Introduction to the Time Value of Money

Transcription

1 Module 3 Introduction to the Time Value of Money David Mannaioni CPCU, CLU, ChFC, CFP 7483

2 This publication may not be duplicated in any way without the express written consent of the publisher. The information contained herein is for the personal use of the reader and may not be incorporated in any commercial programs, other books, databases, or any kind of software or any kind of electronic media including, but not limited to, any type of digital storage mechanism without written consent of the publisher or authors. Making copies of this material or any portion for any purpose other than your own is a violation of United States copyright laws. The College for Financial Planning does not certify individuals to use the CFP, CERTIFIED FINANCIAL PLANNER, and CFP (with flame logo) marks. CFP certification is granted solely by Certified Financial Planner Board of Standards, Inc. to individuals who, in addition to completing an educational requirement such as this CFP Board-Registered Program, have met its ethics, experience, and examination requirements. Certified Financial Planner Board of Standards, Inc. owns the certification marks CFP, CERTIFIED FINANCIAL PLANNER, and federally registered CFP (with flame logo), which it awards to individuals who successfully complete initial and ongoing certification requirements. At the College s discretion, news, updates, and information regarding changes/updates to courses or programs may be posted to the College s website at or you may call the Student Services Center at

3 Table of Contents Study Plan/Syllabus... 1 Learning Activities... 3 Chapter 1: The Importance of the Time Value of Money... 5 Chapter 2: Fundamental Calculator Keystrokes Common Calculator Mistakes Using the Hewlett-Packard 10BII+ Calculator Using the Hewlett-Packard 12C Calculator Chapter 3: Basic Time Value of Money Calculations Capitalization of a Number Future Value of a Single Sum Present Value of a Single Sum Number of Compounding Periods and Interest Rate per Compounding Period Present Value of an Annuity Future Value of an Annuity Periodic Payment or Receipt Chapter 4: Intermediate Time Value of Money Calculations Serial Payments Calculations Involving Single Sums Combined with Annuities Chapter 5: Advanced Time Value of Money Calculations Calculations Involving Unequal Cash Flows Squares, Square Roots, and Nth Roots... 59

4 Amortization Summary Module Review Questions Answers References About the Author Index

5 Study Plan/Syllabus One of the many functions of a personal financial planner is to make appropriate recommendations to clients. The planner must understand that the value of money changes over time, and the changes affect the recommendations that are made. Even if the inflation rate is zero, a dollar received in the future is worth less than a dollar received today, just as an obligation to pay a dollar in the future is less costly than paying a dollar today. This is because a dollar invested wisely will provide some expected positive return. For example, if 100 years ago $100 were invested in an account bearing 10% interest, compounded annually, it would be worth more than $1.3 million today. The compounding (or discounting) of money based on interest is the dynamic force behind the time value of money concept. This module illustrates how to calculate the following time value of money variables: the future value of a single sum or annuity, the present value of a single sum or annuity, the interest rate per compounding period, the number of compounding periods, the periodic payment, the serial payment (or inflation-adjusted payment), the present value of serial payments, and the future value of serial payments. This module also covers the capitalized value of a specific dollar amount. Study Plan/Syllabus 1

6 The chapters in this module include: The Importance of the Time Value of Money Fundamental Calculator Keystrokes Basic Time Value of Money Calculations Intermediate Time Value of Money Calculations Advanced Time Value of Money Calculations* *Note: Material in the Advanced Time Value of Money Calculations chapter will be tested in the Investment Planning course, not in this course. This module focuses on the time value of money (TVM) concepts and applications, including the specific application of TVM principles to life insurance needs. Understanding TVM concepts is more important than knowing which buttons to push on a calculator. In fact, learning the concepts will make it easy to know which buttons to push. Competent financial planning requires knowledge of TVM concepts as well as the ability to communicate them to clients. Upon successful completion of this module, you will be able to solve time value of money problems, as well as be able to explain how the variables in a time value of money problem interact. 2 Introduction to the Time Value of Money

7 Learning Activities Learning Activities Learning Objective 3 1 Calculate the capitalized value of a given income. 3 2 Calculate the future value for a given situation. 3 3 Calculate the present value for a given situation. 3 4 Calculate the number of compounding periods for a given situation. 3 5 Calculate the interest rate per compounding period for a given situation. 3 6 Calculate the periodic payment for a given situation. Readings Module 3, Chapter 2: Fundamental Calculator Keystrokes Chapter 3: Basic Time Value of Money Calculations Module 3, Chapter 3: Basic Time Value of Money Calculations Module 3, Chapter 3: Basic Time Value of Money Calculations Module 3, Chapter 3: Basic Time Value of Money Calculations Module 3, Chapter 3: Basic Time Value of Money Calculations Module 3, Chapter 3: Basic Time Value of Money Calculations Module Review Questions 1, Study Plan/Syllabus 3

8 Learning Objective Readings Learning Activities Module Review Questions 3 7 Calculate the present value for an inflation-adjusted payment. 3 8 Calculate the inflationadjusted payment for a future sum. 3 9 Determine the general result when one parameter in a time value of money calculation is changed. Module 3, Chapter 4: Intermediate Time Value of Money Calculations Module 3, Chapter 4: Intermediate Time Value of Money Calculations Module 3, Chapter 4: Intermediate Time Value of Money Calculations Chapter 5: Advanced Time Value of Money Calculations (optional) Introduction to the Time Value of Money

9 Chapter 1: The Importance of the Time Value of Money Today, most planners use computer software that can model a wide variety of assumptions, and you can have such software complete all of the calculations that will be covered in this course. You may wonder why you need to learn the keystrokes for these calculations, and that is a valid question. In a perfect world, clients would know what they want to accomplish, the appropriate interest rate, funds available, and time frames, and their assumptions would all be realistic. That seldom happens. Clients enter your office and they have no idea whether their goals are achievable. Part of your job in defining the scope of engagement is to define a client s goals, and you must also let clients know when their goals are unrealistic. How do you evaluate whether a goal is realistic before you have done analysis? One of the reasons planners learn to love their calculator is that you can do some rough estimates in a short amount of time that can help you set realistic goals. Imagine that clients enter your office and want to save for their child s college education. They don t know what college costs, they haven t started saving, and they have no idea what they need to save per month. If you are comfortable with the calculator, you can easily walk them through the discussion by providing the clients those estimates. The conversation may go like this. Planner: Have you thought about the type of college you may want to send your child to and how much it will cost? Client: Well, we are thinking an in-state school because of costs but we haven t gone much past that. I think I heard that college was around $20,000 a year last year for costs including room and board. Does that sound right? Planner: We can certainly look up the various costs for in-state schools, but since your son is 5, I think that is a reasonable starting point. In addition to Chapter 1: The Importance of the Time Value of Money 5

10 setting a target amount to accumulate, in order to figure out how much you need to save, we need to set an inflation rate. I happen to know that college costs have increased at about twice the inflation rate for the last several years. With inflation running about 2.25%, that means we would increase college costs around 4.5%. Obviously that makes it more expensive in the future, but it helps give you a sense of what it might cost. Are you okay with my using 4.5% for college inflation? Client: Sure Planner: Well, if I project that forward, instead of 20,000 per year that would require $35,444 for that first year. How much you are earning on the money invested during those four years would impact the calculation. I think assuming just a 5.5% return would be safe. If you did that you would need $139,773 at the start of college to provide that income. Let s just round that to $140,000. Does that amount surprise you? Client: Wow! That sounds like a lot. What would we have to be saving to reach that goal? Planner: Let s see. If we use a 5.5% return, it would require $ or around $620 per month. It could be less if we increase the amount you are saving each year by inflation. I know you said this is important to you. Does this sound like something you could commit to? Client: I know you mentioned there were some other things we might need to be doing, so I m not sure exactly. I know that we have this mutual fund that is worth around $20,000 and we have been putting $150 into it each month. If we put the $20,000 and that $150 per month toward the college fund, how much more would we need to save? Planner: Okay $20,000 today at 5.5% with $150 a month being invested will grow to $74,881. Let s just say $75,000, and you needed around $140,000. Because you have save that amount and are already committing the $150, you only need to come up with an additional $65,000, which means monthly savings of $286 close to $300. I round these numbers because we don t 6 Introduction to the Time Value of Money

11 really know the exact return or inflation, but if you can save this amount, we should be on a good path and can keep adjusting the amounts as we go. Do you think you could find $300 to put toward this goal or should we talk about possibly reducing the goal? Client: I think we can pull off $300. Of course, it depends on what else you tell us we ll need to do. Planner: Well, we will keep working through the issue and identifying solutions and put a plan together that works for you, but I am going to use these assumptions as I work through your plan, okay? In this conversation, you utilized the following learning objectives: 3 2 Calculate the future value for a given situation. 3 3 Calculate the present value for a given situation. 3 6 Calculate the periodic payment for a given situation. 3 7 Calculate the present value for an inflation-adjusted payment. 3 9 Determine the general result when one parameter in a time value of money calculation is changed. If you had not been able to use your calculator to do these calculations, you may have had to spend much more time with the client at a later point. If you waited for the computer analysis and the client was uncomfortable with the monthly savings amount or how large the goal was, you would not have discovered it until you were in the presentation. If you become proficient with the calculator, you can recalculate your own progress toward retirement during commercials! Using your calculator is a required skill for both the CFP Certification Examination and your life as a financial planner. The best planning is useless without a financial goal in mind. Time value of money (TVM) concepts allow you, the planner, to translate goals into dollar Chapter 1: The Importance of the Time Value of Money 7

12 amounts. Further, these concepts permit you to determine the dollar input required to achieve the desired results. We recognize that this calculation is the starting point of a conversation with the client. Other factors may suggest that more or less be invested but clients need to know that a starting point has some validity and isn t just pulled out of the air. TVM calculations require the client and planner to decide upon an estimated investment rate of return and an inflation rate to be used. Since these are certain to be inaccurate for at least some of the years of the planning period, some would argue that any rule of thumb is just as useful. This argument has some merit, but every client has a unique set of circumstances and goals. In reality, people are more likely to follow a plan that is based on their input rather than a rule of thumb that may or may not be appropriate. The TVM concept has many applications in financial planning. For instance, it is used to determine how investment dollars should be applied to best meet financial objectives. It also is used to help calculate education, survivor, and retirement needs for a given client and to determine how to best meet those needs. Additionally, it is used to determine the financial effect of postponing taxes. In insurance planning, time value of money concepts are used to calculate life insurance needs. Of all the concepts that are important to a financial planner, a clear understanding of TVM calculations is imperative. When working through the problems in this module, it is critical that a planner endeavor to understand the concepts behind, and the relationships between the five basic TVM factors: present value [PV], future value [FV], payment [PMT], interest [I/YR], and number of periods [N]. By understanding the relationships between these factors, most problems may be figured logically, freeing students from trying to memorize numerous calculator keystroke sequences. A financial planner who is unable to understand and explain TVM concepts will find it difficult to provide the necessary guidance for his or her clients. Insurance planning, investment planning, income tax planning, retirement planning, education funding, and estate planning all require the use of these basic concepts 8 Introduction to the Time Value of Money

13 and calculations. If extra time is necessary for students to master these concepts, investing that time will prove to be well worth the effort over the long term. The most efficient, accurate means of performing time value of money calculations is through the use of financial calculators or computer programs. Students are required to use financial calculators in completing this CFP Certification Professional Education Program. However, because it is helpful to understand the underlying mathematical principles to adequately use the calculators, the readings for this module include exponential calculation of time value of money problems. The College s modules support the Hewlett-Packard 10BII and 10BII+ calculators, and all primary keystroke instructions refer to these calculators. We have included some Hewlett-Packard 12C calculator keystrokes for basic time value of money calculations. We do not provide additional support for the HP 12C or for any calculators other than the HP 10BII+. You can find online assistance with HP calculators at Before doing any problems in this module, you should become familiar with your calculator by using the owner s manual that accompanies it. If you are completing this module using classroom instruction, please consult with your instructor regarding preferred models before purchasing a calculator. Note: Although excellent tools in business, programmable calculators with alpha-numeric keys (letters on the keys) are not allowed into CFP Certification Examinations or College for Financial Planning educational examinations due to exam security issues. Programmable calculators allowing user-entered programs (e.g., HP 17BII), but without alpha keys are permitted for use on the CFP Certification Examination or the College for Financial Planning s educational exams. However, due to the same security issues these calculators will be examined by proctors before entry to exams is granted. If the calculator is found to contain programming, the student will be barred from taking the exam on the basis of unethical behavior. Chapter 1: The Importance of the Time Value of Money 9

14 Solving Time Value of Money Problems In the long run, understanding time value of money concepts is more important than memorizing calculator keystrokes. In fact, simply memorizing keystrokes will likely lead to frustration and an inability to accurately answer TVM problems. As a simple learning aid, use the table below to separate the known variables from the unknown variable. N I/YR PV PMT FV Number of periods Interest rate per year Present value Payment Future value For instance, suppose a problem reads as follows. A client wants to invest $10,000 for five years. If the investment grows at 8% per year, how much will the investment be worth in five years? The first step in solving this problem is to identify the known and the unknown variables. Using the table above, the problem looks like this: N I/YR PV PMT FV 5 8% $10,000 NA??? Remember that it is important to understand TVM concepts so that when one parameter changes, you will have a good idea of what will happen to the calculation. Understanding the relationship of the inputs is more important than trying to memorize keystrokes. Other Calculators If you are using a calculator such as the HP 12C, where the interest rate on the calculator is the interest rate per period, use the following table. N i PV PMT FV Number of periods Interest rate per period Present value Payment Future value 10 Introduction to the Time Value of Money

15 Relationships Among the Variables Over time, original assumptions used by a planner to solve time value of money problems will change. Clients may receive an unexpected infusion of cash from an inheritance; inflation may be much less or more than anticipated; market returns may be different than expected. All of the variables in a TVM problem can (and most probably will) change. For this reason it is important for financial advisors to understand the relationship among the variables. For instance, if an investment earns a greater return than projected, future deposits toward a given goal can be decreased. Or the goal can be realized sooner than anticipated. On the other hand, if a payment is missed, more time may be required to reach the goal. Perhaps a different investment vehicle with a higher expected rate of return (and thus more risk) will need to be chosen to meet the goal in the same amount of time. Chapter 1: The Importance of the Time Value of Money 11

16 Chapter 2: Fundamental Calculator Keystrokes Common Calculator Mistakes Getting the wrong answers is a common and frustrating experience when learning time value of money calculations. Learning to consider whether the answer showing on the calculator makes sense before accepting it is an important skill to develop. Learning what the common mistakes are and completing a second try when an answer is suspect will help you avoid missing exam questions or worse, giving clients incorrect information. The following are some common calculator mistakes that are worth studying to avoid incorrect answers due to erroneous entries. Clearing the display but not the calculator s memory registers. On most financial calculators, simply pressing the clear [C] key erases only one number the one on the calculator s display screen. The very first step before starting a calculation should be to clear the calculator s memory. If you had been calculating a problem with a payment and the next problem doesn t have a payment, it you don t clear it, the calculator assumes that same payment and you will get a wrong answer. Many additional numbers may be stored in the calculator s memory banks, and the calculator will use them until you tell it otherwise. Be sure to properly clear all numbers from your calculator before you work a new problem. To clear all numbers in memory on the HP 10BII+, press the [SHIFT] key and then the C/[C ALL] key. Most calculators, including the HP 10BII+, store the numbers even if they are turned off, so clearing the memory every time you turn on the calculator is a good habit to get into is. 12 Introduction to the Time Value of Money

17 Note: The manuals of the HP 10B, HP 10BII, and HP 10BII+ calculators call the [SHIFT] key gold, though this key color has been described as orange, copper, pumpkin, and in some cases green; and on the 10BII+ it is red-orange. The keystrokes given here refer to that all-important key as the [SHIFT] key, sometimes identified by the symbols or. It should also be noted that the HP 10BII and the HP 10BII+ are, for all intents and purposes, identical with a very few exceptions; the + has increased memory and offers an additional blue shift key that provides for additional bond calculations (this key will not be required or used during your CFP studies with the College). The terms [SHIFT] or occasionally [gold] key are used throughout this material to refer only to the dark reddish-orange key with the downward pointing arrow on its face as seen in the preceding paragraph. Using the default number of compounding periods. Most financial calculators allow you to preset the number of compounding periods in your calculator. However, once this default number is set, it works in all calculations. You must either mentally adjust the variables for problems that have compounding periods other than the default, or you must use the calculator s built-in functions to set the proper number of compounding periods. Pressing the [SHIFT] key and then holding down the [C ALL] button shows the current setting for compounding periods per year. If you were calculating a problem that required 1 compounding period in the prior question and you are now calculating what would need to be saved on a monthly basis for accumulating $10,000 FV over 5 years using 6% return, your answer may come out $1, rather than $ per month. Logically you know that $1,980 per month just doesn t make sense to get to $10,000 in five years. That is a clue that your compounding period is off. Using the wrong payment mode (beginning or end). Identify whether the first payment occurs at the beginning or end of the first compounding period. Set your calculator accordingly for each new problem. Begin or End modes are a necessary and important consideration only when a payment [PMT] is involved in the calculation. Students get confused with when to use beginning and end. You may find it easy to think that you d prefer to PAY a Chapter 2: Fundamental Calculator Keystrokes 13

18 bill at the end of the month but would rather GET your full paycheck at the beginning of the month. Generally, word problems have to do with this. When is tuition due? At the beginning. When do you want to receive income? At the beginning. Make sure you carefully read the question and know what they are asking for. Entering a rounded number. Answers may vary slightly due to this factor. To minimize any rounding errors when using the financial calculator, especially following the calculation of an adjusted interest rate (Chapter 4), enter the calculated interest rate immediately after calculating it. To do this, press the [i] or [I/YR] key, as appropriate for the calculator being used. Regardless of the number of digits shown on the screen of the HP 10BII+, the calculator will store a 12-digit number. Doing the step above will yield the most accurate answer. If that is inconvenient, always use the interest rate carried out to at least four places to the right of the decimal point. Helpful Hints With Calculators PV = beginning value (one time lump sum) FV = ending value N = number of compounding periods (this variable is not always in years; it also can be weeks, months, quarters, etc.) I/YR = interest rate per year with the HP 10BII+ (for some calculators, i represents the interest rate per compounding period again, not necessarily the interest rate per year) PMT = payment (amount being put in repeatedly) 14 Introduction to the Time Value of Money

19 Using the Hewlett-Packard 10BII+ Calculator Most financial services professionals use financial calculators or computers rather than tables and exponential calculations because of the speed, ease, and accuracy afforded by calculators. Students in the CFP Certification Professional Education Program are required to use a financial calculator for the national examination. This chapter presents basic instructions for using the Hewlett- Packard 10BII+ calculator in computing answers to time value of money problems. However, these instructions should be used only to supplement the owner s manual provided with the calculator, not to replace it. Before discussing the specific keystrokes used to calculate TVM problems, some basic instructions for calculator usage are presented below. 1. The HP 10BII+ has a method for accommodating more functions on its keyboard than there are keys. Most keys have more than one job. The additional functions are written above or below the keys in orange or blue, and they are accessed in a manner similar to creating capital letters on a keyboard, where a shift key is depressed. Most secondary TVM functions on the calculator use the [SHIFT] key (you may see the color as red-orange with a downward pointing arrow, or similar, but it will generally be identified as the [SHIFT] key in these materials). To access a key s alternate function, press the [SHIFT] key (on the far left column, third from the bottom row of the keyboard), then press the desired key. (Unlike using a keyboard, a financial calculator does not need the [SHIFT] key to be held down while the desired key is pressed; it is pressed prior to the desired key.) For example, OFF is a second function of the ON key on the 10BII+. To turn off the calculator, press the [SHIFT] key, then press the [ON] key. The owner s manual shows this procedure as follows: [gold], [OFF]. 2. The HP 10BII+ calculator must be cleared before every problem. There are procedures for clearing all registers and for clearing only some of the registers. Especially when learning how to use the calculator to perform TVM calculations, it is preferable to clear all storage before every problem. Chapter 2: Fundamental Calculator Keystrokes 15

20 The following keystroke sequence is used to clear the HP 10BII+: [SHIFT], [C ALL]. 3. The following HP 10BII+ keys are used for basic TVM calculations: Key Use denotes number of compounding periods per year (accessed through [SHIFT] key) denotes present value (i.e., value before compounding takes place, or after discounting) denotes future value (i.e., value after compounding takes place, or before discounting) denotes periodic payment denotes annual interest rate (Interest per Year) denotes total number of compounding periods (not necessarily years) changes an entered number from a positive to a negative. Utilize this key AFTER you enter your number. automatically multiplies the number of years by the number of compounding periods per year to arrive at the total number of compounding periods (accessed through [SHIFT] key) / used to program calculator for either an OA - ordinary annuity or an AD - annuity due (accessed through [SHIFT] key). The calculator shows no display message if it is set for an ordinary annuity calculation (payments at the end of the period); however, the display will show the letters BEG (lower central region) if it is set for an annuity due. This key toggles between the two options. 4. The calculator always uses the last number entered into a register for a calculation. For example, entering 1,200 and then PV enters 1,200 into the PV register. A new number can be put into the PV register (overwriting the last entry) without clearing the calculator simply by entering it and pressing PV. The new value replaces the old value. To clear a single register without clearing the calculator, enter 0 and then press the key representing the register to be cleared. 5. The appropriate number of decimal places is programmed on the HP 10BII+ by pressing [SHIFT], [DISP], and then the number of decimal places desired 16 Introduction to the Time Value of Money

21 for display. Turning off the calculator or clearing it will not change the number of decimal places. To set four decimal places, the keystrokes are [SHIFT], [DISP], 4. Note: Although the calculator will show only four decimal places, actual calculations are based upon 12 internal digits. Only the final answer is rounded to four decimal places. It is recommended that you display four digits to the right of the decimal. Whenever two dollar values are entered in the calculator as known values, one of the values must be entered as a negative number [+/-] when solving for interest or a number of periods. For instance, when calculating the interest rate for a single sum, the known values are the number of compounding periods, the number of compounding periods per year, the present value, and the future value. The present value or the future value must be entered as a negative, or else the calculator will not be able to perform the calculation ( No Solution will appear in the display). In general, a value representing an outflow is entered as a negative number, whereas a value representing an inflow is entered as a positive. Deposits, investments, or payments are outflows and usually are entered into the calculator as negative values. The [+/-] key, located on the fifth row of keys on the 10BII+, is pressed following entry of the value to change the sign for example, 1,000, [+/-], [PV]. Chapter 2: Fundamental Calculator Keystrokes 17

22 Figure 1: HP 10BII+ Calculator Using the Hewlett-Packard 12C Calculator This chapter presents basic instructions for using the Hewlett-Packard 12C calculator in computing answers to time value of money problems. The HP 12C uses a logic called RPN. You may have noticed that there is no button with an = sign on it. You will need to use the owner s manual to learn how to do basic math functions. Please note that these basic instructions are provided only as a courtesy to HP 12C users. The College materials do not support the use of any calculators other than the HP 10BII+. 18 Introduction to the Time Value of Money

23 Before discussing the specific keystrokes used to calculate TVM problems, some basic instructions for calculator usage are presented below. 1. The HP 12C has a method of accommodating more functions on its keyboard than there are keys. Most keys have more than one job. Additional functions are written above the keys (in gold/orange) or on the lower edge of the keys (in blue). These are accessed in a manner similar to creating capital letters on a typewriter, where a shift key is depressed. To access the functions written in blue on a main key, press the [g] key (also blue), then the main key. (Unlike on a typewriter, on a financial calculator the [g] key does not need to be held down while the main key is pressed; it is pressed prior to the main key.) For example, a quick way to turn an annual rate entered into the calculator into a monthly rate is to press the [g] key, then the [i] key. (This procedure will be given in this guide using the following notation:[g], 12.) To access the functions written in gold, press [f] (also gold) and then the main key. For example, to access the internal rate of return function (IRR), press the [f] key, then the [FV] key. (This will be given in the owner s handbook as:[f],[irr].) 2. The HP 12C calculator must be cleared before every problem. There are procedures for clearing all stored numbers from all registers, and other procedures for clearing only some of the registers. Especially when learning how to use the calculator to perform time value of money calculations, it is preferable to clear all storage before every problem. The following sequence is used to clear the HP 12C:[f],[REG]. This clears all memory, including numbers stored using the STO key. To clear only financial data, the following sequence is used:[f],[fin]. 3. The following HP 12C keys are used for basic TVM calculations: Key Use denotes total number of compounding periods (not necessarily years) denotes periodic interest rate Chapter 2: Fundamental Calculator Keystrokes 19

24 Key Use denotes present value (i.e., value before compounding takes place, or after discounting) denotes future value (i.e., value after compounding or before discounting) denotes periodic payment used to indicate AD - annuity due (Annuity due is accessed using the following keystroke sequence, after which BEGIN will appear in the lower area of the display: g, BEG.) used to indicate OA - ordinary annuity (In the absence of BEGIN appearing in the display, the calculator automatically calculates an annuity problem as if it were an ordinary annuity. If BEGIN appears in the display, it may be eliminated for an ordinary annuity problem by using the following keystroke sequence: g, END. The BEGIN notation should disappear.) 4. The calculator always uses the last number entered into a register for a calculation. For example, entering 1,200 and then pressing [PV] enters 1,200 into the PV register. A new number can be put into the PV register without clearing the calculator simply by entering it and pressing [PV]. The new value replaces the old value. To clear a single register without clearing the calculator, simply enter 0 and then press the key representing the register to be cleared. 5. The appropriate number of decimal places is programmed on the HP 12C by pressing [f] and then the number of decimal places desired. For example, if two decimal places are appropriate, press [f] and then 2. The number of decimal places selected will remain until changed by pressing [f] and then the new number of decimal places desired; clearing the calculator or turning it off will not alter this. You should note that, although the calculator shows only the number of decimal places selected, it retains substantially more digits internally. 20 Introduction to the Time Value of Money

25 Whenever two dollar values are entered into the calculator as known values, one of the values must be entered as a negative number when solving for interest or a number of periods. For instance, when one is calculating the interest rate for a single sum, the known values are the number of compounding periods, the present value, and the future value. Either the present value or the future value must be entered as a negative, or the calculator will not be able to perform the calculation ( Error 5 will appear in the display). In general, a value representing an outflow from the individual is entered as a negative number, whereas a value representing an inflow to the individual is entered as a positive. If the client is depositing, investing, or making payments, these are outflows and usually are entered into the calculator as negative values. The [CHS] key (change sign), which is located near the center of the top row of keys, is depressed following entry of the value. Chapter 2: Fundamental Calculator Keystrokes 21

26 Chapter 3: Basic Time Value of Money Calculations Reading this chapter will enable you to: 3 1 Calculate the capitalized value of a given income. Capitalization of a Number C apitalizing a number is not a true time value of money concept. This process is used to determine the investment needed to provide the desired number of dollars through the use of interest only, leaving the principal sum untouched. It is among the easiest of all calculations that may be used when determining the amount needed to provide a specified income. Testing tip: On the test, you may see what appears to be a TVM question, but it does not include any number of income periods (N) that normally would be required to solve such a problem. Don t assume that the question is in error. Rather, check to see whether it is a capitalization question. Capitalization questions will not provide the number of years during which desired income will be paid. This is known as capital preservation ; or it could be said that one is living off the interest in these cases. The desired income is divided by the assumed interest rate expressed as a decimal. Assume $30,000 is the desired income and the interest rate is 6%. (1) $ 30, = $500, 000 To verify the answer, multiply the capitalized value by the interest rate: (2) $ 500, = $30, Introduction to the Time Value of Money

27 With this approach, an individual with $500,000 in the bank earning 6% could receive $30,000 per year forever. Unfortunately, inflation, which is not factored in when capitalizing a number, gradually would erode the purchasing power of that $30,000. Examples. Solve the three capitalization questions shown below (rounded to the nearest dollar). 1. David Dennison wants to have $40,000 per year at retirement, using only the interest from his invested money. He expects to earn an average of 6.5% annually. How much does he need to have in the bank to provide this annual income? Answer: $615, Charlene Bellingham wants to build up a bank account that will be large enough to provide her with $15,000 interest per year to be used for travel. If she can earn 7% on her money, how much will she need to reach her goal? Answer: $214, When Steve and Marybeth Jones retire, they want a special fund set aside that will provide them with $6,000 interest per year to purchase gifts for their grandchildren. If they earn 5.25% on the money, how much money will have to be in the fund? Answer: $114,286 Reading the next part of this chapter will enable you to: 3 2 Calculate the future value for a given situation. Future Value of a Single Sum The future value of a single sum usually is the easiest time value concept to understand. The term single sum refers to a lump-sum payment or receipt at one point in time. The future value of a single sum is the future amount of an initial deposit when it is compounded for a given number of periods and at a Chapter 3: Basic Time Value of Money Calculations 23

28 given interest rate. Compounding is the process whereby interest is earned upon interest. When a deposit is made, interest is earned on the deposit in the first period; in subsequent periods, interest is earned not only on the original deposit but also on the interest earned in each of the previous compounding periods. Thus, interest is earned on increasing amounts over time. Note: Keystrokes in this and following sections are for the HP 10BII+ calculator. A number of the keystrokes relate to presetting the compounding periods on the HP 10BII+. There are no corresponding keystrokes for the HP 12C calculator. To accomplish the same calculations on the 12C, you must manually adjust the interest rate (i) and term (n) to reflect the appropriate compounding period. Do this by multiplying the stated years by the compounding period (e.g., 5 years, compounded quarterly equals 5 4 = 20 periods), and dividing the annualized interest rate by the same number (e.g., 8%, compounded quarterly = 8 4 = 2%). Tip for the HP 12C: Calculation of the monthly interest rate (i) and period (n) may be simplified by entering the annual amount, pressing the g key, and then the i key or the n key as appropriate. Future value of a single sum. $1,000 is deposited at 8% interest, compounded annually, for three years. What is the future value? Steps Keystrokes 1. Clear calculator. [SHIFT], C ALL 2. Ensure that the number of compounding periods per year is accurate. The number of compounding periods per year is shown in the display when C ALL is held down during the clear function. It will remain in the display until pressure is released from the key. In this case, one period is appropriate. 3. Enter known values, in any order. As indicated previously, the $1,000 represents an outflow to the depositor; therefore, it is entered as a negative using the key. 1, [SHIFT], 1000,, 8, 3, 4. Request unknown value. 24 Introduction to the Time Value of Money

29 The display will show the answer in this case, $1, Future value of a single sum more frequent compounding. $1,000 is invested for five years in an account earning 8% annual interest, compounded semiannually. What will be the value in five years? Steps Keystrokes 1. Clear calculator. [SHIFT], C ALL 2. Ensure that the number of compounding periods per year is accurate. In this case, two periods per year is appropriate. 3. Enter known values, in any order. As stated previously, the value entered for N is the total number of compounding or discounting periods. In this case, compounding occurs twice a year, so the total number of compounding periods is 10. 2, [SHIFT], 1000,, 8, 5, [SHIFT], (or 5,, 2, =, ) 4. Request the unknown value. The display will show the answer in this case, $1, Reading the next part of this chapter will enable you to: 3 3 Calculate the present value for a given situation. Present Value of a Single Sum The present value of a single sum is the present worth of a sum to be received in the future that has been discounted for a given number of periods and at a given interest rate. Present value is determined by reversing the compounding process, also known as discounting. The three known variables used to compute a present value of a single sum are the future value, the discount rate, and the number of discounting periods. Present value concepts are important in comparing the value of a dollar to be received at different points in time. Rather than measuring the sum of a present amount at some future date, present value is concerned with Chapter 3: Basic Time Value of Money Calculations 25

30 determining the current value of a future sum. The interest rate used when determining present value commonly is called the opportunity cost. It represents the annual rate of return that could be earned on money invested today. For example, if an investor can earn 8%, compounded annually, on an investment vehicle, the investor s opportunity cost of not receiving (investing) dollars today is 8%. Present value of a single sum. An individual will receive $1,000 in three years. How much is this worth today if the opportunity cost on investments is 8% annually? Steps Keystrokes 1. Clear calculator. (Keystrokes for clearing the calculator are discontinued for the remainder of the instructions; this procedure should become automatic as a step occurring between each problem.) 2. Ensure that the number of discounting periods per year is accurate (1 P/YR). 3. Enter known values, in any order. 1000, 4. Request the unknown value. 8, 3, The display will show the answer in this case, $ (A negative sign precedes the answer in the calculator display.) Present value of a single sum more frequent compounding. An individual will receive $1,000 in five years and has an opportunity cost of 8% annual interest, compounded monthly. What is the value of this sum today? 26 Introduction to the Time Value of Money

31 Steps Keystrokes 1. Clear calculator. 2. Ensure that the number of discounting periods per year is accurate (12 P/YR). 3. Enter known values, in any order. 1000, 8, 4. Request the unknown value. 5, [SHIFT], (or 5,, 12, =, ) The display will show the answer in this case, $ Reading the next part of this chapter will enable you to: 3 4 Calculate the number of compounding periods for a given situation. 3 5 Calculate the interest rate per compounding period for a given situation. Number of Compounding Periods and Interest Rate per Compounding Period The two previous sections discussed how to calculate the future value and present value of a single sum. Normally, the known variables when determining the future value are (1) interest rate per compounding period, (2) number of compounding periods, and (3) present value. Likewise, the known variables when determining the present value are (1) interest rate per discounting period, (2) number of discounting periods, and (3) future value. However, the unknown variable in a financial planning problem sometimes is the interest rate per compounding (or discounting) period or the number of compounding (or discounting) periods. The number of compounding periods and the interest rate per compounding period are determined in situations that involve progression in time. The number of Chapter 3: Basic Time Value of Money Calculations 27

32 discounting periods and the interest rate per discounting period are determined in situations that involve regression in time. Number of compounding periods. An individual has $1,000 to invest. He wants to accumulate $3,500. He can earn 8% annual interest on investments. How many years will it take to attain his goal? Steps Keystrokes 1. Clear calculator. 2. Ensure that the number of compounding periods per year is accurate (1 P/YR). 3. Enter known values, in any order. 1000,, 3500, 8, 4. Request the unknown value. The display will show the answer in this case, years. If this were asked as a test question, the correct answer would be 17 years. (At the end of 16 years, he would only have $3, ) Remember, interest is accrued but posted to the account at the specified interval (i.e., annually). Number of compounding periods more frequent compounding. An individual invests $1,000 in an account earning an annual rate of 8%, compounded semiannually. He wants to have a total fund balance of $3,670. How many years will it take to achieve his goal? Steps 1. Clear calculator. 2. Ensure that the number of compounding periods per year showing in the display is accurate (2 P/YR). Keystrokes 3. Enter known values, in any order. 1000,, 4. Request the unknown value. 3670, 8, 28 Introduction to the Time Value of Money

33 The display will show the number of compounding periods required for $1,000 to accumulate to $3,670 at 8% annual interest, compounded semiannually in this case, However, the question asks for the number of years that it will take. Therefore, the result must be divided by the number of compounding periods in a year to arrive at the number of years over the term of the investment. In this case, (total compounding periods) is divided by 2 (number of compounding periods in a year) to arrive at the answer, years (keystrokes:, 2, =). Again, the correct answer to the question would be 17 years. Interest rate. An individual has $1,000 to invest. He wants to accumulate $1,470 in five years. What annual rate of interest must be earned for him to accomplish his goal? Steps 1. Clear calculator. 2. Ensure that the number of compounding periods per year is accurate (1 P/YR). Keystrokes 3. Enter known values, in any order. 1000,, 4. Request the unknown value. The display will show the answer in this case, 8.01%. 1470, Interest rate more frequent compounding. An individual invests $1,000 and wants to accumulate $1,470 in five years. Earnings on this investment are compounded quarterly. What annual rate of earnings is required? 5, Chapter 3: Basic Time Value of Money Calculations 29

34 Steps Keystrokes 1. Clear calculator. 2. Ensure that the number of compounding periods per year is accurate (4 P/YR). 3. Enter known values, in any order. 1000,, 1470, 4. Request the unknown value. The display will show the answer in this case, 7.78%. 5, [SHIFT], (or 5,, 4, =, ) Rule of 72 The Rule of 72 provides a guideline for determining how long it will take an investment to double in value or for determining the rate of return required for an investment to double in value. To calculate the number of years required for an investment to double in value, 72 is divided by the annual interest rate. For example, assume an individual invests $1,000 at 8%, compounded annually. He wants to double his investment to $2,000. Using the Rule of 72, divide 72 by the annual interest rate (8). It will take approximately nine years for the investment to double in value. To calculate the interest rate required for an investment to double in value, divide 72 by the number of years. For example, assume an individual will invest $1,000 and leave it in an account for 10 years. He wants to double his investment. Using the Rule of 72, divide 72 by the number of years the investment is held (10). He should earn approximately 7.2% interest, compounded annually, to double his investment in 10 years. To determine how many years are required for an investment to triple, use the Rule of 116 divide 116 by the expected return. For example, an investment earning 8% annually will take 14.5 years to triple. 30 Introduction to the Time Value of Money

35 Reading the next part of this chapter will enable you to: 3 6 Calculate the periodic payment for a given situation. Present Value of an Annuity The preceding material presented techniques for determining the present value and future value of a single sum. However, many financial planning applications involve payments or receipts at regular periodic intervals instead of as a lumpsum payment or receipt. Such a stream of equal periodic payments or receipts occurring at uniform intervals is known as an annuity. Annuities are classified, depending on the timing of payments or receipts, either as an ordinary annuity (OA) or an annuity due (AD). Ordinary annuity payments or receipts are made at the end of each period (in arrears). Mortgage payments, auto note payments, quarterly dividends, and semiannual interest payments are examples of an ordinary annuity. Insurance policy premiums and lease payments are examples of an annuity due, where payments or receipts are made at the beginning of each period (in advance). Present value of an annuity. An individual expects to receive a payment of $1,000 at the end of each of the next three years. If opportunity costs (the amount one might make if invested elsewhere) are 8% annually, what is the annuity worth today? Stated differently; what amount would be required to be deposited today (PV) to insure payments of $1,000 at the end of each year for the next three years. The answer will be the present value of an ordinary annuity (PVOA) and the amount deposited will have one year to generate additional interest earnings to be distributed over the three-year period. Chapter 3: Basic Time Value of Money Calculations 31