Dr. Bernhard Pfaff. The 2nd International R/Rmetrics User and Developer Workshop 29 June 3 July 2007, Meielisalp, Lake Thune, Switzerland

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1 Dr. Bernhard Invesco Asset Management Deutschland GmbH, Frankfurt am Main The 2nd International R/Rmetrics User and Developer Workshop 29 June 3 July 2007, Meielisalp, Lake Thune, Switzerland

2 Contents

3 Overview Cash-portfolio with futures strategy. Long/short positions are allowed. Should cover the major bond and equity markets. Should include a protection mechanism. Fully-hedged against currency risk.

4 The Pieces and Packages Forecasting model, e.g. vars and urca Risk model, e.q. fextremes and QRMlib Linear integer program, e.q. glpk

5 Data Continuous settlement (Wednesdays closing prices) of: Exchange : Instrument DS-Mnemonic Multiple Equity CME: S&P 500 ISPCS USD / IP EUREX: DJ EURO STOXX 50 GEXCS00 10 EUR / IP LIFFE: FTSE 250 LSYCS00 10 GBP / IP OSX: NIKKEI 225 ONACS JPY / IP Bond CBT: 10 YEAR US T-NOTE CTYCS USD / 100 BP EUREX: EURO BUND GGECS EUR / 100 BP LIFFE: LONG GILT LIGCS GBP / 100 BP TSE: 10 YEAR T-BOND JGBCS JPY / 100 BP

6 U.S. Equity USD Price of Future Percent Continuous Returns Density Plot of Returns QQ Plot of Returns Density Ordered Data N = 2334 Bandwidth = Normal Quantiles

7 Eurobloc Equity Price of Future Continuous Returns EUR Percent Density Plot of Returns QQ Plot of Returns Density Ordered Data N = 2356 Bandwidth = Normal Quantiles

8 U.K. Equity GBP Price of Future Percent Continuous Returns Density Plot of Returns QQ Plot of Returns Density Ordered Data N = 2341 Bandwidth = Normal Quantiles

9 Japan Equity JPY Price of Future Percent Continuous Returns Density Plot of Returns QQ Plot of Returns Density Ordered Data N = 2253 Bandwidth = Normal Quantiles

10 U.S. Bond Price of Future Continuous Returns USD Percent Density Plot of Returns QQ Plot of Returns Density Ordered Data N = 2287 Bandwidth = Normal Quantiles

11 Eurobloc Bond Price of Future Continuous Returns EUR Percent Density Plot of Returns QQ Plot of Returns Density Ordered Data N = 2361 Bandwidth = Normal Quantiles

12 U.K. Bond Price of Future Continuous Returns GBP Percent Density Plot of Returns QQ Plot of Returns Density Ordered Data N = 2338 Bandwidth = Normal Quantiles

13 Japan Bond JPY Price of Future Percent Continuous Returns Density Plot of Returns QQ Plot of Returns Density Ordered Data N = 2236 Bandwidth = Normal Quantiles

14 Vector Error-correction Model (VECM) VECM specified in transitory form with one lagged difference. Logarithmic transformation applied. Full sample period from until Pseudo ex ante forecasts starting One-step ahead forecasts are obtained from implied level-var.

15 ERS Tests, Equities variable statistic 1% 5% 10% lags U.S. level U.S. difference Eurobloc level Eurobloc difference U.K. level U.K. difference Japan level Japan difference

16 ERS Tests, Bonds variable statistic 1% 5% 10% lags U.S. level U.S. difference Eurobloc level Eurobloc difference U.K. level U.K. difference Japan level Japan difference

17 KPSS Tests, Equities variable statistic 1% 5% 10% lags U.S. level U.S. difference Eurobloc level Eurobloc difference U.K. level U.K. difference Japan level Japan difference

18 KPSS Tests, Bonds variable statistic 1% 5% 10% lags U.S. level U.S. difference Eurobloc level Eurobloc difference U.K. level U.K. difference Japan level Japan difference

19 Results of Trace Test rank statistic 10% 5% 1% r r r r r r r r = Conclusion: Cointegration rank of r = 2 cannot be rejected.

20 Settings and Quantitative Risk Measures Usage of expected shortfall (ES) numbers. Rolling window of 1000 observations per instrument. Estimates based on EVT (POT-method). Backtest period from to KISS: No combined GARCH & ES forecast. Instead, ES of last trading day used as forecast for today s ES.

21 Overview Long-Positions, 99%-Level Instrument Expected VaR-GPD ES-GPD VaR-Norm Equity U.S Eurobloc U.K Japan Bonds U.S Eurobloc U.K Japan

22 U.S. Equity: VaR & ES (Long, 99%) Continuous return, U.S. Equity ES (GPD) 6 shortfalls VaR (GPD) VaR (Normal) shortfalls 23 shortfalls

23 Eurobloc Equity: VaR & ES (Long, 99%) Continuous return, Eurobloc Equity ES (GPD) 4 shortfalls VaR (GPD) VaR (Normal) shortfalls 18 shortfalls

24 U.K. Equity: VaR & ES (Long, 99%) Continuous return, U.K. Equity ES (GPD) 9 shortfalls VaR (GPD) VaR (Normal) shortfalls 36 shortfalls

25 Japan Equity: VaR & ES (Long, 99%) Continuous return, Japan Equity ES (GPD) 7 shortfalls VaR (GPD) VaR (Normal) shortfalls 28 shortfalls

26 U.S. Bond: VaR & ES (Long, 99%) Continuous return, U.S. Bond ES (GPD) 6 shortfalls VaR (GPD) VaR (Normal) shortfalls 28 shortfalls

27 Eurobloc Bond: VaR & ES (Long, 99%) Continuous return, Eurobloc Bond ES (GPD) 6 shortfalls VaR (GPD) VaR (Normal) shortfalls 25 shortfalls

28 U.K. Bond: VaR & ES (Long, 99%) Continuous return, U.K. Bond ES (GPD) 4 shortfalls VaR (GPD) VaR (Normal) shortfalls 21 shortfalls

29 Japan Bond: VaR & ES (Long, 99%) Continuous return, Japan Bond ES (GPD) 3 shortfalls VaR (GPD) VaR (Normal) shortfalls 25 shortfalls

30 Overview Short-Positions, 99%-Level Instrument Expected VaR-GPD ES-GPD VaR-Norm Equity U.S Eurobloc U.K Japan Bonds U.S Eurobloc U.K Japan

31 U.S. Equity: VaR & ES (Short, 99%) Continuous return, U.S. Equity ES (GPD) 8 exceedances VaR (GPD) VaR (Normal) exceedances 16 exceedances

32 Eurobloc Equity: VaR & ES (Short, 99%) Continuous return, Eurobloc Equity ES (GPD) 7 exceedances VaR (GPD) VaR (Normal) exceedances 13 exceedances

33 U.K. Equity: VaR & ES (Short, 99%) Continuous return, U.K. Equity ES (GPD) 6 exceedances VaR (GPD) VaR (Normal) exceedances 30 exceedances

34 Japan Equity: VaR & ES (Short, 99%) Continuous return, Japan Equity ES (GPD) 5 exceedances VaR (GPD) VaR (Normal) exceedances 14 exceedances

35 U.S. Bond: VaR & ES (Short, 99%) Continuous return, U.S. Bond ES (GPD) 8 exceedances VaR (GPD) VaR (Normal) exceedances 18 exceedances

36 Eurobloc Bond: VaR & ES (Short, 99%) Continuous return, Eurobloc Bond ES (GPD) 9 exceedances VaR (GPD) VaR (Normal) exceedances 16 exceedances

37 U.K. Bond: VaR & ES (Short, 99%) Continuous return, U.K. Bond ES (GPD) 3 exceedances VaR (GPD) VaR (Normal) exceedances 13 exceedances

38 Japan Bond: VaR & ES (Short, 99%) Continuous return, Japan Bond ES (GPD) 5 exceedances VaR (GPD) VaR (Normal) exceedances 18 exceedances

39 Linear integer program Target function Maximize: z = n pi f m i x i, (1) i=1 whereby pi f is the absolute expected price change of the i-th future contract, m i is the corresponding multiple and x i is the integer number of contracts to buy or sell. The expected price changes are denominated in Euro.

40 Linear integer program Restrictions, I Budget: w n p i m i x i. (2) i=1 whereby w is the portfolio wealth. Buffer: p n p i m i r i x i, (3) i=1 whereby p assigns the risk buffer and r i is the risk factor. Equity share: qa 1 1 w n equity j=1 p j m j x j qa 2, (4) whereby qa 1 and qa 2 are the equity bounds.

41 Linear integer program Restrictions, II Bond share: qr 1 1 w n bonds j=1 p j m j x j qr 2 (5) whereby qr 1 and qr 2 are the fixed income bounds. Shorts: qs 1 w n short j=1 p j m j x j, (6) whereby qs is the upper bound on short positions.

42 Assumptions and sample, I Begin of simulation: End of simulation: Risk-free rate: 1-week Euribor. Usage of 99% ES-rate derived from GPD-distribution (POT-method). Scaling of daily ES by 7. No transaction costs and fully hedged.

43 Assumptions and sample, II Initial wealth, w, 1,000,000 Euro. Risk buffer: 90% of the highest wealth amount (high-watermark). Equity share between 10% and 90% of wealth. Bonds share between 10% and 90%. Maximal amount of short positions as high as 50% of wealth. Only trade if expected return is greater than 200 BP above 1 week risk-free rate.

44 Portfolio vs. Cash Portfolio Money Market EUR

45 Cash rate Percentage

46 Equity Contribution U.S. Equity Eurobloc Equity EUR EUR U.K. Equity Japan Equity EUR EUR

47 Cumulated Contribution from Equity Futures 0e+00 1e+05 2e+05 3e+05 4e+05 U.S. Eurobloc U.K. Japan

48 Number of Traded Equity Futures U.S. Eurobloc U.K. Japan

49 Average Return from Equity Futures U.S. Eurobloc U.K. Japan

50 Bond Contribution U.S. Bond Eurobloc Bond EUR EUR U.K. Bond Japan Bond EUR EUR

51 Cumulated Contribution from Bond Futures U.S. Eurobloc U.K. Japan

52 Number of Traded Bond Futures U.S. Eurobloc U.K. Japan

53 Average Return from Bond Futures U.S. Eurobloc U.K. Japan

54 Portfolio Characteristics Sharp ratio: 1.42 Number of futures traded: Average return per single future: 155 EUR. Maximal draw down equities: EUR. Maximal draw down bonds: EUR. Contribution from tactical allocation: EUR. End value of portfolio: EUR.

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