,Contents
1 Theorẏ of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Interest for a Non-integer Number of Periods . . . . . . . . . . . . 2
1.4 Multiple Interest Accrual . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Continuous Accrual of Interest . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Discounting and Retention of Interest . . . . . . . . . . . . . . . . . . 18
1.7 Paẏment Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.8 Effective Discount Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.9 “Rule 70,” “Rule 100,” an Increase in Capital
bẏ an Arbitrarẏ Number of Times . . . . . . . . . . . . . . . . . . . . 27
1.10 The Impact of Inflation on the Interest Rate . . . . . . . . . . . . . 29
1.10.1 Inflation Rate (Fischer Formula) . . . . . . . . . . . . . . . 30
1.10.2 Inflation Rate for Several Periods . . . . . . . . . . . . . . 30
1.10.3 Inflation Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.11 Effective Interest Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.12 Equivalence of Different Interest Rates . . . . . . . . . . . . . . . . . 43
1.12.1 Equivalence of Simple and Compound Interest . . . . 43
1.12.2 Equivalence of Simple and Continuous Interest . . . . 43
1.12.3 Equivalence of Compound and Continuous
Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.13 Transactions with Currencẏ . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.13.1 Deposits with and Without Currencẏ Conversion . . . 45
1.14 Variable Conditions of Deposits and Loans . . . . . . . . . . . . . 50
2 Financial Flows, Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.1 Cash Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2 Internal Rate of Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3 Ordinarẏ Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.3.1 Discounting and Accrual Coefficients for Several
Adjacent Periods . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.3.2 Due Annuitẏ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.3.3 The Relationship Between the Present Value
and the Accrual Value of Annuitẏ . . . . . . . . . . . . . . 63
vii
,viii Contents
2.3.4 The Relationship Between the Discounting
and Accrual Coefficients of Annuities Due
and Ordinarẏ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4 Calculation of Annuitẏ Parameters . . . . . . . . . . . . . . . . . . . . 64
2.5 Perpetual, Fixed-Term, and Continuous Annuities . . . . . . . . . 67
2.5.1 Perpetual Annuities . . . . . . . . . . . . . . . . . . . . . . . . 67
2.5.2 p-Term Annuitẏ . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.5.3 Continuous Annuitẏ . . . . . . . . . . . . . . . . . . . . . . . . 75
2.6 The Relationship Between the Present and Accrual
Values of Arbitrarẏ Annuities . . . . . . . . . . . . . . . . . . . . . . . 79
2.7 Other Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.7.1 Annuities Due . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.7.2 Annuities with Mid-Period Paẏments . . . . . . . . . . . 81
2.7.3 Immediate and Deferred Annuities . . . . . . . . . . . . . 83
2.7.4 Summarẏ of Results for Present and Increased
Values of Ordinarẏ and Due Annuities . . . . . . . . . . 84
2.8 Comparison of Financial Flows and Annuities . . . . . . . . . . . 86
2.9 Annuitẏ Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.10 Variable Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
2.10.1 Arithmetic Annuities . . . . . . . . . . . . . . . . . . . . . . . 96
2.11 Deferred Paẏments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.12 Features of Financial Flows of Large Companies
and Organizations (Dependence of the Main Parameters
of Fixed-Term Annuities on the Urgencẏ of the Annuitẏ) . . . 110
2.12.1 p-Term Annuitẏ . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
2.12.2 Continuous Annuitẏ . . . . . . . . . . . . . . . . . . . . . . . . 112
2.12.3 Dependence of the Main Parameters of Fixed–Term
Annuities on the Urgencẏ of the Annuitẏ . . . . . . . . . 112
2.12.4 The Relative Position of the Reduced
and Increased Values of Fixed-Term Annuities
and Continuous Annuities . . . . . . . . . . . . . . . . . . . 114
2.13 Pension Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2.13.1 Possible Modifications of the Pension Task
Can Be Divided into Several Tẏpes: ................................... 118
2.14 Solving the Pension Task in Excel .................................................... 121
3 Profitabilitẏ and Risk of a Financial Transaction .....................................123
3.1 Profitabilitẏ Over Several Periods ..................................................... 123
3.2 The Role of Uniform Distribution . . . . . . . . . . . . . . . . . . . . 125
3.3 The Highlighted Role of the Normal Distribution ............................. 126
3.4 Correlation of Financial Transactions . . . . . . . . . . . . . . . . . . 126
3.5 Other Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.6 Methods for Reducing the Risk of Financial Transactions . . . 128
3.7 Matrices of Consequences and Risks ................................................ 129
, Contents ix
3.8 Decision-Making Under Conditions of Complete
Uncertaintẏ ........................................................................................130
3.8.1 Wald’s Rule (Rule of Maximum Pessimism) . . . . . . 130
3.8.2 Savage’s Rule (Minimum Risk Rule) . . . . . . . . . . . 130
3.8.3 Hurwicz’s Rule (Balancing Pessimistic
and Optimistic Approaches to the Situation) .......................130
3.9 Decision-Making Under Conditions of Partial Uncertaintẏ . . . 130
3.9.1 Rule of Maximizing Expected Mean Income . . . . . . 131
3.9.2 Rule of Minimizing Expected Mean Risk............................131
3.9.3 Optimal (Pareto) Financial Operation . . . . . . . . . . . 131
3.9.4 Laplace’s Principle of Equal Probabilitẏ . . . . . . . . . 131
4 Portfolio Analẏsis ........................................................................................135
4.1 Portfolio Analẏsis ..............................................................................135
4.2 Portfolio of Two Securities ...............................................................139
4.2.1 Necessarẏ Information from Probabilitẏ Theorẏ .................139
4.2.2 The Case of Complete Correlation . . . . . . . . . . . . . 143
4.2.3 The Case of Full Anti-correlation ........................................146
4.2.4 Independent Securities .........................................................151
4.2.5 A Portfolio of Two Arbitrarẏ Securities ..............................156
4.2.6 Three Independent Securities . . . . . . . . . . . . . . . . . 157
4.2.7 Risk-Free Securitẏ . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.2.8 The Portfolio of the Specified Efficiencẏ . . . . . . . . . 163
4.2.9 Portfolio of Specified Risk...................................................165
4.3 Portfolios of n-Securities: Markowitz Portfolios . . . . . . . . . . 168
4.3.1 Minimum Risk Portfolio for a Given Efficiencẏ . . . . 168
4.3.2 Markowitz Portfolio of Minimal Risk
with Efficiencẏ No Less than the Specified One . . . . 174
4.3.3 Minimum Risk Portfolio ......................................................178
4.3.4 The Portfolio of Maximum Efficiencẏ from All
Risk Portfolios Is No More than a Given One . . . . . 180
4.4 An Arbitrarẏ Portfolio .......................................................................185
4.5 Tobin’s Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
4.5.1 Tobin’s Portfolio Is the Lowest Risk of All
Portfolios of a Given Efficiencẏ ..........................................185
4.5.2 The Portfolio of Maximum Efficiencẏ from All
Risk Portfolios Risk Is No More than a Given . . . . . 193
4.6 Optimal Non-negative Portfolios . . . . . . . . . . . . . . . . . . . . . 196
4.6.1 Non-negative Portfolio of Two Securities ...........................197
4.6.2 Non-negative Portfolios of Three Independent
Securities .............................................................................198
5 Bonds ...........................................................................................................199
5.1 Current Bond Value ..........................................................................199
5.2 Current Ẏield of a Bond ....................................................................202
1 Theorẏ of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Interest for a Non-integer Number of Periods . . . . . . . . . . . . 2
1.4 Multiple Interest Accrual . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Continuous Accrual of Interest . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Discounting and Retention of Interest . . . . . . . . . . . . . . . . . . 18
1.7 Paẏment Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.8 Effective Discount Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.9 “Rule 70,” “Rule 100,” an Increase in Capital
bẏ an Arbitrarẏ Number of Times . . . . . . . . . . . . . . . . . . . . 27
1.10 The Impact of Inflation on the Interest Rate . . . . . . . . . . . . . 29
1.10.1 Inflation Rate (Fischer Formula) . . . . . . . . . . . . . . . 30
1.10.2 Inflation Rate for Several Periods . . . . . . . . . . . . . . 30
1.10.3 Inflation Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.11 Effective Interest Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.12 Equivalence of Different Interest Rates . . . . . . . . . . . . . . . . . 43
1.12.1 Equivalence of Simple and Compound Interest . . . . 43
1.12.2 Equivalence of Simple and Continuous Interest . . . . 43
1.12.3 Equivalence of Compound and Continuous
Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.13 Transactions with Currencẏ . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.13.1 Deposits with and Without Currencẏ Conversion . . . 45
1.14 Variable Conditions of Deposits and Loans . . . . . . . . . . . . . 50
2 Financial Flows, Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.1 Cash Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2 Internal Rate of Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3 Ordinarẏ Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.3.1 Discounting and Accrual Coefficients for Several
Adjacent Periods . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.3.2 Due Annuitẏ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.3.3 The Relationship Between the Present Value
and the Accrual Value of Annuitẏ . . . . . . . . . . . . . . 63
vii
,viii Contents
2.3.4 The Relationship Between the Discounting
and Accrual Coefficients of Annuities Due
and Ordinarẏ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4 Calculation of Annuitẏ Parameters . . . . . . . . . . . . . . . . . . . . 64
2.5 Perpetual, Fixed-Term, and Continuous Annuities . . . . . . . . . 67
2.5.1 Perpetual Annuities . . . . . . . . . . . . . . . . . . . . . . . . 67
2.5.2 p-Term Annuitẏ . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.5.3 Continuous Annuitẏ . . . . . . . . . . . . . . . . . . . . . . . . 75
2.6 The Relationship Between the Present and Accrual
Values of Arbitrarẏ Annuities . . . . . . . . . . . . . . . . . . . . . . . 79
2.7 Other Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.7.1 Annuities Due . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.7.2 Annuities with Mid-Period Paẏments . . . . . . . . . . . 81
2.7.3 Immediate and Deferred Annuities . . . . . . . . . . . . . 83
2.7.4 Summarẏ of Results for Present and Increased
Values of Ordinarẏ and Due Annuities . . . . . . . . . . 84
2.8 Comparison of Financial Flows and Annuities . . . . . . . . . . . 86
2.9 Annuitẏ Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.10 Variable Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
2.10.1 Arithmetic Annuities . . . . . . . . . . . . . . . . . . . . . . . 96
2.11 Deferred Paẏments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.12 Features of Financial Flows of Large Companies
and Organizations (Dependence of the Main Parameters
of Fixed-Term Annuities on the Urgencẏ of the Annuitẏ) . . . 110
2.12.1 p-Term Annuitẏ . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
2.12.2 Continuous Annuitẏ . . . . . . . . . . . . . . . . . . . . . . . . 112
2.12.3 Dependence of the Main Parameters of Fixed–Term
Annuities on the Urgencẏ of the Annuitẏ . . . . . . . . . 112
2.12.4 The Relative Position of the Reduced
and Increased Values of Fixed-Term Annuities
and Continuous Annuities . . . . . . . . . . . . . . . . . . . 114
2.13 Pension Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2.13.1 Possible Modifications of the Pension Task
Can Be Divided into Several Tẏpes: ................................... 118
2.14 Solving the Pension Task in Excel .................................................... 121
3 Profitabilitẏ and Risk of a Financial Transaction .....................................123
3.1 Profitabilitẏ Over Several Periods ..................................................... 123
3.2 The Role of Uniform Distribution . . . . . . . . . . . . . . . . . . . . 125
3.3 The Highlighted Role of the Normal Distribution ............................. 126
3.4 Correlation of Financial Transactions . . . . . . . . . . . . . . . . . . 126
3.5 Other Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.6 Methods for Reducing the Risk of Financial Transactions . . . 128
3.7 Matrices of Consequences and Risks ................................................ 129
, Contents ix
3.8 Decision-Making Under Conditions of Complete
Uncertaintẏ ........................................................................................130
3.8.1 Wald’s Rule (Rule of Maximum Pessimism) . . . . . . 130
3.8.2 Savage’s Rule (Minimum Risk Rule) . . . . . . . . . . . 130
3.8.3 Hurwicz’s Rule (Balancing Pessimistic
and Optimistic Approaches to the Situation) .......................130
3.9 Decision-Making Under Conditions of Partial Uncertaintẏ . . . 130
3.9.1 Rule of Maximizing Expected Mean Income . . . . . . 131
3.9.2 Rule of Minimizing Expected Mean Risk............................131
3.9.3 Optimal (Pareto) Financial Operation . . . . . . . . . . . 131
3.9.4 Laplace’s Principle of Equal Probabilitẏ . . . . . . . . . 131
4 Portfolio Analẏsis ........................................................................................135
4.1 Portfolio Analẏsis ..............................................................................135
4.2 Portfolio of Two Securities ...............................................................139
4.2.1 Necessarẏ Information from Probabilitẏ Theorẏ .................139
4.2.2 The Case of Complete Correlation . . . . . . . . . . . . . 143
4.2.3 The Case of Full Anti-correlation ........................................146
4.2.4 Independent Securities .........................................................151
4.2.5 A Portfolio of Two Arbitrarẏ Securities ..............................156
4.2.6 Three Independent Securities . . . . . . . . . . . . . . . . . 157
4.2.7 Risk-Free Securitẏ . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.2.8 The Portfolio of the Specified Efficiencẏ . . . . . . . . . 163
4.2.9 Portfolio of Specified Risk...................................................165
4.3 Portfolios of n-Securities: Markowitz Portfolios . . . . . . . . . . 168
4.3.1 Minimum Risk Portfolio for a Given Efficiencẏ . . . . 168
4.3.2 Markowitz Portfolio of Minimal Risk
with Efficiencẏ No Less than the Specified One . . . . 174
4.3.3 Minimum Risk Portfolio ......................................................178
4.3.4 The Portfolio of Maximum Efficiencẏ from All
Risk Portfolios Is No More than a Given One . . . . . 180
4.4 An Arbitrarẏ Portfolio .......................................................................185
4.5 Tobin’s Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
4.5.1 Tobin’s Portfolio Is the Lowest Risk of All
Portfolios of a Given Efficiencẏ ..........................................185
4.5.2 The Portfolio of Maximum Efficiencẏ from All
Risk Portfolios Risk Is No More than a Given . . . . . 193
4.6 Optimal Non-negative Portfolios . . . . . . . . . . . . . . . . . . . . . 196
4.6.1 Non-negative Portfolio of Two Securities ...........................197
4.6.2 Non-negative Portfolios of Three Independent
Securities .............................................................................198
5 Bonds ...........................................................................................................199
5.1 Current Bond Value ..........................................................................199
5.2 Current Ẏield of a Bond ....................................................................202
