Testing Distributions of Stochastically Generated Yield Curves Gary G Venter AFIR Seminar September 2003

Guy Carpenter 2 Advantages of Stochastic Generators Deterministic scenarios allow checking risk against specific outcomes Deterministic scenarios allow checking risk against specific outcomes Stochastic generators add dimension of probability of scenarios Stochastic generators add dimension of probability of scenarios Can incorporate full range of reasonably possible outcomes Can incorporate full range of reasonably possible outcomes Each scenario can be a time series of outcomes Each scenario can be a time series of outcomes

Guy Carpenter 3 Testing for Potential Problems of Stochastic Generators Model could miss possible scenarios Model could miss possible scenarios Model could overweight some unlikely scenarios and underweight others – giving unrealistic distribution of results 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 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 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 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 Look for aspects of historical distributions that do not change too much over time

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

Guy Carpenter 5 Example Short-Term Rate Models Usually defined using Brownian motion z t. After time t, z t is normal with mean zero and variance t. Usually defined using Brownian motion z t. After time t, z t is normal with mean zero and variance t. Cox, Ingersoll, Ross (CIR): Cox, Ingersoll, Ross (CIR): dr = a(b - r)dt + sr 1/2 dz In discrete form for a short period: r t – r t–1 = a(b – r t–1 ) + sr t –1 1/2  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 r>b and positive if r b and positive if r<b  Speed of mean reversion given by a – A random component proportional to r 1/2, so variance rts 2 in time t

Guy Carpenter 6 Adding Effects to CIR Mean that is reverted to can be stochastic: Mean that is reverted to can be stochastic: d b = j(q - b)dt + wb 1/2 dz 1 This postulates same dynamics for reverting mean as for r This postulates same dynamics for reverting mean as for r Volatility can be stochastic as well: Volatility can be stochastic as well: d ln s 2 = c(p - ln s 2 )dt + vdz 2 Here Brownian motion in log Here Brownian motion in log Power on r in dz term might not be ½ : dr = a(b - r)dt + sr q dz Power on r in dz term might not be ½ : dr = a(b - r)dt + sr q dz 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). 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 7 Fitting Stochastic Generators If you can integrate out to resulting observed periods you can fit by MLE If you can integrate out to resulting observed periods you can fit by MLE – CIR distribution of r t+T given r t is non-central chi-sq. – f(r t+T |r t ) = ce -u-v (v/u) q/2 I q (2(uv) 1/2 ), where – c = 2as -2 /(1-e -aT ), q=-1+2abs -2, u=cr t e -aT, v=cr t+T I q is modified Bessel function of the first kind, order q I q is modified Bessel function of the first kind, order q – I q (2z)=  k=0  z 2k+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 Can use this for mle estimates of a, b, and s

Guy Carpenter 8 Fitting Stochastic Generators If cannot integrate distribution, some other methods used: If cannot integrate distribution, some other methods used: – Quasi-likelihood – Generalized method of moments (GMM)  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

Guy Carpenter 9 Which Moments to Match? Title of paper developing efficient method of moments (EMM) 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 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 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 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 Actually minimizes distance from zero

Guy Carpenter 10 Andersen-Lund Results Power on r in r-equation volatility somewhat above ½ 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 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 Used US data from 1950’s through 1990’s

Guy Carpenter 11 Getting Yield Curves from Short Rate Dynamics P(T) is price now of a bond paying €1 at time T 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: This is risk-adjusted expected value of €1 discounted continuously over all paths: P(T) = E * [exp(-  r t dt)] P(T) = E * [exp(-  r t dt)] Risk adjustment is to add something to the trend terms of the generating processes 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 The added element is called the market price of risk for the process

Guy Carpenter 12 Testing Generated Yield Curves Want distributions to be reasonable in comparison to history 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 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 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 Look at conditional distributions of spreads given short-term rate

Now for Testing ( Proposed Distributional Test)

Guy Carpenter 14 Three 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 Rate Slope constant but intercept changes each period

Guy Carpenter 16 Possible Tests of Generated Curves Individual scenarios 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 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 17 Five - Year to Three - Year Spreads

Guy Carpenter 18 Spreads in Generated Scenarios 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 Risk Better match on spread

Can also test distribution around the line ( 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 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 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 Short-rate dynamics can be tested by fitting models Yield curve dynamics can be tested with individual generate series Yield curve dynamics can be tested with individual generate series Yield curve distributions tested by conditional distributions of yield spreads given short rate Yield curve distributions tested by conditional distributions of yield spreads given short rate