financial calculator guide

Stalla Review for the CFA® Exams

© 2009 DeVry/Becker Educational Development Corp. All rights reserved. 1

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In accordance with the CFA Institute calculator policy in effect at the time of this writing, CFA candidates are permitted to use one of two approved calculators during the CFA exam. Candidates may choose either the HP12C (HP) or the TIBA2+ (TI). The complete CFA Institute calculator policy can be found on the CFA Institute web site at www.cfainstitute.org. It is extremely important to follow the CFA Institute calculator policy as unauthorized calculators will be confiscated at the exam center. Note that both the platinum and professional models of these calculators are also acceptable.

Many problems that are encountered by financial analysts can be performed most easily by using functions that are built into financial calculators. It is recommended, therefore, that CFA candidates obtain one of the approved financial calculators and learn how to use it to perform these investment and business-related calculations. Both the HP and the TI calculators are capable of performing all the functions required on the CFA exam. The primary differences between the two are the method of keying computations and the cost. The HP uses a unique entry sequence for simple arithmetic where the numbers are keyed first followed by the operator. For example, the equation 2 + 2 would be keyed in as 2, enter, 2, + to arrive at 4. The TI uses the more traditional approach of keying the first number, then the operator, then the last number. For example, 2 + 2 would be keyed as 2, +, 2, = to arrive at 4. The TI has a lower retail price than the HP. It is critical that the candidate be comfortable with the financial calculator. Therefore, if you are more familiar with traditional calculators, you may prefer the TI. Most of the more complex financial functions are performed in virtually the same way for both calculators.

This chapter provides instruction for both currently approved calculators. Most financial calculators utilize the same formats for investment analysis. For that reason, this chapter can be applied to a variety of calculators. Candidates should check CFA Institute publications frequently for any changes in the calculator policy to ensure compliance. While both calculators have a broad range of functions, it is not necessary to learn all of them. Indeed, doing so would waste valuable time that could be put to better use in preparing for the CFA exam itself. Consequently, it is recommended that candidates concentrate on learning only those basic calculator functions that may be useful for the CFA exam. This text is limited to this goal but should not be considered a replacement for the manufacturer's instruction manual.


2 © 2009 DeVry/Becker Educational Development Corp. All rights reserved.

TURNING THE CALCULATOR "ON" AND "OFF"

ON/OFF INSTRUCTIONS

HP12C TIBA2+

Press the On

key to turn the calculator "ON" or "OFF."

When the calculator is "ON," nothing should appear on the on the display screen except numbers. If any other alphanumerics appear on the display screen, they should be purged. Refer to the calculator manufacturer's instruction manual for directions on how to remove specific unwanted alphanumeric characters form the display screen and/or the meaning of such characters.

Press the

ON/OFF key to turn the calculator "ON" or

"OFF."

When the calculator is "ON," nothing should appear on the on the display screen except numbers. If any other alphanumerics appear on the display screen, they should be purged. Refer to the calculator manufacturer's instruction manual for directions on how to remove specific unwanted alphanumeric characters form the display screen and/or the meaning of such characters.



THE KEYBOARD

The table below describes some of the important features of the keyboards of the HP12C and TIBA2+ calculators, as well as the nomenclature that will be used throughout this text in describing keystrokes.

KEYBOARD FEATURES AND KEY NOMENCLATURE

HP12C TIBA2+

Many of the keys on the HP12C (HP) calculator can perform one of three functions. Each function is identified in one of three ways:

1. On the top face of the key itself (in white). 2. On the slanted face of the key (in blue). 3. On the keyboard directly above the key (in

gold).

The function identified on the top face of a key (in white) is activated by simply pressing the key itself.

The function identified on the slanted facing of a key (in blue) is activated by pressing the (blue) key g immediately before pressing the key itself.

The function identified on the keyboard directly above a key (in gold) is activated by pressing the (gold key) f immediately before pressing the key itself.

In this text, the key to be depressed when performing a calculation will be identified by the function it is performing in the context of the problem being solved (which may be depicted on the calculator in white, blue or gold lettering) and not necessarily by the (white) lettering that appears on its top face. For example, consider the following key: PRICE

yx

x

This key can perform one of three functions:

1. It can raise a number (y) to some power (x); 2. It can find the square root of a number (x); 3. It can find the price of a bond.

Most of the keys on the TIBA2+ (TI) are dual function. Each key has a number or function on its face with a different function listed above the key (in white). The function/number on the face of each key is activated by pressing the key itself. The secondary function (in white) is activated by pressing the 2nd key followed by the key directly below the secondary function (in white).

For example, to compute the natural logarithm of 5 depress the following keys:

5 LN and the display will show 1.60944

To compute the antilog (ex) of 1.60944 depress the following keys:

2nd

LN

ex

and the display will show 5.00000 If, in a given problem, this key will be used to calculate the natural logarithm of a number, the keystroke will be denoted as:

LN

If it will be used to find the antilogarithm of a number, the keystrokes will be denoted as:

2nd ex

This notation will be used throughout this text.



HP12C TIBA2+

If, in a given problem, this key will be used to raise a number (y) to the power (x), the keystroke will be denoted as:

yx

If it will be used to find the square root of a number, the keystroke sequence will be denoted as:

g x If it will be used to find the price of a bond, the keystroke sequence will be denoted as:

f PRICE

The TI is designed with two modes of operation, the standard-calculator mode and the prompted-worksheet mode. The standard-calculator mode is used for time value of money (TVM) and standard math calculations. The prompted-worksheet mode is used for more complex applications such as cash flow and bond calculations.

In standard-calculator mode only numbers should appear on the display screen with no other symbols or indicators. In worksheet mode both numbers and prompts will appear at the top of the display. Some examples of these prompts are:

COMPUTE ENTER

If any of these prompts are displayed, this indicates that the calculator is in worksheet mode. It can be returned to standard-calculator mode by pressing 2nd QUIT .



CLEARING THE CALCULATOR DISPLAY AND REGISTERS

It is recommended that the numerical characters that appear on the display screen (and in various memory registers) be cleared before beginning to solve each new problem with a calculator. This is because certain functions store previous answers in the calculator's memory registers and add the latest calculated result to them producing new results. This is useful in some calculator applications, but it can be detrimental at other times. To prevent the answer to one set of calculations from becoming an "input" to a subsequent calculation, it is a good practice to begin every new problem by first clearing the calculator.

CLEARING FUNCTIONS

HP12C TIBA2+

To clear the display and numeric registers (restoring them to zero), use the following keystrokes:

f CLX

The display screen should read 0.000000 (the number of zeros following the decimal point is not important). If any other alphanumeric character(s) are displayed, see your calculator manufacturer's instruction manual.

To clear only the last entry (used to correct a mistaken entry), press the

CLX key.

In standard-calculator mode the CE/C

key will clear the last entry. The entire calculation can be cleared by pressing:

CE/C CE/C

To clear the TVM worksheet and reset the TVM registers (PV, FV, PMT, N, I/Y) to zero press:

2nd CLRTVM

To clear a worksheet while in the prompted worksheet mode, press:

2nd CLR Work

To return to standard-calculator mode, press:

2nd QUIT

Finally, to correct the last numeric digit entry, press the

key.



SETTING THE DECIMAL POINT ACCURACY TO BE DISPLAYED

The amount of decimal point accuracy that is displayed on a calculator can be set. Both the HP12C and the TIBA2+ can display anywhere from zero to nine digits after the decimal point.

SETTING THE DECIMAL POINT ACCURACY

EXAMPLE HP12C TIBA2+

The f key followed by pressing any number from 1 to 9 will set the decimal point accuracy.

The number of decimal places displayed can be set by pressing 2nd

Format followed

by 0 to 9 and the ENTER key. Note: Some models of the TI allow only 8 decimal places.

Set the calculator to display five decimal point accuracy

To display 5-decimal point accuracy, press:

f 5


2nd Format 5 ENTER

Set the calculator to display seven decimal point accuracy.


f 7


2nd Format 7 ENTER



CHANGING THE ALGEBRAIC SIGN OF A NUMBER

In many applications, it is necessary to change the algebraic sign of a number that is entered into the calculator. The table below describes how this is done on the HP and TI calculators.

CHANGING THE ALGEBRAIC SIGN OF A NUMERICAL ENTRY

For example, to display −4:

HP12C TIBA2+

The CHS key changes the sign of the displayed value. To display −4, use the following keystrokes:

4 ENTER CHS

To change the value, −4, back to +4, press:

CHS

The +/− key changes the sign of the displayed value. To display −4, use the following keystrokes:

4 +/−

To change the value, −4, back to +4, press:

+/−



ARITHMETIC FUNCTIONS

The four most common mathematical functions that are performed on a calculator are the simple arithmetic functions of addition, subtraction, multiplication, and division. Presumably, the reader already knows how to use a calculator to perform these common types of calculations. For completeness, however, these functions are briefly explained by the examples found in the table below.

ADDITION


14.1 + 35.35

14.1 ENTER 35.35 +

Answer: 49.45

14.1 + 35.35 =

Answer: 49.45

12.1 + 6.4 + 8.3

12.1 ENTER 6.4 +

8.3 +

Answer: 26.8

12.1 + 6.4 +

8.3 =

Answer: 26.8



SUBTRACTION


34.5 − 13.6

34.5 ENTER 13.6 −

Answer: 20.9

34.5 − 13.6 =

Answer: 20.9

56.3 − 12.1 − 23.5

56.3 ENTER 12.1 −

23.5 −

Answer: 20.7

56.3 − 12.1 −

23.5 =

Answer: 20.7

54.2 + 12.5 − 16.4

54.2 ENTER 12.5 +

16.4 −

Answer: 50.3

54.2 + 12.5 −

16.4 =

Answer: 50.3

43.6 − 13.4 + 12.5

43.6 ENTER 13.4 −

12.5 +

Answer: 42.7

43.6 − 13.4 +

12.5 =

Answer: 42.7

−34.8 + 23.5

34.8 CHS ENTER

23.5 +

Answer: −11.3

34.8 +/− +

23.5 =

Answer: −11.3

45.3 − 58.4

45.3 ENTER 58.4 −

Answer: −13.1

45.3 − 58.4 =

Answer: −13.1

−46.7 − 12.4 +1 18.4

46.7 CHS ENTER

12.4 − 18.4 +

Answer: −40.7

46.7 +/− −

12.4 + 18.4 =

Answer: −40.7



MULTIPLICATION


43.2 × 16.4

43.2 ENTER 16.4 ×

Answer: 708.48

43.2 × 16.4 =

Answer: 708.48

23.4 × −13.8

23.4 ENTER 13.8

CHS

×

Answer: −322.92

23.4 × 13.8

+/−

=

Answer: −322.92

−5.3 × 11.2

5.3 CHS ENTER 11.2

×

Answer: −59.36

5.3 +/− × 11.2

=

Answer: -59.36

−56.1 × −6.8

56.1 CHS ENTER

6.8 CHS ×

Answer: 381.48

56.1 +/− ×

6.8 +/− =

Answer: 381.48

4.5 × 5.2 × 6.3

4.5 ENTER 5.2 ×

6.3 × Answer: 147.42

4.5 × 5.2 ×

6.3 = Answer: 147.42

4.8 × (−1.2) × (4)

4.8 ENTER 1.2 CHS

× 4 ×

Answer: −23.04

4.8 × 1.2 +/−

× 4 =

Answer: −23.04



DIVISION


23.5 ÷ 16.2

23.5 ENTER 16.2 ÷

Answer: 1.4506

23.5 ÷ 16.2 =

Answer: 1.4506

−45.2 ÷ 23.7

45.2 CHS ENTER

23.7 ÷

Answer: −1.9072

45.2 +/− ÷

23.7 =

Answer: −1.9072

34.5 ÷ (−67.3)

34.5 ENTER 67.3 CHS

÷

Answer: −0.5126

34.5 ÷ 67.3 +/−

=

Answer: −0.5126

−45.9 ÷ (−78.5)

45.9 CHS ENTER

78.5 CHS ÷

Answer: 0.5847

45.9 +/− ÷

78.5 +/− =

Answer: 0.5847

4.75 ÷ 12.6

4.75 ENTER 12.6 ÷

Answer: 0.3770

4.75 ÷ 12.6 =

Answer: 0.3770

18.56 ÷ (−3.2)

18.56 ENTER

3.2 CHS ÷

Answer: −5.8000

18.56 ÷

3.2 +/− =

Answer: −5.8000



EXPONENTIAL AND LOGARITHMIC FUNCTIONS

There are seven exponential and logarithmic functions:

1. Raising a number to a power (n = yx).

2. Finding the square of a number (n = x2). Note that this is a special case of raising a number to a power.

3. Taking the square root of a number ( )5n x x= = . Note that this is a special case of raising a

number to a power.

4. Finding the natural logarithm of a number (n = ln x).

5. Finding the common logarithm of a number (n = log x).

6. Finding the natural antilogarithm of a number (which is the same thing as finding the value of ex since, if x = ln n, then n = ex).

7. Finding the common antilogarithm of a number (which is the same thing as finding the value of 10x since, if x = log n, then n = 10x).

In statistics, it is often necessary to find the square and square roots of numbers. The rest of these functions are seldom necessary on the CFA examination, but they do have some applications in finance.

The table below describes, by using examples, how these functional calculations can be performed on the HP and TI calculators.

HP12C TIBA2+

The exponential and logarithmic functions are performed on the HP12C calculator with the following keys: yx x LN ex There are no keys to perform x2, log x, or 10x.

The following keys are used to perform exponential and logarithmic calculations on the TIBA2+ calculator: x2 yx x LN ex There are no keys to perform log x or 10x.



RAISING NUMBERS TO POWERS

EXPLANATION/EXAMPLE HP12C TIBA2+

Compute: (1.06)5

1.06 ENTER 5 yx

Answer: 1.3382

1.06 yx 5 =

Answer: 1.3382

Compute: (1.08).75

1.08 ENTER .75 yx

Answer: 1.0594

1.08 yx .75 =

Answer: 1.0594

Compute: (1.09)-3

1.09 ENTER 3 CHS yx

Answer: 0.7722

1.09 yx 3 +/− =

Answer: 0.7722

Compute: (−3.5)3

3.5 ENTER CHS 3 yx

Answer: −42.875

3.5 +/− yx 3 =

Answer: −42.875

Compute: (1.04).5 Note that ( ).51.04 1.04=

1.04 ENTER .5 yx

Answer: 1.0198

1.04 yx .5 =

Answer: 1.0198



FINDING THE SQUARE OF A NUMBER


Compute: 1.342

1.34 ENTER 2 yx

Answer: 1.7956

1.34 x2

Answer: 1.7956

Compute: (−1.43)2

1.43 CHS ENTER 2 yx

Answer: 2.0449

.43 +/− x2

Answer: 2.0449

TAKING SQUARE ROOTS


Compute: 18.94

18.94 ENTER g x

Answer: 4.3520

18.94 x

Answer: 4.3520

FINDING NATURAL LOGARITHMS


Compute: ln(1.02) 1.02 ENTER g LN

Answer: 0.0198

1.02 LN

Answer: 0.0198



FINDING COMMON LOGARITHMS


Compute: log(206.5)

Note: log n = .4343 ln n

206.5 ENTER g LN

.4343 ×

Answer: 2.3149

Note: log n = .4343 ln n

206.5 LN ×

.4343 =

Answer: 2.3149

FINDING eX (NATURAL ANTILOGARITHMS OF X)


Compute: e3.5

3.5 ENTER g ex

Answer: 33.1155

3.5 2nd ex

Answer: 33.1155

Compute: e−4

4 ENTER CHS g ex

Answer: 0.0183

4 +/− 2nd ex

Answer: 0.0183

Compute: e.5 (which is the same as e )

.5 ENTER g ex

Answer: 1.6487

.5 2nd ex

Answer: 1.6487

Compute: e−5 (which is the same as 1/ e )

.5 ENTER CHS g ex

Answer: 0.6065

.5 +/− 2nd ex

Answer: 0.6065




Compute: 100e.2(4) − -2(.1)

100e.2(4) − 2(.1) = 100e.8 − .2

= 100e.6

Keystroke solution:

.6 g ex

100 ×

Answer: 182.21

100e.2(4) − 2(.1) = 100e.8 − .2

5 100e.6

Keystroke solution:

.6 2nd ex ×

100 =

Answer: 182.21

FINDING COMMON ANTILOGARITHMS


Compute: antilog (2.65)

Note: antilog x = (ex)2.3026

2.65 ENTER 2.3026 ×

g ex

Answer: 446.7

Note: antilog x = (ex)2.3026

2.65 × 2.3026 =

2nd ex

Answer: 446.7

Compute antilog (−1.06)

1.06 CHS ENTER 2.3026

× g ex

Answer: 0.087

1.06 +/− × 2.3026

= 2nd ex

Answer: 0.087



FINANCIAL FUNCTIONS

The most important functions available on a financial calculator are those that are designed to perform:

* Time value of money calculations including future value, present value, and annuity computations.

* Discounted cash flow calculations including net present value and internal rate of return computations.

* Bond calculations including the computation of a bond's price given its yield-to-maturity or its yield-to-maturity given its price.

Time Value of Money Calculations

There are two classes of time value of money calculations:

1. Future/Present Value: Future and present value calculations which use the basic formula for compound interest:

FV = PV (1 + i)n

where: FV is the future value that is produced when a present value sum is invested for n-periods of time at a compound rate of return of i-percent per period.

This equation can be solved for any one of the four variables contained in it if the other three variables are specified. Financial calculators have internal functions that can do this. These are accessed by using the keys that are labeled to denote each of these four variables.

IMPORTANT NOTE: Both the HP and TI calculators use a specific cash flow sign convention that must be followed when performing compound interest calculations. This convention requires that cash inflows into investments and cash outflows from investments must be assigned opposite algebraic signs. Thus, if cash placed into an investment is given a positive (negative) value, then the cash taken out of the investment must be denoted as a negative (positive) value. This has two implications:

a. If the amount of money that is placed into an investment (its present value, "PV") is assigned a positive (negative) value, then the calculator will automatically assign a negative (positive) value to the amount that it calculates. This is the amount that will be returned from the investment at some future time (its future value, "FV") for any given rate of return ("i") and investment time horizon ("n").

b. If both the amount of money that is placed into an investment at the present time ("PV") and the amount that is to be returned from it at some future time ("FV") are entered into the calculator, these two amounts must be entered with opposite algebraic signs; otherwise, an ERROR message will appear on the calculator's display screen.



IMPORTANT NOTE: When used in its normal financial mode, the HP12C calculator is programmed for the standard assumptions that compounding occurs one time per period and that payments are made or received at the end of each compounding period. The TIBA2+ calculator, on the other hand, can be programmed for any number of compoundings per period and for payments or receipts occurring at either the end or the beginning of each compounding period.

Since the candidates who are sitting for the CFA examinations may be using either type of calculator (but the answers that examiners will be expecting are uniform), it is recommended that the TI calculator be programmed so as to generate the same answers as those generated by the HP models. Consequently, it is recommended that TI calculators be set so as to compound one time per period and to assume that payments and receipts are at the end of each compounding period. The TI calculator can be set to do this by using the following keystrokes starting from the standard-calculator mode:

STANDARD SET-UP FOR FINANCIAL FUNCTIONS

HP12C TIBA2+

As mentioned above, the HP calculator is programmed at the manufacturer for compounding once per period and the payments are received at the end of each compounding period. It will retain these settings unless changed by the operator. When the calculator is turned "ON," nothing but numbers should appear on the display screen. If the word "BEGIN" appears on the screen, the calculator is set up to assume that payments and receipts occur at the beginning of each compounding period. This is not the recommended standard set-up. If the word "BEGIN" appears on the display screen, it can be made to disappear by performing the following keystrokes:

g END

The calculator is programmed to assume that compounding occurs once per period.

The TI calculator is set by the manufacturer to function assuming 12 payments per year. It is strongly recommended that candidates change this setting to compounding only once per year by pressing the following keystrokes:

2nd P/Y 1 ENTER 2nd QUIT

When the calculator is turned "ON," nothing but numbers should appear on the display screen. If any worksheet prompts are displayed, press the following keys to place the calculator in standard-calculator mode:

2nd QUIT

If the letters "BGN" appear in the upper right, the calculator is set up to assume that payments and receipts occur at the beginning of each compounding period. This is not the recommended standard set-up. If the letters "BGN" appear on the display screen, it can be reset to end-mode (the letters disappear from the screen) by pressing the following keystrokes:

2nd BGN 2nd SET 2nd QUIT

Once the calculators are set up properly, standard financial problems may be solved using the financial functions. The table below describes how to use the financial functions on the calculators to perform present and future value calculations.



FUTURE/PRESENT VALUE CALCULATIONS

HP12C TIBA2+

The keys that are used to perform future and present value financial calculations are:

n i

PV FV

Values are entered into any three of these variables by pressing the key denoting the appropriate variable immediately after entering its value on the display screen. Entries into the financial variable registers may be made in any order. However, if both the FV and PV variables are entered in the same problem, one of the two must be assigned a negative sign. This is done using the CHS key.

The unknown variable is computed by pressing the key that denotes it after all of the other three variables have been entered.

The keys that are used to perform future and present value financial calculations are:

N I/Y

PV FV

Values are entered into any three of these variables by pressing the key denoting the appropriate variable immediately after entering its value on the display screen. Entries into the financial variable registers may be made in any order. However, if both the FV and PV variables are entered in the same problem, one of the two must be assigned a negative sign. This is done using the +/− key.

The unknown variable is computed by pressing the +/− key followed by the key that denotes it after all of the other three variables have been entered.



(1) Find the future value of $100 invested for 5 years at a 6% interest rate. This problem may be stated as:

PV = 100 i = 6 n = 5

FV = ?

HP12C TIBA2+

To solve this problem, use the following keystrokes:

100 PV 6 i 5 n FV

Answer: $133.82 The display will appear as -133.82 because of the cash flow sign convention.


100 PV 6 I/Y 5 N CPT FV

Answer: $133.82 The display will appear as −133.82 because of the cash flow sign convention.

(2) Find the present value of $500 to be received in 4 years using an 8% discount rate. This problem may be stated as follows:

FV = 500 n = 4 i = 8

PV = ?

HP12C TIBA2+

The following keystrokes will solve the problem:

500 FV 4 n 8 i PV


The following keystrokes will solve the problem:

500 FV 4 N 8 I/Y CPT PV




(3) Find the rate of return that turns $200 into $300 over 5 years. This problem may be restated as:

PV = 200 FV = 300

n = 5 i = ?

HP12C TIBA2+

To solve the problem, use the following keystrokes:

200 PV 300 CHS FV 5 n i

Answer: 8.45%

Note: The reason why the CHS key must be included in the keystroke sequence is that the values in the PV and FV registers must have opposite signs in accordance with the cash flow sign convention. If they do not, an ERROR message will be displayed.

To solve the problem, use the following keystrokes:

200 PV 300 +/− FV 5 N

CPT I/Y

Answer: 8.45% Note: The reason why the +/− key must be included in the keystroke sequence is that the values in the PV and FV registers must have opposite signs in accordance with the cash flow sign convention. If they do not, an ERROR message will be displayed.



(4) Example: If $300 is invested at an interest rate of 6%, how many years will be needed to generate an ending value of $600? This problem can be restated as:

PV = 300 i = 6

FV = 600 n = ?

HP12C TIBA2+

The following keystrokes are used to solve this problem:

300 PV 600 CHS FV

6 i n

Answer: 12 years

Note: Most HP12C calculators round the computed value of "n" up to the nearest whole number when using the financial function keys. The precise answer can be determined as follows:

FV = PV(1 + i )n

600 = 300(1.06)n

(1.06)n = 2

n ln 1.06 = ln 2

1.06 ENTER g LN

Display: 0.058269

2 ENTER g LN

Display: 0.693147

.058269n = .693147

n = .693147/.058269

n = 11.9 years

The following keystrokes are used to solve this problem:

300 PV 600 +/− FV

6 I/Y CPT N

Answer: 11.9 years



Note: The numbers that are entered into the n (or N) and i (or I/Y) registers do not have to represent years and interest rate per year. Instead, the number entered into the n (or N) register represents the number of compounding periods. This can be measured in quarters, months, days, or any other unit of time. Similarly, the numbers entered into the i (or I/Y) register represents the periodic interest rate or interest rate per compounding period—a period that must be measured in the same units as the number entered into the n (or N) register if the standard set-ups recommended here are used.

(5) Find the future value of $100 invested for 5 months at a rate of 0.8% per month. This problem can be stated as:

PV = 100 n = 5 i = .8

FV = ?

HP12C TIBA2+


100 PV 5 n

.8 i FV

Answer: $104.06 (displayed as a negative number because of the cash flow sign convention).


100 PV 5 N

.8 I/Y CPT FV


If the stated interest rate is for a period of time that does not correspond to the same amount of time designated by one compounding period, it must be converted into a rate per comparable unit of time if the calculators are set up in the standard recommended manner. This is done using the bond equivalent convention:

pii

m=

where: ip is the periodic interest rate that is entered into the i (or I/Y) register.

m is the number of compounding periods per year.

i is the nominal interest rate per year.



(6) Find the future value of $100 invested for 33 months at an annual interest rate of 6% compounded monthly.

To do this problem using calculators that are set up in the recommended standard mode, it is necessary to first compute the periodic interest rate which is the interest rate per month using the bond equivalent convention:

p6% per yeari

12 months/year.5% per month

=

=

The problem can then be restated as:

PV = 100 n = 33 months i = .5% per month

FV = ?

HP12C TIBA2+


100 PV 33 n

.5 i FV



100 PV 33 N

.5 I/Y CPT FV


Note that the above keystrokes solve the equation:

( )33

33

FV 100 1.005

.06100 112

=

⎛ ⎞= +⎜ ⎟⎝ ⎠

This is not the same thing as investing $100 for 33 months at a 6% rate per year compounded annually. To solve this problem, n (or N) must be converted into units that correspond to the compounding period of the interest rate:

33 monthsn12 months / year2.75

=

=

years



Then the problem can be restated as:

PV = 100 i = 6% per year n = 2.75 years

FV = ?

HP12C TIBA2+

This problem is solved with the following keystrokes:

100 PV 6 i 2.75 n FV

Answer: $117.42

(displayed as a negative number because of the cash flow sign convention). Note: This answer uses simple interest to calculate the future value of the fractional year as indicated by the following equation:

FV = 100(1.06)2 [1+(.06)(.75)] = 117.42

If it is desired to use compound interest to calculate the future value of the fractional year according to the equation:

FV = 100(1.06)2.75 = 117.38

The following keystrokes may be employed:

STO EEX

The letter "C" will appear on the display screen indicating that the calculator is in the "compound interest" mode. Then proceed with the regular keystroke sequence:

100 PV 6 i 2.75 n FV


To return to the regular mode, press:

STO EEX

and the "C" will disappear from the display screen.


100 PV 6 I/Y 2.75 N

CPT FV




Periodic interest rates calculated according to the bond equivalent yield convention are different from the effective yield convention. While the bond equivalent convention computes the periodic interest rate as:

pii

m=

the effective periodic interest rate is:

( )1/mpi 1 i 1= + −

Applying this concept to the prior example, a 6% nominal interest rate compounded monthly is a .5% per month periodic rate according to the bond equivalent convention:

pi 6%i

m 12.5% per month

= =

=

However, the effective monthly rate is:

( )( )

1/mp

.08333

i 1 i 1

1.06 1.004867.4867% per month

= + −

= −

==

(7) Example: Find the future value of $100 invested at a 6% annual rate compounded monthly using the effective rate convention for 33 months.

This problem can be restated as:

PV = 100 i = (1.06)1/12 − 1 = .4867 n = 33 months

FV = ?

HP12C TIBA2+


100 PV .4867 i 33 n FV



100 PV .4867 I/Y 33 N CPT FV




Alternatively, this problem may be solved according to the equation:

FV = (1.004867)33

HP12C TIBA2+

This equation can be solved by the following keystrokes:

1.004867 ENTER 33 yx 100 ×

Answer: $117.38

The financial functions:

n i PV FV PMT

are generally used when the bond equivalent convention is desired in determining values or yields; the exponential function:

yx

is generally used when the effective yields are employed or desired.

This equation can be solved by the following keystrokes:

1.004867 yx 33 = × 100 =

Answer: $117.38

The financial functions:

N I/Y PV FV PMT

are generally used when the bond equivalent convention is desired in determining values or yields; the exponential function:

yx

is generally used when the effective yields are employed or desired.

2. Annuities: Annuities involve the payment or receipt of a constant amount every period (PMT).

The formulas that are used to compute the future and present value of ordinary annuities are:

( )

( )

n

n

1 i 1FV PMT

i

1 1PV PMTi i 1 i

⎡ ⎤+ −⎢ ⎥=⎢ ⎥⎣ ⎦⎡ ⎤⎢ ⎥= −⎢ ⎥+⎣ ⎦

where: FV is the future value that is produced when a constant payment per period (PMT) is invested for n-periods of time at a compound rate of return of i-percent per period; PV is the present value of a constant stream of payments per period (PMT) that are received for n-periods of time discounted at an interest rate of i-percent per period.

These equations can be solved for any one of the four variables contained in them if the other three variables are specified. Financial calculators have internal functions that can do this which are accessed by using keys that are labeled to denote each of these four variables.

IMPORTANT NOTE: The cash flow sign convention that applies to FV/PV problems also must be followed when performing calculations involving annuities. This convention requires that cash inflows into investments and cash outflows from investments be assigned opposite algebraic signs. Thus, if cash placed into an investment is given a positive (negative) value, then the cash taken out of the investment must be denoted as a negative (positive) value. This has four implications:



a. If the constant payments ("PMT") that are put into an investment for a given number of periods ("n") are assigned positive values, then the calculator will automatically assign a negative value to the amount that it calculates, which will be returned from the investment at some future time (its future value, "FV") for any given rate of return ("i").

b. If the constant stream of funds received from an investment (the payments, "PMT") for each of several ("n") periods are assigned a positive value, then the calculator will automatically assign a negative value to the present value ("PV") of those payments when they are discounted at the discount rate ("i").

c. If both the amount of money that is placed into an investment each period ("PMT") and the amount that is to be returned from it at some future time ("FV") are entered into the calculator, these two amounts must be entered with opposite algebraic signs; otherwise, an ERROR message will appear on the calculator's display screen.

d. If both the amount of money that is to be received from an investment each period ("PMT") and the present value ("PV") of those amounts are entered into the calculator, these two amounts must be entered with opposite algebraic signs; otherwise, an ERROR message will be displayed on the calculator's display screen.

ANNUITY CALCULATIONS

(1) Find the future value of a 5-year annuity consisting of payments of $100 per year invested at an interest rate of 6% per year.

This problem can be stated as:

PMT = 100 n = 5 i = 6

FV = ?

HP12C TIBA2+


100 PMT 5 n 6 i FV


Note: If you fail to obtain this answer, try clearing the financial registers before performing it. The following keystrokes will clear the registers:

f CLX


100 PMT 5 N 6 I/Y CPT FV


Note: If you fail to obtain this answer, try clearing the financial registers before performing it. The following keystrokes will clear the registers:

2nd CLR TVM

CE/C



(2) Find the future value of a $100 per month annuity paid for 24 months and invested at an annualized rate of 6% compounded monthly.

To solve this problem, it is first necessary to determine the periodic interest rate:

p6% per yeari


=

=

Then, the problem can be restated as:

PMT = 100 n = 24 months i = .5%/month

FV = ?

HP12C TIBA2+


100 PMT 24 n .5 i FV

Answer: $2,543.20 (displayed as a negative number because of the cash flow sign convention).


100 PMT 24 N .5 I/Y CPT FV

Answer: $2,543.20 (displayed as a negative number because of the cash flow sign convention).



(3) Find the present value of a $100 per year, 10-year annuity discounted at an 8% rate.


PMT = 100 n = 10 i = 8

PV = ?

HP12C TIBA2+


100 PMT 10 n 8 i PV



100 PMT 10 N 8 I/Y CPT PV


(4) For $10,000, one can buy a 7-year annuity that will pay $2,000 per year for that amount of time. What is the rate of return on the annuity?


PV = 10,000 PMT = 2,000

n = 7 i = ?

HP12C TIBA2+

The following keystrokes solve this problem:

10000 PV 2000 CHS PMT

7 n i

Answer: 9.2% Note: The PV and PMT registers must have values with opposite signs (cash flow convention) or an ERROR message will be displayed.

The following keystrokes solve this problem:

10000 PV 2000 +/− PMT

7 N CPT I/Y

Answer: 9.2% Note: The PV and PMT registers must have values with opposite signs (cash flow convention) or an ERROR message will be displayed.



(5) An investor is putting $2,000 per year into a mutual fund that averages a return of 9% per year. How many years will these payments have to be made before the fund will be worth $100,000?


PMT = 2,000 FV = 100,000

i = 9 n = ?

HP12C TIBA2+

The problem is solved with the following keystrokes:

2000 PMT 100000 CHS FV

9 i n

Displayed answer: 20 years

Note: The algebraic signs assigned to the values entered into the PMT and FV registers must have opposite signs or an ERROR message will be displayed.

Note: The HP12C calculator rounds the solution to "n" up to the nearest whole number (a more precise answer is 19.8 years).

The problem is solved with the following keystrokes:

2000 PMT 100000 +/− FV

9 I/Y CPT N

Displayed answer: 19.8 years

Note: The algebraic signs assigned to the values entered into the PMT and FV registers must have opposite signs or an ERROR message will be displayed.



(6) A person takes out a $200,000 mortgage loan to be repaid with monthly payments over the next 360 months at an annual rate of 8% compounded monthly. What is the monthly payment?

To solve this problem, it is first necessary to compute the periodic interest rate:

p8% per yeari


=

=

The problem can then be restated as:

PV = 200,000 n = 360 i = .66667

PMT = ?

HP12C TIBA2+


200000 PV 360 n .66667 i PMT

Answer: $1,467.53 (displayed as a negative number because of the cash flow convention).


200000 PV 360 N .66667 I/Y

CPT PMT

Answer: $1,467.53 (displayed as a negative number because of the cash flow convention).

All of the examples above were for ordinary annuities. The convention for any ordinary annuity is that payments are made at the end of each period. However, payments are sometimes made at the beginning of payment periods. If this is the case, the annuity formulas change as follows:

( ) [ ]

( )

n

n

1 i 1FV PMT 1 i

i

1 i 1 iPV PMTi i 1 i

⎡ ⎤+ −⎢ ⎥= +⎢ ⎥⎣ ⎦⎡ ⎤+ +⎢ ⎥= −⎢ ⎥+⎣ ⎦

The HP and TI calculators can solve these equations directly by setting them up so as to employ a beginning-of-period payment assumption. These set-ups can be accomplished as shown in the following table.



BEGINNING-OF-PERIOD PAYMENT ASSUMPTION FINANCIAL CALCULATOR SET-UP

HP12C TIBA2+

To put the HP12C calculator into the "BEGIN" mode, use the following keystrokes:

g BEG

The word "BEGIN" will be displayed in small print at the bottom of the display screen. This means that the calculator is set up to assume that all payments or receipts occur at the beginning of each period.

NOTE: THE "BEGIN" MODE IS NOT THE CONVENTIONAL ASSUMPTION FOR FINANCIAL CALCULATIONS; IT SHOULD ONLY BE USED WHEN THE PROBLEM SPECIFICALLY INDICATES THAT PAYMENTS OR RECEIPTS OCCUR AT THE BEGINNING OF EACH PERIOD. ONCE SUCH A PROBLEM IS SOLVED, RETURN THE CALCULATOR TO THE STANDARD "END" MODE SET-UP BY USING THE FOLLOWING KEYSTROKES:

g END

The word "BEGIN" will disappear from the display screen.

To put the TI calculator into the "BEGIN" mode, use the following keystrokes:


The letters "BGN" will be displayed in small print at the upper right of the display screen. This means that the calculator is set up to assume that all payments or receipts occur at the beginning of each period.

NOTE: THE "BEGIN" MODE IS NOT THE CONVENTIONAL ASSUMPTION FOR FINANCIAL CALCULATIONS; IT SHOULD ONLY BE USED WHEN THE PROBLEM SPECIFICALLY INDICATES THAT PAYMENTS OR RECEIPTS OCCUR AT THE BEGINNING OF EACH PERIOD. ONCE SUCH A PROBLEM IS SOLVED, RETURN THE CALCULATOR TO THE STANDARD "END" MODE SET-UP BY USING THE FOLLOWING KEYSTROKES:


The letters "BGN" will disappear from the display screen.



Example: An insurance salesman offers an annuity contract that will pay $100,000 after 20 years if the annuitant pays $2,000 per year with the first payment due immediately. What rate of return is being offered?

Because the first payment is to be made now rather than at the end of the year (with all subsequent payments made one year apart), this is a "Begin Mode" problem of the form:

FV = 100,000 PMT = 2,000

n = 20 i = ?

HP12C TIBA2+

To solve this problem, first set the calculator to the "BEGIN" mode and then perform the usual keystrokes:

g BEG

The word "BEGIN" will appear on the display.

100000 FV 2000 CHS PMT

20 n i

Answer: 8.1%

To get out of the "BEGIN" mode and back to the standard set-up, use the following keystrokes:

g END

The word "BEGIN" should disappear from the display.

To solve this problem, first set the calculator to the "BEGIN" mode and then perform the usual keystrokes:


The letters "BGN" will appear on the display.

100000 FV 2000 +/− PMT

20 N CPT I/Y

Answer: 8.1%

To get out of the "BEGIN" mode and back to the standard set-up, simply repeat first sequence of the above keystrokes.

The letters "BGN" should disappear from the display.



DISCOUNTED CASH FLOW CALCULATIONS

The HP12C and TIBA2+ financial calculators are programmed to compute the Net Present Value (NPV) and the Internal Rate of Return (IRR) of a series of unequal cash flows (simple annuity functions can be used when the cash flows paid or received in every period are equal). These functions may be marginally useful to CFA candidates.

1. Net Present Value (NPV) Calculations: The NPV function is used to compute the net present value of a series of cash flows. It solves the formula:

( ) ( ) ( ) ( )0 1 2 n

0 1 2 n

CF CF CF CFNPV ...

1 i 1 i 1 i 1 i= + + + +

+ + + +

The cash flow sign convention that is employed is to treat cash payments into an investment as negative values and cash receipts out of the investment as positive values. CF0 usually represents the initial cost of the investment and is, therefore, usually assigned a negative algebraic sign.

Given this convention, if the net present value of a stream of cash flows has a positive value, the present value of an investor's assets is enhanced; if it has a negative value, the present value of the investor's assets is diminished.

2. Internal Rate of Return (IRR) Calculations: The IRR function is used to compute the discount rate that makes the net present value of a series of cash flows equal to zero. It is used to calculate the value of "i" (internal rate of return) which solves the following formula for a given set of cash flows (CFjs).

( ) ( ) ( ) ( )31 2 n

0 1 2 3 n

CFCF CF CFCF ... 0

1 i 1 i 1 i 1 i+ + + + + =

+ + + +

The cash flow sign convention that is employed is to treat cash payments into an investment as negative values and cash receipts out of the investment as positive values. CF0 usually represents the initial cost of the investment and is, therefore, usually assigned a negative algebraic sign.

The following tables depict how these classes of problems are solved using the HP and TI calculators.



EXAMPLES OF DISCOUNTED CASH FLOW PROBLEMS

(1) Find the Net Present Value (NPV) of the following cash flow schedule using a 10% discount rate:

Cash Flow Year ( j ) (CFj )

0 $(3,000) 1 (5,000) 2 2,000 3 4,000 4 6,000 5 (1,000)

HP12C TIBA2+

The following keystrokes will solve this problem (do not forget to clear the calculator before starting the problem):

3000 CHS g CF0

5000 CHS g CFj

2000 g CFj

4000 g CFj

6000 g CFj

1000 CHS g CFj

10 i f NPV

Answer: $589.86

Note: The value entered into the CF0 register must be the cash flow of year (0) and all subsequent cash flows must represent sequential years.

If a given year has no cash flow associated with it, it must be entered as a "0" in the proper sequence.

Cash flow functions on the TI are performed using the cash flow worksheet. To enter the cash flow worksheet mode, press CF . It is a good practice to clear the cash flow worksheet registers before each calculation by pressing:

2nd CLR Work

The worksheet starts with the initial cash flow at year0 (CF0). Then each subsequent cash flow (C01, C02, C03, ...) is entered. After each cash flow, the frequency of each can be entered (indicated by F01, F02, etc. on the display screen). The frequency will default to 1. The up and down arrow keys are used to move through the worksheet. For demonstration purposes the following format will be used to indicate moving through a worksheet by pressing the arrow key repeatedly until the display shows the indicated value shown at the right:

...C02

The indicators at the top of the display show the potential functions that can be performed at each input point. See the manufacturer's instruction manual for more details on the worksheet functions.



HP12C TIBA2+

(Continued from previous page)

Once in the calculator is in the cash flow worksheet mode and the worksheet has been cleared, the following keystrokes will solve this problem:

3000 +/− ENTER

5000 +/− ENTER ...C02

2000 ENTER ...C03

4000 ENTER ...C04

6000 ENTER ...C05

1000 +/− ENTER NPV

10 ENTER CPT

Answer: $589.86

Note: The value entered into the CF0 register must be the cash flow of year (0) and all subsequent cash flows must represent sequential years.

If a given year has no cash flow associated with it, it must be entered as a "0" in the proper sequence.

Remember: The notation ...C02 means to press the arrow key repeatedly until the screen displays the value indicated at the right.



(2) Find the internal rate of return (IRR) of a $10,000 investment that is expected to produce the following cash flows:


2 $7,000 3 6,000 4 3,000

Note: This problem must be restated with the following cash flow table:


0 $ −10,000 1 0 2 7,000 3 6,000 4 3,000

This is the form that the cash flow table must take because the calculator requires some cash flow entry for every period ranging from period "0" to period "n" ("n" being 4 years in this problem). The "0"-period cash flow is an outflow of $10,000 which is the amount paid for the investment. The "1"-period cash flow is zero and it must be entered as "0" in the calculator.

HP12C TIBA2+


10000 CHS g CF0

0 g CFj

7000 g CFj

6000 g CFj

3000 g CFj f IRR

Answer: 19%

To solve this problem, place the calculator in cash flow worksheet mode, clear the worksheet, and use the following keystrokes:

10000 +/− ENTER

0 ENTER ...C02

7000 ENTER ...C03

6000 ENTER ...C04

3000 ENTER IRR CPT

Answer: 19%



BOND CALCULATIONS

There are four standard bond calculations that can be performed on the HP12C and TIBA2+ calculators:

1. Computing the PRICE (stated as a percentage of par value) of a conventional (semiannually paying coupon) noncallable bond given its coupon, maturity, and yield-to-maturity.

2. Computing the yield-to-maturity (YTM) of a conventional (semiannually paying coupon) noncallable bond given its coupon, maturity, and price (stated as a percentage of par value).

3. Computing the PRICE (stated as a percentage of par value) of a conventional (semiannually paying coupon) callable bond given its coupon, next call date, call price (as a percentage of par value), and yield-to-call.

4. Computing the yield-to-call (YTC) of a conventional (semiannually paying coupon) callable bond given its coupon, price (stated as a percentage of par value), call price on the next call date (stated as a percentage of par value and assuming the next call date is also a coupon payment date), and the next call date.

In order to use the bond functions of the calculators, it is necessary to understand the accrual, compounding, and dating conventions used by the HP and TI calculators.

Coupon Payment Conventions

Conventional (U.S.) bonds pay coupon interest semiannually. This practice is not followed universally, however. In some countries, coupon interest is paid annually.

The HP calculator's BOND functions are programmed to assume semiannual coupon payments and semiannual compounding using the bond conventional yield calculations; the TI calculator can be set up so as to use annual or semiannual compounding conventions.

Accrued Interest Conventions

U.S. Treasury bonds and notes make coupon payments semiannually and accrue interest based upon the actual number of days between the last coupon payment date and the settlement date when the bond is purchased, relative to the actual number of days between coupon payments. This is called the "actual/actual" accrued interest convention. Conventional (U.S.) corporate and municipal bonds, on the other hand, also pay coupon interest semiannually, but their accrued interest is based upon the assumption that every month has 30 days (therefore, there are 360 days per year). This is called the "30/360" convention.

The HP12C calculator's BOND functions are already programmed in accordance with the conventions used for U.S. Treasury bond and notes (semiannual compounding and coupon interest payments with interest accruing based on the "actual/actual" convention). The TIBA2+ calculator's BOND mode, however, may be set up to use any one of several conventions including:

* Semiannual coupon payment and compounding

* Annual coupon payment and compounding (used for most European bonds)

* Actual/Actual (A/A) accrual

* 30-day months/360-day year (30/360) accrual



In order to obtain the same answers with the TI calculator as will be obtained with the HP model, it is recommended that candidates using the TI model set up its BOND mode to reflect the "actual/actual" accrual and semiannual compounding conventions. This is done using the following keystrokes:

2nd Bond

Scroll through the bond worksheet using the down arrow key four times. The display should show the letters "ACT" for "actual/actual." If the display shows "360," press the following keys so that the letters "ACT" are shown in the display:

2nd SET

Press the down arrow once more to display the number of coupon payments made each year. The display should show "2/Y." This setting indicates that the bond will pay interest twice per year. If the display shows "1/Y," press the following keys so that "2/Y" is displayed.

2nd SET

Dating Conventions

Bond calculations require the inception and maturity dates to be entered into the calculator. The following table shows the date conventions used by the HP and TI calculators.

HP12C TIBA2+

Dates may be entered into the calculator as follows:

MM.DDYYYY

For example, to enter the date April 16, 2002, use the following keystrokes:

04.162002

To enter the date November 2, 2012, use the following keystrokes:

11.022012

Dates may be entered into the calculator as follows:

MM.DDYY

For example, to enter the date April 16, 2002, use the following keystrokes:

04.1602

To enter the date November 2, 2012, use the following keystrokes:

11.0212



1. Calculating the PRICE of Conventional Noncallable Bonds: The table below depicts the keystrokes required to determine the PRICE of a conventional noncallable bond with the HP12C and TIBA2+ calculators.

Example: Find the price of an 8% coupon, 12-year bond whose yield-to-maturity is 7%.

HP12C TIBA2+


8 PMT 7 i 1.012000 ENTER

1.012012 f PRICE

Answer: $108.03

Note: The beginning date that was chosen (January 1, 2000) was arbitrary; the second date was 12 years later because the problem stated that the bond matures in 12 years. Note: When using the calculator's BOND functions, prices must be entered as a percentage of par; i.e., par must be assumed to be 100 and not $1,000.

Note: Certain maturity dates have no corresponding coupon dates six months earlier. Therefore, an ERROR may be encountered if any of the following dates are chosen as the maturity date for a bond: March 31, May 31, August 29, 30, 31, October 31, and December 31. This problem can be corrected by simply adding one day to both the maturity date and the settlement date of the bond. However, even this correction will not work for the maturity dates of August 29 and 30.

Note: By pressing

+

The invoice price that actually would be paid for the bond is displayed. The invoice price is the (quoted) PRICE of the bond plus the accrued interest that a buyer would also have to pay determined via the "actual/actual" convention.

(Continued on following page)

Bond calculations are performed by placing the calculator into the bond worksheet mode by pressing the following keys:

2nd Bond

Note: It is a good practice to clear the worksheet before starting a new calculation. The bond registers can be cleared by pressing:

2nd CLR Work

The display should show "SDT = " indicating the settlement date. Solve the problem by pressing the following keystrokes:

1.0100 ENTER

8 ENTER 1.0112 ENTER

100 ENTER ... YLD =

7 ENTER CPT

Answer: $108.03

Note: The beginning date that was chosen (January 1, 2000) was arbitrary; the second date was 12 years later because the problem stated that the bond matures in 12 years.

Note: When using the calculator's BOND functions, prices must be entered as a percentage of par; i.e., par must be assumed to be 100 and not $1,000.




HP12C TIBA2+

(Continued from previous page.)

Note: In this example, there is no accrued interest since the maturity date is a coupon date. Therefore, the invoice price and the quoted price are the same.

(Continued from previous page.)


Note: By pressing

the accrued interest is displayed. This is the amount of interest earned by the seller since the last coupon date. The calculated bond price plus the accrued interest is the invoice price of the bond.

Note: In this example, there is no accrued interest since the maturity date is a coupon date. Therefore, the invoice price and the quoted price are the same.

Remember: The notation ...C02 means to press the arrow key repeatedly until the screen displays the value indicated at the right.



2. Calculating the Yield-to-Maturity of Conventional Noncallable Bonds: The table below shows how the yield-to-maturity (YTM) of a conventional noncallable bond can be determined given its price and other required parameters.

Example: Find the yield-to-maturity (YTM) of a 9% coupon, 15-year conventional bond priced at 115.

HP12C TIBA2+


9 PMT 115 PV

1.012000 ENTER 1.012015 f YTM

Answer: 7.33%

Note: The beginning date that was chosen (January 1, 2000) was arbitrary; the second date was 15 years later because the problem state that the bond matures in 15 years.

Note: When using the calculator's BOND functions, prices must be entered as a percentage of par, i.e., par must be assumed to be 100 and not $1,000.


Place the calculator into bond worksheet mode by pressing:

2nd Bond

Remember to clear the worksheet before starting a new calculation by pressing:

2nd CLR Work


1.0100 ENTER 9 ENTER

1.0115 ENTER 100 ENTER ... PRI =

115 ENTER CPT

Answer: 7.33%

Note: The beginning date that was chosen (January 1, 2000) was arbitrary; the second date was 15 years later because the problem state that the bond matures in 15 years.

Note: When using the calculator's BOND functions, prices must be entered as a percentage of par, i.e., par must be assumed to be 100 and not $1,000.




3. Calculating the Price of a Conventional Callable Bond Given Its Yield-to-Call: The table below shows how to find the price of a conventional callable bond based upon its yield-to-call and other required parameters.

Find the price of a 10% coupon, 15-year conventional bond that is callable in 5 years at 105 if its yield-to-call is 7%.

This problem can be restated as a cash flow table consisting of 10 semiannual periods with a coupon interest of $5 being paid at the end of each period. Also, at the end of the 10th period (5th year), the holder of the bond would receive the call price of 105 plus the coupon (105 + 5 = 110).

The present value of these cash flows should be calculated using a periodic discount rate (ip) equal to one-half of the yield-to-call:

pi 7%i 3.5%2 2

= = =

The Cash Flow table depicting the facts of this problem is as follows:

Cash Flow Period ( j ) (CFj )

1 $ 5 2 5 3 5 4 5 5 5 6 5 7 5 8 5 9 5

10 110

HP12C TIBA2+

The solution to this problem can be obtained using the following keystrokes:

5 PMT 10 n 3.5 i

105 FV PV

Answer: $116.02

(displayed as a negative value in accordance with the cash flow sign convention).


The solution to this problem can be obtained using the bond worksheet with the following keystrokes:

2nd Bond 2nd CLR Work


1.0105 ENTER 105 ENTER ... YLD =

7 ENTER CPT

Answer: $116.02




HP12C TIBA2+


Alternatively, the problem can be solved using the cash flow functions by using the following keystrokes:

0 g CF0

(the first cash flow is at the end of the first period)

5 g CFj 9 g Nj

(there are nine periods in which a $5 payment is made)

110 g CFj

(the last period has a $110 payment consisting of a $5 coupon plus the $105 call price)

3.5 i f NPV

Answer: $116.02


Alternatively, the problem can be solved using the Time Value of Money (TVM) worksheet.

From the standard-calculator mode press the following keystrokes:

5 PMT 10 N 3.5 I/Y

105 FV CPT PV

Answer: $116.02 Finally, the problem can also be solved using the cash flow worksheet using the following keystrokes:

CF 2nd CLR Work

0 ENTER 5 ENTER

9 ENTER 110 ENTER NPV

3.5 ENTER CPT

Answer: $116.02



4. Calculating the Yield-to-Call of a Conventional Callable Bond Given Its Price: The table below shows how to find the yield-to-call of a conventional callable bond based upon its price and other required parameters.

Example: Find the yield-to-call of a 7% coupon, 12-year bond priced at 108 and callable in two years at 107.

This problem can be restated as a cash flow table consisting of a payment of $108 now (period 0) followed by the receipt of coupon interest of $3.50 being paid at the end of each of the next four periods. Also, at the end of the 4th period (2nd year), the holder of the bond will receive the call price of 107 (107 + 3.5 = 110.5).

The Cash Flow table depicting the facts of this problem is as follows:

Cash Flow Period ( j ) (CFj )

1 $ 3.5 2 3.5 3 3.5 4 110.5

The periodic yield-to-call (ip) is the internal rate of return (IRR) that will make the present value of these cash flows equal to zero. The conventional yield-to-call of the bond is double this periodic yield in accordance with the bond equivalent yield convention.

HP12C TIBA2+

To solve this problem, the cash flow function can be used. This is done by using the following keystrokes:

108 CHS g CF0 3.5 g CFj

3 g Nj

110.5 g CFj f IRR

Answer: 3.02%

This is the periodic yield-to-call (ip).

The yield-to-call is:

2 ip = 2(3.02%)

= 6.04%

To Solve this problem, start from the cash flow worksheet by pressing and clearing it:

CF 2nd CLR Work

Enter the cash flows into the calculator as and solve as follows:

108 +/− ENTER 3.5 ENTER

3 ENTER 110.5 ENTER IRR CPT

Answer: 3.02%

This is the periodic yield-to-call (ip).


2 ip = 2(3.02%)

= 6.04%




HP12C TIBA2+


Alternatively, the solution to this problem can be obtained using the bond worksheet with the following keystrokes:



1.0102 ENTER 107 ENTER ... PRI =

108 ENTER CPT

Answer: 6.04%

Note: Remember to clear the worksheet before each problem as the calculator will retain the call price in the redemption value register.



PRACTICE DRILL

Having described how to use the financial calculator to perform various important generic types of problems, it is recommended that candidates complete the following practice drill in order to gain some "hands-on" experience that will help solidify their calculator skills. The answers are provided in the tables below. These answers assume that every calculation is begun with the calculator registers cleared. If you are using an HP12C calculator, this means that the display screen should appear as follows with no other indicators showing:

0.000000

If you are using an TIBA2+ calculator, this means that the calculator is initially set to the standard-calculator mode with the display screen appearing as follows:

0.000000

The financial functions of the TI should be configured in the recommended standard manner as described below:

* TVM mode: 1 P/YR: END MODE

* BOND mode: ACT...2/Y

FINANCIAL CALCULATOR PRACTICE DRILL

EXAMPLE PROBLEMS

1. A person has $1,000 in a bank account. The bank pays a 5% interest rate. How much will the account be worth in three years if the funds are left undisturbed to compound?

HP12C TIBA2+

1000 PV 5 i 3 n FV

Answer: $1,157.63

1000 PV 5 I/Y 3 N CPT FV

Answer: $1,157.63



2. The earnings of the XYZ Company are currently $2.00 per share. If the earnings per share grow over the next five years at an average annual growth rate of 10%, how much will they be at the end of that time?

HP12C TIBA2+

2 PV 5 n 10 i FV

Answer: $3.22

2 PV 5 N 10 I/Y CPT FV

Answer: $3.22

3. An investor places $12,000 into an investment that pays a 8% rate of return for 5 1/2 years. How much will the investment be worth at the end of that time?

HP12C TIBA2+

12000 PV 8 i 5.5 n FV

Answer: $18,337.21

Note: Some calculators have algorithms that give the answer:

$18,323.65

12000 PV 8 I/Y 5.5 N CPT FV

Answer: $18,323.65

Note: Some calculators have algorithms that give the answer:

$18,337.21



4. If money compounds at a 5% rate, how long will it take it to double?

HP12C TIBA2+

5 i 2 FV 1 CHS PV n

Answer: 15 years

Note that the FV of 2 and the PV of 1 were chosen arbitrarily; any two numbers could have been chosen as long as the FV value was double the PV value.

Note: The HP12C calculator rounds the answer upward to the nearest whole number.

To get a more exact answer, the calculation could be performed by using the equation:

FV = PV (1 + i)n

2 = 1 (1.05)n In 2 = In 1 + n In (1.05)

( )ln2 ln1nln 1.05

−=

2 g LN (=.6931)

1 g LN (=.0000)

1.05 g LN (=.0488)

.6931 0n.0488

−=

= 14.2 years

5 I/Y 2 FV 1 +/− PV CPT N

Answer: 14.2 years

Note that the FV of 2 and the PV of 1 were chosen arbitrarily; any two numbers could have been chosen as long as the FV value was double the PV value.

Note: The HP12C calculator rounds the answer upward to the nearest whole number.



5. An investor puts $25,000 into an investment. At the end of 10 years, the investment sold for $75,000. What is the rate of return on the investment?

HP12C TIBA2+

25000 PV 10 n

75000 CHS FV i

Answer: 11.6%

25000 PV 10 N

75000 +/− FV CPT I/Y

Answer: 11.6%

6. A two-year investment produces an ending value of $15,000 per $10,000 of initial investment. What is the rate of return?

HP12C TIBA2+

2 n 15000 FV

10000 CHS PV i

Answer: 22.5%

2 N 15000 FV

10000 +/− PV CPT I/Y

Answer: 22.5%



7. A stock is expected to pay a dividend of $0, $2, and $4 per share in each of the next three years, respectively. At the end of the third year, the stock is expected to sell for $80 per share. If the stock is currently selling at $50 per share, what is the expected return on a three-year investment in this stock?

HP12C TIBA2+

The cash flow table for this problem is: Year Cash Flow 0 −50 1 0 2 2 3 84

The keystroke solution is:

50 CHS g CF0

0 g CFj 2 g CFj

84 g CFj f IRR

Answer: 20%

The cash flow table for this problem is: Year Cash Flow 0 −50 1 0 2 2 3 84

The keystroke solution is:

CF 2nd CLR Work

50 +/− ENTER

0 ENTER ...C02

2 ENTER ...C03

84 ENTER IRR CPT

Answer: 20%

8. A person puts $10,000 into a mutual fund every year for 40 years. The fund averages a rate of

return of 8.75% per year. How much would the investment be worth at the end of the investment period?

HP12C TIBA2+

10000 PMT 40 n 8.75 i FV

Answer: $3,160,347

10000 PMT 40 N

8.75 I/Y CPT FV

Answer: $3,160,347



9. If $1,000 is put into an annuity every year for 10 years after which time the annuitant receives $15,000, what is the rate of return on the annuity?

HP12C TIBA2+

1000 PMT 10 n

15000 CHS FV i

Answer: 8.73%

1000 PMT 10 N

15000 +/− FV CPT I/Y

Answer: 8.73%

10. In order to accumulate $1,000,000 over a 20-year period, how much must be invested each year in a mutual fund that has an average return of 8.75% per year?

HP12C TIBA2+

1000000 FV 20 n 8.75 i PMT

Answer: $20,102

1000000 FV 20 N

8.75 I/Y CPT PMT

Answer: $20,102



11. An insurance salesman sells a client an investment program that pays a 7% return if the client invests $10,000 every year for 20 years with the first payment due immediately. At the end of 20 years, what will the annuity be worth?

HP12C TIBA2+

g BEG 7 i 10000 PMT

20 n FV

Answer: $438,652

Note: When finished, reset the calculator to the standard END mode by pressing:

g END

This problem is an annuity due, meaning that the payments are received at the beginning of each period. To solve it the calculator must be placed in the Begin mode by pressing:


7 I/Y 10000 PMT

20 N CPT FV

Answer: $438,652

Repeat the first line of keystrokes to return the calculator to end mode. It is strongly recommended that the calculator be returned to end mode after each annuity due problem.

12. An investment is expected to generate a return of $5,000 per year for the next two years. If an investor requires an 8% return on the investment, how much should he be willing to pay for it?

HP12C TIBA2+

5000 PMT 2 n 8 i PV

Answer: $8,916

5000 PMT 2 N 8 I/Y CPT PV

Answer: $8,916



13. An insurance annuity costs $250,000. It will pay $30,000 per year to the purchaser for each of the next 10 years with the first payment beginning in one year. What rate of return is being offered on this annuity?

HP12C TIBA2+

250000 PV 30000 CHS PMT

10 n i

Answer: 3.46%

250000 PV 30000 +/− PMT

10 N CPT I/Y

Answer: 3.46%

14. What is the value of a 9% coupon, 5-year eurodollar bond priced to yield 7.5% (eurodollar bonds pay coupon interest annually.)

HP12C TIBA2+

9 PMT 5 n 7.5 i

100 FV PV

Answer: $106.07

9 PMT 5 N 7.5 I/Y

100 FV CPT PV

Answer: $106.07



15. A $200,000 mortgage is to be repaid in 30 equal annual installments. If the interest rate on the mortgage is 8%, how much should each installment be?

HP12C TIBA2+

200000 PV 30 n 8 i PMT

Answer: $17,765.49

200000 PV 30 N

8 I/Y CPT PMT

Answer: $17,765.49

16. What should the annual payments be on a 10%, 5-year, $25,000 installment loan if the first payment is subtracted from the loan value at the time the loan is extended and every payment is due one year thereafter?

HP12C TIBA2+

g BEG 10 i 5 n

25000 PV PMT

Answer: $5,995.40

Note: Return to END mode by pressing:

g END


10 I/Y 5 N

25000 PV CPT PMT

Answer: $5,995.40

Return to END mode by repeating the first line of keystrokes.



17. What are the monthly payments required to repay a $200,000 mortgage loan over a 30-year period if the loan carries a nominal 12% interest rate?

HP12C TIBA2+

30 years is 360 months and a 12% nominal interest rate is 1% per month.

200000 PV 360 n 1 i PMT

Answer: $2,057.23

30 years is 360 months and a 12% nominal interest rate is 1% per month.

200000 PV 360 N

1 I/Y CPT PMT

Answer: $2,057.23

18. What is the value of a 6% coupon, 5-year conventional bond priced to yield 6.75%?

HP12C TIBA2+

6 PMT 6.75 i 1.012000 ENTER

1.012005 f PRICE

Answer: 96.86 per $100 of par

Enter the Bond worksheet and clear it by pressing:



1.0105 ENTER ...YLD =

6.75 ENTER CPT


Remember: The notation ...YLD = means to press the arrow key repeatedly until the screen displays the value indicated at the right.



19. $2,000 is invested for 10 years at a nominal interest rate of 4.75%. At the end of the investment period how much will the fund be worth if the interest compounds annually?

HP12C TIBA2+

2000 PV 10 n 4.75 i FV

Answer: $3,181.05

2000 PV 10 N

4.75 I/Y CPT FV

Answer: $3,181.05

20. A bank offers an interest rate of 4.7% compounded quarterly. If $2,000 is invested at this rate for 10 years, how much will its ending value be?

HP12C TIBA2+

4.7% per year is 1.175% per quarter; there are 40 quarters in ten years.

2000 PV 1.175 i 40 n FV

Answer: $3,191.23

4.7% per year is 1.175% per quarter; there are 40 quarters in ten years.

2000 PV 1.175 I/Y

40 N CPT FV

Answer: $3,191.23



21. A bank offers an interest rate of 4.6% compounded monthly. If $2,000 is invested at this rate for 10 years, how much will its ending value be?

HP12C TIBA2+

4.6% per year is .383333% per month; there are 120 months in 10 years.

2000 PV .383333 i 120 n FV

Answer: $3,165.35

4.6% per year is .383333% per month; there are 120 months in 10 years.

2000 PV .383333 I/Y

120 N CPT FV

Answer: $3,165.35

22. A bank offers an interest rate of 4.5% compounded continuously. If $2,000 is invested at this rate for 10 years, how much will its ending value be.

The formula for computing the future value (F) of a sum (K) when the interest rate (r) compounds continuously for t years is:

F = Kert

HP12C TIBA2+

Algebraic solution:

F = Kert

= 2000e(.045)(10)

= 2000e.45

Keystroke solution:

.45 g ex 2000 ×

Answer: $3,136.62

Algebraic solution:

F = Kert

= 2000e(.045)(10)

= 2000e.45

Keystroke solution:

.45 2nd ex × 2000 =

Answer: $3,136.62



23. Compute the yield-to-maturity of a 9% coupon, 10-year conventional bond priced at 103.

HP12C TIBA2+

9 PMT 103 PV 1.012000 ENTER

1.012010 f YTM

Answer: 8.55%



1.0110 ENTER ... PRI =

103 ENTER CPT

Answer: 8.55%

24. Compute the price of a 20-year, zero coupon bond priced to yield 7.3%.

HP12C TIBA2+

0 PMT 7.3 i 1.012000 ENTER

1.012020 f PRICE


Alternatively, this problem can be solve using the time value of money functions as follows:

Note: By convention, zero coupon bonds are assumed to compound semi-annually just as coupon bonds do. Therefore, use half the annual yield and twice the number of years to maturity.

100 FV 3.65 i 40 n PV




1.0120 ENTER ... YLD =

7.3 ENTER CPT


Alternatively, this problem can be solve using the time value of money functions as follows:

Note: By convention, zero coupon bonds are assumed to compound semi-annually just as coupon bonds do. Therefore, use half the annual yield and twice the number of years to maturity.

100 FV 3.65 I/Y 40 N CPT PV




25. Compute the yield-to-maturity of a 10-year, zero coupon bond priced at 52.75.

HP12C TIBA2+

0 PMT 52.75 PV 1.012000 ENTER

1.012010 f YTM

Answer: 6.5%




1.0110 ENTER ...PRI =

52.57 ENTER CPT

Answer: 6.5%

26. Compute the yield-to-call of an 8% coupon, 10-year conventional bond priced at 105 that is callable

in 5 years at 108.

HP12C TIBA2+

The cash flow table up to the call date using semiannual compounding is:

Period Cash Flow

0 −105 1 4 2 4 3 4 4 4 5 4 6 4 7 4 8 4 9 4

10 112

Note: The 112 in the last period is the call price of 108 plus the 10th period's coupon of 4.


The cash flow table up to the call date using semiannual compounding is:

Period Cash Flow

0 −105 1 4 2 4 3 4 4 4 5 4 6 4 7 4 8 4 9 4

10 112

Note: The 112 in the last period is the call price of 108 plus the 10th period's coupon of 4.




HP12C TIBA2+


The keystroke solution for the periodic yield-to-call is:

105 CHS g CF0 4 g CFj

9 g Nj

112 g CFj f IRR

The periodic yield-to-call is: 4.047%.


YTC = 2 ip

= 2 (4.047)

= 8.094%


The keystroke solution for the periodic yield-to-call using the cash flow worksheet is:

CF 2nd CLR Work

105 +/− ENTER 4 ENTER

9 ENTER

112 ENTER IRR CPT

The periodic yield-to-call is: 4.047%.


YTC = 2 ip = 2 (4.047) = 8.094%

Alternatively, the problem could be solved using the Bond worksheet as follows:



1.0105 ENTER 108 ENTER ...PRI =

105 ENTER CPT

= 8.094%



27. Compute the yield-to-maturity of an 8% coupon, 10-year bond priced at 105.

HP12C TIBA2+

8 PMT 105 PV 1.012000 ENTER

1.012010 f YTM

Answer: 7.29%




1.0110 ENTER ...PRI =

105 ENTER CPT

Answer: 7.29%

28. Compute the value of a 9-month futures contract on a commodity when its spot price is $4.50,

interest rates are 4% and the cost of storing the commodity is 0.25%.

The formula for calculating the value of a t-year futures contract (F) on a commodity selling at the spot price (S) when interest rates (r) and the storage costs (c) are considered is:

F = S(1 + r + c)t

HP12C TIBA2+

Algebraic solution:

Nine months is .75 of a year. The "cost of carry" is:

r + c = .04 + .0025 = .0425 F = 4.5(1.0425).75

Keystroke solution:

1.0425 ENTER .75 yx 4.5 ×

Answer: $4.6427

Algebraic solution:

Nine months is .75 of a year. The "cost of carry" is:

r + c = .04 + .0025 = .0425 F = 4.5(1.0425).75

Keystroke solution:

1.0425 yx .75 × 4.5 =

Answer: $4.6427



29. An investment manager earns returns of 10%, 15%, -12%, and 14% in each of four years, respectively. What is the time weighted (geometrically computed) average rate of return over the period?

The time weighted rate of return (geometrically computed) is based upon the following formula:

( ) ( )n

nj

j 11 r 1 r

=

+ = +∏

HP12C TIBA2+

Algebraic solution:

(1 + r )4 = (1.10)(1.15)(.88)(1.14)

= 1.269048

r = (1.269048).25 – 1

Keystroke solution:

1.269048 ENTER .25 yx 1 −

Answer: 6.14%

Algebraic solution:

(1 + r )4 = (1.10)(1.15)(.88)(1.14)

= 1.269048

r = (1.269048).25 – 1

Keystroke solution:

1.269048 yx .25 − 1 =

Answer: 6.14%

30. A 12% coupon U.S. Treasury bond matures on November 15, 2023. What will its quoted price have

to be on August 18, 1999 in order for it to be yielding 6.9%?

What would the invoice price of the bond be if it was purchased on August 18, 1999?

HP12C TIBA2+

12 PMT 6.9 i 8.181999 ENTER

11.152023 f PRICE


To get the invoice price, press:

+




11.1523 ENTER ...YLD =

6.9 ENTER CPT

= 159.61 per $100 of par

To see the accrued interest (AI), press:

The display will show 3.10. The invoice price is the computed value plus the accrued interest: 159.61 + 3.10 = 162.71 per $100 of par



31. A 7.25% coupon U.S. Treasury bond maturing on August 15, 2018 is purchased to yield 6.5% on November 3, 1998. What is the quoted price of the bond? What is the accrued interest?

HP12C TIBA2+

7.25 PMT 6.5 i 11.031998 ENTER

8.152018 f PRICE

Price Quote: 108.27 +

Invoice Price: 109.85

Accrued Interest: 109.85 – 108.27

= 1.58


11.0398 ENTER 7.25 ENTER

8.1518 ENTER ...YLD =

6.5 ENTER CPT

= 108.27 per $100 of par

To see the accrued interest (AI), press:

Answer: 1.58.

The invoice price is the price quoted plus the accrued interest:

108.27 + 1.58 = 109.85 per $100 of par

32. A “$25 million jackpot” lottery ticket pays $1,000,000 every year for 25 years. How much is the

after-tax “lump sum” value of this ticket if the discount rate applied to the future payments is 7% and local tax rates combined produce an effective tax rate of 50%?

HP12C TIBA2+

1000000 PMT 25 n 7 i PV

.5 ×

Answer: $5,826,792

1000000 PMT 25 N

7 I/Y CPT PV . × .5 =

Answer: $5,826,792



33. What is the price of an 8% coupon, 15-year bond that is callable in 5 years at 105 if its yield-to-call is 7%?

HP12C TIBA2+

Using semiannual compounding, the coupon is 4 per period, the periodic yield is 3.5% per period and the number of periods until the call date is 10 (five years).

4 PMT 3.5 i 10 n

105 FV PV


Using semiannual compounding, the coupon is 4 per period, the periodic yield is 3.5% per period and the number of periods until the call date is 10 (five years).

4 PMT 3.5 I/Y 10 N

105 FV CPT PV


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