Testing Distributions of Testing Distributions of Stochastically Generated Stochastically Generated Yield Curves Yield Curves

Gary G VenterGary G Venter

AFIR SeminarAFIR Seminar

September 2003September 2003

Guy Carpenter 2

Advantages of Stochastic Advantages of Stochastic GeneratorsGenerators

Deterministic scenarios allow checking risk against Deterministic scenarios allow checking risk against specific outcomesspecific outcomes

Stochastic generators add dimension of probability of Stochastic generators add dimension of probability of scenariosscenarios

Can incorporate full range of reasonably possible Can incorporate full range of reasonably possible outcomesoutcomes

Each scenario can be a time series of outcomesEach scenario can be a time series of outcomes

Guy Carpenter 3

Testing for Potential Problems of Testing for Potential Problems of Stochastic GeneratorsStochastic Generators

Model could miss possible scenariosModel could miss possible scenarios

Model could overweight some unlikely scenarios and underweight Model could overweight some unlikely scenarios and underweight others – giving unrealistic distribution of resultsothers – giving unrealistic distribution of results

Traditional tests look at time series properties of individual Traditional tests look at time series properties of individual scenarios – like autocorrelations, shapes of curves compared to scenarios – like autocorrelations, shapes of curves compared to historical, correlation of short and long term rates and their historical, correlation of short and long term rates and their comparative volatility, and mean reversioncomparative volatility, and mean reversion

Options pricing models test distributions across scenarios by their Options pricing models test distributions across scenarios by their impacts on option pricesimpacts on option prices

For insurer risk models, we propose testing generators by For insurer risk models, we propose testing generators by comparing distributions of yield curves against historicalcomparing distributions of yield curves against historical

Look for aspects of historical distributions that do not change too Look for aspects of historical distributions that do not change too much over timemuch over time

Some Models of the Yield CurveSome Models of the Yield Curve((Then we’ll look at testing)Then we’ll look at testing)

Guy Carpenter 5

Example Short-Term Rate ModelsExample Short-Term Rate Models

Usually defined using Brownian motion zUsually defined using Brownian motion ztt. After time t, z. After time t, ztt is normal with mean zero and variance t.is normal with mean zero and variance t.

Cox, Ingersoll, Ross (CIR):Cox, Ingersoll, Ross (CIR):

dr = a(b - r)dt + srdr = a(b - r)dt + sr1/21/2dz In discrete form for a short dz In discrete form for a short period:period:

rrt t –– rrt–1 t–1 = a(b – r= a(b – rt–1t–1) + sr) + srt –1t –11/21/2

CIR change in interest rate has two components:CIR change in interest rate has two components:– A trend which is mean reverting to b, i.e., is negative if A trend which is mean reverting to b, i.e., is negative if

r>b and positive if r<br>b and positive if r<b Speed of mean reversion given by aSpeed of mean reversion given by a

– A random component proportional to A random component proportional to rr1/21/2, , so variance so variance rtsrts22 in time t in time t

Guy Carpenter 6

Adding Effects to CIRAdding Effects to CIR

Mean that is reverted to can be stochastic:Mean that is reverted to can be stochastic:

d d b = j(q - b)b = j(q - b)ddt + wbt + wb1/21/2ddzz11

This postulates same dynamics for reverting mean as for rThis postulates same dynamics for reverting mean as for r

Volatility can be stochastic as well:Volatility can be stochastic as well:

d d ln sln s22 = c(p - ln s = c(p - ln s22))ddt + vt + vddzz22

Here Brownian motion in logHere Brownian motion in log

Power on r in dPower on r in dzz term might not be ½ : term might not be ½ : dr = a(b - r)dt + srdr = a(b - r)dt + srqqdz dz

CIR with these two added factors fit by Andersen and Lund, CIR with these two added factors fit by Andersen and Lund, working paper 214, Northwestern University Department of working paper 214, Northwestern University Department of Finance, who also estimate the power of r (1/2 for CIR).Finance, who also estimate the power of r (1/2 for CIR).

Guy Carpenter 7

Fitting Stochastic GeneratorsFitting Stochastic Generators

If you can integrate out to resulting observed periods If you can integrate out to resulting observed periods you can fit by MLEyou can fit by MLE– CIR distribution of rCIR distribution of rt+Tt+T given r given rtt is non-central chi-sq. is non-central chi-sq.

– f(rf(rt+Tt+T|r|rtt) = ce) = ce-u-v-u-v(v/u)(v/u)q/2q/2IIqq(2(uv)(2(uv)1/21/2), where), where

– c = 2asc = 2as-2-2/(1-e/(1-e-aT-aT), q=-1+2abs), q=-1+2abs-2-2, u=cr, u=crttee-aT-aT, v=cr, v=crt+Tt+T

IIq q is modified Bessel function of the first kind, order qis modified Bessel function of the first kind, order q

– IIqq(2z)= (2z)= k=0k=0zz2k+q2k+q/[k!(q+k)!], where factorial off /[k!(q+k)!], where factorial off

integers is defined by the gamma functionintegers is defined by the gamma function

Can use this for mle estimates of a, b, and sCan use this for mle estimates of a, b, and s

Guy Carpenter 8

Fitting Stochastic GeneratorsFitting Stochastic Generators

If cannot integrate distribution, some other methods If cannot integrate distribution, some other methods used:used:

– Quasi-likelihoodQuasi-likelihood

– Generalized method of moments (GMM)Generalized method of moments (GMM) E[(3/x) ln x] is a generalized moment, for E[(3/x) ln x] is a generalized moment, for

exampleexample Or anything else that you can take an expected Or anything else that you can take an expected

value ofvalue of Need to decide which moments to matchNeed to decide which moments to match

Guy Carpenter 9

Which Moments to Match?Which Moments to Match?

Title of paper developing efficient method of moments Title of paper developing efficient method of moments (EMM)(EMM)

Suggests finding the best fitting time-series model to the Suggests finding the best fitting time-series model to the time-series data, called the auxiliary modeltime-series data, called the auxiliary model

Scores (partial derivates of log-likelihood of auxiliary Scores (partial derivates of log-likelihood of auxiliary model) are zero for the data at the MLE parameters model) are zero for the data at the MLE parameters

EMM considers these scores, with the fitted parameters of EMM considers these scores, with the fitted parameters of the auxiliary model fixed, to be the generalized moments, the auxiliary model fixed, to be the generalized moments, and seeks the parameters of the stochastic model that and seeks the parameters of the stochastic model that when used to simulate data, gives data with zero scores when used to simulate data, gives data with zero scores

Actually minimizes distance from zeroActually minimizes distance from zero

Guy Carpenter 10

Andersen-Lund ResultsAndersen-Lund Results

Power on r in r-equation volatility somewhat above ½Power on r in r-equation volatility somewhat above ½

Stochastic volatility and stochastic mean reversion Stochastic volatility and stochastic mean reversion are statistically significant, and so are needed to are statistically significant, and so are needed to capture dynamics of short-term ratecapture dynamics of short-term rate

Used US data from 1950’s through 1990’sUsed US data from 1950’s through 1990’s

Guy Carpenter 11

Getting Yield Curves from Short Rate Getting Yield Curves from Short Rate DynamicsDynamics

P(T) is price now of a bond paying P(T) is price now of a bond paying €€1 at time T1 at time T

This is risk-adjusted expected value of This is risk-adjusted expected value of €€1 discounted 1 discounted continuously over all paths:continuously over all paths:

P(T) = EP(T) = E**[exp(-[exp(-rrttdt)]dt)]

Risk adjustment is to add something to the trend terms Risk adjustment is to add something to the trend terms of the generating processesof the generating processes

The added element is called the market price of risk for The added element is called the market price of risk for the processthe process

Guy Carpenter 12

Testing Generated Yield CurvesTesting Generated Yield Curves

Want distributions to be reasonable in comparison to Want distributions to be reasonable in comparison to historyhistory

Distributions of yield curves can be measured by Distributions of yield curves can be measured by looking at distributions of the various yield spreadslooking at distributions of the various yield spreads

Yield spread distributions differ depending on the Yield spread distributions differ depending on the short-term rate: spreads compacted when short rates short-term rate: spreads compacted when short rates are highare high

Look at conditional distributions of spreads given Look at conditional distributions of spreads given short-term rateshort-term rate

Now for TestingNow for Testing((Proposed Distributional Test)Proposed Distributional Test)

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Three Month Rate and 10 – 3 Year SpreadThree Month Rate and 10 – 3 Year Spread

Clear inverse relationship

Mathematical form changes

Five periods selected

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Ten – Three Year Spreads vs Short RateTen – Three Year Spreads vs Short RateSlope constant but intercept changes each periodSlope constant but intercept changes each period

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18

R3M

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0.01096 - 0.272*x1 1960-1968

2 1968-1979

3 1979-1986

4 1986-1995

5 1995-2001

0.0171 - 0.2526*x

0.02485 - 0.2225*x

0.02446 - 0.2957*x

0.01247 - 0.205*x

Guy Carpenter 16

Possible Tests of Generated CurvesPossible Tests of Generated Curves Individual scenariosIndividual scenarios

– Could look at different time points simulated and see if slope and Could look at different time points simulated and see if slope and spread around line is consistent with historical patternspread around line is consistent with historical pattern

– For longer projections – 10 years + – expect some shiftFor longer projections – 10 years + – expect some shift– For 20 year + projections a flatter line would be expected with For 20 year + projections a flatter line would be expected with

greater spread, as in combining periodsgreater spread, as in combining periods

Looking across scenarios at a single timeLooking across scenarios at a single time– Observing points over time can be viewed as taking samples from Observing points over time can be viewed as taking samples from

the conditional distribution of spreads given short ratethe conditional distribution of spreads given short rate– Alternative scenarios can be considered as providing draws from Alternative scenarios can be considered as providing draws from

the same conditional distributionthe same conditional distribution– Distribution of spreads at a time point could reasonably be expected Distribution of spreads at a time point could reasonably be expected

to have the recent inverse relationship to the short rate – same to have the recent inverse relationship to the short rate – same slope and spreadslope and spread

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Five - Year to Three - Year SpreadsFive - Year to Three - Year Spreads

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18

R3M

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

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1 1960-1968

2 1968-1979

3 1979-1986

4 1986-1995

5 1995-2001

0.005539 - 0.134*x0.008277 - 0.1175*x

0.012 - 0.1055*x

0.01246 - 0.1564*x

0.003488 - 0.05258*x

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Spreads in Generated ScenariosSpreads in Generated Scenarios

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18

R3M

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

R5

3

GARP output 05/01 year 4 constant lambdas

0.0063 - 0.0853*x

5 – 3 spreads from Andersen-Lund with a selected market-price of risk

Slope ok, spread too narrow

Same problem for CIR – even worse in fact

Guy Carpenter 19

Add Stochastic Market Price of RiskAdd Stochastic Market Price of Risk

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18

R3M

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

R5

3

GARP output 05/01 year 4 variable lambdas

0.0085 - 0.0973*x

Better match on spread

Can also test distribution around the lineCan also test distribution around the line((Shape of distribution – not just spread)Shape of distribution – not just spread)

Guy Carpenter 21

Distributions Around Trend Line Distributions Around Trend Line Percentiles plotted against t with 33 df

Variable Fixed Historical

Variable looks more like data

But fitted distribution misses in tails for all cases

Test only partially successful

Guy Carpenter 22

SummarySummary

Treasury yield scenarios should be arbitrage-free, and be Treasury yield scenarios should be arbitrage-free, and be consistent with the history of both dynamics of interest rates and consistent with the history of both dynamics of interest rates and distributions of yield curvesdistributions of yield curves

Short-rate dynamics can be tested by fitting modelsShort-rate dynamics can be tested by fitting models

Yield curve dynamics can be tested with individual generate seriesYield curve dynamics can be tested with individual generate series

Yield curve distributions tested by conditional distributions of yield Yield curve distributions tested by conditional distributions of yield spreads given short ratespreads given short rate