About the
Author 17
List of graphical notations and symbols 18
Preface 20
I.1. Who will gain, and what, from this book 20
I.2. A goal and general guiding concept of the book 22
I.3. Foundations of the study 23
I.3.1. Internal rate of return 23
I.3.2. Compounding 23
I.3.3. Shestopaloff’s Linking 24
I.3.4. Other key topics 25
I.4. Organization of the book 27
I.5. A note about references 31
I.6. Acknowledgements 31
Chapter 1
1. Understanding interest rate and rate of return. IRR equation 32
1.1. Interest rate and rate of return 32
1.1.1. Definition of interest rate 32
1.1.2. A general note about mathematical approaches 34
1.2. Introducing compounding 36
1.3. Domain of interest rate. Range of applicability 38
1.3.1. Specifics of negative interest rate 38
1.3.2. The meaning of compounding when the period length is not an
integer value 40
1.4. Computing interest rate 42
1.4.1. Relationship between the period length and interest 42
rate 42
1.4.2. Computing interest rate for shorter or longer periods. Nominal
and effective interest rates 43
1.4.3. Mathematical foundations of interest rate calculations 44
1.4.4. Computing interest rates. Numerical examples 47
1.5. Continuous compounding. 50
1.5.1. Continuous compounding. Definitions 50
1.5.2. Continuous compounding versus discrete compounding. Numerical
example 53
1.5.3. Smooth exponential compounding function versus 55
the piecewise linear function 55
1.6. IRR equation. An inherent relationship of compounding operation and
cash flows 61
1.6.1. How IRR equation accounts for cash transactions 62
inside the period 62
1.6.2. Deriving simple form of IRR equation 64
1.6.3. Noncompounding scenario 66
1.7. Deriving a general form of IRR equation 68
1.7.1. IRR Equation with discrete compounding 68
1.7.2. IRR equation with continuous compounding 70
1.7.3. Generalization of continuous compounding. 71
Force of interest 71
1.7.4. Getting the "look and feel" of IRR equation 73
Chapter 2
Annuities 79
2.1. Deriving an ordinary annuity from the IRR equation 80
2.1.1. Annuity definition and specifics 80
2.1.4. Annuity’s present value 84
2.2. Annuities and withdrawals. Purchasing an annuity 86
2.2.1. Accumulating required amount 87
2.2.2. Finding the number of annuity payments 88
2.2.3. Price of an annuity to receive regular payments 89
2.3. Discussion of annuity’s features 91
2.3.1. Annuity with a negative interest rate. 91
Numerical example 91
2.4. Annuity Due 95
2.4.1. Definition of annuity due 95
2.4.2. Deriving a general formula for an annuity due 96
2.4.3. Finding a lump sum payment 98
2.4.4. Determining the number of payments 99
2.4.5. Calculating a present value 99
2.4.6. Notes with regard to relationship between 100
annuities and IRR equation 100
2.5. Annuities with continuous compounding 102
2.5.1. Ordinary annuity 102
2.5.2. Finding a regular payment 104
2.5.3. Calculating a lump sum payment 105
2.5.4. Finding the number of payments 105
2.5.5. Computing unknown interest rate 106
2.5.6. Finding the present value 106
2.6. Relationship between interest rates for periods with different
lengths 107
2.6.1. Computing interest rate for an arbitrary period length 110
Chapter 3
Mortgages. Deriving mortgage formulas from IRR equation 113
3.1. Defining mortgage in mathematical terms 115
3.1.1. Deriving a mortgage formula for remaining balance 115
3.1.2. Finding payment amount 117
3.1.3. Calculating a number of periods 118
3.1.4. How much can we afford? 119
3.2. Numerical examples, graphs and discussion 121
3.2.1. Interest and principal amounts 121
3.2.2. Influence of payment frequency 124
3.2.3. Weekly and monthly payments versus annual payments 126
3.3. Choosing the right method to calculate an interest rate for shorter
or longer periods 129
3.4. Generalization of mortgage equations 132
3.4.1. Proportionality between the period payment and 132
principal amount 132
3.4.2. Relationship between the interest rate and period length 133
3.5. Mortgages with continuous compounding 136
3.5.1. Mortgage equations for continuous compounding 137
3.5.2. Finding the remaining balance 138
3.5.3 Calculating payment amount 139
3.5.4. Finding a number of payments 139
3.5.5. How much can we afford? 140
3.6. Comparing mortgages with discrete and continuous compounding 143
3.7. Chapter conclusion. Merging mortgages and IRR equation 147
Chapter 4
Introducing return on investment. 148
Relationship between Modified Dietz formula and IRR equation 148
4.1. Overview of problems and general approaches 148
4.1.1. Relationship of rate of return and compounding context 150
4.1.2. IRR equation as the main conceptual foundation of the study 150
4.2. Dietz’s methods 152
4.3. Taylor series expansion of IRR equation 158
4.3.1. Relationship between the Modified Dietz methods and IRR equation
159
4.3.1.1. Progression of methods for calculating of rate of return 161
4.3.1.2. Sibling of the Modified Dietz method 162
4.4. Accuracy of Modified Dietz method when compared to IRR equation 165
4.5. Geometrical interpretation of Modified Dietz equation in relation
to IRR function 169
4.6. Conclusion 172
Chapter 5
Solving IRR equation. New and existing methods. Applications 173
5.1. NewtonRaphson’s method 174
5.2. Number of solutions of IRR equation. 178
The Theorem 178
5.2.1. Available approaches. Transformation of IRR Equation 178
5.2.2. Proof of the Theorem’s Corollary 181
5.2.3. A numerical example of IRR equation with three roots 182
5.3. Introducing new methods for solving IRR equation 185
5.3.1. Linear approximation of IRR equation 185
5.3.2. Quadratic approximations of the IRR equation 187
5.3.3. Quadratic solution at point zero 188
5.3.4. Generalization of iterative algorithms for IRR equation 191
5.3.5. Iteration accuracy 193
5.4. Series reversion method 196
5.4.1. Computing series reversion coefficients 197
5.4.2. Discussion of series reversion method 200
5.5. Application of methods for solving IRR equation in the business
environment 202
5.5.1. Choosing the first approximate value of rate of return for
iterative procedures 202
5.5.2. Factors to be considered when adopting 203
computational methods 203
5.5.2.1. Business need 203
5.5.2.2. Compliance with standards. Relevance to existing methods 204
5.5.2.3. Acceptable level of complexity for the average user 204
5.5.2.4. Sufficiency of computational resources 204
5.5.2.5. Data feeds availability 205
5.5.2.6. Flexibility with regard to future business needs 205
Chapter 6
Computational efficiency of algorithms 206
for solving IRR equation 206
6.1. Software implementation details 206
6.2. Accuracy of methods for computing rate of return 207
6.2.1. Comparison and hierarchy of computational methods from the
accuracy perspective 207
6.2.2. Solutions’ accuracy. Limitations of methods with regard to
business scenarios 210
6.4. Comparing computational performance of different methods 218
6.5. New high performance computational methods for mortgages and
annuities. Computing algorithms for financial calculators 221
6.5.1. IRR equation for mortgages and annuities 222
6.5.2. Specific properties of mortgage IRR function and its
characteristic points 224
6.5.2.1. General form of IRR function. Solution of IRR equation 224
6.5.2.2. Asymptotic behavior of a mortgage IRR function. Characteristic
points 226
6.5.2.3. Transforming a mortgage IRR equation 228
6.5.3. Approximating solution of the IRR equation 230
6.5.4. An accuracy of X and Y approximations, 232
and their computational performance 232
6.5.5. Generalization of X and Y approximations. 234
Aapproximation method 234
6.5.6. Adding CPapproximation 237
6.5.7. Practical implementation of Aapproximation method. Accuracy
evaluation. Numerical examples 238
6.5.7.1. Threshold function 238
6.5.7.2. Power function 240
6.5.7.3. Iterative algorithms 240
6.5.8. Defining an iteration accuracy 242
6.5.9. Conclusion 244
6.6. Series reversion method for mortgages and annuities 245
6.6.1. Mathematical algorithms 245
6.6.2. Computational performance of series reversion algorithms 248
Chapter 7
Climbing to a conceptual level in understanding of rate of return.
Mathematical surprises 249
7.1. Compounding context of the IRR equation 252
7.1.1. Distinguishing direct and reverse problems 253
7.2. Noncompounding nature of Modified Dietz equation 256
7.2.1. Calculating noncompounding rate of return 256
7.2.2. Generalized Modified Dietz equation 257
7.2.3. Comparison of methods’ contexts 258
7.3. Numerical examples demonstrating conceptual differences between the
Modified Dietz and IRR methods 260
7.3.1. Introducing variable transaction time 260
7.3.2. Influence of transaction time to the IRR and Modified Dietz rates
of return. Positive cash flows 260
7.3.3. Converging the time scale 263
7.3.4. Intermediate note about methods’ objectivity 263
7.3.5. Influence of negative cash flows 264
7.3.6. Extreme scenarios with large cash flows 267
7.4. More IRR derivatives. 272
NPV (Net Present Value) method 272
7.5. Modified IRR method (MIRR). 275
Some common misconceptions 275
7.5.1. Enhancing MIRR formula. Arbitrary period lengths 275
7.5.2. Improving MIRR equation. Advantages and new features 277
7.5.3. Inherent relationships of IRR and MIRR 278
7.5.4. Specifics of compounding context of MIRR. Changing MIRR context
280
7.6. Completing discussion on conceptual differences between methods for
calculating rate of return 284
Chapter 8
Shestopaloff’s linking (SL) algorithms for investment performance
measurement and trading. Positioning the geometric linking 287
8.1. Introducing SL method 287
8.1.1. A note on terminology 287
8.1.2. Origins and the present status of SL algorithms 287
8.2. Restrictive nature of geometric linking 290
8.2.1. Applicability domain of geometric linking 290
8.2.2. Mathematical formulation of the geometric linking operation 291
8.2.3. Geometric Linking Theorem 292
8.2.4. Importance of the Geometric Linking Theorem 294
8.3. Shestopaloff’s Linking (SL) concept 297
8.3.1. Linking operation. The origin and general features of SL
algorithms 297
8.4. Shestopaloff’s linking for Modified Dietz formula 301
8.4.1. Linking sequential periods 301
8.4.2. Linking nonsequential periods 306
8.4.3. Linking rates of return for different asset classes within the
same period 307
8.4.4. Numerical example for Shestopaloff’s linking of rates of return
for sequential periods and different assets 308
8.4.5. Discussion of analytical and system design benefits that SL
methods provides 309
8.4.6. Linking assets’ rates of return 312
8.4.7. Data mining 312
8.4.8. Data used in numerical examples 313
8.5. Shestopaloff’s Linking for internal rate of return (IRR) 316
8.5.2. Approximating Shestopaloff’s linking for IRR equation 318
8.6. Computational performance of SL algorithms 322
8.7. Cumulative properties of Shestopaloff’s linking algorithms 325
8.8. SL algorithms and multidimensional analytical research of
investment portfolio 326
8.9. Business considerations with regard to application of investment
performance measurement methods 329
8.9.1. Limitations of existing methods 329
8.9.2. How SL methods can improve the situation 331
8.9.2.1. Diversification of analytical scenarios 332
8.9.2.2. Computational performance 333
8.9.2.3. Enhanced accuracy 333
8.9.2.4. Removing computational redundancy 333
8.9.2.5. Database size 333
8.9.2.6. Optimal data structuring 334
8.9.2.7. Advanced and efficient system design 335
8.9.2.8. Localized changes of data 336
8.9.2.8. Conclusion 336
8.10. Hierarchy of methods for calculating rates of return 338
8.10.1. A brief review of major methods 338
8.10.2. IRR as the most objective method for 340
computing rate of return 340
8.10.3. An inherent relationship of the IRR equation and cash flows 342
8.10.4. Relationship of TWRR and Modified Dietz method 343
8.10.5. Summary of IRR method’s features 346
8.10.6. Schema of hierarchy of methods for calculating rate of return.
347
8.10.7. Some considerations regarding industry standards 348
8.11. Trading. Development of trading strategies using SL algorithms 350
8.11.1. Review of trading methods 350
8.11.2. Optimization of investment strategies 352
8.11.3. Market efficiency from a real life perspective 354
8.11.4. Analytical studies and development of trading strategies 355
8.11.5. Trading on relative values instead of prices. Trading on rate of
return 357
8.11.6. Programming trading. Thinking realistically 359
Chapter 9
Investment attribution analysis 360
9.1. The scope of attribution analysis 361
9.2. Defining the contribution 363
9.3. Arithmetic attribution 366
9.3.1. Main equations of arithmetic attribution 366
9.3.2. Decomposing contribution to relative return using eigenvalues.
Symmetrical arithmetic attribution model (SAA) 373
9.3.3. Removing interaction using symmetry considerations 376
9.3.4. Ideal attribution model 378
9.3.5. Introducing a principle of symmetry 380
9.3.6. Introducing a symmetrical data model. 382
Numerical example 382
9.3.7. Summary of SAA’s attribution formulas 385
9.4. Examining BrinsonFachler’s attribution model 390
9.4.1. BrinsonFachler’s model and noise factors 390
9.4.2. Highlights of symmetrical attribution models 393
9.4.3. Testing the symmetry of the BrinsonFachler’s model 394
9.4.4. Introducing symmetry for a referential 395
arithmetic attribution model (RAA) 395
9.4. Global attribution. Application of the SAA model 401
9.4.1. Global attribution. Definitions 401
9.4.2. Global attribution basics 404
9.4.3. Choosing the base attribution model 406
9.4.4. Introducing a new global attribution model 406
9.4.5. Currency hedging and multiple currency transactions 411
9.5. Geometric attribution 416
9.5.1. Comparison principles 416
9.5.2. The present geometric attribution model 419
9.5.2.1. Summary of the geometric attribution method’s features 423
9.5.3. A new symmetrical geometric attribution model. 424
The Discriminant Theorem 424
9.5.4. Comparing the new symmetrical geometric attribution model with
the present geometric attribution method 433
9.5.5. A ratio validation criterion for developing 436
attribution models 436
9.5.6. Researching the present geometric attribution model from the
perspective of ideal attribution model 437
9.5.7. Reverse transformations 442
9.5.8. Conclusion and general notes about SGA method 442
9.6. Linking attribution parameters for sequential periods 445
9.6.1. Business need to link attribution parameters 445
9.6.2. Input data for linking attribution parameters 446
9.6.3. Linking of geometric attribution parameters 447
9.6.4. Linking of arithmetic attribution parameters 449
9.6.5. Limitations with regard to attribution linking methods. Influence
of methods for calculating rate of return 452
9.6.5.1. Restrictions imposed by geometric linking 452
of rates of return 452
9.6.5.2. The influence of methods for calculating rate of return to the
objectivity of attribution analysis 453
9.6.6. Linking attribution parameters on the basis 454
of rates of return 454
9.7. Concluding the study of attribution models and methods 457
Chapter 10
Measuring Risk 459
10.1. Types of risks and available risk measures 462
10.1.1. Interest rate risk 462
10.1.2. Market risk 463
10.1.2.1. Specific risk 464
10.1.3. Measuring market risk 465
10.1.3.1. Value at risk as a probability based measure 465
10.1.3.2. Expected shortfall 468
10.1.3.3. Volatility 469
10.1.4. Credit risk 470
10.1.5. Operational risk 471
10.2. Estimation of Value at Risk (VaR) 474
10.2.1. Time period 474
10.2.2. Confidence level 477
10.2.3. Computing VaR 480
10.2.3.1. Variance  covariance method (VCV) 481
10.2.3.2. Historical simulation 483
10.2.3.3. Monte  Carlo simulation 483
10.2.3.4. General notes 484
10.3. Introducing volatility 486
10.3.1. Defining volatility via a standard deviation. 487
Discrete variables 487
10.3.2. Standard deviation for continuous random variables 489
10.3.2.1. Specifics of investment data 491
10.3.2.2. An assumption about data independency 492
10.3.3. Defining volatility via the exponential 493
moving average 493
10.4. Sharpe ratio. The role of benchmarks in the risk measurement 496
10.4.1. Importance of Sharpe ratio 499
10.5. Introducing a notion of downside risk 500
10.5.1. The origin and definition of the downside risk 500
10.5.2. Example of downside risk valuation 501
10.6. Implicit embedding of a time period into the risk measures based
on volatility 506
10.7. Other volatility based risk measures 508
10.7.1. Sortino ratio 508
10.7.2. Treynor ratio 509
10.7.2.1. Beta coefficient 509
10.7.2.2. Using beta in Treynor ratio 511
10.7.3. Alpha coefficient 512
10.7.4. Tracking error 513
10.7.5. Information ratio 514
10.7.6. Msquared (ModiglianiModigliani) 515
10.8. Completing the "risky" excursion 519
Chapter 11
Introducing a conceptual framework. 521
Exploring the reality 521
11.1. Let us hit the Road. Numbers, formulas and assumptions
525
11.2. Feeling the numbers 528
11.3. Discussing applicability domain 530
11.4. Consequences of separating methods from their applicability
domains 533
11.5. Mathematical piety and contempt of mathematics as two extreme
approaches 537
11.6. Introducing solutions’ quality 539
11.7. Terms hypothesis and theory 544
11.8. Solutions’ quality in a threedimensional space 545
11.9. Improving the adequacy of models to the reality 548
11.10. Specifics of working environments and their influence to the
development processes 553
11.11. Openness of business and academic environments as a condition of
their optimal development 560
11.11.1. Unbiased acquisition of information as the most important
feature of open systems 560
11.11.2. Amateurs and professionals 563
11.11.3. Some specific features of financial industry 565
11.12. Progress as a story about improving adequacy 568
References 573
Index 577
