A Basic Course in the Theory of Interest and Derivatives ...

[Pages:745]A Basic Course in the Theory of Interest and Derivatives Markets:

A Preparation for the Actuarial Exam FM/2

Marcel B. Finan Arkansas Tech University

c All Rights Reserved Preliminary Draft Last updated

March 24, 2017

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In memory of my parents August 1, 2008 January 7, 2009

Preface

This manuscript is designed for an introductory course in the theory of interest and annuity. This manuscript is suitablefor a junior level course in the mathematics of finance. A calculator, such as TI BA II Plus, either the solar or battery version, will be useful in solving many of the problems in this book. A recommended resource link for the use of this calculator can be found at

. The recommended approach for using this book is to read each section, work on the embedded examples, and then try the problems. Answer keys are provided so that you check your numerical answers against the correct ones. Problems taken from previous exams will be indicated by the symbol . This manuscript can be used for personal use or class use, but not for commercial purposes. If you find any errors, I would appreciate hearing from you: mfinan@atu.edu This project has been supported by a research grant from Arkansas Tech University.

Marcel B. Finan Russellville, Arkansas March 2009

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PREFACE

Contents

Preface

3

The Basics of Interest Theory

9

1 The Meaning of Interest . . . . . . . . . . . . . . . . . . . . . . . 10

2 Accumulation and Amount Functions . . . . . . . . . . . . . . . . 15

3 Effective Interest Rate (EIR) . . . . . . . . . . . . . . . . . . . . 25

4 Linear Accumulation Functions: Simple Interest . . . . . . . . . . 32

5 Date Conventions Under Simple Interest . . . . . . . . . . . . . . 40

6 Exponential Accumulation Functions: Compound Interest . . . . 46

7 Present Value and Discount Functions . . . . . . . . . . . . . . . 56

8 Interest in Advance: Effective Rate of Discount . . . . . . . . . . 63

9 Nominal Rates of Interest and Discount . . . . . . . . . . . . . . 75

10 Force of Interest: Continuous Compounding . . . . . . . . . . . 88

11 Time Varying Interest Rates . . . . . . . . . . . . . . . . . . . . 104

12 Equations of Value and Time Diagrams . . . . . . . . . . . . . . 111

13 Solving for the Unknown Interest Rate . . . . . . . . . . . . . . 118

14 Solving for Unknown Time . . . . . . . . . . . . . . . . . . . . . 127

The Basics of Annuity Theory

155

15 Present and Accumulated Values of an Annuity-Immediate . . . 156

16 Annuity in Advance: Annuity Due . . . . . . . . . . . . . . . . . 170

17 Annuity Values on Any Date: Deferred Annuity . . . . . . . . . 181

18 Annuities with Infinite Payments: Perpetuities . . . . . . . . . . 191

19 Solving for the Unknown Number of Payments of an Annuity . . 199

20 Solving for the Unknown Rate of Interest of an Annuity . . . . . 209

21 Varying Interest of an Annuity . . . . . . . . . . . . . . . . . . . 219

22 Annuities Payable at a Different Frequency than Interest is Con-

vertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

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CONTENTS

23 Analysis of Annuities Payable Less Frequently than Interest is Convertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

24 Analysis of Annuities Payable More Frequently than Interest is Convertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

25 Continuous Annuities . . . . . . . . . . . . . . . . . . . . . . . . 249 26 Varying Annuity-Immediate . . . . . . . . . . . . . . . . . . . . 255 27 Varying Annuity-Due . . . . . . . . . . . . . . . . . . . . . . . . 272 28 Varying Annuities with Payments at a Different Frequency than

Interest is Convertible . . . . . . . . . . . . . . . . . . . . . . 281 29 Continuous Varying Annuities . . . . . . . . . . . . . . . . . . . 294

Rate of Return of an Investment

301

30 Discounted Cash Flow Technique . . . . . . . . . . . . . . . . . 302

31 Uniqueness of IRR . . . . . . . . . . . . . . . . . . . . . . . . . 313

32 Interest Reinvested at a Different Rate . . . . . . . . . . . . . . 320

33 Interest Measurement of a Fund: Dollar-Weighted Interest Rate 331

34 Interest Measurement of a Fund: Time-Weighted Rate of Interest 341

35 Allocating Investment Income: Portfolio and Investment Year

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

36 Yield Rates in Capital Budgeting . . . . . . . . . . . . . . . . . 360

Loan Repayment Methods

365

37 Finding the Loan Balance Using Prospective and Retrospective

Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

38 Amortization Schedules . . . . . . . . . . . . . . . . . . . . . . . 374

39 Sinking Fund Method . . . . . . . . . . . . . . . . . . . . . . . . 387

40 Loans Payable at a Different Frequency than Interest is Convertible401

41 Amortization with Varying Series of Payments . . . . . . . . . . 407

Bonds and Related Topics

417

42 Types of Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

43 The Various Pricing Formulas of a Bond . . . . . . . . . . . . . 424

44 Amortization of Premium or Discount . . . . . . . . . . . . . . . 437

45 Valuation of Bonds Between Coupons Payment Dates . . . . . . 447

46 Approximation Methods of Bonds' Yield Rates . . . . . . . . . . 456

47 Callable Bonds and Serial Bonds . . . . . . . . . . . . . . . . . . 464

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Stocks and Money Market Instruments

473

48 Preferred and Common Stocks . . . . . . . . . . . . . . . . . . . 475

49 Buying Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

50 Short Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

51 Money Market Instruments . . . . . . . . . . . . . . . . . . . . . 493

Measures of Interest Rate Sensitivity

501

52 The Effect of Inflation on Interest Rates . . . . . . . . . . . . . . 502

53 The Term Structure of Interest Rates and Yield Curves . . . . . 507

54 Macaulay and Modified Durations . . . . . . . . . . . . . . . . . 517

55 Redington Immunization and Convexity . . . . . . . . . . . . . . 528

56 Full Immunization and Dedication . . . . . . . . . . . . . . . . . 536

An Introduction to the Mathematics of Financial Derivatives 545 57 Financial Derivatives and Related Issues . . . . . . . . . . . . . 546 58 Derivatives Markets and Risk Sharing . . . . . . . . . . . . . . . 552 59 Forward and Futures Contracts: Payoff and Profit Diagrams . . 556 60 Call Options: Payoff and Profit Diagrams . . . . . . . . . . . . . 568 61 Put Options: Payoff and Profit Diagrams . . . . . . . . . . . . . 578 62 Stock Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 63 Options Strategies: Floors and Caps . . . . . . . . . . . . . . . . 597 64 Covered Writings: Covered Calls and Covered Puts . . . . . . . 605 65 Synthetic Forward and Put-Call Parity . . . . . . . . . . . . . . 611 66 Spread Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . 618 67 Collars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 68 Volatility Speculation: Straddles, Strangles, and Butterfly Spreads634 69 Equity Linked CDs . . . . . . . . . . . . . . . . . . . . . . . . . 645 70 Prepaid Forward Contracts On Stock . . . . . . . . . . . . . . . 652 71 Forward Contracts on Stock . . . . . . . . . . . . . . . . . . . . 659 72 Futures Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . 673 73 Understanding the Economy of Swaps: A Simple Commodity Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 74 Interest Rate Swaps . . . . . . . . . . . . . . . . . . . . . . . . . 693 75 Risk Management . . . . . . . . . . . . . . . . . . . . . . . . . . 703

Answer Key

711

BIBLIOGRAPHY

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