Contents
Preface xi
0 An introduction to the Texas Instruments BA II Plus 1
0.1 Choosing a calculator
0.2 Font convention
0.3 BA II Plus
0.4 Problems, Chapter
1.11 A friendly competition (Constant force of interest)
1.12 Force of interest
1.13 Note for those who skipped Sections (1.11) and (1.12)
1.14 Quoted rates for Treasury bills
1.15 Inflation
1.16 Choice of quotation base for interest rates
1.17 Problems, Chapter 1
2 Equations of value and yield rates 83
2.1 Introduction
2.2 Equations of value for investments involving a single deposit made under compound
2.3 Equations of value for investments with multiple contributions
2.6 Dollar-weighted
2.7 Fund
2.8 Problems, Chapter 2
3 Annuities (annuities certain) 121
3.2 Annuities -
3.3 Annuities - due
3.4 Perpetuities . .
3.5 Deferred annuities and values on any date
3.6 Outstanding loan balances
3.7 Nonlevel annuities .
3.8 Annuities with payments in geometric progression
155
3.9 Annuities with payments in arithmetic progression
158
3.10 Yield rate examples involving annuities
3.11 Annuity symbols for nonintegral terms
3.12 Annuities governed by general accumulation functions
3.13 The investment year method
3.14 Problems, Chapter 3
4 Annuities with different payment and conversion periods 195
4.1 Introduction .
4.2 Level annuities with payments less frequent than each interest period .
4.3 Level annuities with payments more frequent than each interest period
4.4 Annuities with payments less frequent than each interest period
and payments in arithmetic progression
4.5 Annuities with payments more frequent than each interest period and payments in arithmetic progression .
4.6 Continuously paying annuities .
4.7 A yield rate example .
4.8 Problems, Chapter 4
5 Loan repayment 229
5.1 Introduction . .
5.2 Amortized loans and amortization schedules .
229
5.3 The Sinking Fund method .
5.4 Amortized loans with other repayment patterns
245
5.5 Yield rate examples and replacement of capital .
247
5.6 Problems, Chapter 5 . .
6 Bonds 265
6.1 Introduction .
6.2 Bond alphabet soup and the basic price formula
267
6.3 The premium-discount formula
6.4 Other pricing formulas for bonds
6.5 Bond amortization schedules
6.6 Valuing a bond after its date of issue
255
6.7
6.8
6.9 Callable
6.10 Floating-rate
6.11
6.12 Problems, Chapter
7 Stocks and financial markets
7.1 Common
7.2 Brokerage
7.3
7.4
8.9
8.10 Commodity
8.11
8.12
8.14
9
9.1
9.2
9.8
10.4 Inflation risk .
509
10.5 Banks and other financial intermediaries in the retail sector . . 516
10.6 Savings and lending interest rates in the retail sector . . . . . . 518
10.7 Bonds issued by governments and corporations . .
521
10.8 The role of central banks .
10.9 Problems, Chapter 10 .
APPENDICES
A Some useful formulas 533
B Answers to end of chapter problems 541
Bibliography 567
Index 569
526
529
About the Authors 580 An introduction to the Texas Instruments BA II Plus
0.1 CHOOSING A CALCULATOR
0.2 FONT CONVENTION
0.3 BA II PLUS BASICS
0.4 PROBLEMS, CHAPTER 0
0.1 CHOOSING A CALCULATOR
There are many modestly priced financial calculators on the market today. Among these is the Texas Instruments BA II Plus, and we have chosen to integrate instructions regarding its effective use for solving problems. Our choice of calculator for the illustrations is based on the fact that the BA II Plus calculator is very well suited for the tasks we pose, is among the calculators allowed for the SOA/CAS examinations including Financial Mathematics (SOA Exam FM, CAS Exam 2), and Texas Instruments representatives have informed us that they expect to continue selling the BA II Plus calculator for some time to come. You may also follow the instructions using the BA II Plus Professional calculator that was released in 2004. This is an enhanced version of the BA II Plus calculator, and has the same keys. Occasionally we will make notes specific to the BA II
Plus Professional. Although the less expensive Texas Instruments BA-35 calculator performs well for many of the problems addressed, there are some problems (notably those relating to uneven cashflows) where it is deficient. We caution you not to follow our calculator instructions with the BA-35 calculator, since the correct operation is not always what you might guess. For example, the sign with which certain payments need to be entered is not always the same as it would be for the BA II Plus. For those readers who choose to follow along using a calculator other than the BA II Plus calculator, for instance one of the Hewlett Packard financial calculators, perhaps our instructions will still be of some service.
0.2 FONT CONVENTION
Those parts of the text that are aimed at readers using the BA II Plus calculator will be given in sans serif (or boldface sans serif) type. The sans serif font is illustrated in Figure (0.1.1). Focus on the third row of buttons on the BA II Plus Calculator. From left to right, the symbols are N , I/Y , PV , PMT , and FV . FIGURE (0.1.1) Readers who wish to concentrate on the theoretical flow, and not worry
about how to perform calculations on the BA II Plus calculator, may ignore all discussion given in the sans serif font just displayed in Figure (0.1.1). 0.3 BA II PLUS BASICS We will introduce you to the basic features of the BA II Plus calculator only to the extent necessary in order that our directions are clear. For instance, we presume familiarity with standard arithmetic operations of the calculator and how to change a newly entered number on the display if you have entered it incorrectly. It is important that we explain to you the “calculation methods” available on the BA II Plus and BA II Plus Professional calculators; that is, how the calculator interprets a string of keystrokes. The default setting for the calculation method is the so-called chain method; arithmetic operations are done in the order you enter them, unless parentheses indicate instructions to the contrary. So, if you key 4 + 2 × 5 = the calculator will compute (4 + 2) × 5 = 30, and if you press 2 × 5 yx 3 = the calculator will compute (2 × 5)3 = 1,000. If instead, you need to compute 2×53 = 250, this may be accomplished by pressing the sequence of keys 2 × ( 5 yx 3 ) = or 5 yx 3 × 2 = . The
chain calculation method (Chn) is commonly used on financial calculators, and our calculator instructions assume that you have your calculator set for the chain calculation method. Remember, the chain calculation method is the default setting, and you must take action if you prefer your calculator to use the algebraic operating system (AOS) implemented by many scientific calculators. To change the calculation method, you press
2ND FORMAT ↑ 2ND SET 2ND
QUIT . Section 0.3 BA II Plus basics 3 If your calculator is set for AOS, keying 4 + 2 × 5 = results in the calculator computing 4+(2×5) = 14, while pushing the sequence 2 × 5 yx 3 = causes the calculator to find 2 × 53 = 250. Warning: If your calculator is set with the algebraic operating system, adjustments to our keyby-key instructions are needed in some cases. Even if you are accustomed to using a scientific calculator, we encourage you to have your calculator set for the chain calculation method and to use parentheses to achieve the order of operation you desire. There are many times when you need to raise a number to a rational power. For example, you might wish to
calculate (.96) 4 3 . This can be done efficiently by pushing the nine-key sequence • 9 6 yx 4 yx 3 1/x = , or by pressing the nine-key sequence • 9 6 yx ( 4 ÷ 3 ) = . By either method, you should obtain that (.96) 4 3 ≈ .947025437. The keys on the BA II Plus have their primary function printed on the key. Most of the keys also have a secondary function and it is printed above the key, in pale yellow on the BA II Plus calculator and in charcoal gray on the BA II Plus Professional calculator. To access a given secondary function, you first press the key 2ND and then the key whose secondary function you desire. For example, if you wish to reset your calculator to the factory default settings (as is required of candidates on the SOA/CAS exams), you will need to begin by pushing 2ND followed by the key +/− . Even though RESET is the second function on the key +/− , our directions to the reader will be to push 2ND RESET ENTER 2ND QUIT . Our intention is to provide instructions that focus on the function we desire to access rather than the function written directly on the key pushed. To reset your calculator, use 2ND RESET ENTER 2ND QUIT . If
you reset the BA II Plus calculator you clear all ten memories and all worksheet data. The default settings include using angles measured in degrees rather than radians as the arguments of trigonometric functions.1 More importantly, the default format setting is 2formatting. The BA II Plus calculator can 1Trigonometric functions and their inverses are rarely used in this subject, but see Problems (0.3.2) and (1.12.10). The inverse of a trigonometric function is called by keying INV prior to pressing the key sequence for the function; for example INV 2ND TAN accesses tan−1. display at most ten digits (although it stores thirteen-digit numbers), and for α in {0, 1, 2,..., 9}, we say that we have α-formatting if, while not exceeding the ten digit limit, the calculator attempts to display up to α digits to the right of the decimal point. (So with 4-formatting the ten digit number 12345.54321 is displayed as 12345.5432 if it is the result of a calculation, while the twelve-digit number 123456789.543 is displayed as 123456789.5.
Rounding is used, so with 4-formatting the ten digit number 12345.56786 is displayed as 12345.5679 if it
is the result of a calculation, as is 12345.56785; but a resulting 12345.5678499 will be displayed as 12345.5678.) The authors most often prefer to keep their calculators with 9-formatting so that an answer is displayed with maximal possible accuracy. To obtain α-formatting, use 2ND FORMAT α
ENTER
2ND QUIT . Note that you enter α when the calculator screen reads “DEC = ” as it awaits your numerical input.Allowable inputs are in the interval (-.5,9.5), but you might as well enter an integer from {0, 1, 2,..., 9} as the calculator rounds the entered number to an integer. One situation in which you may not want to have 9-formatting is if you are calculating a number of dollars to change hands, particularly if you then use that dollar amount for further computation. To find the correct number of cents and have it stored in the calculator for further calculation, you may use both the FORMAT key and the ROUND key. This technique is demonstrated in the solution to our first (and very elementary) example. EXAMPLE 0.3.1 Problem: Mrs. Juanitez is a savvy shopper. At a drugstore, she finds a binder on sale for $14.97 plus 8.25% sales tax. She
has a mail-in coupon for a $7.00 rebate on the binder, and she charges it on her credit card that refunds 5% of the charged amount to her. The store advertises that the binder is “$7.97 after mail-in rebate.”
Calculate how much the binder will actually cost her assuming that it costs 42 cents to mail in the rebate request (postage and the cost of the envelope).
Solution The drugstore purchase generates a credit card charge of $16.21, since 1.0825 × $14.97 = $16.205025. As stated, Mrs. Juanitez spends $.42 to mail her $7.00 rebate request. She receives the $7.00 rebate and Mrs. Juanitez’s credit card company gives her a rebate of $.81 since .05 × $16.21 = $.8105.
Therefore, Mrs. Juanitez’s total cost for the binder is $16.21+$.42−$7.00−$.81 = $8.82.
To perform this simple calculation on the BA II Plus calculator, first format to two decimal places (if necessary) by pressing 2ND FORMAT 2 ENTER 2ND QUIT . Then press 1 • 0 8 2 5 × 1 4 • 9 7 = 2ND ROUND
STO 0 + • 4 2 = − 7 = − ( RCL 0 × • 0 5 ) = . Your display should now read 8.82, so Mrs. Juanitez’s cost was $8.82. Note that had you pressed the above sequence
of keys except that you omitted 2ND ROUND , your answer would have been $8.81. This is because stored information carries more decimal values than those in the display. The use of 2-formatting and the ROUND key was appropriate in Example (0.1.2) because all rounding was to the nearest integral number of cents. However, you should be careful because sometimes a problem requires you to round up or round down. Note 0.3.2 on rounding: Usually, you wait until the end of the problem to round. The exception is that if money changes hand (for instance, you close a savings account or make a mortgage payment), you must round to an integral number of cents. Whether you should round to the nearest penny, round up, or round down depends on the particular situation. For instance, if you are asked for the smallest deposit you can make today at 3% so as to be able to withdraw $243.25 one year from now and $K deposited today will grow to $K(1.03), you round up $243.25 1.03 ≈ $236.1650485 to $236.17. We have used the notation ≈ to mean “approximately equal to,” and this will be used throughout the text. On the other hand, if you are
asked how much you may withdraw in one year if you deposit $236.17 now to your new account, you compute $236.17(1.03) = $243.2551, and since the balance to the nearest penny is $243.26, you may withdraw $243.26 unless you have a particularly stingy bank that makes you round down to $243.25. (To determine whether the bank can round down, you might have to read the fine print or look at applicable state law, but we will not concern ourselves with this.) As was the case in Example (0.1.2), our calculator instructions will sometimes include storing a partial result in one of the BA II Plus calculator’s ten memories. It is a matter of taste (and habit) which memory is selected. Therefore, although in Example (0.1.2) we specified storage in memory 0, in our directions specific memory numbers will not be given. It is helpful to develop routines as to where you store certain types of intermediate results, but it would be presumptuous of us to impose a particular pattern. To store a number displayed on your calculator in memory m use STO m . To recall a number stored in memory m use RCL m . A useful feature of the BA II
Plus calculator is that you may replace the number stored in register m by its sum with the displayed value by pushing STO + m . Likewise, you may replace the entry in register m by its value minus the displayed value by pushing STO − m , by its value multiplied by the displayed value by pushing STO × m , and by its value divided by the displayed value by pushing STO ÷ m . Pressing STO yx m will result in the value in memory m being raised to the number that was just displayed. If you wish to view the entries in all ten memory registers, it is most efficient to open the Memory worksheet by pressing 2ND MEM . The calculator display will then show “M0 = ”. The number following the equal sign gives the number stored in memory register 0, or as many digits of it as the formatting requires. Go ahead and push ↓ , and the display will include “M1 = ” and the number stored in memory register 1, again as dictated by formatting.
Repeatedly pushing ↓ allows you to cycle through the memory registers. Should you wish to reverse the direction of your cycling, push ↑ rather than ↓ . The memory worksheet is one of twelve “worksheets”
included in the BA II Plus calculator. Five of these we discuss in detail and three others receive at least passing treatment. In general, a worksheet contains registers for storing a set of variables, and the worksheets each have a set of built-in formulas relating how the entries in the worksheet’s registers should be related (although in the case of the memory worksheet, the set of relationships is empty). For instance, the Interest Conversion worksheet has variables NOM, EFF, and C/Y, and if j, i, and m are their respective entries, the formula 1 + i = (1 + j m ) m should hold. The TVM worksheet [discussed in Section (3.2)] has registers that may be filled using the keys N , I/Y , PV , PMT , and FV highlighted in Figure (0.1.1), but usually worksheet registers are accessed using the 2ND key. For instance, the Interest Conversion worksheet is accessed by keying 2ND ICONV
