file:///f /Courses/2010-11/CGA/FA2/06course/m05intro.htm Module 5: Interest concepts of future and present value Overview In this module, you learn about the fundamental concepts of interest and present and future values, as well as ordinary annuities and annuities due. You also become familiar with valuation techniques involving the use of a financial calculator and functions in Excel. Finally, you apply what you have learned by using an Excel spreadsheet to make present value calculations. Test your knowledge Begin your work on this module with a set of test-your-knowledge questions designed to help you gauge the depth of study required. Note: In this module, the solutions to numerical computations are demonstrated using the most common format of data entry for financial calculators. The method of input may differ slightly across brands and models of calculators. Always refer to your owner s manual for specific instruction. This module introduces the following abbreviations: PV present value FV future value PMT the amount of the annuity payment I the interest rate per period N the number of periods BGN you need to set your calculator to compute an annuity due PV, FV, PMT, I, or N =? you should solve for the desired variable? = a number the displayed solution Please note that present values (PVs) are typically entered as a negative number, and that when you solve for PVs, the calculator s output will normally display a negative number as well. The underlying logic here is that a PV represents what you would pay today (a cash outflow) to obtain a sum or sums of money in the future (cash inflows). Outflows are entered as negatives; inflows are entered as positives. Caution: Some financial calculators, such as the Texas Instruments BA II Plus, for example, allow you to specify the number of interest compounding periods per year. Before you start, it is important to check the factory default setting. If the compounding frequency is set at 1 (C/Y = 1), do not change it for this course! The examples in this module assume C/Y = 1 and use the number of periods to maturity, which is consistent with entering data in Excel's financial functions. If the compounding period is other than C/Y = 1, you may need to reset the compounding periods to 1. Refer to your owner s manual for instructions on how to make this change. There are several reasons why the calculator keystrokes are illustrated in this manner: 1. Conceptually, many people find it easier to think in terms of an interest rate per period and the number of periods to maturity. 2. It is consistent with the format for entering the data into Excel. 3. Many students own financial calculators that do not include the added functionality of being able to set the number of compounding periods per year. While the methodology in the examples that follow is of the "interest rate per period" type, you may use whatever method you feel most comfortable with. Topic outline and learning objectives 5.1 Time value of money Describe the concept of the time value of money. (Level 1) 5.2 Basic interest concepts Describe the concept of interest including simple and compound interest, and effective and nominal rates of interest. (Level 1) 5.3 Computing present and future values Compute the present and future value of a single payment, and an annuity (ordinary and due). (Level 1) 5.4 Periodic payments required for present value and future value problems Compute the required periodic payments for a given present value or future value. (Level 1) 5.5 Computing the term Compute the number of periodic payments and the final payment required to eliminate a debt. (Level 1) 5.6 Computer illustration 5.6-1: Value of equipment Design a worksheet to perform time value of money analysis. (Level 1) Module summary Print this module Substantive differences between IFRS and the pre-ifrs CICA Handbook that apply to this module There are not any substantive differences between IFRS and the pre-ifrs CICA Handbook that apply to this module; the governing standards do not mandate how time value of money (TVM) computations are performed. This module deals with TVM concepts. You need to study this material as present values are used extensively in accounting as one way of valuing financial assets and liabilities. While IFRS has various standards that require that present values be used, the computational aspects are well established in finance and are not mandated by either governing body. For example, the IFRS Framework paragraph 100 reads in part as follows: 100 A number of different measurement bases are employed to different degrees and in varying combinations in financial statements. They include the following: (d) Present value. Assets are carried at the present discounted value of the future net cash inflows that the item is expected to generate in the normal course of business. Liabilities are carried at the present discounted value of the future net cash outflows that are expected to be required to settle the liabilities in the normal course of business. Note that the guidance does not tell you how to calculate the present value. file:///f /Courses/2010-11/CGA/FA2/06course/m05intro.htm [22/07/2010 9:15:33 AM]
file:///f /Courses/2010-11/CGA/FA2/06course/m05t01.htm 5.1 Time value of money Learning objective Describe the concept of the time value of money. (Level 1) Required reading Appendix: Interest Concepts of Future and Present Value (located on the OLC), page 395 "Time Value of Money" (Level 1) Please note that this reading is meant to provide you with information pertaining to present and future value concepts, rather than how to compute present values. The reason for this is that while the illustration of principles is sound, the narrative relies on factor tables which are seldom used in practice. The FA2 module notes focus on the use of a financial calculator, and to a lesser degree Excel. To fine tune your skills, we suggest that you use your financial calculator to solve the examples in this appendix. Keep in mind that results may differ slightly because calculators and spreadsheets are more accurate than tables as they take their calculations to more significant decimal places. LEVEL 1 The required reading provides an in-depth study of the time value of money, including the concept of present value (PV). The computation and interpretation of PVs are of interest to accountants, as accounting standards require us to value many liabilities at the present value of the future payment streams. The extent to which PVs are used in accounting will become very evident when you study FA3. Consider Example 5.1-1. At this point, you need not perform any calculations; use logic to decide what the appropriate answer should be. The numerical solutions are provided for you to check after you have mastered the subject matter. Example 5.1-1 1. If $10,000 is deposited in a savings account earning 4% interest compounded annually, how much money will you have at the end of 5 years? 2. If the interest rate is 4% compounded annually, how much would you be willing to pay for a payment of $10,000 to be received in 5 years? Or for a payment of $10,000 to be received in 10 years? 3. If $10,000 is deposited in a savings account, how much money would you have at the end of 5 years if the interest rate is 4% compounded annually versus 6% compounded annually? 4. Will you be willing to pay more today for a payment of $10,000 to be received in 5 years if the interest rate is 4% or 6%? 5. If you deposit $10,000 in a saving account, will you have more money at the end of 5 years if the nominal interest rate of 4% is compounded semi-annually or annually? 6. Will you be willing to pay more today for a payment of $10,000 to be received in 5 years if the interest rate is 4% compounded semi-annually or annually? Solution file:///f /Courses/2010-11/CGA/FA2/06course/m05t01.htm (1 of 2) [22/07/2010 9:15:34 AM]
file:///f /Courses/2010-11/CGA/FA2/06course/m05t01.htm Note: Solutions in this and subsequent exercises have been rounded to whole numbers, which is consistent with the "traditional" practice of rounding to the nearest dollar for expository purposes. In the real world, though, these amounts would be rounded to the nearest cent. 1. You will have $12,167 at the end of 5 years (PV = 10,000, N = 5, I = 4, FV =? = 12,167) and $14,802 at the end of 10 years (PV = 10,000, N = 10, I = 4, FV =? = 14,802). 2. You will be willing to pay $8,219 now to receive $10,000 at the end of 5 years (FV = 10,000, N = 5, I = 4, PV=? = 8,219) and $6,756 to receive $10,000 at the end of 10 years (FV = 10,000, N = 10, I= 4, PV=? = 6.756). 3. As before, you will have $12,167 at the end of 5 years if you earn 4% interest. If the rate increases to 6%, you will end up with $13,382 (PV = 10,000, N = 5, I = 6, FV =?? = 13,382). 4. As before, you will be willing to pay $8,219 now to receive $10,000 in 5 years if the market rate of interest is 4%. If the rate increases to 6%, you will be willing to pay $7,473 (FV = 10,000, N = 5, I = 6, PV =? = 7,473). 5. As before, you will have $12,167 at the end of 5 years if you receive interest at 4% compounded annually. If interest is compounded semi-annually, you will receive $12,190 at the end of 5 years [PV = 10,000, N = 10 (5 2), I = 2 (4/2), FV =? = 12,190] 6. As before, you will be willing to pay $8,219 now to receive $10,000 in 5 years if the market rate of interest is 4% compounded annually. If the interest is compounded semi-annually, you will be willing to pay $8,203 [FV = 10,000, N = 10 (5 2), I = 2 4/2, PV =? = 8,203]. These simple examples illustrate the following important points about the time value of money: The nominal rate of interest refers to the annual stated rate. The effective rate of interest is the rate that you actually end up receiving or paying on an annual basis once the effects of compounding are considered. Unless stated otherwise, all interest rates are quoted on an annual basis. All else being equal: The longer the time to maturity, the greater the maturity value (FV) for a stated PV; conversely, the lesser the PV for a given FV. The higher the rate of interest, the greater the FV for a given PV. Conversely, the lesser the PV for a given FV. The more frequent the compounding of interest, the greater the FV for a stated PV. Conversely, the lesser the PV for a given FV. The relationship between PV and FV, which can be stated as FV = PV(1 + I) n, can be restated as PV = FV/(1 + I) n. These formulas are the basis for the above statements about the time value of money. file:///f /Courses/2010-11/CGA/FA2/06course/m05t01.htm (2 of 2) [22/07/2010 9:15:34 AM]
file:///f /Courses/2010-11/CGA/FA2/06course/m05t02.htm 5.2 Basic interest concepts Learning objective Describe the concept of interest including simple and compound interest, and effective and nominal rates of interest. (Level 1) Required reading Appendix: Interest Concepts of Future and Present Value (located on the OLC), pages 395-396 "Basic Interest Concepts" (Level 1) LEVEL 1 The required reading distinguishes between simple interest and compound interest. As simple interest is rarely used in business, unless stated otherwise, all interest computations are to be calculated on a compound basis. You should assume that the compounding period is annual, unless there is a specific comment to the contrary. An interest period is the period (for example, a month or a year) in which interest is calculated. The method of calculating interest refers to how often the interest is compounded. It is quite common to see lenders compound interest on a daily, weekly, monthly, semi-annual, or annual basis. For computational purposes, accountants are interested in the number of periods (not years) that the investment or loan is to be held and the interest rate per period. This frequently requires converting the nominal interest rate per year into a more appropriate measure. Work through Example 5.2-1 to help you understand this concept. Example 5.2-1 Wittink Company invests $80,000 for six years in an account that pays interest at the rate of 12% per annum. How much money will Wittink receive under each of the following scenarios? Interest is compounded annually Interest is compounded semi-annually Interest is compounded quarterly Solution Interest rate per period Determination of the rate per period Number of periods Determination of the number of periods Fund balance file:///f /Courses/2010-11/CGA/FA2/06course/m05t02.htm (1 of 3) [22/07/2010 9:15:35 AM]
file:///f /Courses/2010-11/CGA/FA2/06course/m05t02.htm Annual compounding 12% 12%/1 compounding period per year = 12% per period n=6 6 1 compounding period per year = 6 PV = 80,000, n = 6, I = 12, FV =? = $157,906 Semi-annual compounding 6% 12%/2 compounding periods per year = 6% per period n=12 6 2 compounding periods per year = 12 PV = 80,000, n = 12, I = 6, FV =? = $160,976 Quarterly compounding 3% 12%/4 compounding periods per year = 3% per period n=24 6 4 compounding period per year = 24 PV = 80,000, n = 24 I = 3, FV =? = $162,624 As you can see, the more frequent the compounding period, the greater the future value. Effective interest rates For comparative purposes, it is necessary to ensure that all nominal (quoted) rates of interest are converted to effective (what you actually pay or receive) rates. Note that interest rates are, by convention, quoted in annual terms with the number of compounding periods referred to, for example, 10%, compounded quarterly. The quoted rate is known as a nominal rate. The effective rate is the annual rate of interest that includes the effects of compounding. Nominal rates can be converted to effective rates using the following equation: Effective rate = [1 + (I n)] n 1, where I = the nominal interest rate and n = the number of compounding periods per year. For example, the effective interest rate for 10% compounded quarterly is: Effective rate = [1 + (I n)] n 1 = (1.025) 4 1 = 10.38% The foregoing equation can be rearranged so as to convert effective rates to nominal rates: Nominal rate = [(1 + I) 1/n 1]n, where I = the effective interest rate and n = the number of compounding periods per year. For example, the 10.38% effective rate derived above is equivalent to a nominal rate of 10% per annum determined as follows: Nominal rate = [(1 + I) 1/n 1]n = [(1.1038) 1/4 1]4 =10% However, an easier method to calculate the effective or nominal rate would be to use the built-in functions of your calculator or a spreadsheet. Equivalent interest rates When the compounding of an interest rate differs from the rate at which the payments occur, the equivalent rate that matches the payment period must be calculated. file:///f /Courses/2010-11/CGA/FA2/06course/m05t02.htm (2 of 3) [22/07/2010 9:15:35 AM]
file:///f /Courses/2010-11/CGA/FA2/06course/m05t02.htm Example 1: What rate of interest compounded quarterly is equivalent to a rate of 9% compounded monthly? First, an annual rate of 9% compounded monthly means that the monthly interest rate is 9%/12 = 0.75%. This translates into an effective rate of (1.0075) 12 1 = 9.3806898%, rounded to 9.3807%. For the effective rate from quarterly compounding to be equivalent to the effective rate from monthly compounding of 9.3807%, the following relationship must exist: (1 + i/4) 4 1 = (1 + 9%/12) 12 1 Therefore (1 + i/4) 4 1 = 9.3807% Through rearranging you get (1 + i/4) 4 = 1.093807 (1 + i/4) = (1.093807) 0.25 i/4 = (1.093807) 0.25 1 i = [(1.093807) 0.25 1] x 4 i = 9.0677% Proof: 9%/12 = 0.75%; (1 +0.75%) 12 1 = 9.3807% 9.0677%/4 = 2.2669%; (1 +2.2669%) 4 1 = 9.3806%* *Very small difference due to rounding Example 2: What rate compounded quarterly is equivalent to an interest rate of 10% compounded semi-annually? In this example, the unknown interest rate (quarterly) will be compounded twice in the space of a half year. If the unknown rate is "i," then the following relationship exists: (1 + i) 2 = (1 +5%) 1 i = (1 + 5%) ½ 1 = 0.024695077 To use this rate in a calculator, you must change it to a percent: 2.46951% keeping at least five decimals. Never round off these equivalent rates to less than four decimal places because the calculations are very sensitive to the interest rate. file:///f /Courses/2010-11/CGA/FA2/06course/m05t02.htm (3 of 3) [22/07/2010 9:15:35 AM]
file:///f /Courses/2010-11/CGA/FA2/06course/m05t03.htm 5.3 Computing present and future values Learning objective Compute the present and future value of a single payment, and an annuity (ordinary and due). (Level 1) Required reading Appendix: Interest Concepts of Future and Present Value (located on the OLC), pages 397-405 (Level 1) LEVEL 1 The required reading details how to compute both the present and future values of a single payment. Our discussion here will be limited to illustrating various methods of accomplishing this. The most common ways to compute PVs and FVs are to use a financial calculator or spreadsheet program such as Excel. While the required reading does illustrate the computation of both present and future values, this topic limits the balance of the discussion to the calculation of present values, because these are what you are most likely to encounter in your accounting career. However, please note that future values do remain examinable. Spreadsheet method Summary of financial functions in Excel Function Purpose =FV(rate, nper, pmt, pv, type) Calculates the future value of an annuity or a present amount =PV(rate, nper, pmt, fv, type) Calculates the present value of an annuity or a future amount =PMT(rate, nper, pv, fv, type) Calculates the payment per period for an annuity =NPER(rate, pmt, pv, fv, type) Calculates the number of interest periods for an annuity Note that you need to specify the present value in the Excel functions as a negative value. For details about Excel, see CT2. Calculator method You should refer to your owner s manual for specific instructions as to the required steps for performing time value of money calculations. FV = $10,000 N = 4 I = 6 PV=? = $7,920.94 Note: 1. The amount of money involved is not negative, but the negative sign represents a "payout" rather than a receipt. That is, you invest (payout) $7,920.94 today (PV) in order to receive $10,000 (FV) in six periods (in this case, six years time). Not all calculators use the "negative" approach. file:///f /Courses/2010-11/CGA/FA2/06course/m05t03.htm (1 of 5) [22/07/2010 9:15:37 AM]
file:///f /Courses/2010-11/CGA/FA2/06course/m05t03.htm 2. Also note that unless the five financial registers (N, I, PV, PMT, and FV) are "zeroed out" before the appropriate amounts are entered, an amount can remain in a register that is not involved in the calculation and the answers may be wrong. To avoid clearing all registers when some numbers are still the same, a "zero" amount should be placed in the register that is not needed in the calculation. In the above case, the PMT does not enter into the calculations (0, PMT). This will ensure that there is no amount in that register. 3. The sequence in which the data is entered into the calculator is irrelevant. Now work through the examples below so as to familiarize yourself with the two methods of computation, but pay particular attention to the use of your calculator. Use your calculator and not someone else's in the examination. Example 5.3-1 Present value of a future amount What is the present value of a single payment of $10,000, which is to be received three years from now using an interest rate of 10% compounded annually? Your known variables are as follows: Future value $ 10,000 Period interest rate 10% Number of periods 3 Calculator method Enter the following on the calculator: Number of periods (N) 3 Period interest rate (I) 10 Future value (FV) 10000 PV =?? = 7,513.15 Spreadsheet method Start your spreadsheet program. Open the file FA2M5E1. Click the sheet tab M5E1. This worksheet has labels pre-entered in column A. Enter appropriate values and formulas in cells B3 to B6. Your completed worksheet should look like this: A B 3 Future value $10,000.00 4 Annual interest rate 10.00% 5 Number of years 3 6 Present value $7,513.15 The formula for the present value amount in cell B6 should be =PV(B4,B5,,B3) file:///f /Courses/2010-11/CGA/FA2/06course/m05t03.htm (2 of 5) [22/07/2010 9:15:37 AM]
file:///f /Courses/2010-11/CGA/FA2/06course/m05t03.htm (Note that there is no payment in this calculation which is indicated by the two commas between B5 and B3. In a worksheet the order is important, unlike a calculator.) Save this worksheet. If you do not obtain the result shown, click the sheet tab for M5E1S and review the formula in cell B6. Annuities An annuity is a fixed payment received at regular intervals for a specific amount of time. An ordinary annuity consists of a series of equal payments (or receipts) that occur at the END of EVERY period. If the payments occur at the end of every "interest period," then the annuity is said to be a "simple ordinary" annuity. If the timing of the payments does not correspond to the interest compounding period, the annuity is said to be a complex annuity. For example, an annuity with quarterly payments at the END of EVERY period and interest that is compounded quarterly would be a "simple ordinary" annuity. If the interest were compounded semi-annually but with quarterly payments, then it would be considered a "complex ordinary" annuity. Basic PV and FV formulas used in calculators apply to "simple ordinary" annuities. A common example of a complex annuity is a mortgage where the stated rate is compounded semi-annually but you make payments monthly, biweekly, or weekly. In this situation, the first step is to calculate the equivalent interest rate that compounds at the same frequency as the payments. To solve that type of problem, the interest rate compounded semi-annually must be converted to an equivalent interest rate compounded quarterly, thus turning a complex annuity into a simple annuity. This conversion was demonstrated earlier. Fortunately, most problems involved in business are "simple" annuities and often the period involved is "annual." Example 5.3-2 Present value of an ordinary annuity You are purchasing an investment that will pay you $2,500 semi-annually for six years (a total of 12 payments). The first payment will be received six months from now. How much should you pay for this investment if the interest rate is 8%, compounded semi-annually? In this case, it is a "simple ordinary" annuity the payments come at the end of every six months and the interest is compounded every six months. The interest rate in this situation is 4% every six months. Calculator method First, confirm that you are in financial mode and that you have fully cleared all the mode registers. Then enter the following data: Number of periods: (N) 12 Payment amount: (PMT) 2,500 Interest rate: (I) 4 PV =?? = 23,462.68 Spreadsheet method Continue with the M5E1 worksheet. This worksheet has labels pre-entered in column A. Enter appropriate values and formulas in cells B8 to B11. Your completed worksheet should look like this: A 8 Periodic payment $2,500.00 B file:///f /Courses/2010-11/CGA/FA2/06course/m05t03.htm (3 of 5) [22/07/2010 9:15:37 AM]
file:///f /Courses/2010-11/CGA/FA2/06course/m05t03.htm 9 Periodic interest rate 4.00% 10 Number of periods 12 11 Present value of annuity $23,462.68 The formula for the present value in cell B11 should be =PV(B9,B10,B8) Compare your result with that shown. If necessary, click the solution sheet tab M5E1S to compare results. If you wish to invest money to receive back $2,500 for 12 payments every half year, then you should invest $23,462.68. On the other hand, if you were to borrow money and pay back $2,500 every six months for 12 payments, you could borrow $23,462.68. Using time lines to calculate annuity due The difference between an ordinary annuity and an annuity due is the timing of the payment. For an ordinary annuity, the payment comes at the end of each interest period, whereas for an annuity due, the payment comes at the beginning of each interest period. You can see this difference by comparing the time line of an ordinary annuity with three annual payments to the time line of an annuity due with three annual payments, as shown in Exhibit 5.3-1. Exhibit 5.3-1 Comparison of an ordinary annuity and an annuity due In the time lines, you can see that the cash flow for an ordinary annuity is made up of three payments starting one period from the initial loan or investment date. In the case of an annuity due, the payments start one period ahead of the ordinary annuity, beginning with the first payment at the initial loan or investment date. The relationship can be expressed as: PV of an annuity due = PV of an ordinary annuity (1 + I) where I = the interest rate per period. Present value of annuity due file:///f /Courses/2010-11/CGA/FA2/06course/m05t03.htm (4 of 5) [22/07/2010 9:15:37 AM]
file:///f /Courses/2010-11/CGA/FA2/06course/m05t03.htm Example 5.3-3 Present value of an annuity due Suppose that you wish to calculate the PV of the investment in the previous example assuming that the first payment will be received immediately. Calculator method Clear the financial mode registers, then enter the following data on the calculator: Mode BGN Number of periods: (N) 6 2 12 Payment amount: (PMT) 2,500 Interest rate per period: (I) 8% 2 4% PV =?? = 24,401.19 Warning: When you use the BGN mode in a calculator, you must be careful to remove it when making "ordinary" annuity calculations. When the BGN mode is engaged, most calculators include "BGN" in the display window, so it is a simple matter to check this. Spreadsheet method Continue with the M5E1 worksheet. Add the following model to the worksheet to calculate the present value of the annuity due. A B 13 Periodic payment $2,500.00 14 Periodic interest rate 4.00% 15 Number of periods 12 16 Present value of annuity $24,401.19 The formula in cell B16 should be =PV(B14,B15,B13,,1) file:///f /Courses/2010-11/CGA/FA2/06course/m05t03.htm (5 of 5) [22/07/2010 9:15:37 AM]
file:///f /Courses/2010-11/CGA/FA2/06course/m05t04.htm 5.4 Periodic payments required for present value and future value problems Learning objective Compute the required periodic payments for a given present value or future value. (Level 1) LEVEL 1 Ordinary annuities Example: A company wishes to borrow $50,000 and make annual payments at the end of each year for eight years. If interest being charged on the loan is 10% compounded annually, how much will each annual payment be? Solution: In this problem, the PV is known and it is the PMTs that need to be calculated. First, decide if the annuity is a simple ordinary annuity or not. In this case, it is; the payments are at the END of every year (annually) and the interest is compounded annually. Enter the following in the calculator: Number of periods (N) 8 Period interest rate (I) 10 Present value (PV) 50,000 PMT =?? = 9,372.20 Caution: If you do not clear all the registers in your calculator before starting this calculation, you may find that there is an amount residing in the one register that you do not use for this calculation (the FV register), and it will give you the incorrect answer. However, sometimes the information in the other registers doesn't change from a previous calculation, so if you clear all the registers you have to re-enter everything you need. You can overcome this by entering "0" in the register that you will not be using (in the FV register) and that will have the effect of "clearing" the unwanted register. The Excel spreadsheet can also be used; however, practice on your calculator is critical because that will be the tool you are allowed in the examination. Annuity due Example: A piece of equipment that cost $40,000 is to be leased, instead of purchased, by quarterly payments in advance for a period of six years. Interest charged on the contract is 12% compounded quarterly. What will the quarterly payments be? Solution: Again, you need to determine if this constitutes an ordinary simple annuity. In this case, it is not because the payments are "in advance," meaning that they are made at the beginning of each period. This, then, is an annuity due. It is a simple annuity because the payments and the compounding are both quarterly, so it is a "simple annuity due." To calculate the required payment using the calculator, make sure the various registers have been cleared and enter the following information: Type of annuity BGN file:///f /Courses/2010-11/CGA/FA2/06course/m05t04.htm (1 of 2) [22/07/2010 9:15:37 AM]
file:///f /Courses/2010-11/CGA/FA2/06course/m05t04.htm Number of periods (N) 24 Period interest rate (I) 3 Present value (PV) 40,000 PMT =?? = 2,293.10 In this case, the PV is positive because you are receiving the benefit of the equipment worth $40,000. As a result, the payments are entered as a negative representing a cash outflow. file:///f /Courses/2010-11/CGA/FA2/06course/m05t04.htm (2 of 2) [22/07/2010 9:15:37 AM]
file:///f /Courses/2010-11/CGA/FA2/06course/m05t05.htm 5.5 Computing the term Learning objective Compute the number of periodic payments and the final payment required to eliminate a debt. (Level 1) LEVEL 1 Example: A company borrowed $100,000 and agreed to make quarterly payments of $10,000 starting in three months time. The applicable interest rate is 10% compounded quarterly. How many full payments will be made and how much will the final payment be? The regular payments form a simple ordinary annuity of "N" payments plus a final payment 3 months after the last full payment. Given the PV and the PMT and INT, solve for N using these steps: Enter the following in the calculator: Payment (PMT) 10,000 Period interest rate (I) 2.5 Present value (PV) 100,000 Future value (FV) 0 Number of payments (N) =?? = 11.65 Entering "0" for the FV removes the necessity of clearing all the registers before entering the new information. There will be 11 full payments and a partial payment made at the end of period 12. To solve for the final payment, the annuity payment and the single payment must be equated with the PV. Most calculators will allow you to calculate a final payment at the same date as the last regular payment, but in this case the last payment will be one period later. Using the calculator, you can calculate the final payment due at point 11 and then add one period of interest to determine what the payment will be at period 12 as follows: Enter the following in the calculator: (This is a situation where the negative sign is important because you have an inflow and an outflow. Whenever there are two or more dollar entries to make, care must be taken as to which amounts are inflows (positive) or outflows (negative) when the calculator uses that concept.) Number of periods (N) 11 Period interest rate (I) 2.5 Present value (PV) 100,000 Payment (PMT) 10,000 Final payment (FV) at 11=?? = 6,374.00 To find the payment one period later, add one period of interest: $6,374.00 x (1 + 2.5%) = $6,533.35 file:///f /Courses/2010-11/CGA/FA2/06course/m05t05.htm [22/07/2010 9:15:38 AM]
file:///f /Courses/2010-11/CGA/FA2/06course/m05t06.htm 5.6 Computer illustration 5.6-1: Value of equipment Learning objective Design a worksheet to perform time value of money analysis. (Level 1) LEVEL 1 In this computer illustration, you use present value calculations to assist in determining the value of equipment to be recorded in the company s books. Material provided A file, FA2M5P1, containing a blank formatted worksheet M5P1 and a solution worksheet M5P1S. Description Suppose you want to buy a new piece of equipment from the manufacturer. The terms and conditions of the purchase plan are as follows: down payment of $10,000 36 monthly payments of $1,500, first payment to be made at the end of the first month a final payment of $4,000 to be made at the end of the 36th month, with the last monthly payment The going interest rate for this type of lease plan is 12% per year compounded monthly. Required Construct a worksheet to calculate the equipment s value to be recorded in the accounting records. Procedure You must calculate the present value of the equipment. There are three components in the purchase plan: the initial down payment, which is a present value the 36 ordinary annuity payments, which you will discount to present values the final payment at the end of the 36th month, which you will discount to present value Make sure to use the same interest rate and compounding periods for both the annuity and final payment computations. The following is a possible layout of your worksheet: Purchase plan Down payment Final payment Monthly payment Annual interest rate file:///f /Courses/2010-11/CGA/FA2/06course/m05t06.htm (1 of 3) [22/07/2010 9:15:39 AM]
file:///f /Courses/2010-11/CGA/FA2/06course/m05t06.htm Monthly interest rate Number of payments Present value of monthly payments Present value of final payment Present value of equipment Save the completed worksheet under your own initials. If you construct your formulas correctly, the present value of the equipment should be $57,956.96. To compare your result with the suggested solution, click the sheet tab M5P1S. Using the calculator There are two distinct calculations that you can either calculate in two separate steps or in one combined step. The two step method is used here for illustrative purposes only. It is obviously much more efficient to compute this in a single step and you are encouraged to do so. Two-step method Present value of the annuity payments of $1,500. Enter the following in the calculator: Number of periods (N) 36 Period interest rate (I) 1 Payment (PMT) 1,500 PV =?? = 45,161.26 PV of lump sum at the end of three years: Enter the following in the calculator: Number of periods (N) 36 Period interest rate (I) 1 Future value (FV) 4,000 PV =?? = 2,795.70 The sum of these two values and the down payment is $10,000 + 45,161.26 + 2,795.70 = $57,956.96. One-step method Number of periods (N) 36 Period interest rate (I) 1 Payment (PMT) 1,500 Future value (FV) 4,000 PV =?? = 47,956.96 The computed value plus the down payment is $10,000 + 47,956.96 = $57,956.96, which is the same answer arrived at in file:///f /Courses/2010-11/CGA/FA2/06course/m05t06.htm (2 of 3) [22/07/2010 9:15:39 AM]
file:///f /Courses/2010-11/CGA/FA2/06course/m05t06.htm the two-step method. file:///f /Courses/2010-11/CGA/FA2/06course/m05t06.htm (3 of 3) [22/07/2010 9:15:39 AM]
file:///f /Courses/2010-11/CGA/FA2/06course/m05summary.htm Module 5 summary Interest concepts of future and present value This module explains the fundamental concepts of interest and present and future values. Ordinary and annuities due are explained. Valuation techniques, including the use of a financial calculator and functions in Excel are demonstrated. Describe the concept of the time value of money. A dollar today is worth more than a dollar received tomorrow because today's dollar can be invested to earn interest. Describe the concept of interest including simple and compound interest, and effective and nominal rates of interest. Interest can be thought of as the rent charged for the use of money. Simple interest is calculated based on the principal amount owing only and not on accrued interest. Simple interest is not commonly used. Compound interest is calculated periodically and is based on the principal amount owing plus any unpaid interest. A nominal rate is the stated rate of interest. An effective rate is the annual rate that you actually pay when the effects of compounding are considered. Compute the present and future value of a single payment, and an annuity (ordinary and due). To determine the future and present values of single payments and annuities, you can use either the calculator or formula method. Refer to the owner's manual for your financial calculator for specific instructions about performing time value of money calculations. Compute the regular payment required for an ordinary annuity and an annuity due. Determine the regular payment to extinguish a debt by equal payment due at the end of each interest period. Determine the regular payment required to accumulate a required future amount. Determine the regular payment required, payable in advance, for an acquisition made on credit. Compute the time taken to extinguish a debt by making fixed payments in the future. Determine the number of full payments plus a smaller last payment required to extinguish a debt. file:///f /Courses/2010-11/CGA/FA2/06course/m05summary.htm [22/07/2010 9:15:40 AM]