Spectral Yield Curve Analysis. The IOU Model July 2008 Andrew D Smith AndrewDSmith8@Deloitte.co.uk
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
Single Factor Yield Stress Models
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
Constant Price Volatility Zero coupon bond price 0.8 0.6 0.4 0.2 Base case Stress test Stress test Stress tests are fixed multiples of the base ZCB price 0 term 5 The IOU Model
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
Yield Curve Stresses Parallel Moves 0.8 0.6 0.4 0.2 0.8.6.4.2 0.8 0.6 0.4 0.2 0 8% 7% 6% 5% 4% 3% 2% 4% 3% 2% % 0% -% % -2% 0% -3% 7 The IOU Model
Constant Price Volatility 0.8 0.6 0.4 0.2 0.4.2 0.8 0.6 0.4 0.2 0 4% 25% 2% 20% 5% 0% 0% 8% 5% 6% 0% -5% 4% -0% 2% -5% 0% -20% -25% 8 The IOU Model
Term Structure of Yield Volatility (relative to 0 year yield) 9 Relative Par Yield Volatility 8 7 6 5 4 3 2 0 Flat Constant P vol Term 9 The IOU Model
Term Structure of Volatility (Log-Log scale) Par Volatility (relative to t=0) 0 0. Gradient - 0 00 Flat Constant P Vol Term (years) 0 The IOU Model
Hull & White s Model 0.8 0.6 0.4 0.2 0.4.2 0.8 0.6 0.4 0.2 0 9% 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
Historic Data Exploration
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
Historic Yield Move Analysis (re-based to start from 5% yield) Zero coupon bonds 0.8 0.6 0.4 0.2 0 Term.8.6.4.2 0.8 0.6 0.4 0.2 0 Par Yield 8% 6% 4% 2% 0% Term 3% 2% % 0% -% -2% -3% The jumps are artefacts of curve fitting 4 The IOU Model
Investigating Estimation Uncertainty (Rolling -year data windows).0% Sterling Annualised Volatility = stdev{weekly changes} * 52 0.8% volatility 0.6% 0.4% 0.2% year rate 0 year rate 0.0% 97 98 99 00 0 02 03 04 05 06 07 year 5 The IOU Model
Annualised Volatility: DEM / EUR.0% 0.8% volatility 0.6% 0.4% 0.2% 0.0% Year rate 0 year rate 97 98 99 00 0 02 03 04 05 06 07 year 6 The IOU Model
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
Correlation between year rate and 0 year rate 00% 80% correlation 60% 40% 20% GBP EUR 0% 97 98 99 00 0 02 03 04 05 06 07 year 8 The IOU Model
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
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
Infinite Factor Yield Stress Models
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
Random Walk Model.2 0.8 0.6 0.4 0.2 Not decreasing Not smooth enough 0.4.2 0.8 0.6 0.4 0.2 H-process is a random walk 0 2% 0% 8% 6% 4% 2% 0% -2% -4% 0% 8% 6% 4% 2% 0% 0-2% 5 0 5 20 25-4% -6% -8% -0% Not differentiable 23 The IOU Model
Integrated Random Walk Model 0.8 0.6 0.4 0.2 Too volatile for long terms.6.4.2 0.8 0.6 0.4 0.2 0 0 9% 8% 7% 6% 5% 4% 3% 2% % 0% 4% 3% 2% % 0% 0 -% 5 0 5 20 25-2% -3% -4% -5% -6% Gradient of H is a random walk 24 The IOU Model
Term Structure of Yield Volatility: Model Comparison 9 8 Relative Volatility 7 6 5 4 3 2 Integrated RW Flat Random walk Constant P vol 0 Term 25 The IOU Model
Term Structure of Volatility (Log-Log Scale) 0 Par Volatility (relative to t=0) 0. Gradient - Gradient -/2 0 00 Gradient +/2 Term (years) Integrated RW Flat Random Walk Constant P Vol 26 The IOU Model
Integrated Ornstein-Uhlenbeck (IOU) Model 0.8 0.6 0.4 0.2 0.4.2 0.8 0.6 0.4 0.2 0 7% 6% 5% 4% 3% 2% 4% 3% 2% % 0% 0 -% 5 0 5 20 25-2% % -3% 0% -4% -5% Gradient of H is an OU process 27 The IOU Model
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
Calibrating the IOU Model
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 0. 30 The IOU Model
Vol of 0 year vs yr yield Measure Correlation between year yield and 0 year yield 0 yr more volatile yr more volatile 0 0. 0 0.2 0.4 0.6 0.8 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
Calibration Values for Major Currencies Vol of 0 year vs yr yield yr more volatile 0 yr more volatile 0 0. 0 0.2 0.4 0.6 0.8 Integrated Random Walk Parallel Shift Random Walk Constant Price Vol JPY DEM USD GBP CHF correlation IOU Model Hull-White 32 The IOU Model
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% 0.056 GBP 0.70% 0.69% 58% 0.64% 0.98% 0.2069 EUR 0.54% 0.6% 66% 0.49% 0.88% 0.0989 JPY 0.2% 0.55% 47% 0.05%.2% 0.049 CHF 0.62% 0.57% 58% 0.58% 0.8% 0.2236 We do not currently know whether the apparently different parameters between currencies reflect inherent economic differences or sampling error. 33 The IOU Model
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
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
Yield Curve Fitting
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
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 0 0 20 30 40 50 t 2 L t 2 38 The IOU Model
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% 0 0 20 30 40 50 39 The IOU Model
Range of Interpolated / Extrapolated Yields 0% 8% 6% 4% 2% Fitted Par Range 0% 0 0 20 30 40 50 40 The IOU Model
Conclusions
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 email AndrewDSmith8@deloitte.co.uk 42 The IOU Model
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