Programming the TI-83 and TI-84 Calculators for Finance

Transcription

1 Programming the TI-83 and TI-84 Calculators for Finance Steven P. Rich, Baylor University The power of the Texas Instrument TI-83 and TI-84 calculators lies in the ability to create programs for them. This paper discusses how to download, install, and use programs I have written for my TI-83 Plus Silver Edition calculator to work with advanced financial models. Equations created to use the TI-83 s built-in math solver allow the user to solve for any unknown variable in the Black-Scholes Option Pricing Model, any unknown variable in the Put-Call Parity model, or any variable in problems involving growing (or non-growing) annuities. Less powerful, but still useful, programs allow the user to solve for put and call values or the present value or future value of (or non-growing) growing annuities. Examples included in the paper allow the user to practice using the programs for the TI-83 and TI-84. Detailed instructions for entering (rather than downloading) my programs are included in appendices. INTRODUCTION The Texas Instruments TI-83 Plus Silver Edition (and the TI-83 Plus) graphing calculator comes with a financial module that allows the user to solve basic time value of money problems. 1 However the true power of the TI-83 calculator stems from being able to program the calculator to work with more advanced financial models. 2 In this paper, I discuss the programs I ve written for the TI-83 calculator that can be used to solve for call values using the Black-Scholes Option Pricing Model, to solve for put values using Put-Call Parity, and to solve for the values of growing annuities. 3 The most powerful programs use the TI-83 s built in math solver. With these solver equations, the user can solve for any unknown variable. For example, with the option solver equations, the user can solve for the value of a call, the implied volatility, or any of the basic variables that determine the value of a call in the Black-Scholes Option Pricing Model. Less powerful but still useful programs prompt the user for input variables and then simultaneously solve for the values of calls and puts, the present value of growing annuities, or the future value of growing annuities. Throughout this paper, I will indicate keys on the calculator with brackets. The default color for keys on the TI-83 Plus Silver Edition is black. For example, [MATH] is the black key labeled Math. Keys that are blue or white will be labeled as such. The [2 nd ] key is yellow and the [ALPHA] key is green. Most keys have a default function as well as a yellow function and a green function. Yellow functions are entered by pressing the [2 nd ] key first and green functions (mostly letters) are entered by pressing the [ALPHA] key first. The rest of the paper is organized as follows. In Section 2, I discuss how to download (from my website), install, and use the solver equations for option (call and put) valuation and for 1 See Czyrnik (2004) for an introduction to the basic financial functions built into the TI-83 calculator. 2 All programs written for the TI-83 calculator will run on the new TI-84. The new TI-84 has is faster and has more memory. 3 Note that the solver equations can also be used for non-growing annuities by setting the growth rate to 0.

2 growing annuity valuation. In Section 3, I discuss how to download (from my website), install, and use the programs that solve for call and put values and that solve for the present value or future value of growing annuities. In Appendix A, I discuss how to enter the solver equations directly into the calculator. In Appendix B, I discuss how to directly enter the programs directly into the calculator. MATH SOLVER 4 The math solver equations can be downloaded from my webpage at: 5 After downloading to your PC, the files can be transferred to a TI-83 Plus using the TI Connect Program. 6 In the TI Connect Program, click on the TI Group Explorer icon and then find and click on the file you want to download to the TI-83. After clicking on the file, click on the Send to Device icon in the TI Group Explorer. 7 Once downloaded into the calculator, the equation solver can be accessed by pressing the [MATH] key then pressing [0]. 8 Loading Equations into the Equation Solver To load the appropriate equation once you are in the equation solver: 1) press the blue up arrow until EQUATION SOLVER eqn:0=y i is displayed, 2) press [VARS], 3) press the right blue arrow key to highlight Y-VARS, 4) press [1] 5) press the white number key of the equation you want to load into solver. For example, if you have loaded the Black-Scholes equation into Y 8 (the slot I ve assigned it to with the equations on my website), get to the equation solver by: 1) pressing [MATH] 2) pressing [0] Then load equation Y 8 (in this case the Black-Scholes equation) into the solver: 1) press the blue up arrow until EQUATION SOLVER eqn:0=y i is displayed, 4 Discussion of the solver equations begins on page 72 of the TI-83 Plus manual. 5 Note that I ve assigned the growing annuity equation to Y 7, the Black-Scholes equation to Y 8, the put-call parity equation to Y 9. 6 For a more complete discussion of downloading files to a TI-83, see Chapter 19 of the TI-83 Plus manual. The discussion of receiving items starts on page The icon shows a calculator with an arrow pointing towards it. The icon only appears after a relevant file is selected. 8 Alternatively, after pressing the [MATH] key, the blue arrow keys can be used to scroll down to Solver. 2

3 2) press [VARS], 3) press the right blue arrow key to highlight Y-VARS, 4) press [1] to select 1:Function 5) press the white number key [8] to load Y 8 into the math solver. Note that if you have previously loaded equation 8 into the solver, then you do not need to reload it. Once an equation is loaded into the solver, you enter values for all of the variables except the one you wish to solve for. Then use the up or down blue arrow keys to move to the variable you wish to solve for and press [ALPHA] then [ENTER]. 9 Black-Scholes Option Pricing Model 10 The Black-Scholes solver equation has 6 variables: V, X, R, S, T, and C. The variables are defined as: V = value of underlying asset today. X = exercise or strike price. S = standard deviation of return on the underlying asset. The value for S must be entered as a decimal rather than as a percentage. R = risk-free interest rate. The rate is an annual rate assumed to be compounded continuously. The value for R must be entered as a decimal rather than as a percentage. T = maturity of option in years. C = value of call. If values are entered for 5 of the variables, then the 6 th can be solved for. Examples 1. Assume that a stock with a standard deviation of returns of 41% has a current price of $35 per share. Assume that a call on this stock that has an exercise value of $30 and that the option expires 180 days from today. Finally, assume that the interest rate on a T-bill maturing 180 days from today is 3.1% per year compounded continuously. What is the value of this call? First, enter the input variables: V = X = R =.031 S =.41 T = 180/365 = (note that if you type in 180[ ]365[ENTER], the TI-83 calculates the result) 9 The [ENTER] key becomes the [SOLVE] key if the green [ALPHA] button is pressed first. 10 Note that the Black-Scholes Option Pricing Model equation must be loaded into the solver before it can be used. As noted earlier, I have assigned the Black-Scholes Option Pricing Model to equation Y 8. 3

4 To solve for the value of the call, move the cursor to C= then press [ALPHA] and then [ENTER] to get an answer of Assume that a call that expires 62 days from today allows you to purchase for $70 a stock currently priced at $65. The rate on a T-bill maturing 62 days from today is 10% per year compounded continuously. What is the implied standard deviation of returns on the stock if the value of the call is $3.30? First, enter the input variables: V = X = R =.1 T = 62/365 = (note that if you type in 62[ ]365[ENTER], the TI-83 calculates the result) C = 3.30 To solve for the standard deviation of returns on the stock, move the cursor to S= then press [ALPHA] and then [ENTER] to get an answer of Thus the implied volatility on the stock is 45.34%. Put-Call Parity 11 The Put-Call Parity solver equation has 6 variables: C, V, X, R, T, and P. The variables are defined as: C = value of call. V = value of underlying asset today. X = exercise price. R = risk-free interest rate. The rate is an annual rate assumed to be compounded continuously. The value for R must be entered as a decimal rather than as a percentage. T = maturity of option in years. P = value of put. If values are entered for 5 of the variables, then the 6 th can be solved for. Example Assume that a stock has a current price of $35 per share. Assume that a call on this stock with an exercise value of $30 that expires 180 days from today has a value of $7.04. Finally, assume that the interest rate on T-bills maturing 180 days from today is 3.1% per year compounded continuously. What is the value of a put on the stock with an exercise price of $30 that expires 180 days from today? 11 Note that the Put-Call Parity equation must be loaded into the solver before it can be used. I have assigned Put- Call parity to equation Y 9. 4

5 First, enter the input variables: C=7.04 V = X = R =.031 T = 180/365 = (note that if you type in 180[ ]365[ENTER], the TI-83 calculates the result) To solve for the value of the put, move the cursor to P= then press [ALPHA] and then [ENTER] to get an answer of Note that if you solve for example 1 from the Black-Scholes section then solve for the value of the put, all of the variables remain in the calculator. As a result, you do not have to re-enter the values. Growing Annuity 12 The growing annuity solver equation has 6 variables: P, C, R, G, N, and F. The variables are defined as: P = present value. C = first periodic cash flow (payment). R = interest rate per period. The interest rate must be entered in decimal rather than percentage form. G = growth rate of payments. The growth rate of the periodic cash flows must be entered in decimal rather than percentage form. N = number of periodic cash flows. F = future value. If values are entered for 5 of the variables, then the 6 th can be solved for. Note, however, that some variables can be entered as 0. For example, to solve for the present value of a single, future cash flow, all variables are set to 0 except for R, N, and F. Note also that when entering values for P, C, and F, signs are important. If inflows/deposits are entered as positive numbers then outflows/withdrawals should be entered as negative numbers. Examples 1. Assume that a year from today you plan to deposit $100 into a savings account that pays an interest rate of 4.5% per year. After the first deposit you plan to make additional deposits each year but plan for each deposit to be 1% larger than the pervious one (the size of your 12 Note that the growing annuity equation must be loaded into the solver before it can be used. I have assigned the growing annuity equation to slot Y 7. 5

6 deposits grow by 1% per year). How much will be in your account after you have made 10 total deposits? First, enter the input variables: P = 0.00 C = R =.045 G =.01 N = 10 To solve for the future value, move the cursor to F= then press [ALPHA] and then [ENTER] to get an answer of Assume you purchase a bond that matures for $1000 five years from today and which pays an annual coupon of $50 per year. What is the value of the bond if the required return on the bond is 6%? First, enter the input variables: C = R =.06 G = 0 N = 5 F = 1000 To solve for the present value, move the cursor to P= then press [ALPHA] and then [ENTER] to get an answer of Assume that you want to save a total of $3000 by 10 years from today in an account that pays an annual rate of 5.2% per year. You currently have $1200 in your savings account and plan to begin making annual deposits into the account a year from today. If your first annual deposit is $75, at what rate will your deposits have to grow to achieve your goal? First, enter the input variables: P = C = R =.052 N = 10 FV = (Note: the - must be entered using the white [(-)] key rather than the blue [-] math operator key.) To solve for the rate at which your deposits must grow, move the cursor to G= then press [ALPHA] and then [ENTER] to get an answer of So the deposits must be increased by 1.38% per year. 6

7 PROGRAMS at: The finance programs I have written for the TI-83 can also be downloaded from my webpage After downloading to your PC, the files can be transferred to a TI-83 Plus using the TI Connect Program. 13 The programs can be run by pressing the [PRGM] key then the number key corresponding to the program you wish to run. Future Value of a Growing Annuity The future value of a growing annuity program prompts you for the following input variables: PV? = initial cash flow or balance today PMT? = 1 st payment in growing annuity N? = number of payments in growing annuity R? = interest rate. The rate must be entered in decimal form rather than percentage form. G? = growth rate for cash flows. The rate must be entered in decimal rather than percentage form. Example 1. Assume that a year from today you plan to deposit $100 into a savings account that pays an interest rate of 4.5% per year. After the first deposit you plan to make additional deposits each year but plan for each deposit to be 1% larger than the pervious one (the size of your deposits grow by 1% per year). How much will be in your account after you have made 10 total deposits? Steps: 1. Press the [PRGM] button. 2. Press the numeric key corresponding to the Future Value of a Growing Annuity program. If you have not renamed the program, it is called FV. 3. Press [ENTER] to run the program. 4. Enter information as prompted: PV?0[ENTER] PMT?100[ENTER] N?10[ENTER] 13 See Section 2 above for a discussion of how to download files from a PC to a TI-83. For a more complete discussion of downloading files to a TI-83, see Chapter 19 of the TI-83 Plus manual. The discussion of receiving items starts on page

8 R?.045[ENTER] G?.01[ENTER] 5. The program displays: FUTURE VALUE: Note that this is the same answer that was found using the math solver program. Present Value of a Growing Annuity The present value of a growing annuity program prompts you for the following input variables: FV? = terminal cash flow in addition to growing annuity cash flows PMT? = 1 st payment in growing annuity N? = number of payments in growing annuity R? = interest rate. The rate must be entered in decimal form rather than percentage form. G? = growth rate for cash flows. The rate must be entered in decimal rather than percentage form. Example 1. Assume you purchase a bond that matures for $1000 five years from today and which pays an annual coupon of $50 per year. What is the value of the bond if the required return on the bond is 6%? Steps: 1. Press the [PRGM] button. 2. Press the numeric key corresponding to the Present Value of a Growing Annuity program. If you have not renamed the program, it is called PV. 3. Press [ENTER] to run the program. 4. Enter information as prompted: FV?1000[ENTER] PMT?50[ENTER] N?5[ENTER] R?.06[ENTER] G?0[ENTER] 5. The program displays: PRESENT VALUE: Note that this is the same answer that was found using the math solver program. Option Valuation The option pricing model program prompts you for the following input variables: ASSET VALUE? = value of underlying asset today. STRIKE? = exercise or strike price. EXPIR.? = maturity of option in years. 8

9 RISK FREE? = risk-free interest rate. The rate is an annual rate assumed to be compounded continuously. The rate must be entered in decimal form rather than percentage form. STD DEV? = standard deviation of return on the underlying asset. The value for STD DEV must be entered as a decimal rather than as a percentage. Example Assume that a stock with a standard deviation of returns of 41% has a current price of $35 per share. Assume that both a call and put on this stock have an exercise value of $30 and expire 180 days from today. Finally, assume that the interest rate on T-bills maturing 180 days from today is 3.1% per year compounded continuously. What is the value of the call and put options on this stock? Steps: 1. Press the [PRGM] button. 2. Press the numeric key corresponding to the Options program. If you have not renamed the program, it is called OPTIONS. 3. Press [ENTER] to run the program. 4. Enter information as prompted: ASSET VALUE?35[ENTER] STRIKE?30[ENTER] EXPIR.? 180[ ]365[ENTER] RISK FREE?.031[ENTER] STD DEV?.41[ENTER] 5. The program displays CALL VALUE: 7.04 and it displays PUT VALUE: 1.58 Note that these are the same answers that were found using the math solver programs. CONCLUSIONS The Texas Instruments TI-83 and TI-84 calculators come with a built-in financial module that allows the user to work basic time value of money problems. However, programs I have written for the TI-83 allow the user to work with options and growing annuities. The most powerful programs written for calculator s math solver allow the user to solve for any unknown variable in the Black-Scholes Option Pricing Model, any unknown variable in the Put-Call Parity model, or any unknown variable in time value of money problems involving growing annuities. Less powerful, but still useful, programs written for the TI-83 allow the user to solve for put and call values, the present value of a growing annuity, or the future value of a growing annuity. In this paper, I discuss how to download, install, and use the programs I have written. Detailed instructions of how to write the programs (rather than download them from my website) are included in the appendices. 9

10 REFERENCES Black, Fisher and Myron Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81 (May-June 1973), p Czyrnik, Kathy, Corporate Finance and the TI-83 Calculator, Working paper presented at 2004 Financial Education Association Annual Meeting in Mystic Seaport, CT. Texas Instruments TI-83 Plus/TI-83 Plus Silver Edition Graphing Calculator Guidebook,

11 Appendix A Equations for Math Solver Equations can be directly entered by pressing the [Y=] key then typing the equation into any of the blank equation slots. 1. Black-Scholes Option Pricing Model a. Equation: Y i = V*normalcdf(-1E99,(ln(V/X)+(R+S^2/2)*T)/(S^2*T)^.5)- X*normalcdf(-1E99,(ln(V/X)+(R-S^2/2)*T)/(S^2*T)^.5)*(e^(-R*T))-C b. Notes: 1) Letters are entered by pressing the green [ALPHA] key then the relevant key on the calculator. For example, V is entered by pressing the green [ALPHA] key then the white [6] key. The white [6] key becomes V when the green [ALPHA] key is pressed first. 1) The function normalcdf must be entered by pressing [2 nd ] then [VARS] then [2]. Note that the [VARS] key becomes the [DISTR] key if the [2 nd ] key is pressed first. 2) The - that is part of -1E99 and R must be entered using the white [(-)] key rather than the blue [-] operator key. 3) The E in -1E99 must be entered as [2 nd ] then [,] rather than as the alphanumeric E. Note that the [,] key becomes the [EE] key if the [2 nd ] key is pressed first. 4) The exponential function e can be entered by pressing either [2 nd ] then [LN] or by pressing [2 nd ] then [ ] then [^]. Note that the [LN] key becomes [e x ] if the [2 nd ] key is pressed first and the [ ] key becomes [e] if the [2 nd ] key is pressed first. 2. Put-Call Parity a. Equation: Y i = C-V+X*(e^(-R*T))-P b. Notes: 1) The - that is part of R must be entered using the white [(-)] key rather than the blue [-] operator key. 2) See note 4 above. 3. Growing Annuity a. Equation: Y i = P + (C/(R-G))*(1-((1+G)/(1+R))^N)+F/(1+R)^N 11

12 Appendix B Programs 1. Future Value of a Growing Annuity a. Program: PROGRAM:FV :Input PV?,P :Input PMT?,C :Input N?,N :Input R?,R :Input G?,G :-(P*(1+R)^N+(C/(R-G))*((1+R)^N-(1+G)^N)) F :Disp FUTURE VALUE:,F b. Notes: 1) A new program is created by pressing [PRGM], pressing the blue right arrow until NEW is displayed, then pressing [1]. The program can be named by pressing the desired letter keys since the alpha lock is automatically engaged when naming the program. 2) The input function is created by pressing [PRGM], selecting I/O then [1]. 3) The quotation marks are created by pressing [ALPHA] then [+]. The blue [+] operator key becomes the [ ] key if the [ALPHA] key is pressed first. 4) The question mark is created by pressing [ALPHA] then the white [-] key. 5) The is created by pressing the [STO ] button. 6) The display function is created by pressing [PRGM], selecting I/O then [3]. 7) A blank space is entered by pressing the [ALPHA] key then the [0] key. 8) A : is created by pressing the [ALPHA] key then the [.] key. 2. Present Value of a Growing Annuity a. Program: PROGRAM:PV :Input FV?,F :Input PMT?,C :Input N?,N :Input R?,R :Input G?,G :-(F/(1+R)^N+(C/(R-G))*(1-((1+G)/(1+R))^N)) P :Disp PRESENT VALUE:,P 12

13 Appendix B Programs (continued) b. Notes: 1) The input function is created by pressing [PRGM], selecting I/O then [1]. 2) The quotation marks are created by pressing [ALPHA] then [+]. The blue [+] operator key becomes the [ ] key if the [ALPHA] key is pressed first. 3) The question mark is created by pressing [ALPHA] then the white [-] key. 4) The is created by pressing the [STO ] button. 5) The display function is created by pressing [PRGM], selecting I/O then [3]. 6) A blank space is entered by pressing the [ALPHA] key then the [0] key. 7) A : is created by pressing the [ALPHA] key then the [.] key. 3. Option Valuation a. Program: PROGRAM:OPTIONS :Input ASSET VALUE?,V :Input STRIKE?,X :Input EXPIR.?,T :Input RISK FREE?,R :Input STD DEV?,S : V*normalcdf(-1E99,(ln(V/X)+(R+S^2/2)*T)/(S^2*T)^.5)-X*normalcdf(- 1E99,(ln(V/X)+(R-S^2/2)*T)/(S^2*T)^.5)*(e^(-R*T)) C :Disp CALL VALUE:,C : C-V+X*(e^(-R*T)) P :Disp PUT VALUE:,P b. Notes: 1) The function normalcdf must be entered by pressing [2 nd ] then [VARS] then [2]. Note that the [VARS] key becomes the [DISTR] key if the [2 nd ] key is pressed first. 2) The - that is part of -1E99 and R must be entered using the white [(-)] key rather than the blue [-] operator key. 3) The E in -1E99 must be entered as [2 nd ] then [,] rather than as the alphanumeric E. Note that the [,] key becomes the [EE] key if the [2 nd ] key is pressed first. 4) The exponential function e can be entered by pressing either [2 nd ] then [LN] or by pressing [2 nd ] then [ ] then [^]. Note that the [LN] key becomes [e x ] if the [2 nd ] key is pressed first and the [ ] key becomes [e] if the [2 nd ] key is pressed first. 5) The input function is created by pressing [PRGM], selecting I/O then [1]. 6) The quotation marks are created by pressing [ALPHA] then [+]. The blue [+] operator key becomes the [ ] key if the [ALPHA] key is pressed first. 13

14 Appendix B Programs (continued) 7) The question mark is created by pressing [ALPHA] then the white [-] key. 8) The is created by pressing the [STO ] button. 9) The display function is created by pressing [PRGM], selecting I/O then [3]. 10) A blank space is entered by pressing the [ALPHA] key then the [0] key. 11) A : is created by pressing the [ALPHA] key then the [.] key. 14