testing distributions of stochastically generated yield curves gary g venter afir seminar september...
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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)
Guy Carpenter 14
Three Month Rate and 10 – 3 Year SpreadThree Month Rate and 10 – 3 Year Spread
Clear inverse relationship
Mathematical form changes
Five periods selected
Guy Carpenter 15
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
-0.020
-0.015
-0.010
-0.005
0.000
0.005
0.010
0.015
0.020
0.025
R1
03
11
11
1
1
1
1 111
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2
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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
Guy Carpenter 17
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
R5
3
11111
111 111
111111
11
1111 111
1111
1111
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11111
1 11111111111
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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
Guy Carpenter 18
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