Transcript Using the Bootstrap in DFA an experiment

Capital Management:
Do You Have the Right Strategy?
Casualty Actuarial Society
1999 Special Interest Seminar
Dynamic Financial
Analysis
Chicago, IL, July 19-20, 1999
Using the Bootstrap in DFA:
An Experiment
William C. Scheel
Swiss Re Investors
Falcon Group
Where Do You Want
To Go Today?

Bootstrap
 What Can It Do for You?
 Uses in DFA…an Illustration
 Pricing Experiment
 It’s Fast. It’s Cool.
The Statistician’s Fractal
The bootstrap is a computerbased method of statistical
inference. No formulas are
needed to answer many statistical
questions.
(I like it already)
Same Method…All Statistics
The BCa confidence interval method
has transformation-respecting
properties. This means that the
procedures are invariant whether we
seek to band the sample mean, median
or some other statistic of interest.
Bootstrap Method

Make Bootstrap Samples (sample with
replacement)
 Calculate statistic using bootstrap samples
 Bias factor is standard normal value
evaluated for the proportion of sample
statistics below average statistic
 Acceleration coefficient from jackknife
samples
Getting a Bootstrap Sample
1.
2.
3.
4.
5.
Make a vector* of uniformly distributed
numbers
Shuffle the vector
Use each element as an offset into the original
data
Evaluate the statistic for each shuffled vector
Repeat steps (1)-(4) about 2,000-5,000 times
*vector has N elements
each with a value of u
1 u  N
Obtaining BCa Confidence
Interval
Evaluate standard
normal at adjusted z
value
Adjusted z value:
 L   pL 
z0 is bias
correction
factor

z0  z 
pL  z0 
1  a z0  z 
zα is standard normal at
α probability
a is acceleration coefficient
Acceleration Coefficient
a
c
6s
3
2
c    fi   
2
i
s    fi   
3
i
f i = the sample statistic calculated
using the ith jackknife sample
What Can It Do for You?

Confidence bands for DFA probability
distributions
 Tighter standard errors of the estimate
 Chance-constrained banding of really
wacko statistics
 Put option pricing
 Getting more out of less
Computer Stuff (fortran)

The technique is a sampling activity and
requires Monte Carlo sampling with
replacement
 Extensive calculations, sorting and scanning
 Ancillary calculations for the statistic for
each bootstrap sample
 Jackknife for the bootstrap samples’
statistics
(You really burn calories
running a bootstrap!)
Excel-fortran Interface

No bootstrap tools generally available
 Microsoft Excel
 Fortran DLL does bootstrap sampling
(Excel is a tad weak for
bootstrap confidence
interval work…you’ll
need fortran or C)
Graphics Palette with
Bootstrap
Applying Bootstrap to DFA
(Let the show begin!)
Experiment with Individual
Claims
The experiment seeks to use
bootstrap results as a convenient and
powerful way to put a confidence
band around the expected incremental
payment emanating from a policy
within a relatively cluttered collection
of risk classes.
(This kinda flopped.
I’ll be back again!)
But,
Central Limit Theorem Using
Bootstrap
 p   i  n
Use bootstrap mean and
standard error for μ and σ.
i
     n
2
p
2
i
2
The number of independent
claims is n.
i
(At last! A use for the Central
Limit Theorem. Well, I’ll
admit I’m stretching it a bit
here!)
P   P  z  P
Bootstrap References

Efron & Tibshirani, Introduction to the
Bootstrap
 Davison & Hinkley, Bootstrap Methods and
Their Application