APPENDIX. Interest Concepts of Future and Present Value. Concept of Interest TIME VALUE OF MONEY BASIC INTEREST CONCEPTS

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1 CHAPTER 8 Current Monetary Balances 395 APPENDIX Interest Concepts of Future and Present Value TIME VALUE OF MONEY In general business terms, interest is defined as the cost of using money over time. Economists prefer to say that interest represents the time value of money. For example, \$100 in hand today will be worth more in one year s time than a second \$100 received one year from today. The assumption is that today s dollars can be put to work earning interest. Thus, today s money has a future value equal to its principal (face amount) plus whatever interest can be earned over the period of time, one year in this case. If the interest rate on money invested for one year is 10%, the future value of today s \$100 principal at the end of 12 months is \$110 (\$100 principal \$10 interest). The \$110 is an amount coming in at the end of 12 months, tomorrow s money, scheduled to be received (or paid) at some future date in this example, one year. Just as today s money has a future value, calculated by adding interest to principal, tomorrow s money has a present value, calculated by subtracting interest. For example, if the interest rate on money invested for one year is 10%, the present value of \$110 a year from now is the principal necessary to invest today to obtain \$110 in a year. We already know that this required principal is \$100, because \$100 (\$100.10) yields \$110 in one year. Suppose, instead, that \$100 is to be received in one year. What is its present value? If the interest rate is again 10%, the present value is the principal needed today to yield \$100 in one year at 10%. This principal amount is \$91, because \$91 plus 10% of \$91 gives (approximately) \$100 in one year (\$91 principal \$9 interest). Accounting involves many applications of the concepts of present and future value. Some of the more prominent applications covered in this book relate to Receivables and payables Bonds Leases Pensions Asset valuation The purpose of this appendix is to provide the concepts necessary to facilitate measurement of the time value of money. Many students have covered this topic in depth in a business mathematics or finance course; this appendix is simply a review of the relevant basics. BASIC INTEREST CONCEPTS Concept of Interest Interest is the excess of resources (usually cash) received or paid over and above the amount of resources loaned or borrowed at an earlier date. The amount loaned or borrowed is called the principal. The cost of the excess resources to the borrower is called interest expense. The benefit of the excess resources to the lender is called interest revenue. To illustrate measurement of interest in a simple situation, assume that the Debont Company borrows \$10,000 cash and promises to repay \$11,200 one period later. The interest on this contract is \$1,200, or 12% of the \$10,000 principal amount borrowed. Interest usually is expressed as a rate per year, such as 12%, although interest is often calculated and accumulated for periods of less than one year, such as monthly, quarterly, or semi-annually. This is called compound interest. Thus, a 12% nominal annual interest rate could be compounded 1% monthly, 3% quarterly, or 6% semiannually. If an interest rate is

2 396 CHAPTER 8 Current Monetary Balances specified with no indication of an interest period less than one year, annual compounding should be assumed. In the case of compound interest, the interest rate and compounding period must be clearly stated. The effective total interest is a function of the principal amount, the interest rate, and the number of interest periods. If compounding is more than once a year, then the stated nominal interest rate understates the effective interest rate, which includes compounding effects, as we shall see. For example, a credit card agreement might describe its terms as 24%, compounded (at 2%) monthly. This understates the effective interest rate. The effective interest rate of 2% compounded monthly is really about 26.8%. CALCULATING SIMPLE INTEREST Business transactions subject to interest must state whether simple or compound interest is to be calculated. Simple interest is the product of the principal amount multiplied by the period s interest rate (a one-year rate is standard). The equation for computing simple interest is Interest amount P (i) (n) where: P Principal i Interest rate per period n Number of interest periods When applied to long-term transactions extending over multiple years, simple interest is based on the principal amount outstanding during the year. If there are no changes caused by repayments or additional borrowing, this will equal the principal outstanding at the beginning of each year. Interest is paid periodically (typically yearly or at the end of the contract) and not added to the principal. Thus, interest is paid only on the initial principal and not on interest accumulated but not yet paid. The yearly interest remains the same. Thus, a threeyear \$10,000 loan at a rate of 10% simple interest incurs \$1,000 of interest per year each year, or \$3,000 total interest, assuming that no instalment payments are made on the principal. CALCULATING COMPOUND INTEREST Compound interest is based on the principal amount outstanding at the beginning of each interest period, to which accumulated interest from previous periods has been added. In compound interest problems, it is assumed that interest is allowed to accumulate rather than being paid (by the borrower) or withdrawn (by the lender). This means that compound interest includes interest on previously computed and recorded interest. When interest periods of less than one year are used, the annual interest rate given must be converted to an equivalent rate for the time period specified for compounding purposes. To demonstrate, we add quarterly compounding to the interest calculation example. The interest rate is now 10% compounded quarterly, or 2.5% (10% 4) per quarter. The first year s interest calculations are as follows: Beginning Ending Quarter Balance Compound Interest Balance 1st \$10, \$ (\$10, %) \$10, nd 10, (\$10, %) 10, rd 10, (\$10, %) 10, th 10, (\$10, %) 11, Total \$ 1, In the above example, quarterly compounding for the first year produces \$1, of interest, \$38.13 more than the \$1,000 resulting from annual compounding. Quarterly compounding of a 10% stated or nominal interest rate is equivalent to an effective annual interest rate of 10.38% (\$1, \$10,000). The effective interest rate is the true interest rate. Semi-annual, quarterly, monthly, weekly, and daily compounding are all in common use; the more compounding, the more interest. Even continuous compounding is possible. However, normal descriptions involve nominal rates. Business transactions are usually described in contracts by their nominal annual interest rate and the compounding period.

3 CHAPTER 8 Current Monetary Balances 397 OVERVIEW OF FUTURE VALUE AND PRESENT VALUE Future value (FV) and present value (PV) pertain to compound interest calculations. Future value involves a current amount that is increased in the future as the result of compound interest accumulation. Present value, in contrast, involves a future amount that is decreased to the present as a result of compound interest discounting. Think of an investment as an example since many investments have finite starting and ending points, they are good illustrations of present and future values. Present value in general refers to dollar values at the starting point of an investment, and future value refers to endpoint dollar values. If the dollar amount to be invested at the start is known, the future value of that amount at the end can be projected, provided the interest rate and number of interest compounding periods are also specified. Similarly, if the dollar amount available at the end of an investment period (future value) is known, the amount of money needed at the start of the investment period (present value) can be determined, again if the interest rate and number of interest compounding periods are known. Present value and future value apply to interest calculations on both single principal amounts and periodic equal payment (annuity) amounts. Single Principal Amount Also known as a lump-sum amount, the single principal amount is based on a one-timeonly investment amount that earns compound interest from the start to the end of the investment time frame. Annuity Amount An annuity is a series of uniform payments (also called rents) occurring at uniform intervals over a specified investment time frame, with all amounts earning compound interest at the same rate. Annuity amounts may take the form of either cash payments into an annuity type of investment or cash withdrawals from an annuity type of investment. An annuity may be an ordinary annuity (or annuity in arrears), where the payments (or receipts) occur at the end of each interest compounding period, or an annuity due, where payments (or receipts) occur at the beginning of each interest compounding period. The difference between an ordinary annuity and an annuity due is illustrated in Exhibit 8A-1 for a four-year annuity. With an ordinary annuity, the first payment occurs one period after the present value is established and the last payment coincides with the determination of future value. For an annuity due, the first payment coincides with the date the present value is established and the last payment occurs one period before the future value is determined. A series of equal payments beginning today an annuity due has a greater present value than the same set of payments beginning one year from now an ordinary annuity. Similarly, a set of payments starting today an annuity due discharges a debt with lower payments than the payments required under an ordinary annuity. Methods to Calculate Present and Future Values There are four methods used to compute future and present values: 1. Make successive interest calculations. 2. Use a formula. 3. Use tables. 4. Use a financial calculator or computer spreadsheet program. All methods will produce the same results, although rounding will produce minor (immaterial) differences. Most accountants prefer to use a financial calculator or computer spreadsheet, since the result is accurate and the application of the technique is not restricted

4 398 CHAPTER 8 Current Monetary Balances EXHIBIT 8A-1 COMPARISON OF AN ORDINARY ANNUITY WITH AN ANNUITY DUE Period (years) X* X X X Ordinary annuity PV X* X X X FV Annuity due PV FV * The four payments are denoted with Xs. to the values displayed in the tables. The vast majority of students have possessed calculators from an early age. Most have mathematical calculators with neat things like tangent and cosine functions, but not interest rate functions. Anyone seriously interested in a career in business, however, should make the investment necessary for a financial calculator. Besides, it s easier to carry around a calculator than a computer, and for most interest calculations, it s also easier to use! If you use electronic means, be sure to write down the key variables interest rate and periods as you go, to leave a trail. It s also a good idea to check your results for reasonableness; wild answers are produced with the speed of light if you push the wrong buttons! Table values will also accurately calculate present and future values. We ll use them in our examples as they are generally applicable. See the Summary of Compound Interest Tables and Formulae at the back of this Volume. VALUES OF A SINGLE PAYMENT Future Value The future value of present amount (denoted as F/P) is the future value of a single payment after a specified number of interest periods (n) when increased at a specified compound interest rate (i). For example, the future value of \$1 left on deposit for six interest periods at an interest rate of 8% per period (F/P, 8%, 6) is: F/P (1 i) n expressed as (F/P, i, n) F/P (1.08) 6 F/P 1.587, or 1.59 rounded The same result can be obtained by using Table I-1, found at the back of this Volume. In the table, first locate the appropriate interest rate column, and then read down the column to the intersecting line representing the number of interest periods involved. The number of interest periods is listed on the left-hand side of the table. Once the correct future value factor is located, multiply it by the principal amount involved. For example, the future value of \$5,000 at 8% for six interest periods is \$7,935, or \$5,000 (1.587). In the tables and formulas, keep an eye on n, the compounding period. It means periods, and a period is not necessarily a year. As well, i must correspond to the length of the interest period. For example, 12% compounded annually for five years corresponds to (F/P, 12%, 5), while 12% compounded quarterly for five years is (F/P, 3%, 20): i 12% 4 quarters per year, and n 5 years 4 quarters per year 20. The results are not the same:

5 CHAPTER 8 Current Monetary Balances 399 \$1 (F/P, 12%, 5) \$1.762 \$1 (F/P, 3%, 20) \$1.806 Present Value of 1 The present value of a future amount (P/F) is the present value of a single payment for a specified number of interest periods (n) at a specific interest rate (i). For example, to find the present value of \$5,000 to be received six interest periods from today at 8% (P/F, 8%, 6), use either a financial calculator, the formula to calculate the present value of 1, or Table I-2, found at the back of this Volume. The algebraic formula is as follows: P/F 1 (1 i) n 1 1 P/F (1.08) P/F.63017, or.63 rounded expressed as (P/F, i, n) Table I-2 produces the same answer. First locate the 8% interest rate column, and then read down the column to the intersecting line representing the number of interest periods involved, 6, found at the left-hand side of the table. Then, all that is left is multiplication: the present value of \$5,000 at 8% for six interest periods is \$3,150.85, or \$5,000 (.63017). Future Value and Present Value of 1 Compared Future values and present values of 1 are the same in one respect: they both relate to a single payment. The future value looks forward from present dollars to future dollars. The present value looks back from future dollars to present dollars. Present value and future value, for a given i and n, are reciprocals: F/P 1 and P/F 1 P/F F/P In our example so far, we have determined that: a. The future value of \$1 invested at 8% for six periods is \$1.59 (rounded). b. The present value of \$1 discounted at 8% for six periods is \$.63. The reciprocal relationship is as follows: For (a): For (b): Typical Examples 1. A company buys a machine on 1 January 20X1; the payment terms state that the \$40,000 invoice price is not due until the end of 20X2. The going interest rate is 8% compounded annually, but there is no interest added over time; therefore, the conclusion is that part of the \$40,000 invoice price is really interest. What is the real cost of the machine? Interest has to be recorded separately! Solution: Cost \$40,000 (P/F, 8%, 2) \$40, \$34,294 Interest \$40,000 \$34,294 \$5,706

6 400 CHAPTER 8 Current Monetary Balances 2. An employee earned \$30,000 per year in 20X1, and will retire after another 25 years. If salaries are expected to increase at the rate of 4% per year, how much will the employee be earning after 25 more years? This information is often needed to calculate pension entitlements. Solution: Salary \$30,000 (F/P, 4%, 25) \$30, \$79, A company receives a loan of \$32,000 on 1 January 20X2, and will have to repay \$41,946 on 31 December 20X5. What is the effective interest rate if interest is compounded annually? Solution: \$32,000 \$41,946 (P/F, i, 4) (P/F, i, 4) \$32,000 \$41,946 (P/F, i, 4).7629 The interest rate, i, is found by looking along the 4-year row of Table I-2: the interest rate is 7%. 4. A company wishes to invest \$45,811 today so that it will have \$100,000 to repay a loan in the future. If interest rates are 10% compounded semi-annually, how long will the investment have to be left in the account? Solution: \$45,811 \$100,000 (P/F, 5%, n) (P/F, 5%, n) \$45,811 \$100, From the 5% interest column of Table I-2, we can find the value of , which shows us that the number of semi-annual interest periods is 16, which comes to eight years. In cases 3 and 4, the accuracy of the solution depends on finding the right numerical value in the table. The exact number you are looking for will seldom be there; you must interpolate between the closest two numbers in the table. Since compound interest functions are not linear, the interpolated solution will always be an approximation. The answer? Buy a financial calculator that will find the exact solution for you very quickly! VALUES OF AN ANNUITY Future Value of an Ordinary Annuity The future value of an ordinary annuity (or annuity in arrears) (F/A) is the future value of a series of payments (or receipts) in equal dollar amounts being made over a specified number of equally spaced interest periods (n) at a specified interest rate (i). Unless otherwise stated, all annuities are assumed to be ordinary annuities, meaning that every payment occurs at the end of the interest period. The future value of an ordinary annuity can be determined by compounding each payment separately, and then adding the results, or by adding the interest factors for each number of periods over the entire stream of payments. For example, the future value of an annuity of \$1,000 paid at the end of each year for three years, with interest compounding annually at 6% is rounded: Future amount \$1,000 [(F/P, 6%, 2) (F/P, 6%, 1) 1] \$1,000 [ ] \$1, \$3,184 The first payment (at the end of year 1) accumulates interest for two years, the second payment accumulates interest for one year, and the final payment earns no interest because it is made at the end of the three-year period. To avoid the necessity of accumulating the individual periodic interest factors, interest tables provide the cumulative compound factor. These are shown in Table I-3, found at the

7 CHAPTER 8 Current Monetary Balances 401 back of this Volume. In Table I-3, the value for three periods at 6% is , the same as that derived above (rounded). Therefore, the future amount of the stream of \$1,000 payments can be determined directly from Table I-3: Future amount = \$1,000 (F/A, 6%, 3) \$1, \$3, The same F/A factor could be obtained by formula, which is shown at the top of the table. Present Value of an Ordinary Annuity The present value of an ordinary annuity (P/A) is today s equivalent dollar amount of a series of payments (or receipts) made over a predetermined time frame. The present value of an ordinary annuity can be determined using a formula or an appropriate table (Table I-4). To find the present value of an ordinary annuity of \$5,000 invested at 8% for six interest periods (P/A, 8%, 6) using Table I-4 is \$23,114.40, or \$5, The present value of an annuity is also the sum of the present values of the individual payments. For example, the present value (P/A, 8%, 6), which was just determined to be , is the sum of the first six entries in the 8% column of Table I-2. Future Value of an Annuity Due The best way to understand an annuity due is to compare its cash flows with those of an ordinary annuity. Exhibit 8A-2 shows two time lines that compare the future value of an ordinary annuity (F/A) of 1 with the future value of an annuity due (F/AD) of 1 for the same interest rate and annuity period. EXHIBIT 8A-2 COMPARISON OF FUTURE VALUE OF AN ORDINARY ANNUITY WITH AN ANNUITY DUE (F/A, 10%, 3) versus (F/AD, 10%, 3) Future value of an ordinary annuity (F/A) payment at the end of each annuity period (Table I-3) \$1.00 \$1.21 \$1.00 \$1.10 \$1.00 \$3.31* (F/A) \$1.00 Present Period 1 Period 2 Period 3 Future value of an annuity due (F/AD) payment INSERT at Exhibit the beginning 8A-2 of each annuity period (Table I-5) \$1.00 \$1.33 \$1.00 \$1.21 \$1.00 \$1.10 \$3.64** (P/A) Present Period 1 Period 2 Period 3 * The first payment is compounded for two periods, the second payment for one period, and the third payment for none because the payment is at the end of the period. ** The first payment is compounded for three periods, the second payment for two periods, and the third payment for one period.

8 402 CHAPTER 8 Current Monetary Balances The future value of the ordinary annuity illustrated in Exhibit 8A-2 involves three payments but only two interest periods. The annuity due involves three payments and three interest periods. For each of the three values illustrated in the last column of Exhibit 8A-2, F/A (1 i) F/AD. This relationship means that if the F/A is known, it can be multiplied by (1 i) to determine the F/AD value for the same i and n. (F/AD, 10%, 3) or (F/A, 10%, 3)(1.10) 3.31(1.10) Table I-5 gives the values for the future value of an annuity due. The algebraic formula is shown at the top of the table. Present Value of an Annuity Due Exhibit 8A-3 shows two time lines that compare the present value of an ordinary annuity (P/A) with the present value of an annuity due (P/AD) of \$1 for the same interest rate and number of payments. The present value of the ordinary annuity illustrated in Exhibit 8A-4 involves three payments and three discounting periods. The annuity due involves three payment periods but only two discounting periods. The annuity due is discounted for one less period than the ordinary annuity, so the ordinary annuity amount is less than the corresponding annuity due amount by a factor of 1 (1 i); therefore, P/A (1 i) P/AD. This relationship EXHIBIT 8A-3 COMPARISON OF PRESENT VALUE OF AN ORDINARY ANNUITY WITH AN ANNUITY DUE (P/A, 10%, 3) versus (P/AD, 10%, 3) Present value of an ordinary annuity (P/A) payment at the end of each annuity period (Table I-4) \$0.75 \$1.00 \$0.83 \$1.00 \$0.91 \$1.00 (P/A) \$2.49* Present Period 1 Period 2 Period 3 Present value of an annuity due (P/AD) payment at the beginning of each annuity period (Table I-6) \$0.83 \$1.00 \$0.91 \$1.00 \$1.00 (P/AD) \$2.74** \$1.00 Present Period 1 Period 2 Period 3 * The first payment is discounted for one period, the second for two periods, and the third for three periods. ** The first payment is not discounted because there is no lapse of time. The second payment is discounted for one period and the third for two periods.

9 CHAPTER 8 Current Monetary Balances 403 means that a known P/A value can be multiplied by (1 + i) to yield its corresponding P/AD value. More conveniently, Table I-6, found at the back of this Volume, gives the values for the present value of an annuity due. Financial calculators give the option of automatically computing an annuity as either an ordinary annuity or an annuity due. They also permit rapid calculation when the unknown quantity in the equation is the amount of the annuity payments, that is, when i is known, n is known, and the desired present value (or future value) is known. Underlying Assumptions for Annuities It is important to bear in mind that there are several assumptions underlying the use of annuity calculations: the amount of each payment is the same throughout the entire stream of annuity payments; the payments are equally spaced (e.g., monthly, quarterly, or annually); the interest rate is stable; and the periods used for compounding interest coincide with the payment periods (e.g., annual payments and annual compounding rather than annual payments but semi-annual compounding). If any of these conditions does not exist, then a more labourious calculation is required. Typical Examples 1. A lease agreement is structured so that it requires a payment of \$6,000 every quarter for four years, with the first payment due at the beginning of the first quarter. The interest rate is 8%, compounded quarterly. What is the present value of the payment steam? (This will be capitalized if the lease is a capital lease.) Solution: Present value \$6,000 (P/AD, 2%, 16) \$6, \$ 83, Assume the same lease as in (1), except that payments are due at the end of each quarter. Solution: Present value \$6,000 (P/A, 2%, 16) \$6, \$ 81, A company is required to place \$10,000 in a bank account every six months for 10 years; the bank account pays 4% interest, compounded semi-annually. How much will be in the bank account after 10 years if the payments are made at the end of each six-month period? Solution: Balance \$10,000 (F/A, 2%, 20) \$10, \$ 242, Repeat example (3), except assume that payments are made at the beginning of each sixmonth period. Solution: Balance \$10,000 (F/AD, 2%, 20) \$10, \$ 247, A company borrows \$56,000 on 1 January 20X1, and is required to make end-of-year payments for eight years. Interest of 8% is compounded annually; what is the payment? Solution: The present value (P) of the payments must be \$56,000 at 8% interest. The annual payment (represented as A) is the unknown in the equation:

10 404 CHAPTER 8 Current Monetary Balances P \$56,000 \$56,000 A (P/A, 8%, 8) A \$56,000 (P/A, 8%, 8) \$56, \$9, A company borrows \$100,000 on 1 January 20X1 and is required to make quarterly payments of \$6,258 at the beginning of each quarter for five years. What is the interest rate? Solution: \$100,000 \$6,258 (P/AD, i, 20); (P/AD, i, 20) \$100,000 \$6, Using Table I-6 along the 20-year line, 2 1/2% per quarter, or 10%, compounded quarterly. Using Multiple Present and Future Values Some transactions require application of two or more future or present value amounts. These more complex problems require careful analysis. Two cases are given below to illustrate the application of multiple future and present values. Case A. Deferred Annuity. A deferred annuity occurs in two phases: (1) capital is invested over a period to accumulate maximum interest compounding and principal growth and (2) the principal is paid out in uniform amounts until the total accumulated principal is exhausted. Investment during the accumulation phase may be in the form of either periodic payments or a lump-sum payment at the beginning. During the second phase, while withdrawals are being distributed to the annuitant, the remaining principal continues to earn interest. To illustrate, assume that on 1 January 20X1, Fox Company invests in a \$100,000 deferred annuity for the benefit of an employee, George Golf, who was injured while at work. The terms call for Fox Company to make an immediate \$100,000 lump-sum payment, which will earn interest at 11% for four years, the capital accumulation phase of the annuity. Then, beginning on 1 January 20X5, when George Golf retires, the annuity will be paid to him in five equal annual instalment payments. First, the fund grows at 11% for four years. At that time, the fund will equal \$100,000 (F/P, 11%, 4) \$100,000 (1.518) \$151,800 Second, the fund is used in total to pay Golf a five-year annuity beginning 1 January 20X5. The fund is assumed to continue to earn 11% until the last payment is made. The fund will be used up by the payments, so Annuity (P/AD, 11%, 5) \$151,800 Annuity \$151,800 Annuity \$151, \$ 37,002 Case B. Annuity and Lump Sum. Explo Company is negotiating to purchase four acres of land containing a gravel deposit that is suitable for development. Explo Company has completed a survey that provides the following reliable estimates: Expected net cash revenues over life of resource: End of 20X2 \$ 5,000 End of 20X3 to 20X6 (per year) 30,000 End of 20X7 to 20X10 (per year) 40,000 End of 20X11 (last year resource exhausted) 10,000 Estimated sales value of four acres after exhaustion of gravel, net of land restoration costs (end of 20X11) 2,000 What is the maximum amount Explo Company could offer on 1 January 20X2 for the land, assuming that Explo requires a 12% after-tax return on the investment? We will assume that all amounts are measured at year-end and that the above amounts are net of income tax.

11 CHAPTER 8 Current Monetary Balances 405 EXHIBIT 8A-4 CASH FLOWS FOR EXPLO CO. EXAMPLE PV at End of year cash flows 1/1/20X2 20X2 20X3 20X4 20X5 20X6 20X7 20X8 20X9 20X10 20X11 PV a \$5,000 PV b \$30,000 \$30,000 \$30,000 \$30,000 PV c \$40,000 \$40,000 \$40,000 \$40,000 PV d \$12,000 This case requires computation of the present value of the future expected cash inflows. The amount that the company would be willing to pay is the sum of the present values of the net future cash inflows for the various years. The calculation is complex because both single payments and annuities are involved. Because equal but different future cash inflows are expected for years 2 to 5 and years 6 to 9, two annuities may be calculated. Because the cash inflows are assumed to be received at year-end, the annuities are ordinary. The cash flows are depicted graphically in Exhibit 8A-4. This case is best solved in several steps in which the cash flows are separated into single payments and annuities and each expressed in present value terms. Thus: 1. \$5,000 (P/F, 12%, 1) \$5, \$ 4, \$30,000 (P/A, 12%, 4) (P/F, 12%, 1) \$30, , \$40,000 (P/A, 12%, 4) (P/F, 12%, 5) \$40, , (\$10,000 \$2,000) (P/F, 12%, 10) \$12, ,864 \$158,625 Explo should offer no more than \$158,625 for the properties. Note that the present value of the annuity shown in equation (2) is first calculated as of 31 December 20X2, as \$30,000 (P/A, 12%, 4). It is then discounted for one period using (P/F, 12%, 1). The present value of the annuity shown in equation (3) is first calculated as of 31 December 20X6, as \$40,000(P/A, 12%, 4) and is then discounted to 31 December 20X1, using the value for (P/F, 12%, 5). There are several other approaches to manipulating present values to provide this result all result in the same answer and are perfectly acceptable.

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