit(Y
Let SStotal = SSreg+SSerr
SStotal = Total sum of squares
= [Y-mean(Y)]’ -mean(Y)]
[Y mean(Y [Y mean(Y
[Y
SSreg = Regression sum of squares
= {fit(Y)-mean[fit(Y)]}’ Y)-mean[fit(Y)]}
{fit(Y mean[fit(Y {fit( mean[fit(Y
{fit(Y
SSerr = Residual sum of squares
= {Y-mean[fit(Y)]}’ -mean[fit(Y)]}
{Y mean[fit(Y {Y mean[fit(Y
{Y
28

29. Review of OLS (continued)
§ The coefficient of determination is:
R2=SSreg/SStotal
§ R2 compares explained variance with total variance
§ Higher R2 indicates regression fits data better
§ Be cautious not to over fitting the model
§ Use modified R2 to avoid spurious regression
29