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Calculate Bond Present Value Using Sharp EL738 and EL733a
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Key Notations in This Manual
Key operations are described in this manual as follows: To specify log To specify 1 To specify xy :. h. : 1 or 1.. : i V. . ?. Q i Z. J..
To specify CLR-D: To specify ENT : To specify Z : To specify DATA :
Functions that are printed in orange above the key require
. to be pressed rst before the key.
Number entry examples are shown with ordinary numbers
(i.e., 100 will be indicated instead of 0).
To specify a memory function (printed in green on/above the
key), press i rst.
Functions that are printed in black adjacent to the keys are
effective in specic modes. Using the. and i keys Press s. t i A x , 10. . t and i A mean you have to press. followed by ) key and i followed by * key.
Notes: The multiplication instruction and alphabetic letter X are distinguished as follows: Key Display Multiplication instruction x Alphabetic letter X X
Examples in this manual are performed using default settings (e.g., SET UP menu items) unless values are otherwise assigned.
Preparing to Use the Calculator
Before using your calculator for the rst time, you must reset (initialize) it.
Resetting the calculator
Press the RESET switch located on the back of the calculator with the tip of a ball-point pen or similar object. Do not use an object with a breakable or sharp tip. After resetting the calculator, the initial display of the NORMAL mode appears.
Resetting the Calculator In Case of Difculty
Caution: The RESET operation will erase all data stored in memory and restore the calculators default setting. In rare cases, all the keys may cease to function if the calculator is subjected to strong electrical noise or heavy shock during use. If pressing any of the keys (including s) has no effect, reset the calculator.
See the above procedure. Note: Pressing. k and 1 = will also erase all data stored in memory and restore the calculators default setting.
Calculator and Display Layout
Display screen Power ON/OFF and Clear key Cursor keys MODE key
Key operation keys SET UP key
Display screen: The calculator display consists of a 12-character dot matrix character line and a 12-digit 7-segment character line (10-digit mantissa and 2-digit exponent). Power ON/OFF and Clear key: Turns the calculator ON. This key also clears the display. To turn off the calculator, press., then c. Key operation keys:.: Activates the second function (printed in orange) assigned to the following key. i: Activates the memory (printed in green) assigned to the following key. SET UP key: Displays the SET UP menu to select the display notation, angular unit, depreciation method and date format. Cursor keys: Move the cursor. MODE key: Switches between NORMAL and STAT modes.
Symbol Equation/ variable name display
Memory clear key
Press. k to display the menu.
To clear all (A-H, M, X-Z, ANS, TVM variables, listed nancial variables, cash ow data, and STAT), press or 0 =. To RESET the calculator, press or 1 =. The RESET operation erases all data stored in memory, and restore the calculators default settings.
Editing and Correcting an Entry
In a menu, such as the SET UP menu, use g or y to select a number (the selected number will blink), then press =. If you need to scroll up or down the screen, use z or i. In nancial calculations, such as bond calculations, press i or z to move through the variables (items).
After obtaining an answer, pressing g brings you to the end of the equation and pressing y brings you to the beginning. Press g or y to move the cursor. Press. g or. y to jump the cursor to the beginning or end of the equation.
Insert and overwrite modes in the equation display
This calculator has two editing modes: insert mode (default), and overwrite mode. Pressing. d switches between the two modes. A triangular cursor indicates an entry will be inserted at the cursor, while the rectangular cursor indicates existing data will be overwritten as you make entries. To insert a number in the insert mode, move the cursor to the place immediately after where you wish to insert, then make the desired entry. In the overwrite mode, data under the cursor will be overwritten by the number you enter. This mode setting will be retained until you press. d or RESET the calculator.
Changing = into =
Procedure = Key operation s 3 = Display
Enter the playback function.
Switch to overwrite. d mode. Change 15 to y y and move the cursor to 3. Change to insert mode. Change 3 to 13 and calculate.d
An error will occur if an operation exceeds the calculation ranges, or if a mathematically illegal operation is attempted. When an error occurs, pressing g or y automatically moves the cursor to the place in the equation/number where the error occurred. Edit the equation/number or press s to clear the equation. For details, see page 76.
This calculator has 11 temporary memories (A-H and X-Z), one independent memory (M) and one last answer memory (ANS). It also has various variables for use in nancial calculations and statistical calculations.
Memory use in each mode for memory calculations
Mode NORMAL STAT
A-H, X-Z M ANS
TVM Listed financial Statistical variables*1 variables *2 variables *3
*1 *2 *3
N, I/Y, PV, PMT, FV All nancial variables, except for TVM variables x, sx, x, n, x, x2, y, sy, y, y, y2, xy, r, a, b, c
Temporary memories (A-H, X-Z) Press g and the variable key to store a value in memory. Press f and the variable key to recall a value from the memory. To place a variable in an equation, press i and the variable key. Independent memory (M) In addition to all the other features of temporary memories, a value can be added to or subtracted from an existing memory value. Press s g M to clear the independent memory (M). Last answer memory (ANS) The calculation result obtained by pressing = or any other calculation ending instruction (including storing and recalling operations) is automatically stored in the last answer memory. Listed nancial variables are automatically stored in the last answer memory by displaying the variable and the value. TVM variables TVM variables can be recalled using f in the same way as temporary memories. It is not necessary to press g to store a value. Listed nancial variables Financial variables are specic to the type of calculation they are used in. For example, the variable N is available to the TVM solver but not to discounted cash ow analysis calculations. If you want to carry a value from a variable over into a different type of calculation, use one of the following methods: Last answer memory (ANS): Within the original calculation, display the variable and value that you wish to carry over. The value is automatically entered into last answer memory. Press s to exit the calculation (the listed nancial variables will disappear from the screen), and press i / to bring up the value from the previous calculation. M-D-Y (D-M-Y) 1 and M-D-Y (DM-Y) 2 are not stored in last answer memory. Variables common to both calculations: If the value that you wish to carry over is held in a variable that exists in both types
of calculation (for example, both bond calculations and the TVM solver use the variable I/Y), you can retrieve the value simply by switching calculation types and bringing up the variable. Statistical variables Statistical data is not entered into variables. Statistical variables are the results of the calculation of statistical data. Therefore, you cannot enter values directly into statistical variables. After calculation, however, you can use the values held in statistical variables in subsequent calculations. Note: Use of f or i will recall the value stored in memory using up to 14 digits. Memory calculations
24 (8 2) = (8 2) 5 =
Key operation s8x2gM i M = iMx5= sgM
$1503:Mx 3 h +)$250:M2=M1+h )M25% fMx5.% M
$1 = 110 26,510 = $? $2,750 = ? r = 3 cm (rY) 2 r = ?
110 g Y f Y = 2750 x f Y = 3gY. t i Y*.;=
8 ( 4 + 6 ) = = 2.4.(A) 4+x i / + 3 (A) + 60 (A) = i / =
* Entry of the multiplication procedure is omitted between and a variable.
The following nancial functions are available. Use NORMAL mode to perform nancial calculations. TVM (Time Value of Money) solver: Analyze equal and regular cash ows. These include calculations for mortgages, loans, leases, savings, annuities and contracts or investments with regular payments. Amortization calculations: Calculate and create amortization schedules using values stored in the TVM solver. Discounted cash ow analysis: Analyze unequal cash ows and calculate NPV (net present value) and IRR (internal rate of return). Bond calculations: Solve bond prices or yields to maturity with accrued interest. Depreciation calculations: Obtain depreciation base values using three types of calculation methods. Conversion between APR and EFF: Interest rates can be converted between APR (annual, or nominal percentage rate) and EFF (effective interest rate). Day and date calculations: Calculate dates and the number of days between dates. Percent change/Compound interest calculations: Calculate percent change (increase or decrease) and compound interest rates. Cost/Sell/Margin/Markup calculations: Calculate cost, selling price and margin/markup. Breakeven calculations: Calculate breakeven points (quantity) using xed costs, variable costs per unit, unit prices, and prot.
Variables used in nancial calculations
Financial calculations use multiple variables. By entering known values into variables, you can obtain unknown values. Variables used in nancial calculations are categorized into the following two types, depending on the entry method. TVM variables: Variables that are used in the TVM solver. These include N, I/Y, PV, PMT and FV. You can store, recall or calculate values directly using the corresponding keys. Listed nancial variables: Variables that are organized into lists in different categories. These variables can be accessed using the z/i cursor keys in each calculation. P/Y and C/Y in the TVM solver are of this type of variable.
For calculation only
For entry or calculation Calculated automatically
Notes: During nancial calculation, the word calculating! will be displayed on the screen. You can press s at this time to cancel the calculation. Calculation-only and automatically calculated variables have no default values. The symbol will be displayed if the value of the displayed variable has not been calculated yet (for variables that can be calculated).
This calculator assumes interest is compounded periodically in nancial calculations (compound interest). Compound interest accumulates at a predened rate on a periodic basis. For example, money deposited in a passbook saving account at a bank accumulates a certain amount of interest each month, increasing the account balance. The amount of interest received each month depends on the balance of the account during that month, including interest added in previous months. Interest earns interest, which is why it is called compound interest. It is important to know the compounding period of a loan or investment before starting, because the whole calculation is based on it. The compounding period is specied or assumed (usually monthly).
Cash ow diagrams
The direction of arrows indicates the direction of cash movement (inow and outow) with time. This manual uses the following cash ow diagrams to describe cash inows and outows. Inflow (+) Present value (PV) Time Cash flow Payment (PMT) Outflow () Future value (FV).
TVM (Time Value of Money) Solver
Analyze equal and regular cash ows. These include calculations for mortgages, loans, leases, savings, annuities, and contracts or investments with regular payments. Note: Discounted cash ow analysis can be done using unequal cash ows (see page 37). An amortization schedule can be calculated using the information stored in the TVM solver (see page 33).
Variables used in the TVM solver
Variable N I/Y PV PMT FV P/Y C/Y Corresponding variable key N f v u T.w Description Total number of payments Interest rate per year Present value Payment Future value Number of payments per year Default value 1
Calculate the total num- 20. < N ber of payments and store in N. Enter the present value. 56000 v Enter payment. , 440 u
(-44)~PMT ~FV I/Y=
Enter the future value.
Calculate the annual interest rate.
Answer: The annual interest rate is 7.17%. Note: If you make a mistake, press L to erase the number and enter the correct number to continue. After pressing the TVM variable key, you must re-enter values from the beginning.
Calculating basic loan payments
Calculate the quarterly payment for a $56,000 mortgage loan at 6.5% compounded quarterly during its 20-year amortization period. PV = $56,000 I/Y = 6.5% FV = 0. PMT = ? N = years = 80
Procedure Set all the variables to default values. Key operation.b Display
Make sure ordinary annuity is set (BGN is not displayed). Set the number of payments per year to 4.w4Q
Conrm the number of i compounding periods per year. Quit the P/Y and C/Y set- s tings. Calculate the total number of payments and store in N. Enter the present value.
20. < N
56000 v 0T 6.5 f
56~PV ~FV 6.5~I/Y PMT=
Enter the annual interest rate. Calculate the quarterly payment.
Answer: The quarterly payments are $1,255.86.
Calculating future value
You will pay $200 at the end of each month for the next three years into a savings plan that earns 6.5% compounded quarterly. What amount will you have at the end of period if you continue with the plan? FV = ? PV = 0 I/Y = 6.5% (quarterly). PMT = $200 N = years = 36
PMT = $200
Set the number of com- i 4 Q pounding periods per year to 4. Quit the P/Y and C/Y settings. s
Calculate the total num- 3. < N ber of payments and store in N. Enter the present value. 0 v , 200 u
(-2)~PMT 6.5~I/Y FV=
Enter the annual interest rate. Calculate the future value.
Answer: You will have $7,922.19 at the end of the three-year period.
Calculating present value
You open an account that earns 5% compounded annually. If you wish to have $10,000 twenty years from now, what amount of money should you deposit now? FV = $10,000
N = 20 years PV = ?
I/Y = 5%
Key operation.b Display
Calculate and create amortization schedules using values stored in the TVM solver. Note: Prior to using amortization, you need to enter values into TVM variables.
Variables used in amortization
Variable AMRT P1 AMRT P2 BALANCE PRINCIPAL INTEREST Description Start of payment (nth time) End of payment (nth time) Remaining balance after payment Principal paid Interest paid over the specied periods Default value
BALANCE, PRINCIPAL and INTEREST are calculated automatically, so no default values are set. AMRT P1 and AMRT P2 must be between 1 and 9,999.
Refer to page 19 for basic variable operations. 1. Press s to clear the display. Make sure the calculator is in NORMAL mode. All the TVM solver variables retain their previously entered values. If you wish to clear all the data, press. b. 2. Select ordinary annuity or annuity due using. ". 3. Enter the appropriate numeric values for the variables used in the TVM solver. Conrm the values of N, I/Y, PV, PMT, FV, P/Y and C/Y. 4. Press * to use amortization calculation. 5. Enter a value for AMRT P1 and press Q.
6. Press i, enter a value for AMRT P2 and press Q. 7. Display values for BALANCE, PRINCIPAL and INTEREST by pressing i once for each. Each value is calculated automatically.
8. Press i to calculate the next period of the amortization schedule. 9. Repeat steps 5 to 7 above. If you press @ during AMRT P1 and AMRT P2 entry, the values for the next period of payment will be automatically calculated and displayed. To end amortization calculations, press s. Pressing s during entry will clear the value entered.
Calculating mortgage payments and generating an amortization schedule
1. Calculate the monthly payment of a 20-year loan with a loan amount of $90,000 and a 5.45% APR.
Make sure ordinary annuity is set (BGN is not displayed). Set TVM solver variables and calculate payment. w 12 Q s 20. < N 90000 v 0 T 5.45 f @ u
Answer: The monthly payment is $616.56. Now generate an amortization schedule for the rst 5 years of the loan. If the rst payment is in August, the rst year has 5 payment periods and the following years have 12 payment periods each. 2. Calculate the amortization schedule for the rst year.
Procedure Key operation Display Change to amortization * 1 Q calculation and enter 1 (August) for the starting payment. Enter 5 (December) for the ending payment. Display the remaining balance. i5Q
AMRT P1= AMRT P2=
Procedure Display the principal paid. Display the interest paid.
Key operation i
3. Calculate the amortization schedule for the second year.
Procedure Key operation Display Change amortization i6Q schedule to the second year and enter 6 (January) for the starting payment. Enter 17 (December) i 17 Q for the ending payment. Display the remaining balance. Display the principal paid. Display the interest paid. i
Change the rst cash , 30000 J ow value from 25,000 to 30,000. Change the frequency of 5000 from 2 to 1. Add a new data set (6000) immediately before 5000.
iiii CF N3= iii1J. e 6000 J
To conrm the corrections, press. z to jump to the rst data item and press i to browse through each data item.
Variables used in discounted cash ow analysis
Variable RATE (I/Y) NET_PV Description Internal rate of return (IRR) Net present value (NPV) Default value 0
The variable RATE (I/Y) is shared by the variable I/Y. NET_PV is for calculation only and has no default value. The BGN/END setting is not available for discounted cash ow analysis.
NPV and IRR
The calculator solves the following cash ow values: Net present value (NPV): The total present value of all cash ows, including cash paid out (outows) and cash received (inows). A protable investment is indicated by a positive NPV value. Internal rate of return (IRR): The interest rate that gives a net present value of zero.
Refer to page 19 for basic variable operations. 1. Press s to clear the display. Make sure the calculator is in NORMAL mode.
2. Enter cash ow data. Refer to page 37 for instructions on entering cash ow data. 3. Press. < to begin discounted cash ow analysis. If a previously entered cash ow value is displayed, press s to exit and then press. <. 4. To nd NPV or IRR, do the following: To obtain NPV: Enter the interest rate (discounted rate) into RATE(I/Y) and press Q. Move to NET_PV and calculate by pressing i and @. To obtain IRR: Press @ to calculate IRR (RATE(I/Y)). Note: If Error 5 is displayed in step 4, or if you want to nd another IRR, enter an estimated value into RATE(I/Y) and calculate again in step 4.
Solving for unequal cash ows
Your company pays $12,000 for a new network system, and expects the following annual cash ows: $3,000 for the rst year, $5,000 for the second to fourth years, and $4,000 for the fth year. At what IRR does the net present value of the cash ows equal zero? $3,000 $5,000 $5,000 $5,000 $4,000
$12,000 1. Enter cash ow data.
Setting the day-count method
You can toggle between the actual calendar (365 days plus leap years) and a 360-day calendar (12 months of 30 days each) using. &. The actual calendar is set by default (360 is not displayed). The calendar range is from January 1, 1901 to December 31, 2099.
Refer to page 19 for basic variable operations. 1. Press s to clear the display. Make sure the calculator is in NORMAL mode. 2. Select bond calculations by pressing #. To end bond calculations, press s. If you press s during entry, any entered values will be cleared.
3. Change the day-count setting, if necessary, by pressing. &. 4. Enter the coupon rate (%) into COUPON (PMT) by entering the value and pressing Q. 5. Enter the redemption value into REDEMPT (FV) by pressing i, entering the value, and pressing Q. 6. Enter the date of bond purchase into M-D-Y 1 (or D-M-Y 1) by pressing i, entering the date, and pressing Q. For date entry, refer to page 47, Entering dates. 7. Enter the redemption date into M-D-Y 2 (or D-M-Y 2) by pressing i, entering the date, and pressing Q. For date entry, refer to page 47, Entering dates. 8. Enter the number of coupon payments per year into CPN/Y (N) by pressing i, entering the value, and pressing Q. 9. To nd bond price or yield to maturity, do the following: To obtain bond price (PRICE (PV)): Enter annual yield (%) into YIELD(I/Y) by pressing i, entering the value, and pressing Q. Move to PRICE (PV) and calculate by pressing i and @. Display the accrued interest (ACCU INT) by pressing i. The accrued interest is calculated automatically. To obtain yield to maturity (YIELD (I/Y)): Move to PRICE (PV) and enter the bond price by pressing i i, entering the value, and pressing Q. Move to annual yield, YIELD (I/Y) and calculate by pressing z @. Display the accrued interest (ACCU INT) by pressing i i. The accrued interest is calculated automatically.
Calculating bond price and accrued interest A $100, 20-year, 6.5% coupon bond is issued to mature on August 15, 2023. It was sold on November 3, 2006 to yield the purchaser 7.2% compounded semiannually until maturity. At what price did the bond sell? Also calculate the accrued coupon interest.
A pseudo-random number, consisting of three signicant digits from zero up to 0.999, can be generated by pressing. | 0 =. To generate further random numbers in succession, press = for each number. Press s to exit. To display numbers with three signicant digits, set the display notation settings to oating point.
To simulate a die roll, a random integer between 1 and 6 can be generated by pressing. | 1 =. To generate further random numbers in succession, press = for each number. Press s to exit.
To simulate a coin ip, 0 (heads) or 1 (tails) can be randomly generated by pressing. | 2 =. To generate further random numbers in succession, press = for each number. Press s to exit.
An integer between 0 and 99 can be generated randomly by pressing. | 3 =. To generate further random numbers in succession, press = for each number. Press s to exit.
Example Key operation Display Pick a random number s. | 0 between zero and 9.99. x 10 =
The result will not be the same each time this operation is performed.
Calculation results are internally obtained in scientic notation with a mantissa of up to 14 digits. However, because calculation results are displayed in the form designated by the display notation and the number of decimal places indicated, a displayed calculation result may differ from that held internally. By using the modify function (the 2nd function of the + key), the internal value is converted to match that of the display, so that the displayed value can be used without change in subsequent operations.
Example = ANS ANS 9 = [FIX, TAB = 2] Key operation s589= x 9 = *1 589=. l x 9 = *2 Result
5.9 0.56 9
Statistical calculations can be performed in STAT mode. The symbol will be visible if you are in STAT mode. There are seven sub-modes within STAT mode, corresponding to each of the functions below:
Key operation m10 m11 m12 m13 m14 m15 m16 Sub-mode SD: Single-variable statistics LINE: Linear regression QUAD: Quadratic regression EXP: Exponential regression LOG: Logarithmic regression PWR: Power regression INV: Inverse regression Display
Stat Stat 1 Stat 2 Stat 3 Stat 4 Stat 5 Stat 6
Entering statistical data
Before you can perform statistical calculations, you will need to enter your data. In STAT mode, use > and J (the Q key) to enter the following: For a single-variable data set: Value J Value > frequency J (To enter multiples of the same value) For a two-variable data set: x value > y value J x value > y value > frequency J (To enter multiples of the same x and y values) Note: Before entering data, clear any previously entered data from memory by doing either of the following: Press. b in STAT mode. Switch sub-modes within STAT mode.
Single-variable statistical calculations
Section results only.
Linear regression calculations
Section and results, except for coefcients c. The estimate of y for a given x (estimate y) and the estimate of x for a given y (estimate x) can also be found.
Quadratic regression calculation
Section and results, and coefcients a, b, and c in the quadratic regression formula (y = a + bx + cx2). In quadratic regression calculations, no correlation coefcient r can be obtained.
The estimate of y for a given x (estimate y) and the estimate of x for a given y (estimate x) can also be found. When there are two x values, the COMP symbol will appear. Press @ to switch between x values.
Exponential regression, logarithmic regression, power regression, and inverse regression calculations
Section and results, except for coefcients c. The estimate of y for a given x (estimate y) and the estimate of x for a given y (estimate x) can also be found. Because the calculator converts each formula into a linear regression before actual calculation takes place, it obtains all statistics except coefcients a and b from converted data rather than from entered data.
Variables n x sx x x x y sy y y y
Content Number of samples Mean of samples (x values) Sample standard deviation (x values) Population standard deviation (x values) Sum of samples (x values) Sum of squares of samples (x values) Mean of samples (y values) Sample standard deviation (y values) Population standard deviation (y values) Sum of samples (y values) Sum of squares of samples (y values) Sum of products of samples (x, y) Coefcient of regression equation Coefcient of regression equation Coefcient of quadratic regression equation Correlation coefcient
xy a b c r
Use i or f to perform a variable calculation in STAT mode.
Single-variable statistical calculation
Key operation m10. b Result
DATA 50 x= x = n= x =
95 J 80 J J
DATA SET=100 DATA SET=200 DATA SET=300 DATA SET=400 DATA SET=500
75 > 3 J
50 J f= f2 fn fU f[ f4.;= ( 95 - i
n= = =
sx = sx2 =
(95 x ) =)8i 10 + 50 = sx 4 x 10 + 50
2 Example DATA x y 15 a= b= r= sx = sy = x=3 y = 40 25
Linear regression calculation
Key operation m11 Result
2>5J J 12 > 24 J 21 > 40 > 3 J 15 > 25 J fa fb fr f4 f5 3.? 46. 9
a= b= r= Sx= Sy= 3y 46x
y = ? x = ?
3 Example DATA x 15 a= b= c= x = 10 y = 22 y 71
Key operation m12 Result
12 > 41 J 8 > 13 J 5>2J 23 > 200 J 15 > 71 J fa fb fa
DATA DATA DATA DATA DATA
a= b= c= 10y
10. ? 22. 9 @ @
000 SET=100 SET=200 SET=300 SET=400 SET=-2449 963
Financial Calculation Formulas
PMT, PV, FV, N
If PMT 0 or N 1 then nd i using the following equations: Error + Error i 1
Calculations (for PV, PMT, and i, see the TVM solver) END INT(1) = ROUND(PV i) BGN INT(1) = 0 PRN(1) = INT(1) + ROUND(PMT) BAL(1) = PRN(1) + PV INT(m) = ROUND(BAL(m 1) i) PRN(m) = INT(m) + ROUND(PMT) BAL(m) = PRN(m) + BAL(m 1) ROUND(NUM): If a display notation tab setting has been chosen, NUM is rounded and truncated to the specied number of places after the decimal point. Results
Error AMRT P1 > AMRT P2
Discounted cash ow analysis
If PMT = 0 then
RATE(I/Y) where i = 100 , CFimax = the maximum data set number IRR is obtained as i, which satises NPV = 0 in the above equations.
If N = 1 then
In its bond calculations, this calculator conforms to rules set up by the book titled Standard Securities Calculation Methods, by Jan Mayle, Securities Industry Association, 1993. Bond calculation is based on the following rules: 1. Whenever the redemption date happens to be the last day of a month, coupons are also paid on the last days of months. For example, if coupon payments are semi-annual and the redemption date is September 30, coupon payments occur on March 31 and September 30. 2. If coupons are to be paid twice a year and the redemption date is set to August 29, 30, or 31, coupon payments for February occur on the 28th (29th for leap years). 3. The Odd Coupon is not supported. 4. All data stored or calculated for bonds are assumed to be positive values. Negative values in any of the variables used by bond calculations will cause errors. The formulas used for bond calculations are shown using the following variable denitions: TD: Total number of days in the coupon period that begins with the coupon date previous to the settlement date and ends with the rst coupon date after the settlement date. (On the 360-day calendar, TD is 180 for semi-annual coupon and 360 for annual coupon.) PD: The number of days preceding the settlement date in the coupon period described above. (see Day and date calculations) FD: The number of days following the settlement date in the coupon period described above (in TD). FD = TD PD NP: The number of whole coupon periods between the settlement date and the redemption date (rounded up to the next highest whole number, if necessary).
An error will occur in a statistical calculation if: The absolute value of an intermediate or calculation result is equal to or greater than 1 10100. The denominator is zero. An attempt is made to nd the square root of a negative number. No solution exists for a quadratic regression calculation.
where MAR = MARGIN, MU = MARK UP
Errors and Calculation Ranges
An error will occur if an operation exceeds the calculation ranges, or if a mathematically illegal operation is attempted. When an error occurs, pressing g or y automatically moves the cursor back to the place in the equation where the error occurred. Edit the equation or press s to clear the equation. Note: If an error occurs during the automatic calculation of a listed nancial variable, pressing s, g, or y displays the rst variable in the function.
Error codes and error types
Syntax error (Error 1): An attempt was made to perform an invalid operation. Ex. 2 + - 5 = During the editing or insertion of cash ow or statistical data, a value was entered but. ? was pressed before J. Calculation error (Error 2): The absolute value of an intermediate or nal calculation result equals or exceeds 10100. An attempt was made to divide by zero. The calculation ranges were exceeded while performing calculations. There was a nancial calculation error, such as: an error listed in Financial Calculation Formulas (see pages 72 75) occurred an attempt was made to nd I/Y when PV, PMT N, and FV contain all negative or all positive values an attempt was made to nd IRR when the cash ow data contains all negative or all positive values an attempt was made to nd YIELD in a bond calculation when any of COUPON, REDEMPT, or PRICE contain negative values Depth error (Error 3): The available number of buffers was exceeded. (There are 10 buffers for numeric values and 24 buffers for calculation instructions). The number of entered cash ow and statistics data items combined exceeded 100.
Equation too long (Error 4): The equation exceeded its maximum input buffer (160 characters). An equation must be shorter than 160 characters. No solution (Error 5): The iteration limit was exceeded while calculating one of the following values in an overly complex problem: I/Y (TVM solver) IRR (Discounted cash ow analysis)*1 YIELD (Bond calculations)
Time Value of Money
Ordinary Annuity and Annuity Due Formula
There are two types of annuity payments. An annuity that is paid at the end of the compounding periods is called an ordinary annuity, whereas an annuity that is paid at the beginning of the compounding periods is called an annuity due.
The Ordinary Annuity Calculation Formula
The ordinary annuity formula, as shown in the document Derivation of the Annuity Calculation Formula is
Rn 1 PV R = a R 1
PV = The original loan amount (occasionally the notation K is used).
n = The number of compounding periods constituting the duration of the or loan.
PV R n = The future value (FV) of K after n compounding periods. i = The interest rate per compounding period.
a = The annuity
i R = 1 + 100
Expressing the interest rate as a decimal fraction results in
R = (1 + i )
The graphical illustration of the ordinary annuity payment is shown in Fig. 1, where a is the annuity payment. The present value
usually represented by a loan amount, occurs at the beginning of the 1st compounding period. The future value
occurs at the end of the last compounding period.
Compounding Period 1 Compounding Period 2 Compounding Period 3 Compounding Period 4 Compounding Period n
a a a a a FV
The Annuity Due Calculation Formula
The annuity due formula indicates that the annuities are paid at the beginning of the compounding periods, as illustrated in Fig. 2, where a is the annuity payment. The present value
usually represented by a loan amount, occurs at the beginning of the 1st compounding period, as does the 1st annuity payment. The future value
occurs only at the end of the last compounding period, since the last annuity must remain invested for the duration of the loan period. Fig. 1
PV a a a a a FV
Future Value of the Loan
Since the loan will be paid off at the beginning of the last compounding period, the *future value
is defined as
*Although the annuities are paid at the beginning of the compounding periods, the future value of the loan is still calculated based on the present (initial) value, despite the fact that the 1st annuity is paid simultaneously with the initiation of the loan.
The difference lies in the fact that the interest is compounded for one compounding period less, resulting in a somewhat lesser total debt (future value of the loan) as compared to an ordinary annuity.
resulting in the annuity due calculation formula is
Rn 1 PV R n 1 = a R 1
Note that the right hand side of the equations 1 and 2 are identical. The reason is that the number of annuity payments remains the same whether the type of annuity is an ordinary annuity or an annuity due. The left hand side of the equations are, however different. If the annuities are paid at the beginning of the compounding periods, the present value
is compounded one period less than compared to an ordinary annuity loan. Consequently, the compounding of interest takes place only for compounding periods, since the 1st annuity payment was paid simultaneously with obtaining the loan. Therefore, the borrower can be charged interest only for
( n 1) ( n 1)
compounding periods, whereas the lender must accept that the annuity payments remain invested through the duration of the loan in order to attain the future value
Test of the Equation 2
A. Using a SHARP EL-738 Business/Financial calculator. Assume that a person secures a loan with the following data:
PV = 10000 i = 5 percent APR
n = 20
cp = compounding period = year
The annuity loan is an annuity due loan. What is the annuity payment per year?
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Performing the required calculations using a SHARP EL-738 Business/Financial calculator results in an *annuity of
a = 764.22
* Note that the notation a for annuity usually is denoted PMT on financial calculators.
B. Manual Calculation Inserting the values in the example in the equation 2 results in
Rn 1 PV R n 1 = a R 1 1.10000 1.05201 = a 1.
a= 10000 1.05201 (1.05 1) 1.0520 1
10000 2.526950195 0.05 1.653297705
The identical test results indicate that the equation 2 is correct.
Derivation of the Annuity Due Formula
Based on the cash flow analysis in Fig. 2, the interest rate for the loan amount, represented by the variable present value
is compounded for
compounding periods. Thus, the future value
of the loan is
FV = PV R n1
The sum of the future values
of the annuities including compound interest is obtained as follows: The 1st annuity
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periods, resulting in
FVa1 = a1 R n 1
The 2nd annuity
FVa2 = a2 R n 2
etc. Finally, the last annuity
FVan = an R n n
The sum of all future values of the annuities is
FVatot = a1 R n 1 + a2 R n 2 + a3 R n 3 +. + an R n n
a1 = a2 = a3 =. = an = a
Thus, one obtains
FVatot = a R n 1 + a R n 2 + a R n3 +. + a R 0
FVatot = PV R n1
PV R n 1 = a R n1 + R n 2 + R n 3 +. + R 0
Multiplying the equation 15 by the factor
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( R ) PV R n1 = ( R ) a ( R n1 + R n 2 + R n 3 +. + R 0 )
PV R n = a R n R n 1 R n 2 . R1
Adding the equations 15 and 17 results in
PV R n 1 = a R n1 + R n 2 + R n 3 +. + R 0 PV R n
( = a (R
R n 1 R n 2
) . R )
(15) (17) (18) (19) (20)
PV R n 1 PV R n = a R 0 a R n PV R n1 (1 R ) = a 1 R n
1 R n PV R n 1 = a 1 R Rn 1 PV R n 1 = a R 1
If the annuity due formula (see the equation 21)
Rn 1 = a R 1
is multiplied by the factor
Rn 1 R PV R n 1 = R a R 1 R n 1 PV R n = a R R 1
Thus, the annuity due formula is obtained by multiplying the ordinary annuity formula by the factor
for the applicable compounding period, depending on whether the interest rate is expressed as a decimal fraction or as the actual percentage value.
A person has secured a 5-year home renovation loan of $10000 at a 5 percent annual interest rate (APR). The annuities are to be paid annually, at the beginning of each compounding period (year). What is the annuity amount ? Alternative 1: Calculate the annuity using a SHARP EL-738 Business /Financial calculator The annuity is
a = 2199.76
Alternative 2: Manual calculation of the annuity Inserting the values in the example in the equation 21
1.10000 1.0551 = a 1.
10000 1.054 0.05 1.607.55 0.276281562
a = 2199.75
Both calculation results are identical, indicating that the equation 2 is correct.
A 30-year mortgage of $125000 with an annual interest rate (APR) of 6.5 percent is paid at the beginning of each quarter. What is the annuity amount ? Alternative 1: Calculate the annuity using a SHARP EL-738 Business /Financial calculator Turn on the EL-738: ON/C Reset all memory banks:
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Set the payments per year: 2ndF P/Y 4 ENT
Exit the P/Y function: ON/C Enter loan data: 125000 PV 6.5 I/Y 120 N COMP PMT
The calculator returns the value
a = 2336.44
While financial calculators usually use the notation PMT (payment) for annuity, the shorter notation a is preferred during manual calculations. Alternative 2: Manual calculation of the annuity Inserting the values in the example in the equation 21
Note that the interest rate per compounding period is
6.5 = 1.625 4
i = 0.01625
expressed as a decimal fraction. The number of compounding periods is
n = = 120
This results in
1.125000 1.016251201 = a 1.
a= 125000 1.016251201 0.01625 1.01625120 1
13830.24 5.919377615 a = 2336.44 a=
Again, both calculation results are identical, indicating that the equation 2 is correct.
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