Financial Mathematics for Actuaries (Updated Edition)

Contents About the Authors v 
Preface to the Updated Edition vii 
Preface to the 1st Edition ix 
List of Mathematical Symbols xi 


Chapter 1 Interest Accumulation and Time Value of Money 1 1.1 Accumulation Function and Amount Function 2 
1.2 Simple and Compound Interest 2 
1.3 Frequency of Compounding 5 
1.4 Effective Rate of Interest 8 
1.5 Rates of Discount 12 
1.6 Force of Interest 17 
1.7 Present and Future Values 20 
1.8 Equation of Value 26 
1.9 Summary 29 
Exercises 29 


Chapter 2 Annuities 39 2.1 Annuity-immediate 40 
2.2 Annuity-due 43 
2.3 Perpetuity, Deferred Annuity and Annuity Values at Other Times 45 
2.4 Annuities under Other Accumulation Methods 49 
2.5 Payment Periods, Compounding Periods and Continuous Annuities 51 
2.6 Varying Annuities 57 
2.7 Term of Annuity 62 
2.8 Summary 66 
Exercises 66 


Chapter 3 Spot Rates, Forward Rates and the Term Structure 73 3.1 Spot and Forward Rates of Interest 74 
3.2 The Term Structure of Interest Rates 79 
3.3 Present and Future Values Given the Term Structure 80 
3.4 Accumulation Function and the Term Structure 86 
3.5 Summary 92 
Exercises 92 


Chapter 4 Rates of Return 99 4.1 Internal Rate of Return 100 
4.2 One-period Rate of Return 105 
4.3 Rate of Return over Multiple Periods 110 
4.4 Portfolio Return 115 
4.5 Short Sales 118 
4.6 Crediting Interest: Investment-year Method and Portfolio Method 119 
4.7 Inflation and Real Rate of Interest 121 
4.8 Capital Budgeting and Project Appraisal 123 
4.9 Summary 129 
Exercises 130 


Chapter 5 Loans and Costs of Borrowing 141 5.1 Loan Balance: Prospective and Retrospective Methods 142 
5.2 Amortization 146 
5.3 Sinking Fund 148 
5.4 Varying Installments and Varying Interest Rates 154 
5.5 Comparison of Borrowing Costs 158 
5.6 Flat Rate Loan and Flat Rate Discount Loan 161 
5.7 Borrowing Costs and Reference Rates 164 
5.8 Summary 167 
Exercises 168 


Chapter 6 Bonds and Bond Pricing 181 6.1 Basic Concepts 182 
6.2 Bond Evaluation 183 
6.3 Bond Amortization Schedule 187 
6.4 Valuation between Coupon-payment Dates 192 
6.5 Callable Bonds 195 
6.6 Bond Pricing under a General Term Structure 198 
6.7 Summary 200 
Exercises 200 


Chapter 7 Bond Yields and the Term Structure 207 7.1 Some Simple Measures of Bond Yield 208 
7.2 Yield to Maturity 208 
7.3 Par Yield 213 
7.4 Holding-period Yield 215 
7.5 Discretely Compounded Yield Curve 218 
7.6 Continuously Compounded Yield Curve 222 
7.7 Term Structure Models 226 
7.8 Summary 231 
Exercises 232 


Chapter 8 Bond Management 239 8.1 Macaulay Duration and Modified Duration 240 
8.2 Duration for Price Correction 246 
8.3 Convexity 249 
8.4 Some Rules for Duration 250 
8.5 Immunization Strategies 254 
8.6 Some Shortcomings of Duration Matching 268 
8.7 Duration under a Nonflat Term Structure 269 
8.8 Passive versus Active Bond Management 273 
8.9 Summary 273 
Exercises 274 


Chapter 9 Stochastic Interest Rates 285 9.1 Deterministic Scenarios of Interest Rates 286 
9.2 Random-scenario Model 287 
9.3 Independent Lognormal Model 289 
9.4 Autoregressive Model 294 
9.5 Dynamic Term Structure Model 296 
9.6 An Application: Guaranteed Investment Income 297 
9.7 Summary 300 
Exercises 301 


Appendix A Review of Mathematics and Statistics 307 A.1 Exponential Function 307 
A.2 Logarithmic Function 307 
A.3 Roots of a Quadratic Equation 308 
A.4 Arithmetic Progression 308 
A.5 Geometric Progression 308 
A.6 Some Derivatives 309 
A.7 Integration by Part 309 
A.8 Taylor Series Expansion 309 
A.9 Binomial Expansion 310 
A.10 Expected Value and Variance of a Random Variable 310 
A.11 Mean and Variance of Sum of Random Variables 310 
A.12 Uniform Distribution 311 
A.13 Normal Distribution 311 
A.14 Lognormal Distribution 312 


Appendix B Answers to Selected Exercises 313 Index 325