120,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley Series in Probability and Statistics
- Sprache: Englisch
An Introduction to Machine Learning in Finance, With Mathematical Background, Data Visualization, and R Nonparametric function estimation is an important part of machine learning, which is becoming increasingly important in quantitative finance. Nonparametric Finance provides graduate students and finance professionals with a foundation in nonparametric function estimation and the underlying mathematics. Combining practical applications, mathematically rigorous presentation, and statistical data analysis into a single volume, this book presents detailed instruction in discrete chapters that allow readers to dip in as needed without reading from beginning to end. Coverage includes statistical finance, risk management, portfolio management, and securities pricing to provide a practical knowledge base, and the introductory chapter introduces basic finance concepts for readers with a strictly mathematical background. Economic significance is emphasized over statistical significance throughout, and R code is provided to help readers reproduce the research, computations, and figures being discussed. Strong graphical content clarifies the methods and demonstrates essential visualization techniques, while deep mathematical and statistical insight backs up practical applications. Written for the leading edge of finance, Nonparametric Finance: * Introduces basic statistical finance concepts, including univariate and multivariate data analysis, time series analysis, and prediction * Provides risk management guidance through volatility prediction, quantiles, and value-at-risk * Examines portfolio theory, performance measurement, Markowitz portfolios, dynamic portfolio selection, and more * Discusses fundamental theorems of asset pricing, Black-Scholes pricing and hedging, quadratic pricing and hedging, option portfolios, interest rate derivatives, and other asset pricing principles * Provides supplementary R code and numerous graphics to reinforce complex content Nonparametric function estimation has received little attention in the context of risk management and option pricing, despite its useful applications and benefits. This book provides the essential background and practical knowledge needed to take full advantage of these little-used methods, and turn them into real-world advantage. Jussi Klemelä, PhD, is Adjunct Professor at the University of Oulu. His research interests include nonparametric function estimation, density estimation, and data visualization. He is the author of Smoothing of Multivariate Data: Density Estimation and Visualization and Multivariate Nonparametric Regression and Visualization: With R and Applications to Finance.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 965
Veröffentlichungsjahr: 2018
Ähnliche
Table of Contents
Cover
Title Page
Copyright
Preface
Chapter 1: Introduction
1.1 Statistical Finance
1.2 Risk Management
1.3 Portfolio Management
1.4 Pricing of Securities
Part I: Statistical Finance
Chapter 2: Financial Instruments
2.1 Stocks
2.2 Fixed Income Instruments
2.3 Derivatives
2.4 Data Sets
Chapter 3: Univariate Data Analysis
3.1 Univariate Statistics
3.2 Univariate Graphical Tools
3.3 Univariate Parametric Models
3.4 Tail Modeling
3.5 Asymptotic Distributions
3.6 Univariate Stylized Facts
Chapter 4: Multivariate Data Analysis
4.1 Measures of Dependence
4.2 Multivariate Graphical Tools
4.3 Multivariate Parametric Models
4.4 Copulas
Chapter 5: Time Series Analysis
5.1 Stationarity and Autocorrelation
5.2 Model Free Estimation
5.3 Univariate Time Series Models
5.4 Multivariate Time Series Models
5.5 Time Series Stylized Facts
Chapter 6: Prediction
6.1 Methods of Prediction
6.2 Forecast Evaluation
6.3 Predictive Variables
6.4 Asset Return Prediction
Part II: Risk Management
Chapter 7: Volatility Prediction
7.1 Applications of Volatility Prediction
7.2 Performance Measures for Volatility Predictors
7.3 Conditional Heteroskedasticity Models
7.4 Moving Average Methods
7.5 State Space Predictors
Chapter 8: Quantiles and Value-at-Risk
8.1 Definitions of Quantiles
8.2 Applications of Quantiles
8.3 Performance Measures for Quantile Estimators
8.4 Nonparametric Estimators of Quantiles
8.5 Volatility Based Quantile Estimation
8.6 Excess Distributions in Quantile Estimation
8.7 Extreme Value Theory in Quantile Estimation
8.8 Expected Shortfall
Part III: Portfolio Management
Chapter 9: Some Basic Concepts of Portfolio Theory
9.1 Portfolios and Their Returns
9.2 Comparison of Return and Wealth Distributions
9.3 Multiperiod Portfolio Selection
Chapter 10: Performance Measurement
10.1 The Sharpe Ratio
10.2 Certainty Equivalent
10.3 Drawdown
10.4 Alpha and Conditional Alpha
10.5 Graphical Tools of Performance Measurement
Chapter 11: Markowitz Portfolios
11.1 Variance Penalized Expected Return
11.2 Minimizing Variance under a Sufficient Expected Return
11.3 Markowitz Bullets
11.4 Further Topics in Markowitz Portfolio Selection
11.5 Examples of Markowitz Portfolio Selection
Chapter 12: Dynamic Portfolio Selection
12.1 Prediction in Dynamic Portfolio Selection
12.2 Backtesting Trading Strategies
12.3 One Risky Asset
12.4 Two Risky Assets
Part IV: Pricing of Securities
Chapter 13: Principles of Asset Pricing
13.1 Introduction to Asset Pricing
13.2 Fundamental Theorems of Asset Pricing
13.3 Evaluation of Pricing and Hedging Methods
Chapter 14: Pricing by Arbitrage
14.1 Futures and the Put–Call Parity
14.2 Pricing in Binary Models
14.3 Black–Scholes Pricing
14.4 Black–Scholes Hedging
14.5 Black–Scholes Hedging and Volatility Estimation
Chapter 15: Pricing in Incomplete Models
15.1 Quadratic Hedging and Pricing
15.2 Utility Maximization
15.3 Absolutely Continuous Changes of Measures
15.4 GARCH Market Models
15.5 Nonparametric Pricing Using Historical Simulation
15.6 Estimation of the Risk-Neutral Density
15.7 Quantile Hedging
Chapter 16: Quadratic and Local Quadratic Hedging
16.1 Quadratic Hedging
16.2 Local Quadratic Hedging
16.3 Implementations of Local Quadratic Hedging
Chapter 17: Option Strategies
17.1 Option Strategies
17.2 Profitability of Option Strategies
Chapter 18: Interest Rate Derivatives
18.1 Basic Concepts of Interest Rate Derivatives
18.2 Interest Rate Forwards
18.3 Interest Rate Options
18.4 Modeling Interest Rate Markets
References
Index
End User License Agreement
Pages
c1
iii
iv
xiii
xiv
1
2
3
4
5
6
7
8
9
11
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
193
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
297
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
419
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
673
674
675
676
677
678
679
680
681
682
683
684
685
Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 2: Financial Instruments
Figure 2.1
S&P 500 index
. (a) Daily closing prices of S&P 500 and (b) daily returns.
Figure 2.2
S&P 500 and Nasdaq-100 indexes
. (a) The prices of S&P 500 (black) and Nasdaq-100 (red). The prices are normalized to start at value one. (b) A scatter plot of the daily returns of S&P 500 and Nasdaq-100.
Figure 2.3
S&P 500, US Treasury 10 year bond, and 1 month bill
. (a) The cumulative wealth of S&P 500 (black), 10 year bond (red), and 1 month bill (blue). The cumulative wealths are normalized to start at value one. (b) A scatter plot of monthly returns of S&P 500 and 10 year bond.
Figure 2.4
US Treasury bill rates and 10 year bond yields
. (a) Treasury bill rates (blue). (b) Yields of 10 year Treasury bond (red).
Figure 2.5
10 year US Treasury bond
. (a) Daily yields of the 10 year US Treasury bond and (b) daily returns of the bond.
Figure 2.6
S&P 500 components
. (a) Time series of the normalized prices of the components. (b) A scatter plot of , where are the 95% empirical quantiles of the negative returns, and are the annualized sample means of the returns.
Chapter 3: Univariate Data Analysis
Figure 3.1
Empirical distribution functions
. (a) Empirical distribution functions of S&P 500 returns (red) and 10-year bond returns (blue); (b) zooming at the lower left corner.
Figure 3.2
Left and right tail plots
. (a) The left tail plot for S&P 500 returns; (b) the right tail plot. The red curve shows the theoretical Gaussian curve and the blue curves show the Student curves for the degrees of freedom ν = 3–6.
Figure 3.3
Smooth tail plots
. The gray scale images show smooth tail plots of a collection of stocks in the S&P 500 index. The red points show the tail plots of the S&P 500 index. (a) A smooth left tail plot; (b) a smooth right tail plot.
Figure 3.4
Regression plots which are linear for exponential tails: S&P 500 daily returns
. (a) Left tail with (black), (red), and (blue); (b) right tail with (black), (red), and (blue).
Figure 3.5
Fitting of parametric families for data that is linear for exponential tails
. The data points are from left tail of S&P 500 daily returns, defined by the th empirical quantile with . (a) Fitting of exponential distributions; (b) fitting of Pareto distributions.
Figure 3.6
Regression plots which are linear for Pareto tails: S&P 500 daily returns
. (a) Left tail with (black), (red), and (blue); (b) right tail with (black), (red), and (blue).
Figure 3.7
Fitting of parametric families for data that is linear for Pareto tails
. The data points are from left tail of S&P 500 daily returns, defined by the th empirical quantile with . (a) Fitting of exponential distributions; (b) fitting of Pareto distributions.
Figure 3.8
Empirical quantile functions
. (a) Empirical quantile functions of S&P 500 returns (red) and 10-year bond returns (blue); (b) zooming to the lower left corner.
Figure 3.9
Histogram estimates
. (a) A histogram of historically simulated S&P 500 prices. A graph of kernel density estimate is included. (b) A histogram of historically simulated call option pay-offs.
Figure 3.10
Kernel density estimates of distributions of asset returns
. (a) Estimates of the distribution of S&P 500 monthly returns (blue) and of US 10-year bond monthly returns (red); (b) estimates of S&P 500 net returns with periods of 1–5 trading days (colors black–green).
Figure 3.11
Normal and log-normal densities
. Shown are a normal density (black) and a log-normal density (red) of the distribution of the stock price , when . In panel (a) , which equals 20 trading days, and in panel (b) years.
Figure 3.12
Parameter estimates for various return horizons
. The maximum likelihood estimates of (a) and (b) as a function of the return horizon in trading days.
Figure 3.13
Distribution of estimates
and
. (a) A kernel density estimate and a histogram of the distribution of ; (b) the estimates of the distribution of . The maximizers of the kernel estimates are indicated by the blue lines.
Figure 3.14
Excess distributions
. (a) The density function of -distribution with degrees of freedom five. The green, blue, and red vectors indicate the location of quantiles for , , and . (b) The right excess distributions for .
Figure 3.15
Exponential model for S&P 500 daily returns: Regression fits
. Panel (a) considers the left tail and panel (b) the right tail. We show the regression data and the fitted regression lines for / (blue), / (green), and / (red).
Figure 3.16
Pareto model for S&P 500 daily returns: Regression fits
. Panel (a) considers the left tail and panel (b) the right tail. We show the regression data and the fitted regression lines for / (blue), / (green), and / (red).
Figure 3.17
Exponential model for S&P 500 daily returns: Parameter estimates
. Panel (a) shows estimates of and panel (b) shows estimates of , as a function of . Red and blue: the maximum likelihood estimates; pink and green: the regression estimates; red and pink: the left tail; blue and green: the right tail.
Figure 3.18
Exponential model for S&P 500 daily returns: Tail plots with maximum likelihood
. Panel (a) shows the left tail plots and panel (b) shows the right tail plots. The red and green points show the observed data and the black lines show the exponential fits with , , , and .
Figure 3.19
Pareto model for S&P 500 daily returns: Parameter estimates
. Panel (a) shows estimates of and panel (b) shows estimates of as a function of . Red and blue: the maximum likelihood estimates; pink and green: the regression estimates; red and pink: the left tail; blue and green: the right tail.
Figure 3.20
Pareto model for S&P 500 daily returns: Tail plots with maximum likelihood
. Panel (a) shows the left tail plots and panel (b) shows the right tail plots. The red and green points show the observed data and the black curves show the fits with , , , and .
Figure 3.21
Gamma model for S&P 500 daily returns: Parameter estimates
. Panel (a) shows estimates of and panel (b) shows estimates of , as a function of . Red: the left tail; blue: the right tail.
Figure 3.22
Gamma model for S&P 500 daily returns: Tail plots with maximum likelihood
. Panel (a) shows the left tail plots and panel (b) shows the right tail plots. The red and green points show the observed data and the black lines show the fits with , , , and .
Figure 3.23
Generalized Pareto model for S&P 500 daily returns: Parameter estimates
. Panel (a) shows estimates of and panel (b) shows estimates of , as a function of . Red shows the estimates for the left tail, and blue shows them for the right tail.
Figure 3.24
Generalized Pareto model for S&P 500 daily returns: Tail plots with maximum likelihood
. Panel (a) shows the left tail plots and panel (b) shows the right tail plots. The red and green points show the observed data and the black curves show the fits with , , , and .
Figure 3.25
Weibull model for S&P 500 daily returns: Parameter estimates
. Panel (a) shows estimates of and panel (b) shows estimates of , as a function of . Red shows the estimates for the left tail, and blue shows them for the right tail.
Figure 3.26
Weibull model for S&P 500 daily returns: Tail plots with maximum likelihood
. Panel (a) shows the left tail plots and panel (b) shows the right tail plots. The red and green points show the observed data and the black curves show the fits with , , , and .
Figure 3.27
Density estimates of the distribution of Hill's estimates
. (a) Distribution of the left tail index; (b) the right tail index. Hill's estimates are calculated for the 312 stocks and kernel estimates are calculated from 312 estimated values of . There is an kernel estimate for each .
Figure 3.28
A scatter plot of estimates of
. We show a scatter plot of points for the stocks in the S&P 500 components data. The red line shows the points with .
Figure 3.29
Simulated i.i.d. time series
. We have simulated 10,000 observations. (a) Student's -distribution with degrees of freedom ; (b) Student's -distribution with degrees of freedom ; (c) Gaussian distribution. The mean of the observations is zero and the standard deviation is equal to the standard deviation of the S&P 500 returns.
Chapter 4: Multivariate Data Analysis
Figure 4.1
Linear and Spearman's correlation, together with volatility
. (a) Time series of moving average estimates of correlation between S&P 500 and Nasdaq-100 returns, with linear correlation (blue) and Spearman's rho (yellow); (b) we have added moving average estimates of the standard deviation of S&P 500 (black solid) and Nasdaq-100 (black dashed).
Figure 4.2
Tail coefficients for S&P 500 and Nasdaq-100 returns
. (a) Tail coefficients as a function of for lower tail coefficients (red) and as a function of for upper tail coefficients (blue); (b) time series of moving average estimates of lower tail coefficients.
Figure 4.3
Scatter plots
. Scatter plots of the net returns of S&P 500 and Nasdaq-100. (a) Original data; (b) copula transformed data with marginals being standard normal.
Figure 4.4
Smooth scatter plots
. Scatter plots of the net returns of S&P 500 and Nasdaq-100. (a) Original data; (b) copula transformed data with marginals being standard normal.
Figure 4.5
Correlations of DAX 30
. (a) An image of the correlation matrix for DAX 30; (b) correlations for DAX 30 with multidimensional scaling.
Figure 4.6
Gaussian and Student densities
. (a) Contour plot of the Gaussian density with marginal standard deviations equal to one and correlation 0.5; (b) Student density with degrees of freedom 2 and correlation 0.5.
Figure 4.7
Gaussian copulas
. Perspective plots of the densities of the Gaussian copula with the correlation (a) and (b) . The margins are uniform on .
Figure 4.8
Student copula with standard Gaussian margins
. Contour plots of the densities of the Student copula with degrees of freedom (a) 2 and (b) 4. The correlation is .
Figure 4.9
Gumbel copula
. Contour plots of the densities of the Gumbel copula with , , and . The marginals are standard Gaussian.
Figure 4.10
Clayton copula
. Contour plots of the densities of the Clayton copula with , , and . The marginals are standard Gaussian.
Chapter 5: Time Series Analysis
Figure 5.1
Removing a trend: Differences of logarithms
. (a) S&P 500 prices; (b) logarithms of S&P 500 prices; (c) differences of the logarithmic prices.
Figure 5.2
Removing a trend: Differencing
. (a) Differences of S&P 500 prices over 65 years; (b) differences over 4 years; (c) differences over 100 days.
Figure 5.3
Removing a trend: Subtracting a moving average
. (a) A times series of squared returns and a moving average of squared returns (red); (b) squared returns minus the moving average of squared returns.
Figure 5.4
Stochastic trend
. (a) Prices of S&P 500 over 100 days; (b) simulated random walk of length 100, when the initial value is 0.
Figure 5.5
S&P 500 autocorrelation
. (a) The sample autocorrelation function of S&P 500 returns for ; (b) the sample autocorrelation function for absolute returns. The red lines indicate the 95% confidence band for the null hypothesis of i.i.d process.
Figure 5.6
Time varying Hill's estimator
. (a) Left tail index; (b) right tail index. The black curves show sequentially calculated Hill's estimates, the blue curves show the time localized estimates with and the yellow curves have .
Figure 5.7
Time varying regression estimator
. Time series of estimates of the tail index are shown. (a) Left tail index; (b) right tail index. The black curves show sequentially calculated regression estimates, the blue curves show the time localized estimates with and the yellow curves have .
Figure 5.8
The definition of a MA
()
process
. (a) MA() process; (b) MA() process.
Figure 5.9
Moment generating functions under GARCH
. We show functions , where (a) and (b) . The case is with black, is with red, and is with blue.
Figure 5.10
GARCH(1,1) residuals: Tail plots
. (a) Left tail plot; (b) right tail plot. The red curves show the standard normal distribution function, and the blue curves show the Student distributions with degrees of freedom , , and .
Figure 5.11
GARCH(1,1) estimates versus Heston–Nandi estimates
:
and
. (a) A scatter plot of ; (b) a scatter plot of , where and are estimates in the GARCH() model, and and are estimates in the Heston–Nandi model.
Figure 5.12
GARCH(1,1) estimates versus Heston–Nandi estimates
:
and
. (a) A scatter plot of , where are estimates in the GARCH() model, and are estimates in the Heston–Nandi model. Panel (b) shows a histogram of estimates of in the Heston–Nandi model.
Figure 5.13
S&P 500 scatter plots of absolute returns
. (a) Scatter plot of points ; (b) scatter plot of points .
Figure 5.14
Simulated GARCH
returns and S&P 500 returns
. (a) A time series of simulated returns from a GARCH model; (b) the time series of S&P 500 returns.
Figure 5.15
S&P 500 returns
. The 10 smallest returns are shown in red and the 10 largest returns are shown in green.
Chapter 6: Prediction
Figure 6.1
Risk-free rate and moving averages
. (a) Moving averages of S&P 500 monthly gross returns with small (red) and large (blue); (b) moving averages of 10-year bond monthly gross returns with small (red) and large (blue). The black time series shows the 1-month T-bill rate.
Figure 6.2
Looking for times with similar states
. The time periods similar to the current state in terms of the dividend price ratio are shown with green. (a) ; (b) .
Figure 6.3
Default spread
. (a) Time series of the default spread; (b) time series of the differences.
Figure 6.4
TED spread
. (a) Time series of TED spread; (b) time series of the differences.
Figure 6.5
VIX index
. (a) Time series of the VIX index; (b) time series of the differences.
Figure 6.6
Term spread
. (a) The time series of term spread; (b) the time series of the differences of term spread.
Figure 6.7
Dividend price ratio
. (a) Time series of the dividend price ratio; (b) time series of the differences.
Figure 6.8
Purchasing managers index
. (a) Time series of the PMI; (b) time series of the differences.
Figure 6.9
S&P 500 trend
. (a) Time series of the S&P 500 trend; (b) time series of the differences.
Figure 6.10
S&P 500 returns for various horizons
. Panel (a) shows the returns , defined in (6.32). Panel (b) shows the times series . The black time series show 1-month returns, the red time series show 1-year returns, and the green time series show 5-year returns.
Figure 6.11
S&P 500 return autocorrelations for various horizons and lags
. (a) We show autocorrelations , where lag is (black), (red), and (blue). The -axis shows return horizon . (b) We show autocorrelations for lags , for horizons (black), (red), and (green).
Figure 6.12
of linear regression when predicting S&P 500 returns
. Predictors are the dividend price ratio (red), the term spread (green), and both the dividend price ratio and the term spread (black). Panel (a) shows from regression (6.33) when predicting month returns, and Panel (b) shows from regression (6.34) when predicting 1-month returns.
Figure 6.13
Time series of predictions and realized values
. (a) Prediction horizon of 1 year; (b) prediction horizon of 5 years. The black time series show the realized values , the green time series show the predictions when the predictor is dividend price ratio, the yellow time series show the predictions when the predictor is term spread, and red time series show the predictions when the predictors are both dividend price ratio and term spread.
Figure 6.14
Dividend price ratio as a predictor: Scatter plots and regression functions
. (a) Prediction horizon of 1 year; (b) prediction horizon of 5 years. Scatter plots show the points . The pink lines show the fitted regression functions.
Figure 6.15
Term spread as a predictor: Scatter plots and regression functions
. (a) Prediction horizon of 1 year; (b) prediction horizon of 5 years. Scatter plots show the points . The pink lines show the fitted regression functions.
Figure 6.16
Ten-year bond returns for various horizons
. Panel (a) shows the returns , defined in (6.32). Panel (b) shows the times series . The black time series show 1-month returns, the red time series show 1-year returns, and the green time series show 5-year returns.
Figure 6.17
Ten-year bond autocorrelations for various horizons and lags
. (a) We show autocorrelations , where lag is (black), (red), and (blue). The -axis shows return horizon . (b) We show autocorrelations for lags , for horizons (black), (red), and (green).
Figure 6.18
of linear regression when predicting 10-year bond returns
. Predictors are the dividend price ratio (red), the term spread (green), and both the dividend price ratio and the term spread (black). Panel (a) shows from regression (6.33) when predicting -month returns and Panel (b) shows from regression (6.34) when predicting 1-month returns.
Figure 6.19
Time series of predictions and realized values
. (a) Prediction horizon of 1 year; (b) prediction horizon of 5 years. The black time series show the realized values , the green time series show the predictions when the predictor is dividend price ratio, the yellow time series show the predictions when the predictor is term spread, and red time series show the predictions when the predictors are both dividend price ratio and term spread.
Figure 6.20
Dividend price ratio as a predictor: Scatter plots and regression functions
. (a) Prediction horizon of 1 year; (b) prediction horizon of 5 years. Scatter plots show the points . The pink lines show the fitted regression functions.
Figure 6.21
Term spread as a predictor: Scatter plots and regression functions
. (a) Prediction horizon of 1 year; (b) prediction horizon of 5 years. Scatter plots show the points . The pink lines show the fitted regression functions.
Chapter 7: Volatility Prediction
Figure 7.1
S&P 500 volatility process
. The time series of estimated volatility in the GARCH() model.
Figure 7.2
Distribution of S&P 500 volatility predictions
. (a) A tail plot and (b) a kernel density estimate computed from . The blue lines show the annualized sample standard deviation, and the red lines show the annualized unconditional standard deviation.
Figure 7.3
Error criterion
. Function is shown for .
Figure 7.4
Comparison of ARCH
()
and GARCH(1,1)
. (a) Function . (b) Time series . The prediction horizons are (black with “a”) and (red with “b”).
Figure 7.5
Sequential and GARCH(1,1) standard deviations
. The black curve shows the sequentially computed sample standard deviations and the blue curve shows the sequentially computed GARCH() stationary standard deviations.
Figure 7.6
Comparison of the sequential sample variance and GARCH(1,1) predictors
. (a) Shown is the function . (b) Shown are the time series of the differences of the cumulative sums of prediction errors. We show the cases (black), (red), (green), and (blue).
Figure 7.7
Comparison of EWMA and GARCH(1,1) using MSPE
. Shown are the ratios . (a) The -axis shows values of smoothing parameter . The symbols “a,” “b,” and “c” correspond to , , and . (b) The -axis shows values of prediction horizon . The symbols “1,”“2,”“3,” and “4” correspond to the smoothing parameters , , , and .
Figure 7.8
Comparison of EWMA and GARCH(1,1) using CSPE
. Shown are the time series of the differences of the cumulative sums of prediction errors. (a) The complete time series; (b) the beginning of the time series. We show the cases (black), (red), and (blue).
Figure 7.9
Comparison of smoothing parameters of EWMA
. Shown are the time series of the differences of the cumulative sums of prediction errors. (a) The first part of the time series; (b) the second part of the time series. The smoothing parameter takes values (black), (red), (blue), (dark green), (turquoise), and (pink).
Figure 7.10
Comparison of asymmetric EWMA and GARCH(1,1) using MSPE
. Shown are the ratios . (a) ; (b) . The skewness parameter takes values (black with “a”), (red with “b”), and (blue with “c”).
Figure 7.11
Linear prediction with moving averages
. (a) Shown are functions . (b) Shown are time series for smoothing parameter . The prediction horizon is (black with “a”) and (red with “b”).
Figure 7.12
Linear prediction with moving averages
. Shown are time series . Panel (a) shows the beginning of the period and panel (b) shows the end of the period. The prediction horizon is (black) and (red).
Figure 7.13
Linear prediction with past squared returns
. (a) Shown are functions . (b) Shown are time series for smoothing parameter . The prediction horizon is (black with “a”) and (red with “b”).
Figure 7.14
Kernel prediction
. (a) Shown are functions . (b) Shown are time series for smoothing parameter . The prediction horizon is (black with “a”) and (red with “b”).
Figure 7.15
Kernel prediction
. Shown are the time series for smoothing parameter . Panel (a) shows the beginning of the period and panel (b) shows the end of the period. The prediction horizon is (black) and (red).
Figure 7.16
Kernel prediction: Leverage effect
. The estimated regression function is visualized using (a) a contour plot and (b) a perspective plot.
Figure 7.17
Kernel prediction: Leverage effect
. (a) Slices for several values of . (b) Slices for several values of .
Chapter 8: Quantiles and Value-at-Risk
Figure 8.1
Definition of quantiles
. (a) Definition by distribution function; (b) definition by density function.
Figure 8.2
Loss functions for quantile estimation
. Loss function in (8.16) with (black solid line) and with (red dashed line).
Figure 8.3
Definition of empirical quantiles
. (a) The empirical distribution function; (b) the first half of the function, indicated by the black vectors in panel (a). The red vectors indicate the location of the th empirical quantile for .
Figure 8.4
Extrapolating and interpolating the empirical distribution
. (a) The black vertical line shows the empirical th quantile for . The red vertical line shows the Pareto quantile, and the red curve shows the fitted Pareto distribution. (b) Kernel distribution function estimates. The blue curve has smoothing parameter chosen by the normal reference rule. The black curve is oversmoothing and the green curve is undersmoothing.
Figure 8.5
Kernel quantile estimator
. (a) The implied distribution function estimates. The smoothing parameter is (red), (blue), and (green). (b) The quantile estimates as a function of smoothing parameter. We estimate the th quantile for (purple), (violet), and (black).
Figure 8.6
Kernel estimates of conditional quantiles
. (a) Conditional quantile estimates for the levels , when the smoothing parameter is ; (b) estimates for the level when the smoothing parameters are . A contour plot of a kernel estimate of the density of is also shown.
Figure 8.7
Quantile estimator performance
. We plot function , as defined in (8.8). Empirical quantiles (red), Gaussian quantiles (black), and Student quantiles (blue). The green lines show level fluctuation bands. (a) Level is close to zero; (b) is close to one.
Figure 8.8
Quantile estimator performance: Logarithmic
-
axis
. We plot function , as defined in (8.8). Empirical quantiles (red), Gaussian quantiles (black), and Student quantiles (blue). The green lines show level fluctuation bands. (a) Level is close to zero; (b) is close to one.
Figure 8.9
Quantile estimator performance: Slices of probability differences
. We show when for the empirical quantiles (red) and for the Gaussian quantile estimates (black). Panel (a) shows slices in (8.12) for and for . Panel (b) shows slices in (8.13) for and for .
Figure 8.10
Quantile estimator performance: Vectors over time intervals
. We show vectors joining points and , as defined in (8.14), when the time intervals are defined in (8.15) for and . (a) Empirical quantiles (red) and Gaussian quantiles (black); (b) empirical quantiles (red) and Student quantiles (blue).
Figure 8.11
Performance of quantile estimators: Expected loss
. We show functions (black) and (blue). (a) Range and (b) .
Figure 8.12
Performance of quantile estimators: Cumulative losses
. We show time series in (8.35). (a) and (b) .
Figure 8.13
GARCH(1, 1) quantiles
. The Figure shows the time series of estimated th quantiles with for the S&P 500 returns data. The quantiles are estimated with the GARCH(1, 1) method with the residual distribution being the standard normal.
Figure 8.14
Performance of GARCH(1, 1) quantile estimators: Probability differences
. (a) Functions are shown for . (b) Functions are shown for . The residuals are the standard normal (black) and the -distribution with degrees of freedom 12 (blue), degrees of freedom 5 (red), and the empirical distribution (purple).
Figure 8.15
Performance of GARCH(1, 1) quantile estimators: Slices of probability differences
. We show when for the empirical residuals (purple) and for the Gaussian residuals (black). Panel (a) shows slices in (8.12) for and for . Panel (b) shows slices in (8.13) for and for .
Figure 8.16
Performance of GARCH(1, 1) quantile estimators: Loss function
. Functions (a) : (b) . The residuals in are the standard normal (black) and the -distribution with degrees of freedom 12 (blue) and degrees of freedom 5 (red).
Figure 8.17
Performance of GARCH(1, 1) quantile estimators: Cumulative losses
. We show time series in (8.40). (a) ; (b) . The residuals of are the standard normal (black) and the -distribution with degrees of freedom 12 (blue) and degrees of freedom 5 (red).
Figure 8.18
EWMA
quantile estimator
:
Mean losses
. The mean loss is shown as a function of smoothing parameter . (a) ; (b) . The residual distributions are the standard Gaussian (black with “2”), the Student with degrees of freedom (red with “3”), degrees of freedom (blue with “4”), and the empirical (purple with “1”).
Figure 8.19
EWMA
quantile estimator
:
Mean losses for Student residuals
. The mean loss is shown as a function of smoothing parameter . (a) and (b) . The residual distributions are the Student distributions with degrees of freedom (black with “1”), (red with “2”), and (blue with “3”). The horizontal lines show the mean losses for GARCH(1, 1) volatility.
Figure 8.20
EWMA
quantile estimator
:
Smoothing parameter selection for Student residuals
. Panel (a) shows the curves for and panel (b) shows the curves for the cases . The distribution of the residuals is Student with degrees of freedom equal to . The smoothing parameters are shown with the colors black, red, blue, and purple.
Figure 8.21
EWMA
quantile estimator
:
Smoothing parameter selection for Gaussian residuals
. Panel (a) shows the curves for and panel (b) shows the curves for the cases . The distribution of the residuals is the standard normal. The smoothing parameters are shown with the colors black, red, blue, and purple.
Figure 8.22
EWMA
quantile estimator
:
Selection of residual distribution using probability differences
. (a) The curves for . (b) The curves for . The residual distributions standard normal, standard -distribution with degrees of freedom 12, degrees of freedom 5, and the empirical distribution are shown with the colors blue, red, black, and green.
Figure 8.23
Performance of EWMA
quantile estimators
:
Selection of residual distribution using loss function
. Functions for (a) and (b) . The residuals in are the standard normal (black) and the -distribution with degrees of freedom (blue) and degrees of freedom (red).
Figure 8.24
Performance of EWMA quantile estimators: Cumulative losses
. We show time series in (8.41). (a) and (b) . The residuals are the standard normal (black) and the -distribution with degrees of freedom 12 (blue), degrees of freedom 5 (red), and the empirical distribution (purple).
Figure 8.29
Kernel versus GARCH quantile estimator: Cumulative losses
. We show time series in (8.42). (a) and (b) .
Figure 8.25
Kernel quantile estimator
:
Mean losses
. The mean loss is shown as a function of smoothing parameter . (a) and (b) . The residual distributions are the standard Gaussian (black with “2”), the Student with degrees of freedom (red with “3”), degrees of freedom (blue with “4”), and the empirical (purple with “1”).
Figure 8.26
Kernel quantile estimator
:
Mean losses for Student residuals when
is close to 0
. The mean loss is shown as a function of smoothing parameter . (a) and (b) . The residual distributions are the Student distributions with degrees of freedom (black with “1”), (red with “2”), and (blue with “3”). The horizontal lines show the mean losses for GARCH(1, 1) volatility.
Figure 8.27
Kernel quantile estimator
:
Mean losses for Student residuals when
is close to 1
. The mean loss is shown as a function of smoothing parameter . (a) and (b) . The residual distributions are the Student distributions with degrees of freedom (black with “1”), (red with “2”), and (blue with “3”). The horizontal lines show the mean losses for GARCH(1, 1) volatility.
Figure 8.28
Kernel quantile estimator
:
Empirical residual for many
. (a) The curves for . (b) The curves for . The smoothing parameter of the kernel estimator is (black), (red), and (blue).
Figure 8.30
Exponential model
. (a) Ratios as a function of the multiplier . (b) The expected loss as a function of multiplier . The quantile level takes values (black), (red), and (blue).
Figure 8.31
Exponential model
. (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss when the multiplier of the quantile estimator is . The multiplier is (violet), (purple), (dark green), (pink), and (yellow).
Figure 8.32
Pareto model
. (a) Ratios as a function of the multiplier . (b) The expected loss as a function of multiplier . The quantile level takes values (black), (red), and (blue).
Figure 8.33
Pareto model
. (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss when the multiplier of the quantile estimator is . The multiplier is (violet), (purple), (dark green), and (pink).
Figure 8.34
Gamma model
. (a) Ratios as a function of the multiplier . (b) The expected loss as a function of multiplier . The quantile level takes values (black), (red), and (blue).
Figure 8.35
Gamma model
. (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss when the multiplier of the quantile estimator is . The multiplier is (violet), (purple), and (dark green).
Figure 8.36
Generalized Pareto model
. (a) Ratios as a function of the multiplier . (b) The expected loss as a function of multiplier . The quantile level takes values (black), (red), and (blue).
Figure 8.37
Generalized Pareto model
. (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss when the multiplier of the quantile estimator is . The multiplier is (violet), (purple), and (dark green).
Figure 8.38
Weibull model
. (a) Ratios as a function of the multiplier . (b) The expected loss as a function of multiplier . The quantile level takes values (black), (red), and (blue).
Figure 8.39
Weibull model
. (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss when the multiplier of the quantile estimator is . The multiplier is (violet), (purple), and (dark green).
Figure 8.40
Exponential model
: . (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss. The values of are (black with “1”), (blue with “2”), (violet with “3”), and (dark green with “4”).
Figure 8.42
Exponential model
. (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss when the multiplier of the quantile estimator is , and corresponds to the empirical quantile. The smoothing parameter is (orange), (purple), and (dark green). In panel (a) the red curve corresponds to the empirical quantile.
Figure 8.41
Exponential model
: . (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss. The values of are (black with “1”), (blue with “2”), (violet with “3”), and (dark green with “4”).
Figure 8.43
Gamma model
: . (a) Function , where is the estimate of the exceedance probability; (b) function , where is the estimated loss. The values of are (black with “1”), (blue with “2”), (violet with “3”), and (dark green with “4”). The red horizontal lines show the performance of empirical quantiles.
Figure 8.45
Gamma model
. (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss when the multiplier of the quantile estimator is , and corresponds to the empirical quantile. The smoothing parameter is (orange), (purple), and (dark green). In panel (a) the red curve corresponds to the empirical quantile.
Figure 8.44
Gamma model
: . (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss. The values of are (black with “1”), (blue with “2”), (violet with “3”), and (dark green with “4”). The red horizontal lines show the performance of empirical quantiles.
Figure 8.46
Generalized Pareto model
: . (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss. The values of are (black with “1”), (blue with “2”), (violet with “3”), and (dark green with “4”). The red horizontal lines show the performance of empirical quantiles.
Figure 8.48
Generalized Pareto model
. (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss when the multiplier of the quantile estimator is , and corresponds to the empirical quantile. The smoothing parameter is (orange), (purple), and (dark green). In panel (a) the red curve corresponds to the empirical quantile.
Figure 8.47
Generalized Pareto model
: . (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss. The values of are (black with “1”), (blue with “2”), (violet with “3”), and (dark green with “4”). The red horizontal lines show the performance of empirical quantiles.
Figure 8.49
Weibull model
: . (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss. The values of are (black with “1”), (blue with “2”), (violet with “3”), and (dark green with “4”). The red horizontal lines show the performance of empirical quantiles.
Figure 8.51
Weibull model
. (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss when the multiplier of the quantile estimator is , and corresponds to the empirical quantile. The smoothing parameter is (orange), (purple), and (dark green). In panel (a) the red curve corresponds to the empirical quantile.
Figure 8.50
Weibull model
: . (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss. The values of are (black with “1”), (blue with “2”), (violet with “3”), and (dark green with “4”). The red horizontal lines show the performance of empirical quantiles.
Figure 8.52
Exponential model
: . (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss. The values of are (black with “1”), (blue with “2”), (violet with “3”), and (dark green with “4”). The red horizontal lines show the performance of empirical quantiles.
Figure 8.53
Exponential model
: . (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss. The values of are (black with “1”), (blue with “2”), (violet with “3”), and (dark green with “4”). The red horizontal lines show the performance of empirical quantiles.
Figure 8.54
Exponential model
. (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss when the multiplier of the quantile estimator is , and corresponds to the empirical quantile. The smoothing parameter is (orange), (purple), and (dark green). In panel (a) the red curve corresponds to the empirical quantile.
Figure 8.55
Gamma model
: . (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss. The values of are (black with “1”), (blue with “2”), (violet with “3”), and (dark green with “4”). The red horizontal lines show the performance of empirical quantiles.
Figure 8.56
Gamma model
: . (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss. The values of are (black with “1”), (blue with “2”), (violet with “3”), and (dark green with “4”). The red horizontal lines show the performance of empirical quantiles.
Figure 8.57
Gamma model
. (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss when the multiplier of the quantile estimator is , and corresponds to the empirical quantile. The smoothing parameter is (orange), (purple), and (dark green). In panel (a) the red curve corresponds to the empirical quantile.
Figure 8.58
Generalized Pareto model
: . (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss. The values of are (black with “1”), (blue with “2”), (violet with “3”), and (dark green with “4”). The red horizontal lines show the performance of empirical quantiles.
Figure 8.59
Generalized Pareto model
: .
Gamma model
: . (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss. The values of are (black with “1”), (blue with “2”), (violet with “3”), and (dark green with “4”). The red horizontal lines show the performance of empirical quantiles.
Figure 8.60
Generalized Pareto model
. (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss when the multiplier of the quantile estimator is , and corresponds to the empirical quantile. The smoothing parameter is (orange), (purple), and (dark green). In panel (a) the red curve corresponds to the empirical quantile.
Figure 8.61
Weibull model
: . (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss. The values of are (black with “1”), (blue with “2”), (violet with “3”), and (dark green with “4”). The red horizontal lines show the performance of empirical quantiles.
Figure 8.62
Weibull model
: . (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss. The values of are (black with “1”), (blue with “2”), (violet with “3”), and (dark green with “4”). The red horizontal lines show the performance of empirical quantiles.
Figure 8.63
Weibull model
. (a) Function , where is the estimate of the exceedance probability. (b) Function , where is the estimated loss when the multiplier of the quantile estimator is , and corresponds to the empirical quantile. The smoothing parameter is (violet), (purple), and (dark green). In panel (a) the red curve corresponds to the empirical quantile.
Chapter 9: Some Basic Concepts of Portfolio Theory
Figure 9.1
Being long and short a stock
. The blue lines show the gross return of being long a stock for and as a function of the stock price. The red line shows the gross return of being short a stock. Shown are the functions , where is the previous value of the stock.
Figure 9.2
Comparison of distributions
. Shown are two return densities, where the red distribution has a higher risk and a higher return than the black distribution. It is not obvious which return distribution should be preferred.
Figure 9.3
Two correlated assets
. A contour plot of function is shown. The function is equal to the variance of the portfolio when the portfolio components have variance one, correlation , and the weight of the portfolio components are and .
Figure 9.4
Ranking distributions with the expected utility: Different means
. (a) Four density functions of gross return ; (b) the four density functions of . The expectations are marked with vertical vectors.
Figure 9.5
Ranking of distributions with expected utility: Different variances
. (a) Four density functions of return ; (b) the density functions of . The expectations are marked with vertical vectors.
Figure 9.6
Utility functions
. (a) Power utility functions (9.28) for risk aversion values , , and ; (b) exponential utility functions (9.29) for risk aversion values , , and .
Figure 9.7
Expected utility as a function of mean and standard deviation
. We show contour plots of functions , where follows a normal distribution . (a) ; (b) with .
Figure 9.8
Approximation of logarithmic utility
. The black curve shows log-utility , the blue curve shows the linear approximation, the red curve shows the quadratic approximation, the green curve shows the third-order approximation, and the yellow curve shows the fourth-order approximation.
Figure 9.9
Portfolio selection: S&P 500 and Nasdaq-100
. (a) Functions (9.32) for S&P 500 (black) and Nasdaq-100 (red); (b) functions (9.33) for (blue) and (green).
Figure 9.10
Portfolio selection: Risk-free rate and S&P 500
. (a) Functions (9.34) for (black), (red), (blue), and (green); (b) functions (9.35) for (purple) and (dark green).
Figure 9.11
First-order stochastic dominance
. The black distribution dominates the red distribution. (a) Density functions; (b) distribution functions.
Figure 9.12
Second-order stochastic dominance
. The black distribution dominates the red distribution. (a) Density functions; (b) distribution functions; (c) functions , , where and are the distribution functions.
Chapter 10: Performance Measurement
Figure 10.1
Confidence intervals for the S&P 500 Sharpe ratio
. (a) Confidence intervals corresponding to a coverage probability in . The -axis shows the range of possible values of the Sharpe ratio and the -axis shows the coverage probabilities of the confidence intervals. The yellow vertical vector indicates the point estimate of the Sharpe ratio and the red vertical vectors show the confidence interval with coverage. (b) The ratios as a function of smoothing parameter, where is the estimator assuming zero autocorrelation, whereas assumes autocorrelation.
Figure 10.2
Drawdown
. (a) Drawdown time series for S&P 500 (red) and 10-year bond (blue); (b) time series (red) and the cumulative wealth (orange) for S&P 500.
Figure 10.3
Alphas and betas of S&P 500 components
. (a) A scatter plot of ; (b) linear functions .
Figure 10.4
Time series of cumulative wealths
. (a) The -axis shows the cumulative wealth; (b) the -axis has a logarithmic scale. We show the cumulative wealth of S&P 500 (red), 10-year bond (blue), and 1-month bill (black).
Figure 10.5
Time series of relative cumulative wealth of 10-year bond
. We compare 10-year bond to S&P 500 and to 1-month bill. (a) The wealth ratio , where is the wealth of the 10-year bond, is the wealth of S&P 500 (green), and is the wealth of 1-month bill (purple). Panel (b) shows time series .
Figure 10.6
Time series of relative cumulative wealth of S&P 500
. We compare S&P 500 to 10-year bond and to 1-month bill. (a) The wealth ratio , where is the wealth of S&P 500, is the wealth of 10-year bond (green), and is the wealth of 1-month bill (purple). Panel (b) shows time series .
Figure 10.7
Sharpe ratios for every period: Contour plots
. We show contour plots of function , defined in (10.36). (a) Sharpe ratios of S&P 500; (b) Sharpe ratios of US Treasury 10-year bond.
Figure 10.8
Sharpe ratios for every period: Image plots
. The bright yellow shows the time periods where the Sharpe ratio is high and the red color shows the time periods where the Sharpe ratio is low. (a) S&P 500; (b) 10-year bond.
Figure 10.9
Sharpe ratios for every period: Level sets
. We show a level set in (10.39) with blue color. (a) Function is the Sharpe ratio of S&P 500; (b) function is the Sharpe ratio of US Treasury 10-year bond. The blue regions show the time periods for which the Sharpe ratio is above the usual value, and the red color shows when it is below the usual value.
Figure 10.10
Time series of Sharpe ratios: Slices
. (a) A slice at time shows the Sharpe ratio computed with the data starting at and ending , where (red), (blue), (green), and (black). (b) A slice at time shows the Sharpe ratio computed with the data starting at and ending , where (black), (red), (blue), and (green).
Figure 10.11
Time series of Sharpe ratios
. (a) Sharpe ratios of S&P 500; (b) Sharpe ratios of US 10-year bond. The black curves show the Sharpe ratios, the green curves show the means of the excess returns, and the blue curves show the standard deviations of the excess returns. The time series at time show Sharpe ratios computed with the data starting at and ending . The violet horizontal lines show the Sharpe ratios over the complete time period.
Figure 10.12
Time series of certainty equivalents
. (a) Certainty equivalents of S&P 500; (b) certainty equivalents of US Treasury 10-year bond. The green curves show the case of risk aversion , the red curves have , and the purple curves have . Time series at time show certainty equivalents computed with the data starting at and ending .
Chapter 11: Markowitz Portfolios
Figure 11.1
Markowitz bullets: Portfolios of two risky assets when correlation varies
. (a) Shown are long-only portfolios that can be obtained from two risky assets when correlation between the risky assets varies between and . (b) Shorting is allowed.
Figure 11.2
Markowitz bullets: Portfolios of three risky assets
. (a) Long-only portfolios that can be obtained from three risky assets. (b) Portfolios when shorting is allowed.
Figure 11.3
Markowitz bullet: Long-only portfolios and leveraging
. Panel (a) shows long-only portfolios for two risky assets and the risk-free investment. Panel (b) shows portfolios for two risky assets and the risk-free investment when the weight of the risk-free investment is allowed to be negative, which means the borrowing is allowed.
Figure 11.4
Markowitz bullet: Shorting and leveraging
. Panel (a) shows portfolios from two risky assets, and from two risky assets and the risk-free investment. Panel (b) shows portfolios from two risky assets and a risk-free investment when the weight of the risk-free investment is allowed to be negative, so that borrowing is possible.
Figure 11.5
Markowitz bullet: Uncorrelated assets
. Markowitz bullets are shown for an increasing number of assets: blue curve shows portfolios from two risky assets, the green area portfolios from three risky assets, and the yellow area portfolios from four risky assets, when the risky assets are uncorrelated.
Figure 11.6
Markowitz bullet: S&P 500 components
. (a) A scatter plot of annualized sample standard deviations and means of excess returns of a collection of stocks in the S&P 500 index. (b) A kernel density estimate of the distribution of the Sharpe ratios.
Figure 11.7
S&P 500 and Nasdaq-100: Markowitz weights
. The time series of the weights for the S&P 500. (a) The unrestricted weights and (b) the long-only weights. The risk aversion parameter takes values (black, red, blue, green, and orange).
Figure 11.8
S&P 500 and Nasdaq-100: Sharpe ratios, means, and standard deviations
. (a) The Sharpe ratios as a function of ; (b) the annualized means; (c) the annualized standard deviations. The black line with labels “1” is obtained when the unrestricted weights are used and the green line with labels “2” is obtained when the long-only weights are used.
Chapter 12: Dynamic Portfolio Selection
Figure 12.1
Trend following with the previous 1-month return: Cumulative wealth
. (a) The cumulative wealth of the trend following portfolio with 10-year bond (purple) and S&P 500 (black); (b) logarithmic scale. The red curves show the cumulative wealth of S&P 500 and the blue curves show the cumulative wealth of 10-year bond.
Figure 12.2
Trend following with the previous 1-month return: Wealth ratios
. We show wealth ratios . (a) The ratio compares S&P 500 trend following to the benchmark (black) and 10-year bond trend following to the benchmark (purple); (b) the ratio compares 10-year bond trend following to the S&P 500 trend following.
Figure 12.3
Trend following with the previous 1-month return: Times series of Sharpe ratios
. (a) S&P 500 and (b) 10-year bond.
Figure 12.4
Trend following with the previous 1-month return: Level sets of Sharpe ratios
. We show with blue the time periods where the Sharpe ratio of trend following was better than the Sharpe ratio of the benchmark. (a) S&P 500 and (b) 10-year bond.
Figure 12.5
Trend following with moving averages
. (a) Sharpe ratios of trend following strategies with S&P 500 (black curve) and 10-year bond (purple curve); (b) certainty equivalent is shown for the same portfolios as in panel (a). The -axis shows the smoothing parameter of the moving average. The red horizontal line shows the Sharpe ratio of S&P 500 and the blue horizontal line shows the Sharpe ratio of 10-year bond.
Figure 12.6
Trend following with moving averages: Wealth ratios
. (a) Following the trend of S&P 500 and (b) following the trend of 10-year bond. We show the ratios of the cumulative wealth to cumulative wealth of the benchmark. The smoothing parameters of moving averages are (black), (red), (blue), and (green).
Figure 12.7
Expected returns determined by economic indicators: Sharpe ratios
. Sharpe ratios of portfolios with S&P 500 (black curve) and 10-year bond (purple curve). (a) The prediction method of (6.35) and (b) the prediction method of (6.36). The -axis shows the prediction horizon. The red horizontal line shows the Sharpe ratio of S&P 500 and the blue horizontal line shows the Sharpe ratio of 10 year bond.
Figure 12.8
Expected returns determined by economic indicators: Cumulative wealth
. (a) The cumulative wealth of the portfolio whose weights are chosen according to the expected returns, when the risky asset is 10-year bond (purple) and S&P 500 (black); (b) logarithmic scale. The red curves show the cumulative wealth of S&P 500 and the blue curves show the wealth of 10-year bond.
Figure 12.9
Expected returns determined by economic indicators: Wealth ratios
. (a) The ratio of the wealth of trend following with S&P 500 to the wealth of S&P 500 (black), and the ratio of the wealth of trend following with 10-year bond to the wealth of 10-year bond (purple); (b) The ratio of the wealth of trend following with S&P 500 to the wealth of trend following with 10-year bond.
Figure 12.10
Markowitz portfolios with moving averages: Sharpe ratios
. (a) Sharpe ratios of trend following strategies with S&P 500 (black curve) and 10-year bond (purple curve). The -axis shows the smoothing parameter of the moving average. The red horizontal line shows the Sharpe ratio of S&P 500 and the blue horizontal line shows the Sharpe ratio of 10 year bond. (b) Sharpe ratios of trend following strategies with S&P 500 when both expected returns and volatility are estimated using moving averages.
Figure 12.11
Markowitz portfolios with moving averages: Wealth ratios
. (a) Trading with S&P 500 and (b) trading with 10-year bond. We show the ratios of the cumulative wealth to cumulative wealth of the benchmark. The smoothing parameters of moving averages are (black), (red), (blue), and (green).
Figure 12.12
Markowitz portfolios with moving averages: Sharpe ratios when short selling and leveraging are allowed
. Sharpe ratios of trend following strategies as a function of smoothing parameter when the risky asset is (a) S&P 500; (b) 10-year bond. The red horizontal line show the Sharpe ratio of S&P 500 and the blue horizontal line show the Sharpe ratio of 10 year bond.
Figure 12.13
Sharpe ratios as a function of prediction horizon when the expected returns are determined by economic indicators: Markowitz criterion
. Sharpe ratios of portfolios with S&P 500 (black curve) and 10-year bond (purple curve). (a) The prediction method of (6.35) and (b) the prediction method of (6.36). The -axis shows the prediction horizon. The red horizontal line shows the Sharpe ratio of S&P 500 and the blue horizontal line shows the Sharpe ratio of 10 year bond.
Figure 12.14
Trend following with the previous 1-month return and two risky assets: Cumulative wealth
. (a) The cumulative wealth of the trend following portfolio (black); (b) logarithmic scale. The red curves show the cumulative wealth of S&P 500 and the blue curves show the wealth of 10-year bond.
Figure 12.15
Trend following with the previous 1-month return and two risky assets: Wealth ratios
. (a) The wealth of the two asset trend following is divided by the S&P 500 wealth (dark green) and 10-year bond wealth (orange); (b) the wealth of the two asset trend following is divided by the wealth of S&P 500 trend following (dark green) and the wealth of 10-year bond trend following (orange).
Figure 12.16
Trend following with moving averages
. (a) Sharpe ratios of trend following strategies (black curve); (b) certainty equivalent is shown for the same portfolios as in panel (a). The -axis shows the smoothing parameter of the moving average. The red horizontal line shows the Sharpe ratio of S&P 500 and the blue horizontal line shows the Sharpe ratio of 10 year bond.
Figure 12.17
Expected returns determined by economic indicators: Sharpe ratios
. Sharpe ratios of managed portfolios. (a) The prediction method of (6.35) and (b) the prediction method of (6.36). The -axis shows the prediction horizon. The red horizontal line shows the Sharpe ratio of S&P 500 and the blue horizontal line shows the Sharpe ratio of 10 year bond.
Figure 12.18
Expected returns determined by economic indicators: Cumulative wealth
. (a) The cumulative wealth of the portfolio whose weights are chosen according to the expected return (black); (b) logarithmic scale. The red curves show the cumulative wealth of S&P 500 and the blue curves show the wealth of 10-year bond. The wealth is normalized to have value one at the beginning.
Figure 12.19
Markowitz portfolios with two risky assets and moving averages: Sharpe ratios
. (a) Sharpe ratios as a function of the smoothing parameter of the means and , for the smoothing parameter of the variances in the range 1–1000; (b) Sharpe ratios as a function of the smoothing parameter of the variances and , for the smoothing parameter of the means in the range 0.1–1000.
Figure 12.20
Markowitz portfolios with two risky assets and moving averages: Sharpe ratios
. (a) Sharpe ratios as a function of the smoothing parameter of the variances and , for the smoothing parameter of the covariance in the range 1–1000; (b) Sharpe ratios as a function of the smoothing parameter of the covariance , for the smoothing parameter of the variances in the range 1–1000.
Figure 12.21
Markowitz portfolios with two risky assets: Moving averages when short selling and leveraging are allowed
. (a) Sharpe ratios as a function of the smoothing parameter of the means and ; (b) Sharpe ratios as a function of the smoothing parameter of the variances and . The curve with labels “1” leverage, labels “2” indicate leverage , and labels “3” indicate leverage . The red horizontal line shows the Sharpe ratio of S&P 500 and the blue horizontal line shows the Sharpe ratio of 10-year bond.
Figure 12.22
Markowitz portfolios with two risky assets and moving averages: Wealth
. (a) The cumulative wealth of the Markowitz portfolio (black), S&P 500 (red), and 10-year bond (blue); (b) the ratios of the wealth of the Markowitz portfolio to the wealth of S&P 500 (purple) and to the wealth of 10-year bond (orange).
Figure 12.23
Regression on economic indicators: Sharpe ratios
. (a) Sharpe ratios as a function of the prediction horizon, for values –1000 of the smoothing parameter of the variances; (b) Sharpe ratios as a function of the smoothing parameter of the variances, for prediction horizons –120.
Figure 12.24
Regression on economic indicators: Ratios of the Sharpe ratios
. The Sharpe ratios of (6.35) are divided by the Sharpe ratios of (6.36). (a) The ratios as a function of the prediction horizon, for values –1000 of the smoothing parameter of the variances; (b) the ratios as a function of the smoothing parameter of the variances, for prediction horizons –120.
Figure 12.25
Regression on economic indicators: Sharpe ratios
. (a) The Sharpe ratios as a function of the smoothing parameter of the estimator of the variances, for values
