Spectral Yield Curve Analysis. The IOU Model July 2008 Andrew D Smith

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1 Spectral Yield Curve Analysis. The IOU Model July 2008 Andrew D Smith AndrewDSmith8@Deloitte.co.uk

2 Presentation Overview Single Factor Stress Models Parallel shifts Short rate shifts Hull-White Exploration of Yield Curve Data Why graduate historic correlation matrices? Infinite Factor Stress Models Random walk Integrated random walk Integrated Ornstein-Uhlenbeck (IOU) Calibrating the IOU Model Business applications Problems where IOU gives new answers Yield Curve Fitting Quantifying interpolation / extrapolation error Conclusions 2 The IOU Model

3 Single Factor Yield Stress Models

4 Yield Curve Stresses Parallel Moves Par yield 8% 7% 6% 5% 4% 3% Stress test Stress test Base case Stress test Stress test 2% % 0% term 4 The IOU Model

5 Constant Price Volatility Zero coupon bond price Base case Stress test Stress test Stress tests are fixed multiples of the base ZCB price 0 term 5 The IOU Model

6 Four Ways to Look at Yield Curve Moves Zero coupon bond price P t H function P t stress = P t base * H t Par Yield g t = g t (P +P 2 + +P t ) + P t H gradient H t = dh t / dt 6 The IOU Model

7 Yield Curve Stresses Parallel Moves % 7% 6% 5% 4% 3% 2% 4% 3% 2% % 0% -% % -2% 0% -3% 7 The IOU Model

8 Constant Price Volatility % 25% 2% 20% 5% 0% 0% 8% 5% 6% 0% -5% 4% -0% 2% -5% 0% -20% -25% 8 The IOU Model

9 Term Structure of Yield Volatility (relative to 0 year yield) 9 Relative Par Yield Volatility Flat Constant P vol Term 9 The IOU Model

10 Term Structure of Volatility (Log-Log scale) Par Volatility (relative to t=0) 0 0. Gradient Flat Constant P Vol Term (years) 0 The IOU Model

11 Hull & White s Model % 4% 8% 7% 6% 5% 4% 3% 2% -2% % -3% 0% -4% 3% 2% % 0% -% Multiples of the same exponentially decreasing function. The IOU Model

12 Historic Data Exploration

13 One Year Yield Changes: 0 Examples from Past 0 years Yield Shift 3% 2% % 0% -% -2% -3% 09/06/2004 4/07/2004 0/2/999 29/09/999 /08/999 6/02/2005 Real data shows tangles and shape changes, unlike idealised one-factor models. Term 6/06/999 09/09/998 27/0/999 08/04/998 3 The IOU Model

14 Historic Yield Move Analysis (re-based to start from 5% yield) Zero coupon bonds Term Par Yield 8% 6% 4% 2% 0% Term 3% 2% % 0% -% -2% -3% The jumps are artefacts of curve fitting 4 The IOU Model

15 Investigating Estimation Uncertainty (Rolling -year data windows).0% Sterling Annualised Volatility = stdev{weekly changes} * % volatility 0.6% 0.4% 0.2% year rate 0 year rate 0.0% year 5 The IOU Model

16 Annualised Volatility: DEM / EUR.0% 0.8% volatility 0.6% 0.4% 0.2% 0.0% Year rate 0 year rate year 6 The IOU Model

17 Historic Volatility Estimates.2% Volatility.0% 0.8% 0.6% 0.4% 0.2% USD history GBP history EUR history JPY history CHF history 0.0% Term 7 The IOU Model

18 Correlation between year rate and 0 year rate 00% 80% correlation 60% 40% 20% GBP EUR 0% year 8 The IOU Model

19 Historic Correlation Matrices USD GBP EUR 00% 94% 89% 82% 7% 63% 94% 00% 99% 94% 84% 77% 89% 99% 00% 98% 90% 84% 82% 94% 98% 00% 96% 9% 7% 84% 90% 96% 00% 98% 63% 77% 84% 9% 98% 00% 00% 88% 82% 73% 58% 46% 88% 00% 98% 90% 74% 60% 82% 98% 00% 96% 82% 68% 73% 90% 96% 00% 92% 80% 58% 74% 82% 92% 00% 95% 46% 60% 68% 80% 95% 00% 00% 94% 89% 8% 66% 54% 94% 00% 98% 93% 80% 68% 89% 98% 00% 97% 86% 75% 8% 93% 97% 00% 94% 85% 66% 80% 86% 94% 00% 96% 54% 68% 75% 85% 96% 00% These correlation matrices are based on 0 years of weekly historical yield moves. Shaded data points will later be used for calibration. 9 The IOU Model

20 Raw Historic Correlation Estimates or Graduated Formula? Raw Estimate Advantages Transparent Retains all aspects of historic data Easy to implement mechanically. Graduation Advantages Lower sampling error, fewer parameters estimated Abstract from jump effects of curve fitting; correct artefacts Continuous time output Smooth results: stress tests, capital requirements Easy parameter comparison between time periods and economies Historic data awkward in form (par curves), more convenient to model ZCB prices. 20 The IOU Model

21 Infinite Factor Yield Stress Models

22 Counting Model Factors Single factor model If we know one point on the yield curve, we can construct all the others We can hedge a 50 year liability with -year and 5-year bonds Two-factor model If we know two points on the yield curve, we can construct all the others We can hedge a 50 year liability with -year, 5-year & 0-year bonds Three-factor model If we know three points on the yield curve, we can construct all the others How many factors in real markets? Portfolio construction may use a small number of factors, but more are needed for risk analysis We now look at some infinite factor models Infinite factors need not mean infinitely many parameters 22 The IOU Model

23 Random Walk Model Not decreasing Not smooth enough H-process is a random walk 0 2% 0% 8% 6% 4% 2% 0% -2% -4% 0% 8% 6% 4% 2% 0% 0-2% % -6% -8% -0% Not differentiable 23 The IOU Model

24 Integrated Random Walk Model Too volatile for long terms % 8% 7% 6% 5% 4% 3% 2% % 0% 4% 3% 2% % 0% 0 -% % -3% -4% -5% -6% Gradient of H is a random walk 24 The IOU Model

25 Term Structure of Yield Volatility: Model Comparison 9 8 Relative Volatility Integrated RW Flat Random walk Constant P vol 0 Term 25 The IOU Model

26 Term Structure of Volatility (Log-Log Scale) 0 Par Volatility (relative to t=0) 0. Gradient - Gradient -/ Gradient +/2 Term (years) Integrated RW Flat Random Walk Constant P Vol 26 The IOU Model

27 Integrated Ornstein-Uhlenbeck (IOU) Model % 6% 5% 4% 3% 2% 4% 3% 2% % 0% 0 -% % % -3% 0% -4% -5% Gradient of H is an OU process 27 The IOU Model

28 Describing the Ornstein-Uhlenbeck Process 4% 3% stdevσ S Reversion α stdevσ L 2% % 0% -% -2% -3% -4% -5% H 0 H t ~ N H s ( 2 0, σ ) S ( α ( t s) ( 2α ( t s) ) 2 ~ N e H, e σ ) s L (s t) 28 The IOU Model

29 Calibrating the IOU Model

30 Remember Relative Volatility of 0 year vs -year yields 0 Vol of 0 year vs yr yield yr more volatile 0 yr more volatile Integrated Random Walk Parallel Shift Random Walk Constant Price Vol Par Volatility (relative to t=0) 0 0. These are all limiting cases of the IOU Model 0 00 Term (years) Integrated RW Flat Random Walk Constant P Vol The IOU Model

31 Vol of 0 year vs yr yield Measure Correlation between year yield and 0 year yield 0 yr more volatile yr more volatile Integrated Random Walk Parallel Shift Random Walk Constant Price Vol correlation IOU Model Hull-White The shaded region shows the range of volatilities and correlations achievable with different parameter choices under the IOU model. 3 The IOU Model

32 Calibration Values for Major Currencies Vol of 0 year vs yr yield yr more volatile 0 yr more volatile Integrated Random Walk Parallel Shift Random Walk Constant Price Vol JPY DEM USD GBP CHF correlation IOU Model Hull-White 32 The IOU Model

33 Possible Estimated Parameters v( yr par) v(0 yr par) correlation σ S σ L α Currency USD 0.79% 0.97% 7% 0.7%.57% GBP 0.70% 0.69% 58% 0.64% 0.98% EUR 0.54% 0.6% 66% 0.49% 0.88% JPY 0.2% 0.55% 47% 0.05%.2% CHF 0.62% 0.57% 58% 0.58% 0.8% We do not currently know whether the apparently different parameters between currencies reflect inherent economic differences or sampling error. 33 The IOU Model

34 Historic Versus Fitted Volatility: Failure to Get the Hump.2% Volatility.0% 0.8% 0.6% 0.4% 0.2% 0.0% Exact fit for t= and t=0 USD history USD fit GBP history GBP fit EUR history EUR fit JPY history JPY fit CHF history CHF fit Term 34 The IOU Model

35 Historic vs Fitted Par Correlations (terms,2,3,5,0,20) Historic Fitted USD 00% 94% 89% 82% 7% 63% 94% 00% 99% 94% 84% 77% 89% 99% 00% 98% 90% 84% 82% 94% 98% 00% 96% 9% 7% 84% 90% 96% 00% 98% 63% 77% 84% 9% 98% 00% 00% 97% 92% 84% 7% 58% 97% 00% 98% 92% 78% 64% 92% 98% 00% 97% 84% 70% 84% 92% 97% 00% 93% 79% 7% 78% 84% 93% 00% 92% 58% 64% 70% 79% 92% 00% GBP EUR 00% 88% 82% 73% 58% 46% 88% 00% 98% 90% 74% 60% 82% 98% 00% 96% 82% 68% 73% 90% 96% 00% 92% 80% 58% 74% 82% 92% 00% 95% 46% 60% 68% 80% 95% 00% 00% 94% 89% 8% 66% 54% 94% 00% 98% 93% 80% 68% 89% 98% 00% 97% 86% 75% 8% 93% 97% 00% 94% 85% 66% 80% 86% 94% 00% 96% 54% 68% 75% 85% 96% 00% 00% 95% 87% 75% 58% 45% 95% 00% 97% 86% 68% 53% 87% 97% 00% 94% 76% 60% 75% 86% 94% 00% 88% 7% 58% 68% 76% 88% 00% 88% 45% 53% 60% 7% 88% 00% 00% 96% 9% 8% 66% 52% 96% 00% 98% 90% 74% 60% 9% 98% 00% 96% 8% 66% 8% 90% 96% 00% 9% 75% 66% 74% 8% 9% 00% 9% 52% 60% 66% 75% 9% 00% Our graduation underestimates the extent to which very long par yields are correlated with shorter yields. 35 The IOU Model

36 Yield Curve Fitting

37 Why Fit Yield Curves? Market yields observed at discrete intervals Valuation applications need discount bond price as a continuous function of maturity Not a graduation problem we want to hit all input data exactly Consider two techniques Variational approach Bayesian approach 37 The IOU Model

38 Variational Approach: Fitting Relative to a Base Curve 7% 6% 5% 4% 3% 2% % 0% Choose H t = P fit t / Pbase t so H t is smooth Intuitively, we want H t and H t to be small Mathematically, we minimise roughness, defined by: 0 α 2 2σ L 2 2 σ 2σ 2 2 σ σ 2ασ 2 S L ( H ) + H H + ( H ) t S L t Spot Par Forward dt t 2 L t 2 38 The IOU Model

39 Posterior Mean, based on IOU Prior 7% 6% 5% 4% 3% 2% % Spot Par Forward The posterior mean ZCB curve minimises the variational problem. Therefore, we can interpret the IOU-Bayes approach as finding a smooth curve that passes through market price data. 0% The IOU Model

40 Range of Interpolated / Extrapolated Yields 0% 8% 6% 4% 2% Fitted Par Range 0% The IOU Model

41 Conclusions

42 Conclusions Multiple factor models (preferably infinite-factor) are important for understanding exposures to yield curve risk Improve understanding of risks associated with interpolation and extrapolation when interest rate markets are incomplete Graduate historic volatility and correlation to reduce sampling error and curve-fit artefacts Possible future improvements Avoid negative yields or ZCB prices Better capture patterns in historic volatility and correlation matrices Multiple horizon models Efficient algorithms needed to implement high factor models in business risk calculations Is there a more general connection between the Bayesian posterior mean and the solution to a variational problem? This is a new model and the associated white paper is likely available later this month. If you would like a copy, please AndrewDSmith8@deloitte.co.uk 42 The IOU Model

43 This document is confidential and prepared solely for your information. Therefore you should not, without our prior written consent, refer to or use our name or this document for any other purpose, disclose them or refer to them in any prospectus or other document, or make them available or communicate them to any other party. No other party is entitled to rely on our document for any purpose whatsoever and thus we accept no liability to any other party who is shown or gains access to this document. Deloitte & Touche LLP is a limited liability partnership registered in England and Wales with registered number OC and its registered office at Stonecutter Court, Stonecutter Street, London EC4A 4TR, United Kingdom. Deloitte & Touche LLP is the United Kingdom member firm of Deloitte Touche Tohmatsu ('DTT'), a Swiss Verein whose member firms are separate and independent legal entities. Neither DTT nor any of its member firms has any liability for each other's acts or omissions. Services are provided by member firms or their subsidiaries and not by DTT. 43 The IOU Model

44 Member of Deloitte Touche Tohmatsu

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