An Introduction to the Mathematics of Finance

1. Introduction1.1 The Concept of Interest1.2 Simple Interest1.3 Compound Interest1.4 Some Practical IllustrationsSummary2. Theory of Interest Rates2.1 The Rate of Interest2.2 Nominal Rates of Interest2.3 Accumulation Factors2.4 The Force of Interest2.5 Present Values2.6 Present Values of Cash Flows2.7 Valuing Cash Flows2.8 Interest Income2.9 Capital Gains and Losses, and TaxationSummaryExercises3. The Basic Compound Interest Functions3.1 Interest Rate Quantities3.2 The Equation of Value3.3 Annuities-certain: Present Values and Accumulations3.4 Deferred Annuities3.5 Continuously Payable Annuities3.6 Varying Annuities3.7 Uncertain PaymentsSummaryExercises4. Further Compound Interest Functions4.1 Interest Payable pthly4.2 Annuities Payable pthly: Present Values and Accumulations4.3 Annuities Payable at Intervals of Time r, Where r > 14.4 Definition of for Non-integer Values of nSummaryExercises5. Loan Repayment Schedules5.1 The General Loan Schedule5.2 The Loan Schedule for a Level Annuity5.3 The Loan Schedule for a pthly Annuity5.4 Consumer Credit LegislationSummaryExercises6. Project Appraisal and Investment Performance6.1 Net Cash Flows6.2 Net Present Values and Yields6.3 The Comparison of Two Investment Projects6.4 Different Interest Rates for Lending and Borrowing6.5 Payback Periods6.6 The Effects of Inflation6.7 Measurement of Investment Fund PerformanceSummaryExercises7. The Valuation of Securities7.1 Fixed-Interest Securities7.2 Related Assets7.3 Prices and Yields7.4 Perpetuities7.5 Makeham’s Formula7.6 The Effect of the Term to Redemption on the Yield7.7 Optional Redemption Dates7.8 Valuation between Two Interest Dates: More Complicated Examples7.9 Real Returns and Index-linked StocksSummaryExercises8. Capital Gains Tax8.1 Valuing a Loan with Allowance for Capital Gains Tax8.2 Capital Gains Tax When the Redemption Price or the Rate of Tax Is Not Constant8.3 Finding the Yield When There Is Capital Gains Tax8.4 Optional Redemption Dates8.5 Offsetting Capital Losses Against Capital GainsSummaryExercises9. Term Structures and Immunization9.1 Spot and Forward Rates9.2 Theories of the Term Structure of Interest Rates9.3 The Discounted Mean Term of a Project9.4 Volatility9.5 The Volatility of Particular Fixed-interest Securities9.6 The Matching of Assets and Liabilities9.7 Redington’s Theory of Immunization9.8 Full ImmunizationSummaryExercises10. An Introduction to Derivative Pricing: Forwards and Futures10.1 Futures Contracts10.2 Margins and Clearinghouses10.3 Uses of Futures10.4 Forwards10.5 Arbitrage10.6 Calculating the Forward Price10.7 Calculating the Value of a Forward Contract Prior to Maturity10.8 Eliminating the Risk to the Short PositionSummaryExercises11. Further Derivatives: Swaps and Options11.1 Swaps11.2 Options11.3 Option Payoff and Profit11.4 An Introduction to European Option Pricing11.5 The Black–Scholes Model11.6 Trading Strategies Involving European OptionsSummaryExercises12. An Introduction to Stochastic Interest Rate Models12.1 Introductory Examples12.2 Independent Annual Rates of Return12.3 The Log-Normal Distribution12.4 Simulation Techniques12.5 Random Number Generation12.6 Dependent Annual Rates of Return12.7 An Introduction to the Application of Brownian MotionSummaryExercises