Testing Distributions of
Stochastically Generated
Yield Curves
Gary G Venter
AFIR Seminar
September 2003
Advantages of Stochastic
Generators

Deterministic scenarios allow checking risk against
specific outcomes

Stochastic generators add dimension of probability of
scenarios

Can incorporate full range of reasonably possible
outcomes

Each scenario can be a time series of outcomes
Guy Carpenter
2
Testing for Potential Problems
of Stochastic Generators

Model could miss possible scenarios

Model could overweight some unlikely scenarios and
underweight others – giving unrealistic distribution of results

Traditional tests look at time series properties of individual
scenarios – like autocorrelations, shapes of curves compared
to historical, correlation of short and long term rates and their
comparative volatility, and mean reversion

Options pricing models test distributions across scenarios by
their impacts on option prices

For insurer risk models, we propose testing generators by
comparing distributions of yield curves against historical

Look for aspects of historical distributions that do not change
too much over time
Guy Carpenter
3
Some Models of the Yield Curve
(Then we’ll look at testing)
Example Short-Term Rate Models

Usually defined using Brownian motion zt. After time t, zt
is normal with mean zero and variance t.

Cox, Ingersoll, Ross (CIR):
dr = a(b - r)dt + sr1/2dz In discrete form for a short
period:
r t – r t–1 = a(b – r t–1) + sr t –11/2

CIR change in interest rate has two components:
– A trend which is mean reverting to b, i.e., is negative if
r>b and positive if r<b
Speed of mean reversion given by a
– A random component proportional to r1/2, so variance
rts2 in time t

Guy Carpenter
5
Adding Effects to CIR

Mean that is reverted to can be stochastic:
d b = j(q - b)dt + wb1/2dz1

This postulates same dynamics for reverting mean as for r

Volatility can be stochastic as well:
d ln s2 = c(p - ln s2)dt + vdz2

Here Brownian motion in log

Power on r in dz term might not be ½ : dr = a(b - r)dt + srqdz

CIR with these two added factors fit by Andersen and Lund,
working paper 214, Northwestern University Department of
Finance, who also estimate the power of r (1/2 for CIR).
Guy Carpenter
6
Fitting Stochastic Generators

If you can integrate out to resulting observed periods
you can fit by MLE
– CIR distribution of rt+T given rt is non-central chi-sq.
– f(rt+T|rt) = ce-u-v(v/u)q/2Iq(2(uv)1/2), where
– c = 2as-2/(1-e-aT), q=-1+2abs-2, u=crte-aT, v=crt+T

Iq is modified Bessel function of the first kind, order q
– Iq(2z)=
Sk=0z2k+q/[k!(q+k)!], where factorial off
integers is defined by the gamma function

Can use this for mle estimates of a, b, and s
Guy Carpenter
7
Fitting Stochastic Generators

If cannot integrate distribution, some other methods
used:
– Quasi-likelihood
– Generalized method of moments (GMM)



Guy Carpenter
E[(3/x) ln x] is a generalized moment, for
example
Or anything else that you can take an expected
value of
Need to decide which moments to match
8
Which Moments to Match?

Title of paper developing efficient method of moments
(EMM)

Suggests finding the best fitting time-series model to the
time-series data, called the auxiliary model

Scores (partial derivates of log-likelihood of auxiliary
model) are zero for the data at the MLE parameters

EMM considers these scores, with the fitted parameters of
the auxiliary model fixed, to be the generalized moments,
and seeks the parameters of the stochastic model that
when used to simulate data, gives data with zero scores

Actually minimizes distance from zero
Guy Carpenter
9
Andersen-Lund Results

Power on r in r-equation volatility somewhat above ½

Stochastic volatility and stochastic mean reversion
are statistically significant, and so are needed to
capture dynamics of short-term rate

Used US data from 1950’s through 1990’s
Guy Carpenter
10
Getting Yield Curves from Short
Rate Dynamics



P(T) is price now of a bond paying €1 at time T
This is risk-adjusted expected value of €1 discounted
continuously over all paths:
P(T) = E*[exp(-r tdt)]

Risk adjustment is to add something to the trend terms
of the generating processes

The added element is called the market price of risk for
the process
Guy Carpenter
11
Testing Generated Yield Curves

Want distributions to be reasonable in comparison to
history

Distributions of yield curves can be measured by
looking at distributions of the various yield spreads

Yield spread distributions differ depending on the
short-term rate: spreads compacted when short rates
are high

Look at conditional distributions of spreads given
short-term rate
Guy Carpenter
12
Now for Testing
(Proposed Distributional Test)
Three Month Rate and 10 – 3 Year Spread
Clear inverse relationship
Mathematical form changes
Five periods selected
Guy Carpenter
14
Ten – Three Year Spreads vs Short Rate
Slope constant but intercept changes each period
0.025
0.020
0.015
R103
0.010
0.005
0.000
1 1960-1968
0.01096 - 0.272*x
2 1968-1979
0.0171 - 0.2526*x
44 4
3 1979-1986
4
4 1986-1995
4
44
44 44
4
4
4 444
5 1995-2001
4
4
4
4
4
4
4
4
42
2 22
44
44
442
33 3
2
3
22
3
3 333
33
4
2
2222 4
3
4
4
3
4
4
3
4
22
4
2 4
4
44 2 4
3
3 3 3
4
3
2 44
33
3
445224 224
33 3
4 4444
244
4
4
2
3 3
3
33
2
3 3
4
44
1
24
2 22 22
4
1
4
22
3
3
2
1
3
111
1
3
2
4
5
2
1
1
2
11 1
2 2
2
11
4 4
5
1
1
3 33
22 5 2 5
111
11
55
55
44
55
1
422
2 25
3
5554224224
1
5
2
11111 1 1
2
2
2
5
2
2
3
3 3
5
1
5
5
22 2
11
3
1 11
3
1
55
25
1 111 1
2
5
25
4
5
2
5
2
1
5
2
2
44
5
1 1
5
11
33
5555555552 5
1
1
5
55
1
1111 1
2 2 2
1
5
1
1
1
5
1
1 1
5 25
552 2 52
22 24
2
1
5
4222
3
4
55
11 1
1
2
3
222 22
44
1
1 5
44
2 2
3
5 2252 2222 2 2 22 44
3
3
3
1
1 11
22
44
3
22
1
1 111
1
1 11 1
1
1
-0.005
22
5 25
2 5
5
5
55 5
1 5
2
1
1
2 244
3
24 33
4
3
2
2
2
2
22 2
2
2
2
2
2
22
2
2 2
2
2
3
0.02485 - 0.2225*x
0.02446 - 0.2957*x
0.01247 - 0.205*x
3
3 3
3
3
3
3
3
3
3 3 3
3
3
3
3
33
3
3
33
-0.010
3
3
3
-0.015
-0.020
0.00
0.02
Guy Carpenter
0.04
0.06
0.08
0.10
R3M
0.12
0.14
0.16
0.18 15
Possible Tests of Generated Curves

Individual scenarios
– Could look at different time points simulated and see if slope
and spread around line is consistent with historical pattern
– For longer projections – 10 years + – expect some shift
– For 20 year + projections a flatter line would be expected
with greater spread, as in combining periods

Looking across scenarios at a single time
– Observing points over time can be viewed as taking samples
from the conditional distribution of spreads given short rate
– Alternative scenarios can be considered as providing draws
from the same conditional distribution
– Distribution of spreads at a time point could reasonably be
expected to have the recent inverse relationship to the short
rate – same slope and spread
Guy Carpenter
16
Five - Year to Three - Year Spreads
0.025
1 1960-1968
2 1968-1979
0.020
3 1979-1986
4 1986-1995
5 1995-2001
0.015
R53
0.010
0.005
0.000
4
4444
44
4 4
44
4
44
4
4 4
4
4
4 444
4
4
44442442
333 3 333 3 3
4
2 22 22 2
24
33333 3 3
22
2 24
24
333
2 2222
3
4
22 2
333
3
24
3 3
4
2 2
3
4
4
4
2
2
3
4
2
2
2
11
4
3
22 2 2
4
44
2
3 33 3 3
1
4 44444 4 4
3
2
1
45
42244
11 1 11
111
1
4
4
3
3
2
3
4
2
4
2
3
2
11
1
4
4
5
4
11 1 11111
44
2
3
2 22
3 3 33
1
2 1 5 22
4454
24
1
225
2 244 4 4 3 3
11 1 1 1
5
5
55
5
1
1 11 1
5
5
25
5
22
1
55224
222
22
11 1
2
55
1
5
525
5
225
5
25
2
11
15
5
22255
244244
1
1
5
2
222 2
5
1 1
15
2
1
1
2
3
5
1
2
2
2
2
2
2
11
1 1
5
1
1
2
4
5
5
5
2
1
11
1
1
5
2
5
2
2
1
52 5
222 2
1 11 5
24
44
24
5
51 5
224
2 3
2
22
11 5 2 2 5 22 2
2
4
1 5
5
11
11
1
1
1
1
2
5
55
52 2
5
555
22
2
2
2
4422 4244444 3
2
2
2 2
2
2
3
333
3
2
2
3
0.005539 - 0.134*x
0.008277 - 0.1175*x
0.012 - 0.1055*x
0.01246 - 0.1564*x
0.003488 - 0.05258*x
3
3
3
333
3 33
3 3
3
3 3 3
3
33
3
33 3
33
3
3
3
-0.005
3
3
-0.010
-0.015
-0.020
0.00
Guy Carpenter
0.02
0.04
0.06
0.08
0.10
R3M
0.12
0.14
0.16
0.18
17
Spreads in Generated Scenarios
GARP output 05/01 year 4 constant lambdas
0.025
0.020
5 – 3 spreads from Andersen-Lund with a selected
market-price of risk
0.015
Slope ok, spread too narrow
0.0063 - 0.0853*x
R53
0.010
0.005
Same problem for CIR – even worse in fact
0.000
-0.005
-0.010
-0.015
-0.020
0.00
0.02
Guy Carpenter
0.04
0.06
0.08
R3M
0.10
0.12
0.14
0.16
0.18
18
Add Stochastic Market Price of Risk
GARP output 05/01 year 4 variable lambdas
0.025
Better match on spread
0.020
0.0085 - 0.0973*x
0.015
R53
0.010
0.005
0.000
-0.005
-0.010
-0.015
-0.020
0.00
Guy Carpenter
0.02
0.04
0.06
0.08
0.10
R3M
0.12
0.14
0.16
0.18
19
Can also test distribution around the line
(Shape of distribution – not just spread)
Distributions Around Trend Line
Percentiles plotted against t with 33 df
Historical
Variable 
Fixed 
Variable  looks more like data
But fitted distribution misses in tails for all cases
Guy Carpenter
Test only partially successful
21
Summary

Treasury yield scenarios should be arbitrage-free, and be
consistent with the history of both dynamics of interest rates and
distributions of yield curves

Short-rate dynamics can be tested by fitting models

Yield curve dynamics can be tested with individual generate series

Yield curve distributions tested by conditional distributions of yield
spreads given short rate
Guy Carpenter
22