Finance 2400 / 3200 / 3700 Lecture Notes for the Fall semester 2017 V.4 of Bite-size Lectures on the use of your Hewlett-Packard HP-10BII Financial Calculator Sven Thommesen 2017 Generated on 6/9/2017
USING THE HP-10BII FINANCIAL CALCULATOR This document is an attempt to summarize some essentials about the use of your financial calculator. If this document is not clear enough, or extensive enough, you are well advised to consult the manufacturer s documentation (e.g. the 144-page User Guide supplied by Hewlett-Packard.) As computer geeks like to say: when all else fails, RTFM! NOTE: this document does not cover the additional functions introduced in the HP- 10BII(+). 2
HP-10BII CALCULATOR LECTURES: TABLE OF CONTENTS Lecture 1: Lecture 2: Lecture 3: Lecture 4: Lecture 5: Lecture 11: Lecture 12: Lecture 13: Lecture 21: Lecture 22: Lecture 24: Lecture 26: Lecture 29: Lecture 31: Lecture 32: Lecture 33: Lecture 34: Lecture 35: Lecture 39: The Basics Display Formats and Scientific Notation Internal Representation of Numbers and Rounding Algebraic Mode. Chain Calculations. Using Parentheses. RPN Mode. Chain Calculations. [Not in the HP-10BII.] Clearing the Calculator of old data Calculator Setup Functions More Setup Functions TVM functions or Cash Flow functions: which do I use? Using the TVM functions Calculating NPV and IRR using the TVM functions The TVM functions: Amortization over a set of payments A note on amortization and rounding The Cash Flow functions The Cash Flow functions: Individual cash flows (using CFj) The Cash Flow functions: Repeated cash flows (using Nj) The Cash Flow functions: Cash flows with holes Cash flow data registers in the HP-10BII Caution: Data entry using TVM and CFLO functions. Lecture 61: Other Calculator Functions: y x Lecture 62: Other Calculator Functions: roots Lecture 63: Other Calculator Functions: compounding. EAR vs. nominal i. Lecture 64: Continuous compounding: e x. Lecture 71: Lecture 81: Lecture 91: Lecture 92: Macaulay s Duration (D). Special Bond Functions. [Not in the HP-10BII.] Limitations on TVM data entry in the HP-10BII Limitations on Cash Flow data entry in the HP-10BII 3
LECTURE 1 CALC A NOTE ON SYMBOLS USED IN THIS MANUAL Notice that we have a small problem: standard computer fonts do not have the division symbol that appears on your calculator. So in this document I m using the usual symbol / instead. Also, for multiplication you may find me using either x or *. THE BASICS Notice that the calculator holds 8 rows of 5 keys each, for a total of 40 keys. Each key has a number or function imprinted on the top in white. This is the key s primary or unshifted function. In addition, some keys have additional functions imprinted on the bottom in yellow, and some keys also have functions written in purple above them. SHIFT KEYS You ll see two keys in the left column, one yellow and one purple. These are shift keys that allow us to execute those functions written in yellow and purple on the keys. If you press the yellow key (which we ll call YSHIFT) you see the word SHIFT appear in the display. (The calculator manual calls this displayed word an annunciator.) This means that whatever key you hit next, the function that will be executed will be the one written on that key in yellow. The purple key works the same way. We ll refer to the purple key as PSHIFT. The annunciator for that key is STATS, since the purple functions are all related to statistics. We won t need these functions for purposes of dealing with financial problems. Note: if you hit a shift key by mistake, you can un-shift it by hitting the same key again. 4
LECTURE 2 CALC NUMBER DISPLAY FORMATS Out of the box, your calculator will display all numbers with two decimals, and using the American convention for the decimal point. So for example, the number one million will be displayed as 1,000,000.00. You can switch the calculator to the European convention by executing YSHIFT./, If you do, the same number will now display as 1.000.000,00. Execute the same function a second time to toggle back to the American format. You can also change the number of decimals displayed after the decimal point. Just hit YSHIFT DISP n where n is the number of decimals you want. Try it: key in 1 / 7 = and then use the above function to change the number of decimals. The largest number of decimals the calculator will display is 9. You can also ask the calculator to display only as many decimals as needed, by executing YSHIFT DISP. The resulting calculator behavior may take a little getting used to. SCIENTIFIC NOTATION If the result of a calculation is either too small or too large to fit in the display, the calculator will switch to scientific display format. It will display the number in the format of ±a.bbbbbbbe±ccc where En means times 10 n. So E3 means times 1000 while E-3 means times 1/1000 or times 0.001 (Since this is a financial calculator, it is not possible to set it to scientific display mode permanently.) NOTE: When necessary, you can enter very big or very small numbers into the calculator using scientific notation. To enter the number -1.24*10-40 for example, you would enter 1. 2 4 +/- YSHIFT E +/- 4 0 =. Try it! The display should read: -1.2400000E 40. 5
LECTURE 3 CALC INTERNAL REPRESENTATION OF NUMBERS Regardless of how you choose to display numbers, every number is represented internally in the calculator in scientific mode, with 12 significant digits plus a 3-digit exponent (power of 10). You can see the full precision representation of any number currently in the display by executing YSHIFT DISP =. For as long as you hold down the = key, the calculator will display all 12 digits of the number (without the decimal point). When you release the = key, the calculator goes back to the display format it is set for. ROUNDING Since most financial problems involve dollars and cents, you will likely set your calculator to display 2 decimals. Any result you calculate will be DISPLAYED in a rounded format, where standard rounding rules are used to determine the last decimal shown. HOWEVER, the number is still represented internally with its full precision! This can sometimes throw off the results of calculations involving many time periods. You can round off (the internal representation of ) the number displayed to the number of decimals shown in the display by executing YSHIFT RND. Try it: set the display to 9 decimals. Calculate 1/7. Set your display to show only 3 decimals. Execute YSHIFT RND. Now set your display back to 9 decimals. See? The last 6 decimals have been set to zero. 6
LECTURE 4 CALC NORMAL OPERATION: ALGEBRAIC MODE The HP-10BII calculator operates, like most other calculators, in so-called Algebraic mode(*), meaning you pretty much enter problems in the order you would write them down. So to calculate 1+2 you enter simply: 1 + 2 = and the calculator displays the result (3). You can now do something with the result: for example, enter x 7 = and the result is the expected 21. (*) HP has other calculators that operate differently, in so-called reverse Polish mode or RPN. Most of HP s advanced scientific calculators, as well as the HP12C financial calculator, can operate in this mode. I find this RPN mode to be preferable because it is more flexible for chain calculations, but you have to re-wire your brain to think in RPN to make efficient use of it ;) CHAIN CALCULATIONS If you need to add many numbers together, you do not need to hit the = key for each step. You can just enter a sequence of operations, with the = function to display the final result: 1 + 2 + 4.5 + 8.3 + 7 = should give the result 22.8. USING PARENTHESES: ORDER OF PRECEDENCE Say you want to calculate the value of this expression: 2 4 (3*7). 1.5 You happily enter a sequence of operations into your calculator: 4 + 3 x 7 2 / 1.5 =. What would the result be? Your calculator says 31.333. But is that correct? No! Note that the calculator cannot see what the expression looks like that you are trying to calculate, so it just executes operations in the order you type them in, using the intermediate results as it goes along. In this case, it is as if you tried to calculate (4+3)*7-2 ((4+3)x7 2) / 1.5 or -------------- 1.5 Was that what you wanted? If not, you need to TELL the calculator which operations belong together, by entering PARENTHESES where needed. As a minor annoyance, HP has chosen to make the parentheses shifted operations: YSHIFT ( and YSHIFT ). The correct sequence of operations is: 4 + ( 3 x 7 ) ( 2 / 1.5 ) = which gives the result 23.667. A good rule to follow is: when in doubt, add parentheses. 7
LECTURE 5 CALC NORMAL OPERATION: REVERSE POLISH (RPN) MODE [The HP-10BII is not capable of operating in this mode.] CHAIN CALCULATIONS (n/a) NO NEED FOR PARENTHESES: ORDER OF PRECEDENCE (n/a) Say you want to calculate this expression: 2 4 (3*7). 1.5 8
LECTURE 11 CALC CLEARING THE CALCULATOR Before attacking a given problem, it is always a good idea to clear the calculator of data from previous problems. Executing C will clear the last number entered. Executing YSHIFT C-ALL will clear all financial registers. [And YSHIFT CLΣ will clear all statistical registers, but we aren t concerned with those here.] (Notice that when you execute YSHIFT C-ALL the calculator briefly displays how many payments per year it is set for. We ll discuss that setting further down.) CORRECTING DATA ENTRY: BACKSPACE If you inadvertently key in an incorrect digit while entering data, you can erase it by hitting the BACKSPACE (left-arrow) key. 9
LECTURE 12 CALC PREPARATION: CALCULATOR SETUP FUNCTIONS Before we start tackling a specific problem using the calculator, we need to get the calculator ready. 1. CLEARING THE CALCULATOR Before starting a new problem, clear the calculator of data from previous problems: YSHIFT C-ALL The calculator will briefly display how many payments/year it has been set for. (See below.) 2. BEGIN/END: REGULAR ANNUITIES VS. ANNUITIES DUE For those problems which involve annuities due, set the calculator to BEGIN mode: YSHIFT BEG/END (the annunciator BEGIN should show.) For most problems (regular annuities), the calculator should be set to END mode. YSHIFT BEG/END (the BEGIN annunciator goes away.) 10
LECTURE 13 CALC PREPARATION: MORE CALCULATOR SETUP 3. NUMBER OF PAYMENTS PER YEAR The calculator needs to be told how many payments per year your problem assumes: nn YSHIFT P/YR This is because it assumes that any interest rate or discount rate you enter with I/YR is a nominal yearly rate, and it needs to be able to calculate internally what rate that translates into per payment period. Also, when you ask the calculator to figure the IRR (for cash flows) or I/YR (for TVM problems), it uses P/YR to translate from its internal payment period rate to a yearly rate which is what is reported. NOTE: There is an alternative way to use your calculator, and that is to set the calculator to 1 payment per year and leave it there. If you do this, all interest rates entered must be, and rates calculated by the calculator will be, rates per payment period. You then have to translate yearly rates into payment period rates manually on data entry, and you have to translate the calculator s output results into yearly rates yourself. The instructor for the particular class you are taking will likely have a preference as to which way is better. My general advice on this topic is: you re better off not lying to your calculator! 4. COMPOUNDING FREQUENCY The HP-10BII assumes that interest will be calculated and added with the same frequency as payments are made, i.e. compounding frequency equals payment frequency [entered with P/YR]. This is the normal practice for most loans and financial instruments in U.S. financial markets. Other financial calculators [such as the TI-BAII+] may allow you to specify a compounding frequency (C/YR) that is different from the payments frequency (P/YR). 11
LECTURE 21 CALC TVM FUNCTIONS or CASH FLOW FUNCTIONS: WHICH DO I USE? Your financial calculator has two sets of functions to deal with sets of cash flows: the general cash flow functions (CFj, Nj, IRR, I/YR, NPV), and the more specialized TVM functions (N, I/YR, PV, PMT, FV). How do you decide which set of functions to use for a specific problem? In principle, any cash flow project can be handled using the cash flow functions. If you use those functions for data entry, however, you are limited to calculating the NPV and the IRR for the project. There are some circumstances when you must use the cash flow functions: if the sequence of payments consists of cash flows of varying sizes or directions, or the cash flows are not evenly spaced out (there are holes in the sequence), then you must use the cash flow functions. However, if you have a financial project that consists of at most: (a) one unique cash flow (payment) at the beginning of time, (b) one set of cash flows (payments) of equal size, spaced out equally over time, and (c) possibly one large payment at the end of the project (at the same time as the last regular payment), then you may use the TVM functions. The typical mortgage loan is an example of such a project. For projects where you use the TVM functions, you can calculate not only the NPV (PV) and the IRR (I/YR) for the project, but also the ending balance (FV), the size of the repeating cash flow (PMT) and the number of payments involved (N). 12
LECTURE 22 CALC USING THE TVM FUNCTIONS The Time Value of Money (TVM) functions Your calculator has buttons on the top row for: N = the number of payments or compounding periods I/YR = the nominal yearly interest rate PV = present value (the amount borrowed or invested) PMT = the size of the recurring payments FV = future value (remaining balance, or a balloon payment) You enter data for 4 of these 5 items, and ask the calculator to calculate the 5 th one. The most obvious case: given PV, N, and I/YR (and setting FV = 0), calculate PMT. Example: you borrow $300,000 to buy a house, at 6% over 30 years. Enter PV=300000, N=360 (the number of monthly payments), I/YR=6.0, and FV=0 (because we want to pay the house off.) Then calculate PMT to find the size of the required monthly payment (answer: -1,798.65.) Example: you save $500 per month for your retirement during the last 20 years of your working life. How much is in the bank when you retire, assuming a return of 9% after tax? Enter N=240, I/YR=9.0, PV=0, PMT=-500, and calculate FV (answer: $333,943.43). Example: you want to pay yourself $5,000 per month out of savings for at least 20 years after you retire. How much do you have to have in the bank the day you retire to afford this? Assume an interest rate of 8%. Solution: Enter FV=0, PMT=5000, N=240, I/YR=8.0; calculate PV (answer: $597,771.46). One wrinkle: if the payment schedule requires you to make payments more often than once a year (for example, once a month), you must tell the calculator how many payments/year you have. How to do this differs between calculators. (See Lecture 13.) Note: If the planned future loan balance is not zero, we refer to the loan as a balloon loan and the remaining loan balance (that last big payment) as a balloon payment. Your calculator handles this without any problem just enter the size of the balloon payment as FV instead of 0 (and be sure to enter it as a negative number!) Note: to calculate the NPV of a series of future equal size payments, enter the other variables and solve for PV. To calculate the IRR (API) of a loan transaction, enter N, PV, PMT, FV and solve for I/YR. 13
LECTURE 24 CALC CALCULATING NPV AND IRR USING THE TVM FUNCTIONS Calculating NPV and IRR for a given financial project is easy if you use the Cash Flow (CFLO) functions [CFj, Nj] to do data entry. (See CALC lectures 31-34.) If the project is in the form of an annuity, and you prefer to use the TVM functions instead (see CALC lecture 21 for when you can do so), the following notes apply: 1. Data entry in general: if the project has a non-zero final balance to be paid (e.g. a balloon payment on a mortgage), make sure this final balance is entered into FV with a negative sign! 2. Calculating IRR: If you have entered the relevant values for N, PV, PMT, and FV, then calculating I/YR will give you the IRR. [Note: If you have set P/YR to its true value, the IRR displayed will be the yearly rate. If instead you have chosen to leave the calculator set to 1 P/YR, the interest rate shown will be the periodic rate. If so, you must multiply this rate by however many payment periods per year the project actually has to find the yearly interest rate.] 3. Calculating NPV: using the TVM functions, the calculator can be used to calculate the net present value of a set of future cash flows only, e.g. the fair market price of a coupon bond. To do so, you enter data for N, I/YR, PMT, and FV, and calculate PV. As you can see, there is nowhere to put the value of a time-zero cash flow (CF0). So what you have to do is to calculate the PV of all the future cash flows, then manually subtract any cash flows that take place at the beginning of the project from that PV to get NPV for the project as a whole. 14
LECTURE 26 CALC THE TVM FUNCTIONS: AMORTIZATION OVER A SET OF PAYMENTS Amortization Some calculators will allow you to examine the payment schedule for a loan after the loan data have been input and/or calculated. You typically input two numbers N1 and N2, representing starting and ending payment numbers, and then calculate: INT = total interest paid in those payments; PRIN = total amount applied to principal; BAL = the remaining loan balance after the last payment (N2) For the HP-10BII, you do the following: First, enter the necessary data for a TVM problem (BEGIN or END mode, P/YR, N, I/YR, PV, FV, and PMT). Then: TO AMORTIZE A RANGE OF PAYMENTS, say payments #61 through #66, you enter: 61 INPUT 66 YSHIFT AMORT The calculator displays: PER 61-66 to confirm your selection. Now hit the = key several times, and in sequence the calculator will display for you PRIN $xxx INT $xxx BAL $xxx [the amount applied to principal out of the specified group of payments] [the amount of interest paid out of the specified group of payments] [the remaining principal after the last of the specified payments] If you keep hitting = the calculator will cycle through the information again. If you now execute YSHIFT AMORT again, the calculator will calculate the above data for the next 6 payments (that is, for however many payments you last specified). If after entering the original loan data you just execute YSHIFT AMORT without specifying a range of payments, the calculator will start with payment #1 and take the number of payments from what you entered for P/YR (usually 12 for a mortgage). TO AMORTIZE A SINGLE PAYMENT you enter for example: 25 INPUT YSHIFT AMORT for payment #25 only. You exit out of the amortization functions and back to normal calculator operations by hitting the BACKSPACE key. 15
LECTURE 29 CALC ABOUT AMORTIZATION AND ROUNDING Assume we are given the following mortgage loan: loan amount $300,000; interest rate 6.5%; loan term 30 years. What is the required monthly payment? Set END mode, 12 P/YR, N=360, I/YR=6.5, PV=300,000; calculate PMT $-1,896.20. If we now amortize the first 12 payments of this loan, we get: INT = $- 19,401.27 PRIN = $- 3,353.18 BAL = $ 296,646.82 But recall what we said above (in Lecture 3) about rounding. The calculator calculates the required monthly payment with full internal precision (12 digits). If you calculate the payment again, then set the calculator to display all available digits (YSHIFT DISP 9), you wil see that the payment that was calculated is: $-1,896.20407048. And the amortization numbers are based on somebody making that exact payment every month. But in the real world, actual payments have to be rounded to the nearest penny ($0.01). So: after calculating the above payment, let us round it off: YSHIFT. 2 then YSHIFT RND then hit PMT to store the new value. Since this payment is slightly smaller (by less than a penny) than the exact one needed, we will end up with a slight balance at the end: hit FV to display the ending balance of $-4.50 this is money we still owe the bank, so it would get added to the last payment. If we now amortize the first 12 payments, we get: INT = $- 19,401.27 BAL = $- 3,353.13 BAL = $ 296,646.87 As you can see, the new balance is off by a few pennies. The difference would get bigger further into the loan repayment schedule. (If we were to increase PMT by a penny, we would get an ending balance of +$6.56 by which we could reduce the last payment.) For the reasons discussed above, most real-world loans end up with an odd-sized final payment. 16
LECTURE 31 CALC THE CASH FLOW FUNCTIONS The cash flow functions The cash flow functions of your calculator allow you to calculate the net present value (NPV) of an arbitrary sequence of cash flows or payments, and the internal rate of return (IRR) for a cash flow project. Data entry consists of entering data for the cash flows one after the other, observing the cash flow sign convention: money you receive is entered as a positive amount, money you pay out is entered as a negative amount. Definitionally, CF0 = any amount paid or received today, at the beginning of time (e.g. a loan amount) CF1 = the first cash flow, taking place at the end of the first time period CF2 = the second cash flow, taking place at the end of the second time period etc. You can now calculate the net present value (NPV) of the project, which is equal to the sum of the net present values of each individual cash flow. You need to provide the calculator with a suitable discount rate or required rate of return. We have: NPV CF0 CF1 CF2 CF n 0 1 2 (1 i) (1 i) (1 i) (1 i) n The decision rule then is: any project whose NPV > 0 is worth doing, while NPV < 0 means you are getting a yield that is too low (that is, less than the required rate.) You can also calculate the project s internal rate of return (IRR), which is the interest rate(s) at which NPV = 0. Note that there needs to be at least one sign reversal in the project cash flows for the calculator to be able to find an answer for IRR; if there is more than one sign reversal there may be more than one possible IRR, and you have to use NPV calculations instead for project evaluation. Advanced calculators may be able to plot NPV as a function of the interest rate. 17
LECTURE 32 CALC THE CASH FLOW FUNCTIONS: INDIVIDUAL CASH FLOWS To use the cash flow functions, first clear the calculator of old data: YSHIFT C-ALL Then, enter cash flows one by one (with the appropriate sign), followed by the CFj function. Example: you participate in a project where you contribute one million dollars today, and then receive yearly payback amounts of $500k, $600k, and $700k, respectively. 1. First, set the calculator to one payment per year: 1 YSHIFT P/YR. 2. Data entry would be: 1 0 0 0 0 0 0 +/- CFj [annunciator: 0 for cash flow #0] 5 0 0 0 0 0 CFj [annunciator: 1 ] 6 0 0 0 0 0 CFj [annunciator: 2 ] 7 0 0 0 0 0 CFj [annunciator: 3 ] 3. You can now calculate the internal rate of return (IRR) for the project: YSHIFT IRR/YR gives the result: 33.87%. Keep in mind that the reported IRR is the yearly rate, based on the number of payments per year you have set. If you inadvertently had left your calculator set for 12 payments per year, the reported IRR would be 406.5%! So be careful to set P/YR to the correct value for the project at hand. NOTE: IRR will be negative if the project pays back fewer dollars than what you contributed. PAY ATTENTION TO THE SIGN. 4. We can also calculate the net present value (NPV) for the whole project. To do that, we need to tell the calculator which interest rate to use when discounting future cash flows. Let us say that for this project, we have an alternative project that could have yielded a 25% return, so we want to use 25% as the required rate of return: 2 5 I/YR YSHIFT NPV gives the result of $142,400.00. If instead we had used 35% as our discount rate, the NPV would be $-15,902.05. NOTE: NPV will be negative if the project is earning a return smaller than the discount rate you entered. PAY ATTENTION TO THE SIGN. 18
LECTURE 33 CALC THE CASH FLOW FUNCTIONS: REPEATED CASH FLOWS (GROUPS) In some cases, the same cash flow or payment is repeated a number of times. We can then simplify data entry quite a bit! Data entry in such a case consists of entering the following data for the project: CF0 = any amount paid or received today, at the beginning of time CF1 = the size of the first cash flow($$) N1 = the number of times CF1 is repeated CF2 = the size of the second cash flow ($$) N2 = the number of times CF2 is repeated etc. As you see, it is possible to save on data entry by entering a repetition factor Nj for cash flows that repeat (i.e. the same dollar amount is paid or received for several periods.) Let us change the previous example a little: you contribute $5 million today, then you get back 5 yearly payments of $500k each, 4 yearly payments of $600k, and finally 3 yearly payments of $700k. First, clear the calculator of old data, and set it to 1 payment per year. Then: 5 0 0 0 0 0 0 +/- CFj [annunciator: 0 for cash flow #0] 5 0 0 0 0 0 CFj 5 YSHIFT Nj [annunciator: 1 ] 6 0 0 0 0 0 CFj 4 YSHIFT Nj [annunciator: 2 ] 7 0 0 0 0 0 CFj 3 YSHIFT Nj [annunciator: 3 ] Now calculate IRR to get a result of 5.1994%. At a discount rate of 5%, we get an NPV of $60,546.31. At a discount rate of 15%, we get an NPV of $-2,017,940.60. Note that you cannot enter a repetition factor for the first payment. CF0 is always unique. So if the project requires you to contribute $1.25 million per year for 4 years, data entry must be: 1 2 5 0 0 0 +/- CFj [ 0 ] 1 2 5 0 0 0 +/- CFj 3 Nj [ 1 ] etc. 19
LECTURE 34 CALC THE CASH FLOW FUNCTIONS: CASH FLOWS WITH HOLES The cash flow functions are used when we have a project where the cash flows are of different sizes, OR when they are not spaced out evenly. Sometimes you will encounter projects that have holes in the sequence of cash flows. If, in our example above, those last 3 payments of $700k each come 3 years after the last $600k payment, with nothing happening inbetween, then that is as if there were two intervening cash flows of size $0. NOTE: it is useful in these cases to draw the cash flow diagrams to make sure you are getting the spacing right To make sure the calculator understands which payments take place when (so that the correct discount factor can be applied to each payment), we need to tell it about those zero cash flows. Data entry would look like this: 5 0 0 0 0 0 0 +/- CFj [ 0 ] 5 0 0 0 0 0 CFj 5 YSHIFT Nj [ 1 ] 6 0 0 0 0 0 CFj 4 YSHIFT Nj [ 2 ] 0 CFj 2 YSHIFT Nj [ 3 ] 7 0 0 0 0 0 CFj 3 YSHIFT Nj [ 4 ] etc. 20
LECTURE 35 CALC CASH FLOW DATA REGISTERS The HP-10BII calculator has room for a total of 15 groups of cash flows, with one value for CFj and one value for Nj for each group. After you clear the calculator with YSHIFT C-ALL, the table looks like this: j CFj Nj 0 0.00 1 1 0.00 1 13 0.00 1 14 0.00 1 After you enter the data from Lecture 34 above, the table would look like this: j CFj Nj 0-5,000,000 1 1 500,000 5 2 600,000 4 3 0 2 4 700,000 3 5 0 1 6... 14 0 1 When the calculator estimates NPV or IRR, it assumes that all (groups of) cash flows that we have entered happen right after each other with no gaps, which is why we had to insert cash flow group #3 (CFj=0, Nj=2) to make sure the last cash flow group was treated correctly. 21
LECTURE 39 CALC CAUTION: DATA ENTRY USING THE TVM AND THE CFLO FUNCTIONS Imagine that we have a coupon bond with 5 years left until the maturity date. How do we handle this as a project using the financial calculator? Say the bond is a $1,000 face value bond with a coupon rate of 9%. Thus the semi-annual coupon payments are $45. Let the current market interest rate be 6%. You were just offered this bond for $1,090. We can now ask two questions: - what is the fair market value (FPV) of this bond? - if I purchase the bond at the offered price, what will my yield to maturity (YTM) be? Using the TVM functions: Here, we can use the FV register to hold the value of the principal repayment (as you can see, it is entirely similar to a balloon payment on a mortgage.) We set the calculator to 2 P/YR. Then we enter: N = 10, I/YR = 6, PMT = 45, and FV = 1,000 so we can calculate PV which is the fair market value of the bond. [$-1,127.95] Or, we enter PV = -1,090 and calculate I/YR which is then the IRR or YTM [=6.84%] The implicit assumption of the TVM functions is that the cash flow in FV takes place at the same time as the last of the payments entered with (N, PMT). Using the CFLO functions: Enter the purchase price as CF0 = -1,090, the semi-annual coupon payments as (CF1 = 45.00, N1 = 9), and the last cash flow (CF2) is equal to $1,000 plus the last coupon, or CF2=$1,045 and N2=1. If you now calculate the IRR for the project, this will be the YTM. [You should get 6.84%.] If instead you enter CF0 = 0, you can enter the market rate (6%) as I/YR, and calculate the NPV for the project. This is then the fair present value of the bond s future cash flows, or the fair market price. [$1,127.95 in this case.] The important point about CFLO data entry is this: since the last coupon payment is made at the same time as the repayment of principal, the total resulting payment must be entered as a single cash flow. If you entered (CF1 = 45, N1 = 10) and (CF2 = 1,000) this would be interpreted by the calculator as if the repayment of principal took place one period (here, 6 months!) later than the last coupon payment. Since this is not in fact what happens, you get incorrect results for NPV and IRR. 22
LECTURE 61 CALC OTHER CALCULATOR FUNCTIONS: Y X Calculating y x Your calculator has a function key to calculate powers: YSHIFT y x. This function requires two numbers to operate on: the base number (y) and the power (x). As an example, say you need to calculate (3.5) 3. You key in: 3. 5 YSHIFT y x 3 = and get the answer 42.875. You can (and should) use parentheses where needed for either y or x. For example: i you want to know the value of (1 ) mn where i=6%, m=4, and n=10. You key in: m 1 + YSHIFT (. 0 6 / 4 YSHIFT ) YSHIFT y x YSHIFT ( 4 x 1 0 YSHIFT ) = and your answer should be 1.814. NOTICE: after you key in YSHIFT y x or YSHIFT ( the display shows the word PEND. This annunciator indicates that you have entered some data, which the calculator has stored internally, and that the calculator is waiting for further input before executing whatever operation is needed. When you are done (after hitting = in the case of y x, or after hitting YSHIFT ) in the case of parentheses, the PEND annunciator goes away. LIMITATIONS: there are none. Either x or y may be whole numbers or fractional numbers, either positive or negative. 23
LECTURE 62 CALC OTHER CALCULATOR FUNCTIONS: ROOTS Calculating square roots Your calculator has a separate function for calculating square roots: YSHIFT x. So: 9 YSHIFT x yields the answer 3. (Notice there s no need to hit the = button.) And: 3. 6 8 YSHIFT x yields the answer 1.9183. However: 2. 5 +/- YSHIFT x gives you an error message, since it is not mathematically possible to extract a square root from a negative number. [As long as we stay out of the realm of complex numbers.] You clear the error by hitting the C function. Calculating general roots To calculate other roots than the square root, we need to resort to a little bit of subterfuge. As you may recall from math class, 1 x y y x so for example 10 0.1 3.7 3.7 1.1398. To extract the 4 th root of a number, we just raise that number to the power of ¼ = 0.25. And so on. Or, to extract the 7 th root of PI: You key in 3. 1 4 1 5 9 YSHIFT y x YSHIFT ( 1 / 7 YSHIFT ) = and get the answer 1.1779. 24
LECTURE 63 CALC OTHER CALCULATOR FUNCTIONS: COMPOUNDING Special functions: from nominal i to EAR Your calculator has special functions for compounding problems look at the top row. To find out what EAR you would earn if the nominal interest rate is 7% and we have weekly compounding, enter the following data: 52 YSHIFT P/YR 7 YSHIFT NOM% then calculate the answer: YSHIFT EFF% yields 7.2458. Special functions: from EAR back to nominal i You can go the other way too: to earn the EAR we just calculated, what would the nominal yield have to be if we had only monthly compounding? Just change P/YR to 12, then calculate EFF%. You should get 7.0157. The greatest practical compounding frequency would be daily (P/YR = 365). LIMITATION: The HP-10BII will allow you to set P/YR to a value no larger than 999. 25
LECTURE 64 CALC OTHER CALCULATOR FUNCTIONS: CONTINUOUS COMPOUNDING Continuous compounding: from nominal i to EAR Can we use the special top-row functions on the calculator to estimate EAR for continuous compounding? Not exactly, but perhaps close enough. Note that this particular calculator (the HP-10BII) is limited to 999 P/YR. So let s say i=12%. We get: a) Set P/YR=999 and NOM%=12. Calculate EFF% = 12.74887262%. b) Use the exact formula: EAR = e i 1 = 12.74968516%. The difference shows up in the 5 th digit (3 rd decimal). If you require greater accuracy than that, you need to use the exact formula for continuous compounding. Continuous compounding: from EAR to nominal i Using the exact formula (i = ln(1 + EAR)), we can get the nominal yield if EAR is 8.74% with continuous compounding: YSHIFT ( 1 +. 0 8 7 4 YSHIFT ) YSHIFT LN gives the result 0.08378953. If instead you enter 8.74 as EFF% and 999 P/YR, we get NOM% = 8.3793%. 26
LECTURE 71 CALC MACAULAY S DURATION D The HP-10BII financial calculator does not have a special function to calculate D. We therefore have to calculate it manually. While it is easy to calculate the NPV of the whole project using either the TVM or CFLO functions, calculating the t * NPV(CFt)) for each cash flow regrettably must be done manually one by one and the answers added up. See the examples in the TVM lectures (#101-103) for how to calculate the duration for a bond and for a mortgage. 27
LECTURE 81 CALC SPECIAL BOND FUNCTIONS The HP-10BII financial calculator does not have specialized bond functions. Some other financial calculators do. 28
LECTURE 91 CALC LIMITATIONS ON TVM DATA ENTRY IN THE HP-10BII 1. There are no built-in limitations on the values you can enter for N, I/YR, PV, PMT, or FV. (But to make sense, N ought to be a positive integer greater than zero, and we don t usually work with negative interest rates.) 2. The BEG/END toggle has only two possible values no limitations there ;) 3. This calculator limits P/YR to a maximum value of 999. This is good enough to approximate continuous compounding to about 4 decimal places. 29
LECTURE 92 CALC LIMITATIONS ON CASH FLOW DATA ENTRY IN THE HP-10BII 1. The HP-10BII is limited to a total of 15 (groups of) cash flows (CF0.. CF14). The first group is limited to a single cash flow (i.e. N0 = 1 always.) If you need to work with large projects that have more cash flows than this, you need to purchase a better financial calculator! (Or use a spreadsheet ) 2. The HP-10BII limits Nj to a maximum of 99 repetitions per cash flow group. (Other HP calculators, and the TI-BAII+, do not have this limitation.) This forces us to do a bit of extra data entry in some cases: Say you save $100 per month for 40 years, then you receive $300 per month for 20 years from your pension plan after you retire. What is your return (IRR) to this project? Without the limitations, you would enter CF0=0, (CF1 = -100.00 and N1 = 480), (CF2 = 300.00 and N2 = 240). You could then calculate the IRR (answer: 1.32% per year.) Instead, we are forced to do this: 0 CFj [ 0 ] 1 0 0 +/- CFj 90 YSHIFT Nj [ 1 ] 1 0 0 +/- CFj 90 YSHIFT Nj [ 2 ] 1 0 0 +/- CFj 90 YSHIFT Nj [ 3 ] 1 0 0 +/- CFj 90 YSHIFT Nj [ 4 ] 1 0 0 +/- CFj 90 YSHIFT Nj [ 5 ] 1 0 0 +/- CFj 30 YSHIFT Nj [ 6 ] 3 0 0 CFj 90 YSHIFT Nj [ 7 ] 3 0 0 CFj 90 YSHIFT Nj [ 8 ] 3 0 0 CFj 60 YSHIFT Nj [ 9 ] You see the principle: keep entering the same dollar value for CFj until the corresponding Nj s add up to the number we need (480 in this case). [I find it easier to use Nj=90 in these cases rather than the maximum of 99.] 30