Financial Mathematics CT1

01 Cashflow models......Page 36
0 Introduction......Page 37
1 Cashflow process......Page 38
2.1 A zero-coupon bond......Page 39
2.2 A fixed interest security......Page 41
2.3 An index-linked security......Page 42
2.5 An equity......Page 44
2.6 An annuity certain......Page 46
2.8 A repayment loan (or mortgage)......Page 47
3 Specimen exam questions......Page 49
Chapter 1 Summary......Page 51
02 The time value of money......Page 56
0 Introduction......Page 57
1.1 Simple interest......Page 59
1.2 Compound (effective) interest......Page 61
1.3 Accumulation factors......Page 63
1.4 The principle of consistency......Page 64
2 Present values......Page 66
3 Discount rates......Page 67
3.1 Simple discount......Page 68
3.2 Compound (effective) discount......Page 70
3.3 Discount factors......Page 71
4.1 Effective rate of interest......Page 73
4.2 Effective rate of discount......Page 74
5 Equivalent rates......Page 75
Chapter 2 Summary......Page 79
Chapter 2 Solutions......Page 81
03 Interest rates......Page 86
1 Nominal rates of interest and discount......Page 87
1.1 Nominal rates of interest......Page 88
1.2 Accumulating and discounting using nominal interest rates......Page 91
1.3 Nominal rates of discount......Page 92
1.4 Accumulating and discounting using nominal discount rates......Page 96
2.1 Derivation from nominal interest convertible pthly......Page 98
2.2 Accumulating and discounting using the force of interest......Page 101
2.3 Derivation of δ from nominal discount convertible pthly......Page 103
3.1 An alternative way of considering nominal interest convertiblepthly......Page 106
3.2 An alternative way of considering nominal discount convertiblepthly......Page 107
3.3 An alternative way of considering force of interest......Page 108
4.1 Formal definition......Page 110
4.2 Relationship to constant force of interest......Page 114
4.3 Present values......Page 115
Chapter 3 Summary......Page 118
04 Real and money interest rates......Page 128
0 Introduction......Page 129
1 Definition of real and money interest rates......Page 130
3 Usefulness of real and money interest rates......Page 133
Chapter 4 Summary......Page 136
05 Discounting and accumulating......Page 140
0 Introduction......Page 141
1.1 Discrete cashflows......Page 142
1.2 Continuously payable cashflows (payment streams)......Page 144
2 Valuing cashflows......Page 149
2.1 Constant interest rate......Page 152
2.2 Payment st......Page 154
2.3 Sudden changes in interest rates......Page 157
3 Interest income......Page 159
4 Exam-style question......Page 162
Chapter 5 Summary......Page 165
06 Level annuities......Page 172
0 Introduction......Page 173
1.1 Payments made in arrear......Page 175
1.2 Payments made in advance......Page 177
2 Accumulations......Page 180
3 Continuously payable annuities......Page 181
4.1 Present Values......Page 183
4.2 Accumulations......Page 188
4.3 Annuities payable pthly where p < 1......Page 189
5 Non-integer values of n......Page 190
6 Perpetuities......Page 194
Chapter 6 Summary......Page 197
07 Deferred and increasing annuities......Page 204
0 Introduction......Page 205
1.1 Annual payments......Page 206
1.2 Continuously payable annuities......Page 207
1.4 Non-integer values of n......Page 208
2.1 Annual payments......Page 209
2.2 Continuously payable annuities......Page 214
2.3 Decreasing payments......Page 217
3.1 Irregular payments......Page 218
3.3 Compound increasing annuities......Page 220
4 Exam-style question......Page 225
Chapter 7 Summary......Page 229
08 Equations of value......Page 244
0 Introduction......Page 245
1.1 The theory......Page 246
1.2 Solving for an unknown quantity......Page 248
1.3 Example applications......Page 256
2.1 Probability of cashflow......Page 258
2.2 Higher discount rate......Page 259
Chapter 8 Summary......Page 261
09 Loan schedules......Page 269
An example......Page 270
Introduction......Page 272
The theory......Page 273
Retrospective loan calculation......Page 275
Calculating the interest and capital elements......Page 278
The loan schedule......Page 280
Instalments payable more frequently than annually......Page 281
Introduction......Page 299
Estimating cashflows......Page 300
Accumulated value......Page 303
Net present values......Page 305
Internal rate of return......Page 307
The comparison of two investment project......Page 310
Different interest rates for lending and borrowing......Page 313
Payback periods......Page 315
Other considerations......Page 319
Measurement of investment performance......Page 324
Money-weighted rate of return......Page 325
Time-weighted rate of return......Page 327
Linked internal rate of return......Page 330
Chapter 10 Summary......Page 339
Fixed interest government borrowings......Page 351
Cashflows......Page 352
Tax......Page 353
Security, marketability and return......Page 354
Government bills......Page 357
Characteristics of corporate debt......Page 358
Debentures......Page 360
Eurobonds......Page 361
Certificates of deposit......Page 362
Ordinary shares......Page 364
Preference shares......Page 366
Convertibles......Page 367
Property......Page 369
Derivatives......Page 371
Futures......Page 372
Range of futures......Page 373
Margin......Page 374
Bond futures......Page 375
Short interest rate futures......Page 376
Options......Page 377
Timing......Page 378
Interest rate swaps......Page 379
Currency swaps......Page 380
Characteristics......Page 381
Chapter 11 Summary......Page 382
12 Elementary compound interest problems......Page 391
Calculating the price......Page 393
No tax......Page 395
Income tax......Page 396
Capital gains tax......Page 398
Offsetting capital losses against capital gains......Page 403
The indexation of capital gains......Page 404
Optional redemption dates......Page 405
Uncertain income securities......Page 409
Equities......Page 410
Property......Page 412
Real rates of interest......Page 415
Calculating real yields using an inflation index......Page 416
Calculating real yields given constant inflation assumptions......Page 418
Payments related tothe rate of inflation......Page 421
The effects of inflation......Page 422
Index-linked bonds......Page 426
Chapter 12 Summary......Page 433
13 Arbitrage and forward contracts......Page 443
Introduction......Page 445
Why do we assume “No Arbitrage”?......Page 448
Introduction......Page 449
Calculating the forward price for a security with no income......Page 450
Calculating the forward price for a security with fixed cash
income......Page 451
Calculating the forward price for a security with known
dividend yield......Page 453
Hedging......Page 456
No interest or dividend income......Page 458
Special case – no income......Page 459
Note......Page 462
Chapter 13 Summary......Page 467
14 Term structure of interest rates......Page 474
Discrete time spot rates......Page 476
Discrete time forward rates......Page 479
Continuous time forward rates......Page 482
Instantaneous forward rates......Page 484
Why interest rates vary over time......Page 486
The theories......Page 487
Liquidity Preference......Page 489
Market Segmentation......Page 490
Par yields......Page 491
Interest rate risk......Page 493
Effective duration......Page 494
Duration......Page 495
Convexity......Page 498
Redington’s conditions......Page 501
Practice questions......Page 507
15 Stochastic interest rate models......Page 525
Preliminary remarks......Page 527
Fixed interest rate model......Page 528
Varying interest rate model......Page 529
The log-normal distribution......Page 540
Appendix – basic statistical results......Page 547
Expectation......Page 548
Lognormal distribution......Page 549
Chapter 15 Summary......Page 553
Part 1 – Questions......Page 560
Development Questions......Page 562
Exam-style Questions......Page 570
Part 1 – Solutions......Page 575
1 Development Questions......Page 601
Part 2 – Solutions......Page 613
1 Development Question......Page 645
2 Exam-style Questions......Page 649
Part 3 – Solutions......Page 657
1 Development Questions......Page 683
2 Exam-style Questions......Page 687
Part 4 – Solutions......Page 697
Part 5 – Revision Questions......Page 731