DISCLAIMER Copyright: 2014

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1 DISCLAIMER: This publication is intended for EDUCATIONAL purposes only. The information contained herein is subject to change with no notice, and while a great deal of care has been taken to provide accurate and current information, UBC, their affiliates, authors, editors and staff (collectively, the "UBC Group") makes no claims, representations, or warranties as to accuracy, completeness, usefulness or adequacy of any of the information contained herein. Under no circumstances shall the UBC Group be liable for any losses or damages whatsoever, whether in contract, tort or otherwise, from the use of, or reliance on, the information contained herein. Further, the general principles and conclusions presented in this text are subject to local, provincial, and federal laws and regulations, court cases, and any revisions of the same. This publication is sold for educational purposes only and is not intended to provide, and does not constitute, legal, accounting, or other professional advice. Professional advice should be consulted regarding every specific circumstance before acting on the information presented in these materials. Copyright: 2014 by the UBC Real Estate Division, Sauder School of Business, The University of British Columbia. Printed in Canada. ALL RIGHTS RESERVED. No part of this work covered by the copyright hereon may be reproduced, transcribed, modified, distributed, republished, or used in any form or by any means graphic, electronic, or mechanical, including photocopying, recording, taping, web distribution, or used in any information storage and retrieval system without the prior written permission of the publisher.

2 LESSON 1 Finance Fundamentals I Note: Selected readings can be found under "Lesson 1" on your course website Assigned Reading 1. UBC Real Estate Division CPD 151 Real Estate Finance Basics. Vancouver: UBC Real Estate Division. Lesson 1: Finance Fundamentals I Recommended Reading 1. HP10BII: Introduction to the Calculator and Review of Mortgage Finance Techniques 2. HP10BII Computer Based Training. Learn how to use the HP 10BII calculator by using our new interactive training tool. This will show you the steps on the calculator that are required to solve real estate finance problems. Please note that this will link to the Hewlett Packard website and to view this application, users must have Macromedia Shockwave Flash Player installed. To download this free plug-in, visit the Macromedia website and follow the on-screen instructions. 3. Math Review Kit. An overview of math and algebra calculations. Provided with permission from CGA Canada. 4. Excel tutorials. Microsoft: Various tutorials ranging from easy to advanced for Excel 2007, 2010, 2013, and excel for Mac FX aspx Learnfree.org: Provides quite basic instructions for various versions of Excel (2002, 2007, 2010, etc.). Explains the basic with screenshots and video tutorials. Some lessons include cell basics, creating complex formulas, formatting, and creating tables. Learning Objectives After completing this lesson, the student should be able to: 1. explain the difference between simple and compound interest; 2. define and differentiate between a variety of interest rate types, including nominal, periodic, effective, and equivalent rates of interest; 3. discuss the underlying mathematics for financial analysis and apply this in structuring solutions to problems using both a financial calculator and spreadsheet software; 4. solve for the specific calculations necessary in lump sum problems, including present value (PV), future value (FV), timeframe (N), or interest rate (I/YR); and 5. convert equivalent interest rates based on different compounding frequencies. 1.1

3 Lesson 1 Instructor's Comments This lesson outlines the basic tools and techniques required for real estate mathematics. The basis for all financial analysis is interest rates. Since these can be quoted and applied in many different ways, it is crucial to be fluent in working with the varying definitions and also to be able to apply these properly in mathematical solutions. As a result, much of this initial lesson focuses on interest rate explanations and calculations. This includes the difference between nominal and periodic rates, plus the calculations necessary to convert equivalent interest rates where the compounding frequencies differ. Financial Fluency Mathematics is often described as another form of language. Like learning a new language, this requires understanding the grammar and then a lot a memorization and repetition until you can use it without really thinking about it. For financial analysis, the mathematics involves the use of formulas. These are usually pre-programmed into a financial calculator or into a spreadsheet; therefore, most financial analysts do not work with the formulas directly. However, in all cases, the interest rate must be correctly applied, or the calculator/spreadsheet will not operate accurately. To be successful in financial analysis, you must be fluent in your understanding of how interest rates are stated and in how to work between the different types as required for the financial tool you are working with. The financial calculations in this lesson will start out with somewhat simple problems, such as interest accrual loans and investments involving the present and future value of a single cash flow. The calculations will advance to more complicated examples in Lessons 2 and 3. Much like the construction of a high-rise, you need a solid foundation before you can start building the upper floors. This lesson focuses on the foundations of financial analysis. Math Formulas and Financial Calculators Finance calculations involve complicated mathematical formulas. If you are comfortable with algebra, these formulas can be solved manually and using any calculator. However, financial calculators and spreadsheets have the formulas pre-programmed into them, making financial calculations much easier. Once you have a programmed tool, whether a calculator or computer software, the financial problems become less about mathematics and more about structured problem solving: in general, you need only specify the inputs for the problem, and the calculator/computer will do the mathematics for you. In this course we will use the Hewlett Packard (HP) 10BII/10BII+ calculator to demonstrate analytical techniques. There is nothing particularly special about this particular calculator; it is a standard financial calculator that is able to carry out the necessary calculations and not a lot more. There are a variety of preprogrammed financial calculators on the market, some of which perform more sophisticated calculations or have greater programming capacity. Students in Real Estate Division courses are welcome to choose other calculators if they prefer, but it is up to the student to ensure that the alternate calculator will perform all necessary calculations and to determine its operations. As well, for Real Estate Division courses with proctored final examinations, it is important to understand that only silent, cordless, hand-held calculators that are not both alphanumeric and programmable are permitted in the examination room. The problems covered in this course will initially provide both mathematical formulas and calculator steps. In subsequent problems, this will be abbreviated to provide calculator steps only. As well, the use of computer spreadsheets is becoming increasingly important in financial analysis, more or less a mandatory tool in contemporary business. In recognition of this, the lessons will also provide the formulas and steps for the equivalent analysis in Microsoft Excel at key points. 1.2

4 Real Estate Finance Basics The HP10BII/10BII+ has some unique features, so some basic orientation will help avoid difficulties. 1. Setting Up Your Calculator The HP 10BII+ has two O (shift) keys. One is orange (for financial functions); the other is blue (for statistical functions). To access the financial functions on the calculator, students should always use the ORANGE O (shift) key. All functions that are activated by the ORANGE O (shift) key are located on the lower half of each of the calculator keys, and are also labeled in ORANGE. We do not use the BLUE O (shift) key in this course. Decimal Places: The HP10BII/II+ allows you to set the number of decimal places displayed using the ORANGE O then DISP key. It is best to display more decimal places than you need, so you are working with the most accurate numbers possible. O DISP 9 will display 9 digits for all problems (and in fact, shows 9 zeros, even if not significant, such as ). You may also use what is known as "floating decimals", or SHIFT DISP. However, it also displays small and large numbers in scientific notation or exponential functions, e.g., 4/19 = E-2. This same value set to 9 decimal places changes the display to The use of floating decimal places gives slightly more accuracy, but you have to be comfortable working with the exponents (E-n). This is your personal preference, as the final decimal place does not significantly affect calculations. Note that the calculator uses its full accuracy of up to 15 decimal places regardless of the level of display chosen. For ease of presentation, in each of the examples presented in this course, the calculator is programmed to display a "fixed decimal point" set to six decimal places. This is accomplished by turning the calculator ON, pressing the ORANGE O then DISP and then the 6 key. You will see on your display screen. However, note that on the display portion of calculator steps in the lessons, we will not show the display with zeros when they do not impact the result (and are mathematically insignificant). 2. Using Your Calculator to Solve Questions The internal operation of the HP 10BII/10BII+ calculator requires that all financial calculations have at least one positive and one negative cash flow. This means that at least one of the PV, FV, and PMT keys will have to be shown and/or entered as a negative amount. Generally, cash flowing in is positive, while cash flowing out is negative. For example, in mortgage loan problems, the borrower receives loan funds at the beginning of the term (cash in, so a positive amount) and pays back the loan funds either during or at the end of the term (cash out, so negative amounts). In this type of problem, PV will be shown/entered as a positive, while PMT and FV will be shown/entered as negatives. When entering a negative amount the +/- key is used, not the - key. Similarly, from an investor's perspective, the initial investment is paid out (cash out, so negative amount) and the investor receives money in the future (cash in, so positive amount). In this type of problem, PV will be shown/entered as a negative, while PMT and FV will be shown/entered as positives. Summary Borrower's Perspective Investor's Perspective PV + PV B PMT B PMT + FV B FV + (continued on the following page) 1.3

5 Lesson 1 (continued) 3. Clearing Information on Your Calculator Note that if you enter an incorrect number on the screen, it can be cleared by pushing C once or by pressing the key to delete the last entered digit. If you enter an incorrect number into any of the six financial keys, N, I/YR, PMT, PV, FV, or P/YR, it can be corrected by re-entering the desired number into that key. You can verify what information is stored in each of the above financial keys by pressing RCL and then the corresponding financial key you are interested in. For example, if you obtained an incorrect solution for a financial problem, you can check what is stored in N by pressing RCL N; I/YR by pressing RCL I/YR, etc. The HP 10BII/10BII+ calculator has a "constant memory". This means that whatever is stored in the keys remains there until it is expressly changed (even when the calculator is turned off), unless the C ALL function is used or the batteries are removed. 4. Troubleshooting i. Please be aware that the HP 10BII/10BII+ calculator has both Begin and End modes. The Begin mode is needed for annuity due calculations, or those which require payments to be made "in advance". For example, lease payments are generally made at the beginning of each month, not at the end. On the other hand, interest payments are almost always calculated at the end of each payment period, or "not in advance". These types of calculations each require a different setting on the calculator. When your calculator is set in Begin mode, the bottom of the display screen will show BEGIN or BEG. If BEG is not on your display screen, your calculator must be in End mode, as there is no annunciator for this mode. In this course, there are minimal calculations which require your calculator to be in Begin mode, so your calculator should be in End mode at all times. You should not see the BEGIN or BEG annunciator on your calculator's display for most of the calculations in this course. To switch between modes, press O BEG/END. ii. Please be aware that if the calculator is displaying a comma (,) instead of a decimal place (.), this can be fixed by pressing./, or,/.. Alert! Students may wish to view the "Introduction to the HP10BII/10BII+ Calculator" online video tutorial found under "Tutorial Assistance" and "Course Materials" on the Course Resources webpage. As well, students can consult the HP 10BII/10BII+ owner's manual for more information. 1.4

6 Real Estate Finance Basics Calculations and Solutions to Math Problems Calculator Steps: In developing this course, we have illustrated the mathematical solutions for only some problems, i.e., providing the solution based on formulas. For most problems, the calculator steps are illustrated with the Hewlett-Packard HP10BII/10BII+ financial calculator. Spreadsheet Applications: To emphasize the real-world nature of many of these calculations, we have provided spreadsheet applications for many of these, in Microsoft Excel format. You can find the Excel spreadsheet "CPD 151 Lesson 1 Solutions" under "Online Readings" on the Course Resources webpage. Interest Rates The General Characteristics of Interest In a general sense, interest is rent on borrowed capital, where the level of rent per dollar borrowed for each type of investment is a function of general economic conditions plus the costs and risks associated with the specific investment. However, in the discussion of the mathematics of real estate finance, such macroeconomic factors are taken as given: interest is discussed in a very pragmatic sense as the cost of borrowing and/or the revenue from lending or investing. The focus of analysis in this context is upon the effects of different patterns of interest calculation and principal repayment on the costs and benefits of mortgage lending and borrowing and other forms of real estate investment. The basic concept of valuation of financial instruments focuses upon the relationship between when interest must be paid and when principal must be repaid. In the case of simple interest, interest is paid or earned each period on the original principal amount only, but not on any interest charged or paid. However, with compound interest, the interest charged changes with the interest that accumulates over time. Compound interest means that interest is charged (or earned) on interest (as well as on the principal amount). The essential difference between simple and compound interest is that simple interest is based on the principal amount only, whereas with compound interest, the interest charged changes with the interest that accumulates over time. Note that compound interest need not be actually paid to the lender on a periodic basis; unpaid interest may be added to the debt and itself earn interest. It is important to remember that interest rates are generally stated as the rate that prevails for a full year, regardless of whether the loan is for a very short period of time, exactly a year, or a very long period of time, and regardless of whether simple or compound interest is charged. Also, in most cases, interest is charged only at the end of some period of time over which the borrower has use of the funds advanced under the loan agreement. Thus, interest rates will be stated as a certain percentage per annum not in advance. This differs from charges for the use of space, such as rent, which are levied in advance of the use. Simple Interest Simple interest is interest that is paid only on the amount of principal borrowed, but not on any interest charged or paid. Therefore, the amount of interest due (I) at the term of the loan is dependent only upon the interest rate, the amount of principal borrowed, and the length of the loan period. 1 The simple interest rate (r) is expressed as a percentage of the simple interest interest that is paid only on the amount of principal borrowed, but not on any interest charged or paid 1 As is shown in the next section, the amount of interest paid when compound interest is charged is a function of these three factors plus the number of compounding periods during the term of the loan. 1.5

7 Lesson 1 principal borrowed (P), where that percentage would be charged for the use of funds for exactly one year. The principal borrowed is expressed as a dollar figure, and cannot change during the term of a simple interest loan, as no payments of principal or interest are made during the term. The length of the loan period (t) is measured as the number of full (or elapsed) days between the initiation of the loan and its termination. Thus, if a loan is made one day, and repaid the next, although two calendar days are involved, only one day has elapsed: interest would be charged for one day only. Loans and Investments A Matter of Perspective: Most of the examples in this section focus on borrowing scenarios evaluating the interest cost for borrowers, the periodic payments required on a loan, or what is owing at the end of a loan term. However, these could all just as easily be considered from a lender or investor s perspective evaluating the yield on investment based on cash flow received back from it. Every investment has two parties involved on either side of the transaction, for every borrower there must also be a lender and whether it is money paid out or money received, the financial math is the same! The following illustration is representative of simple interest calculations and serves to introduce a number of items relating to standard symbols used in financial analysis. Illustration 1.1 Consider a loan in the amount of $100,000 at 5% per annum simple interest where the loan is outstanding for two years. Calculate the amount of interest owing at the end of the loan and the total amount owed at the end of the loan. Solution The amount of interest owing at the end as found as: I = P r t where P = amount of principal borrowed r = simple interest rate t = term (in years) I = $100, I = $10,000 The amount owing at the end of the term is: A = I + P A = $10,000 + $100,000 A = $110,000 For simple interest calculations, only the basic arithmetic functions of the calculator are used. Alternatively, one could obtain the same answer by expressing the loan term in days rather than years (I = P r number of days in loan number of days in year). In this example, I = $100, = $10,

8 Real Estate Finance Basics Compound Interest: The Basis for Interest Accrual Loans and Investments The preceding section dealt with real estate loans/investments where simple interest was charged: the formulas presented are only applicable in the infrequent cases where such an arrangement is made. However, the formulas do provide the basis for the analysis of the much more common practice of real estate loans/investments where compound interest is charged. The remainder of this material is devoted to the technical aspects of real estate finance using compound interest scenarios. In this section, the basic concepts of compound interest calculations are presented as the foundation for the analysis of real estate financing. Compound interest occurs when interest is charged or calculated on interest during the life of a loan. This occurs when interest (and some principal) is paid when interest is calculated (as in the case of amortizing loans) or is added to the principal and therefore, earns interest (as in the case of interest accrual loans). We will discuss interest accrual loans in this section and deal with amortizing loans in a later section. compound interest interest calculated on the initial principal and the accumulated interest; interest on interest interest accrual loan debt which is paid off as one lump sum, including principal plus accumulated compound interest An interest accrual loan is one on which no payments of interest and no repayments of principal are made or received during the life of the loan. The full amount of principal and all interest that accumulates during the term are payable when the term expires. A loan with interest charged during the life of the loan which is added to the principal amount and, in turn, earns interest over the remainder of the contract, is called an accrual loan. In effect, the lender is actually lending the borrower additional amounts of money equal to the amount of interest due during the life of the loan. Illustration 1.2 Consider a $45,000 interest accrual loan on which interest is to be charged at the rate of 6% at the end of each six-month period for a 2-year term. Calculate the amount owed at the end of the loan. Solution By the end of six months, the borrower will owe $47,700, the sum of the $45,000 principal plus $2,700 interest ($45,000 6%). At the end of the first year, the borrower will owe $50,562 which is broken down as follows: $45,000 Principal amount borrowed + 2,700 Interest during first six months ($45,000.06) = 47,700 Owing during second six months + 2,862 Interest during the second six months ($47,700.06) = $50,562 Total amount owing The outstanding balance (amount owing to the lender) on this loan has grown from $45,000 when it was advanced to $50,562 by the end of one year, and it continues to grow over time to $56, at the end of the term as shown by Figure

9 Lesson 1 Figure 1.1: Outstanding Balance on an Interest Accrual Loan Since the lender receives no payments before the maturity of the loan, the entire $45,000 original investment is at risk throughout the term of the loan. The income to the lender, which is the interest on the monies advanced, is also at risk throughout the entire term. For these reasons, almost all interest accrual loans are written for short terms, one year or less (note that a two-year term is used in this example, for illustration purposes). Many more examples of compound interest calculations will be provided later in this lesson. In fact, given compound interest is the basis of most contemporary financial analysis, nearly all of the remaining calculations in the course focus on compound interest! However, before we continue to illustrate further examples, we must first deal with one additional aspect of working with interest rates: the difference between nominal and periodic rates. Nominal and Periodic Interest Rates Interest rates can be stated in many different ways. In order to accurately work with interest rates in financial analysis, we must first ensure that we are using the correct rate and using it properly. Compound interest rates are generally expressed on an annual basis for convenience and to follow convention. However, within the year, the compounding period may vary, e.g., daily, weekly, monthly, quarterly, semi-annual, and annual compounding periods. These will result in significant differences in the amounts of interest owing or earned. Therefore, it is not enough to simply state "a rate of x% per year" it is crucial to also specify the compounding period for the interest rate. 1.8

10 Real Estate Finance Basics The annual interest rate quoted for compound interest is referred to as the nominal interest rate per annum. The nominal rate is represented mathematically as jm: or where: j i m m i jm m j m = nominal interest rate compounded "m" times per year m = number of compounding periods per annum i = interest rate per compounding period, or periodic interest rate The nominal interest rate (jm) is always expressed as a certain percentage per year compounded a specific number of times during the year (m). The "m" could be 365 for daily compounding, 12 for monthly compounding, or 52 for weekly compounding. The periodic rate is the rate per compounding period, such as the rate per day, per week, per month, or per half-year. Consider the nominal rate of 12% per annum, compounded monthly, not in advance 2. nominal interest rate an interest rate quoted as a rate per annum, with a stated compounding frequency; equals the interest rate per compounding period multiplied by the number of compounding periods periodic rate an interest rate per compounding period The nominal rate would be expressed as: j 12% 12 The periodic rate would be expressed as: j12 i 12% 1% Thus, if an interest rate is stated as 12% per annum, compounded monthly (not in advance), this indicates that there are 12 compounding periods (m) per annum and that the interest is 1% (i = jm m) in each monthly compounding period. This can be illustrated using a time diagram 3 as shown below: 2 "Not in advance" refers to the fact that the amount of interest accruing over the compounding period is calculated at the end of the compounding period, so that the borrower pays the interest at the end (or, not in advance) of the compounding period. Almost all rates of interest are calculated "not in advance". Therefore, the statement "not in advance" is frequently not used, and the interest rate would be quoted as 12% per annum, compounded monthly. Unless it is explicitly stated to be otherwise, students may assume that all interest rates are "not in advance". 3 Time diagrams are shown as a horizontal line representing time. The present value is at the left (time 0) and the future value is at the right. In financial arrangements, time is measured by compounding periods, and so 12 monthly compounding periods are shown along the "time" line. 1.9

11 Lesson 1 j 12 = 12% nominal interest rate (per year) m = monthly periods per year i = 1% periodic interest rate (monthly period) As a second example, if the nominal rate is 6% per annum, compounded semi-annually, not in advance, this would be expressed as: j 6% 2 The periodic rate would be expressed as: i j2 6% 2 2 3% Thus, if an interest rate is stated as 6% per annum, compounded semi-annually (not in advance), this indicates that there are two compounding periods (m) per annum and that the interest is 3% (i = jm m) in each semi-annual compounding period. This can be illustrated using a time diagram as shown below: j 2 = 6% nominal interest rate (per year) m = 2 12 monthly periods per year i = 3% periodic interest rate (semi-annual period) Nominal and Periodic Rates The HP10BII/II+ calculator has pre-programmed financial interest rate keys that require nominal rates. TheI/YR and/or NOM% keys require a rate per year entered in nominal form (e.g., 5% not 0.05). Furthermore, the calculator needs to know the associated compounding frequency (P/YR). Alternatively, the mathematical formulas, other calculators, and Excel require interest rates to be stated as a periodic interest rate (i). This varies for every calculator and software package and that is why you must be fluent in working between these interest rate formats! For a nominal interest rate "j", by definition you know it is a rate per year. However, without the "m" specified, you do not know the compounding frequency. Therefore, to state "the interest rate is a nominal rate of 6%" is incomplete. You must know the compounding frequency in order to carry out further calculations. 1.10

12 Real Estate Finance Basics Examples of nominal interest rates: i d x 365 = j 365 i w x 52 = j 52 i mo x 12 = j 12 i q x 4 = j 4 i sa x 2 = j 2 i a x 1 = j 1 You may have noticed that the final line shows that the interest rate ia and j1 are equal. This is an annual interest rate with annual compounding, also known as the effective annual interest rate. This rate is the most common basis for comparing interest rates on loans/investments, as it effectively accounts for all complications in trying to compare rates with different compounding. The effective annual rate is found in EFF% key on the HP10BII/II+ calculator. We will use this key extensively later in this lesson, when carrying out equivalent interest rate calculations. Effective annual rates (and the EFF% key) will be used in a future section to find equivalent interest rates. Similarly, periodic rates must also have the frequency of compounding stated. The following shorthand notation is used in this course to indicate the frequency of compounding for periodic rates: For example: i d represents an interest rate per daily compounding period i w represents an interest rate per weekly compounding period i mo represents an interest rate per monthly compounding period i q represents an interest rate per quarterly compounding period i sa represents an interest rate per semi-annual compounding period i a represents an interest rate per annual compounding period i d = j i w = j i mo = j i q = j 4 4 i sa = j 2 2 i a = j 1 1 effective annual interest rate an annual interest rate that is compounded annually (j 1 ) Exercise 1 will help you become more familiarity with periodic interest rates, compounding frequency, nominal rates, and the interrelationship between them. Remember, it is crucial that you be fluent in moving between these rates, in order to be successful in the more complicated financial problems you will experience later. 1.11

13 Lesson 1 Exercise 1 The tables below represent a sample of interest rates. Complete the tables by entering the appropriate values for the question marks for either the periodic rate, the number of compounding periods, or the nominal rate. Question Periodic Rate i Number of Compounding Periods per Year (m) Nominal Rate (j m = i m) SAMPLE (a) (b) (c) (d) (e) (f) (g) SAMPLE (h) (i) (j) (k) (l) (m) (n) 3% 0.625% 1.275% 4% 3.4% 6% 2.5% % j 2 = 5% j 12 = 6% j 4 = 10% j 365 = 8% j 52 = 13% j 1 = 4% j m = 10% j m = 4.5% m =? m =? m =? m =? j 2 = 6% j 12 =? j 4 =? j 1 =? j 2 =? j 2 =? j m = 10% j m = 13% i sa = 2.5% i mo =? i q =? i d =? i w =? i a =? i sa = 5% i mo = 0.375% Abbreviated Solution: (a) j 12 = 7.5% (b) j 4 = 5.1% (c) j 1 = 4% (d) j 2 = 6.8% (e) j 2 = 12% (f) m = 4 (g) m = 365 (h) i mo = 0.5% (i) i q = 2.5% (j) i d = % (k) i w = 0.25% (l) i a = 4% (m) m = 2 (n) m =

14 Real Estate Finance Basics Self-Help Exercise: Nominal and Periodic Rates In order to become fluent with nominal and periodic rates, you need practice. Here s an exercise you can complete on your own. Make a list of nominal rates using a variety of compounding and your choice of numbers: Nominal Periodic Nominal j 2 = 12%? j 4 = 8%? j 52 = 7%? Solve for the periodic rates (as shown in the second column): Nominal Periodic Nominal j 2 = 12% i sa = 6% j 4 = 8% i q = 2% j 52 = 7% i w = Now cover up the first column and work backwards from periodic to nominal. Nominal Periodic Nominal i sa = 6%? i q = 2%? i w = ? Uncover the first column and your first and last columns should match. Do more practice on your own until you are comfortable with this nominal/periodic relationship. Nominal Periodic Nominal j 2 = 12% i sa = 6% j 2 = 12% j 4 = 8% i q = 2% j 4 = 8% j 52 = 7% i w = j 52 = 7% Illustrations of Compound Interest Calculations Now that we have the terminology of interest rates clarified, we can proceed to calculations involving compound interest. For our first example of the nature of compound interest, we will complete a problem quite similar to the example covered in the Interest Rates section. Illustration 1.3 A commercial enterprise has arranged for an interest accrual loan whereby the $10,000 amount borrowed is to be repaid in full at the end of one year. The borrower has agreed, in addition, to pay interest at the rate of 8% per annum, compounded annually on borrowed funds and this interest is to be calculated and paid at the end of the loan's one-year term. Calculate the amount of interest due and the total amount owing at the end of the term of the loan. 1.13

15 Lesson 1 Solution Given that the borrower owes $10,000 throughout the year, the amount of interest owing at the end of the one-year term is calculated as follows: Interest owing = Principal borrowed interest rate per interest calculation period (in this example, interest is calculated per annual compounding period) = $10,000 8% = $10, Thus, the amount of interest owing at the end of the one year term is $800. The total amount owing at the end of the one year term of this interest accrual loan would be the principal borrowed ($10,000) plus the interest charged ($800) or $10,800. This illustration introduces a number of very important definitions and concepts. Financial analysts use short form abbreviations for the loan amount, interest rates and other mortgage items. In this shorthand notation, the following symbols are used: PV = Present value: the amount of principal owing at the beginning of an interest calculation period; FV = Future value: the amount of money owing in the future; i = Interest rate per compounding period; the fraction (or percentage) used to calculate the dollar amount of interest owing; I = Interest owing, in dollars, at the end of an interest calculation (compounding) period; and n = Number of compounding periods contracted for. The amount of interest owing can be mathematically calculated as follows: I = PV i I = $10,000 8% I = $10, I = $800 The total amount owing at the end of the one-year term would be: FV = PV + I FV = $10,000 + $800 FV = $10, = 800 Equals interest owing = 10,800 Equals total owing For illustrations in this course, the HP10BII/II+ calculator is set to six decimal places by pressing O DISP 6. However, for ease of presentation, on the display portion of calculator steps, we will not show the display with zeros when they do not impact the result (and are mathematically insignificant). 1.14

16 Real Estate Finance Basics This example can be illustrated on a time diagram: The calculations in this illustration are extremely simple and probably do not need a calculator, but it is a good opportunity to introduce its use. For illustrations throughout the text, we will provide each step in the calculation, noting which button to press, what appears on the screen (display), and comments to explain each new step as it is encountered. Excel calculations will also be demonstrated for many illustrations. You can find the associated Excel file under "Online Readings" section of the Course Resources webpage. The Power of Spreadsheets By using a spreadsheet like Excel, we can solve financial problems either by programming in math formulas manually or by the use of Excel s pre-programmed financial formulas. For this simple example, we identify the loan amount (PV), interest rate, and length of loan. We then calculate the amount of interest and future value amount. If we specify the numbers in separate cells, and then program the cell coordinates into the formulas, it allows us the flexibility to easily change a number and then recalculate the future value. For example, if we change the loan amount to $5,000, we change that one cell (B4) and then the interest amount will change to $400 and future value to $5,400. This is a simple example of the dynamic nature of spreadsheets which can be very useful in working with highly complex, sophisticated analyses. In this simple example, note that the answer is the same as would result if this was simple interest or compound interest. This is because there is only one term in the loan. Let's see what happens when the loan is for multiple terms. Illustration 1.4 Assume that the commercial borrower in Illustration 1.3 arranged for another loan which was similar in all respects, except that the contract specified a term of three years. Calculate the amount owing at the end of the 3-year term of the interest accrual loan. Solution The amount owing at the end of the three year term is calculated in a series of steps based on the analysis presented in Illustration 1.1. The amount owing at the end of the first compounding period (in this case, at the end of the first year) must be calculated. This amount was calculated to be $10,800 (the $10,000 originally borrowed plus $800 interest). As a result, the amount owing during the second year (or second annual compounding period) would be $10,800. The amount owing at the end of the second year of the contract can be calculated in the same way as in Illustration 1.3. I = PV i I = $10,800 8% I = $10, I = $864 Thus, the amount of interest that is charged (but not paid) at the end of the second year of the loan term is 1.15

17 Lesson 1 $864. As no payments (of principal or interest) are due until the end of the loan term, this amount is added to the balance owing at the end of year two. FV = PV + I FV = $10,800 + $864 FV = $11,664 The total amount owing during the third year of the contract would be $11,664. The amount owing at the end of three years can be calculated in a similar fashion: I = PV i I = $11,664 8% I = $11, I = $ FV = PV + I FV = $11,664 + $ FV = $12, This calculation above can be summarized on a time diagram, similar to Illustration 1.3: The previous calculations show that the steps involved in determining the amount owing on an interest accrual loan can be time-consuming. Therefore, to reduce the number of calculations involved in interest accrual loans, the financial calculator has been pre-programmed to perform this function. It was noted earlier that the amount owing at the end of year one could be determined by first calculating the amount of interest at the end of the first (annual) compounding period, then adding this amount to the principal borrowed. The formulas used to perform these calculations were I = PV i and FV = PV + I. By using these relationships in a repetitive fashion, the amount owing at the end of the term of an interest accrual loan could be determined. However, there is a faster way to do these calculations. In Illustration 1.4 you could have determined the amount owing at the end of the first year by calculating 108% of the amount owing during the year. This would involve simply multiplying the amount owing during the year by 108% (or 1.08) to get $10,800, the product of multiplying $10,000 by The amount owing at the end of the second year will be the amount owing during the second year ($10,800), multiplied by 108% (or 1.08); i.e., $11,664. Finally, and recognizing that the borrower owes $11,664 during the third year, the amount owing at the end of the third year can be calculated by multiplying this value by 108% (or 1.08) with the result of $12, To summarize, the problem can be solved as follows: Amount owing at end of year one = $10, = $10,800 Amount owing at end of year two = $10, = $11,664 Amount owing at end of year three = $11, = $12,

18 Real Estate Finance Basics A simpler calculation recognizes that for each annual compounding period the principal outstanding is multiplied by 1.08: FV = $10, FV = $12, Therefore, in determining the amount owing at the end of the term of an interest accrual loan, the principal amount originally borrowed can be multiplied by one plus the rate of interest per compounding period (expressed as a decimal) for the number of compounding periods during the contract term. To simplify the analysis even further, standard mathematical notation can be used which represents the value of one plus the rate of interest per compounding period as (1+i). A superscript indicates that a number has been raised to a power (or multiplied by itself some number of times) and the relationship may be restated as follows: FV = $10,000 (1.08) 3 or in more general terms: FV = PV (1 + i) n where FV = Future value (or amount owing in the future) PV = Present value (or original amount borrowed) i = interest rate per compounding period expressed as a decimal n = number of compounding periods in the loan term Any financial or scientific calculator can do these kinds of repetitive multiplications, also known as exponential calculations. The steps below show how the calculator can be used to determine the amount owing on the loan by using the exponential function O y x 3 = raised to the power of = 12, Total amount owed at end of year three The use of the exponential key (y x ) reduces the number of repetitive calculations required in analyzing interest accrual loans. You only need to determine the value of (1 + i) n at the appropriate rate of interest (expressed as a decimal) and for the appropriate number of compounding periods and then multiply the result by the principal amount borrowed. This type of exponential analysis is the basis for many algebra calculations; therefore, this capability is programmed into all scientific or business calculators. 1.17

19 Lesson 1 Caution! Interest Rates: Percent or Decimal? The HP10BII/II+ financial calculators require interest rates to be entered in percent form (e.g., 3.5), not decimals (e.g., 0.035), and as a rate per year (nominal rate). When using math formulas directly, you must instead specify interest rates in decimal form, not percentage form, and as a rate per period. Excel similarly requires decimal expressions of interest rates in periodic form. For example, if the calculator uses 8% in nominal (annual) percentage format, Excel and the math formula solution would use However, in Excel you may format the cells to show in percentage terms, in which case Excel will automatically switch to a decimal for the calculation. Be careful in Excel that you have specified your interest rates correctly if you specify a rate as 8, but don t set it as a percent, Excel may translate this as 800%! When using the HP 10BII/10BII+ calculator, the above formula, FV = PV (1 + i) n, must be slightly modified to consider nominal interest rates. Recall that a periodic rate is equal to the nominal rate divided by the compounding frequency. Thus the formula becomes: FV = PV (1 + j m /m) n where j m = nominal interest rate per annum m = compounding frequency n = number of compounding periods in the loan term This modified version is necessary because this calculator only works with nominal interest rates. This formula for interest accrual loans has been pre-programmed into the mortgage finance keys of the HP 10BII/10BII+ calculator. These keys are: I/YR O P/YR N PV FV PMT Nominal interest rate per year (jm) B entered as a percent amount (not as a decimal) "Periods per year" (m) B this indicates the compounding frequency of the nominal rate in I/YR and is located below the PMT key Number of compounding or payment periods in the financial problem - this number will be expressed in the same frequency as P/YR (in other words, if P/YR is 12, then N will represent the number of months) Present value Future value after N periods Payment per period B this is expressed in the same frequency as P/YR and N (i.e., if N is months, PMT represents the payment per month) Illustration 1.4 is once again illustrated below with a time diagram, but with a new feature added. The cash flows are placed along the horizontal line with an arrow representing positive or negative cash flows. An "up arrow" represents a positive cash flow (money received), while a "down arrow" represents a negative cash flow (money paid out). To calculate this problem with the HP 10BII/10BII+ calculator, a number should be entered and then "labelled" appropriately. For example, the loan in this example has a three-year term, so "3" should be entered and then "N" pressed in order to enter a value of 3 as the number of compounding periods during the term. By entering a number and then labelling it, you can enter the information in any order. 1.18

20 Real Estate Finance Basics 8 I/YR 8 Enter nominal interest rate per year 1 O P/YR 1 Enter compounding frequency 3 N 3 Enter number of compounding periods 0 PMT 0 No payments during term PV 10,000 Enter present value 4 FV -12, Computed future value 5 This is the same answer as that calculated with either of the two approaches shown earlier, but with much less work needed. Future Value Calculations in Excel You can solve Illustration 1.4 using Excel by either using the math formula or the pre-programmed financial functions. With the pre-programmed FV function, you enter the periodic rate, the loan or investment period (NPER), payment (0), PV (the loan amount), and type of loan (0). The type of loan is a value representing the timing of the payment. If payments occur at the beginning of a period, the type is 1 and if the payments occur at the end of a period, the type is 0. Since there is no payment in this calculation, the type is 0 or can be omitted as 0 is the default option. Notice that the result shows as a negative, like the calculator solution. Note also that the compounding frequency of the interest rate must match the periods used in specifying the loan or investment period (NPER) if not, then an interest rate conversion is required more on this topic later in this lesson. Recapping our progress: so far we have illustrated the difference between simple and compound interest and shown that compound interest is the standard method for financial investments. Compound interest allows "interest on the interest", which is an important consideration for investors making long-term loans. The calculations shown so far have illustrated how compound interest works, in leading to a higher future value for a loan or investment than would accrue in a simple interest scenario. We will now proceed to illustrate further present value and future value calculations, from a variety of real estate finance and investment contexts. Note that these calculations all involve the future and present values of a single payment at some point in time, which is called a lump sum amount (or single cash flow). The introduction of payments during the loan or investment period adds further complexity this will be examined in detail in Lesson 2. 4 Note that the borrower receives the cash; therefore, it is entered as a positive amount. 5 This will be paid out by the borrower; therefore, it is a negative amount. 1.19

21 Lesson 1 Structuring Present Value and Future Value Problems Consider the interest accrual formulas already covered: FV = PV (1 + i) n or FV = PV (1 + j m /m) n If the values of any four of the five variables (FV, PV, j m, m, and n) are known, one may directly calculate the value of the fifth variable. However, you need the following conditions for the above relationship (or the financial calculator) to be used: 1. The present value must occur at the beginning of the first compounding period. 2. The future value must occur at the end of the last (or n th ) compounding period. 3. Interest rates must be expressed as a rate of interest per compounding period when solving using the exponent key y x, or as a nominal rate per year when using the calculator's pre-programmed functions. 4. There can be no payments (made or received) during the term other than the present value and future value, i.e., PMT = 0. These conditions are summarized in the time diagram below: Calculation of Future Value A common interest accrual problem involves calculating the future value of a lump sum. When you receive (or invest) a lump sum cash flow today and need to find the value of this cash flow with accrued interest at some point in the future, it is known as the future value of a lump sum. The following illustrations demonstrate how to use the pre-programmed feature of the HP 10BII/10BII+ calculator to solve these types of problems. Excel spreadsheet solutions are provided under "Online Readings" on the Course Resources webpage. Illustration 1.5 A residential subdivision developer has purchased a 30-acre section of land for $400,000. The firm feels that land values in the area will increase at the rate of 1% per month over the next three years. If this rate of increase in property values proves to be correct, calculate the value of the parcel at the end of the three years. Solution The objective of this problem is to calculate the future value of $400,000 at the end of 3 years, where the rate of interest is 1% per monthly compounding period. This problem can be solved using the exponential y x key. However, using the pre-programmed financial features, the solution may be done as follows. The number of compounding periods during the investment period must be identified. The periodic rate of interest was given as 1% per month and the investment horizon is three years; this means that the number of compounding periods is 36 (12 3 = 36) and the nominal interest rate per year is j12 = 12% (1% 12 = 1.20

22 Real Estate Finance Basics 12%). The development firm will receive no income from the property during the three year investment horizon and wants to know the likely future value of the parcel: FV = PV (1 + i) n FV = $400,000 (1 + 1%) 36 FV = $400,000 ( ) 36 The financial arrangement involved may be illustrated by a time diagram: Notice that the present value is shown as a negative in this example. This represents the funds needed to purchase the land, which is an outflow of cash for the developer. The future value which will be calculated represents the money received by the developer for selling the land in three years, which is a cash inflow and a positive amount = I/YR 12 Enter nominal interest rate per year 12 O P/YR 12 Enter compounding frequency 3 12 = N 36 Enter number of compounding periods 0 PMT 0 No payments during term /- PV -400,000 Enter present value FV 572, Computed future value If property values rise at a rate of 1% per month, and the current value of the property is $400,000, the property will have a value of $572, at the end of 3 years. Excel Note You must use the periodic rate (monthly rate for this example) for either the formula solution or the preprogrammed FV function in Excel. Note that since this illustration is examined from the investor s perspective, we enter the amount invested (present value) as a negative (cash outflow) and the future value is a positive (cash inflow). Keep in mind that you could reverse these, with PV as a positive and FV as a negative; it makes no difference mathematically! 1.21