1 Page 1 of 23 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 popular Texas Instruments BA II Plus have the added functionality of allowing you to specify how many interest compounding periods there are per year. The default setting for the BA II Plus is 12 compounding periods per year whereas the examples that follow are based on the assumption that you have set the number of 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.
2 Page 2 of 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. Learning objectives 5.1 Describe the concept of the time value of money. (Level 1) 5.2 Describe the concept of interest including simple and compound interest, and effective and nominal rates of interest. (Level 1) 5.3 Compute the present and future value of a single payment, and an annuity (ordinary and due). (Level 1) 5.4 Design a worksheet to perform time value of money analysis. (Level 1) 5.1 Time value of money Learning objective Describe the concept of the time value of money. (Level 1) Required reading present_value_appendix.html Appendix: Interest Concepts of Future and Present Value, (located on the OLC) page 395 "Time value of money" (Level 1) 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. 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 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.
3 Page 3 of 23 Example If $10,000 is deposited in a savings account earning 4% interest compounded annually, will you have more money at the end of 5 years or 10 years? 2. If the interest rate is 4% compounded annually, will you be willing to pay more for a payment of $10,000 to be received in 5 years, or $10,000 to be received in 10 years? 3. If $10,000 is deposited in a savings account, will you have more money at the end of 5 years if the interest rate is 4% compounded annually or 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 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=? = ). 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:
4 Page 4 of 23 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 once the effects of compounding are considered. 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. 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 "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-annually, or annually 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 to help you understand this concept.
5 Page 5 of 23 Example Wittink Company invests $80,000 for 6 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 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= compounding periods per year = 12 PV = -80,000, n = 12, I = 6, FV =? = $160,958 Quarterly compounding 3% 12%/4 compounding periods per year = 3% per period n= 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 rate that you actually pay when the effects of compounding are taken into account is the effective rate. 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:
6 Page 6 of 23 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. 5.3 Computing present 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 (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
7 Page 7 of 23 =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, Now work through the examples below so as to familiarize yourself with the two methods of computation. Example Present value of a future amount What is the present value of a single payment of $10,000, which is to be received 3 years from now using an interest rate of 10% compounded annually? Your known variables are: 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) PV =?? = -7, 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:
8 Page 8 of 23 The formula for the present value amount in cell B6 should be =PV(B4,B5,,B3) A 3 Future value $10, Annual interest rate 10.00% 5 Number of years 3 6 Present value -$7, Save this worksheet. If you do not obtain the result shown, click the sheet tab for M5E1S and review the formula in cell B6. B Example Present value of an ordinary annuity You are purchasing an investment that will pay you $2,500 semi-annually for 6 years (a total of 12 payments). The first payment will be received 6 months from now. How much should you pay for this investment if the interest rate is 8%, compounded semi-annually? 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) 2500 Interest rate: (I) 4 PV =?? = -23, 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, Periodic interest rate 4.00% 10 Number of periods Present value of annuity -$23, B
9 Page 9 of 23 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. 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 Exhibit 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 Example 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
10 Page 10 of 23 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] Payment amount: [PMT] 2500 Interest rate per period: [I] 8% 2 4% PV =?? = -24, Spreadsheet method Continue with the M5E1 worksheet. Add the following model to the worksheet to calculate the present value of the annuity due. The formula in cell B16 should be =PV(B14,B15,B13,,1) A 13 Periodic payment $2, Periodic interest rate 4.00% 15 Number of periods Present value of annuity -$24, B 5.4 Computer illustration 5.4-1: 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.
11 Page 11 of 23 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, 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 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, To compare your result with the suggested solution, click the sheet tab M5P1S.
12 Page 12 of 23 Module 5 self-test Question 1 Computer question Trunet Company has $55,000 in excess cash that it wishes to invest for a five-year period. After analyzing several investment options, Trunet narrowed its choice to three GICs (Guaranteed Investment Certificates). As a CGA, you were asked to help the management of Trunet determine which of the investment options yields the best result. The options are GIC A: 5.5% annual interest rate, compounded annually GIC B: 5.25% annual interest rate, compounded semi-annually GIC C: 5% annual interest rate, compounded monthly Required Use the partially completed worksheet in the file stest05q01.xls to determine which of the GICs has the highest maturity value (future value). a. Analyze the results from your completed worksheet and identify which GIC has the highest maturity value. What is the value of that GIC? Show the spreadsheet formula used to calculate it. b. Explain the results by discussing the effect of interest compounding. Calculate the effective interest rate of each GIC. Procedure 1. Open the file stest05q01.xls. Note that the worksheet contains three sections dealing with each of the three GICs. 2. For GIC A, move to cell B12 and enter a formula to calculate the interest rate per period. 3. In cell B13, enter the number of interest periods. 4. In cell B14, build a formula that will calculate the future value of the investment based on the amount of the initial investment found in cell B4, and on the interest rate per period and the number of periods calculated in cells B12 and B13, respectively. The formula should make use of absolute cell references where necessary to enable copying to other cells. 5. In cell B15, calculate the effective interest rate. 6. Repeat steps 2 to 5 for GICs B and C, copying formulas from the section for GIC A whenever possible.
13 Page 13 of Save your worksheet, and print your results. 8. Display and print the formulas. 9. Answer parts (a) and (b). Solution Question 2 Multiple choice a. Which of the following statements best describes the relationship between an annuity due and an ordinary annuity? 1. The payments will be the same in both situations in order to accumulate $10,000 in three years. 2. The payment for the ordinary annuity will be equal to the payment for the annuity due plus one period of interest. 3. The interest paid on a purchase would be the same if the amount of purchase is paid with an annuity due as it would be with an ordinary annuity. 4. The principal outstanding will be the same after the same number of payments in each case. b. What is the best definition of the present value? 1. The value of a sum of money today 2. The value of a sum of money at the beginning of the period of a money transaction 3. The accumulation of principal and interest over time 4. The value of goods that can be bought today after considering inflation c. If the interest rates are changed from 8% compounded annually to 8% compounded monthly, which of the following would be true? 1. The future value of a series of dollar values would decrease. 2. The present value of a single dollar figure would increase. 3. The present value of a series of dollar figures would decrease. 4. The present value of an annuity of equal payments would increase. d. Which of the following would provide the greatest growth rate for an investment? 1. 8% compounded monthly % compounded annually % compounded quarterly % compounded semi-annually e. A piece of equipment can be leased over five years. Which statement describes how the asset and liability should be valued for this lease? 1. The total of all the payments to be made on the lease 2. The price of the same asset as found in the local store 3. The present value of all the payments discounted at the rate of interest as stated in the lease agreement
14 Page 14 of The present value of all the future payments, but excluding the first payment, made at the time of signing the lease f. Which of the following statements would best describe a deferred annuity? 1. An annuity that is to be received some time in the future and is reported on the balance sheet as deferred revenue 2. A series of equal payments that start immediately 3. A series of equal payments that have been deposited in an account in the past and are being left to accumulate more interest 4. A series of equal payments to pay off a debt where the first payment does not take place for several periods g. When the interest compounding period does not coincide with the payment frequency in an annuity, an adjustment has to be made before the present and future value formulae or tables can be used. Which of the following adjustments must be made? 1. The total of the payments in a year must be divided up in the same frequency as the interest conversion. 2. The given interest rate must be converted into the equivalent interest rate compounded at the same frequency as the payments are made. 3. The payments made in a year must be totalled and at the same time the interest should be converted to the equivalent annual rate. 4. The given annual nominal rate must be divided by the number of payments per year. h. Which will grow to the largest amount in five years time? 1. A single deposit of $5,000 at the rate of interest of 7.5% compounded quarterly 2. A deposit of $2,000 immediately and a further $3,000 in two years time, both earning 8.5% compounded monthly 3. A quarterly deposit of $300 at the end of each quarter at 7.5% compounded quarterly 4. $1100 at the beginning of each year at 8.2% compounded annually i. How long would it take a deposit of $2,000 to reach $5,000 if it could earn interest at 9% compounded annually? years years years years j. What interest rate compounded quarterly is equivalent to 8.5% compounded semi-annually? % % % % k. A loan of $20,000 has to be repaid by monthly payments for a period of four years. If interest charged is 12% compounded monthly, what will be the monthly payments? 1. $ $470.83
15 Page 15 of $ $ l. An asset can be obtained with a capital lease for 4 years with quarterly payments of $4,000 starting immediately. The interest included in the lease is 14% compounded quarterly. What value should be placed on the asset? 1. $36, $48, $50, $64, m. What rate of interest compounded quarterly was earned on an investment of $6,000 that earned interest of $1, over a two-year period? 1. 2% 2. 8% % 4. 9% n. Over the last 20 years, at the end of each month you have been saving $100 in a special savings account that has paid a fixed rate of 6% compounded monthly. You have just made the last deposit and are now considering withdrawing monthly amounts of $800 at the end of each month. If the interest remains at 6% compounded monthly, how long will it be before the account is exhausted? 1. 2 years and 6 month 2. 5 years and 1 months 3. 5 years and 9 months 4. 6 years and 1 month o. What is the present value of $3,000 in two years time and $5,000 in four years time if money is worth 10% compounded quarterly? 1. $5, $5, $6, $8, p. Twenty years ago, you made a single deposit of $30,000 to a savings account. The balance in your account today is $70,000. If interest was compounded quarterly, what was the effective annual rate of interest that you earned on the account? % % % % q. You have made a deposit of $6,000 to a savings account semi-annually for the past five years. The first deposit was made on January 1, It is now January 1, 2007 and the balance in your account is $85,000. If interest was compounded semi-annually, what was the effective annual rate of interest that you earned on the account? %
16 Page 16 of 23 Solution Question % % % In terms of the time value of money, explain what is meant by a. Future value b. Present value c. Compounding Solution Question 4 a. The market value of a bond is calculated as the present value of the future benefits. If a $10,000 bond pays interest of $500 every six months and returns the $10,000 in 10 years time, what is the market value if interest required on such an investment is 9% compounded semi-annually? b. John has just won a lottery in which he can receive $250,000 now or $2,000 at the end of each month for the next 25 years. If John can invest the lump sum at 9% compounded monthly, which option should he take? c. Company XYZ is planning on a large expansion which requires $20 million. This can be funded by issuing bonds that carry interest of 10% payable annually and also by depositing into a sinking fund (a savings account) to earn 6% effective, sufficient monies at the end of every year for 20 years, accumulating $20 million in 20 years time. How much altogether does company XYZ have to pay out each year for this money? d. What is the effective interest rate charged on a loan of $50,000 that requires payments of $7,791 every year for 10 years? Solution Question 5 A company is having a cash flow problem and is due to make a payment of $500,000 today, January 1, 20X2. They wish to renegotiate the payment to delay the payment for two years and then make three annual payments starting in three years time on January 1, 20X5. The lender does not wish to wait for two years before getting any money, so the lender requests $50,000 immediately and $50,000 in one year s time. The rest can be repaid according to the company s wishes. Required a. If the interest to be charged is 12% effective, calculate the annual payments under each alternative. b. Prepare a debt amortization schedule for each alternative.
17 Page 17 of 23 c. Give the journal entries for each alternative up to and including the January 20X5 payment. Assume that the company s reporting year end is December 31. Solution Question 6 A loan negotiated with the bank for $400,000 is to be repaid by quarterly payments starting in three months time and finishing in five years time. If the interest rate being charged is 9% compounded quarterly, how much will those payments be? Solution Question 7 A car that costs $22,000 can be purchased for $2,000 down on January 1, 20X1, with monthly payments of $500 starting one month after the purchase. Interest being charged is 6% compounded monthly. Required a. How long will it be before the purchaser actually owns the car? b. How much will the final payment be? Hint: The last partial payment will be made exactly one month after the last payment of $500. Solution Self-test - Content Links Self-test 5 Question 1 Computer solution a. Of the three GICs, GIC A (5.5%, compounded annually) has the highest maturity value of $71,