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

2. 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

3. 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

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

5. 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: rt –rt–1 = a(b – rt–1) + srt –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

6. 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).

7. 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

8. Fitting Stochastic Generators • 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

9. 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

10. 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

11. 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(-rtdt)] • 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

12. 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

13. Now for Testing(Proposed Distributional Test)

14. Three Month Rate and 10 – 3 Year Spread Clear inverse relationship Mathematical form changes Five periods selected

15. Ten – Three Year Spreads vs Short RateSlope constant but intercept changes each period

16. 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

17. Five - Year to Three - Year Spreads

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

19. Add Stochastic Market Price of Risk Better match on spread

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

21. Distributions Around Trend Line Percentiles plotted against t with 33 df Fixed  Historical Variable  Variable  looks more like data But fitted distribution misses in tails for all cases Test only partially successful

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 • 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