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.
1 3. Time value of money We will review some tools for discounting cash flows. Simple interest 2 With simple interest, the amount earned each period is always the same: i = rp o where i = interest earned
5.1 Simple and Compound Interest Question 1: What is simple interest? Question 2: What is compound interest? Question 3: What is an effective interest rate? Question 4: What is continuous compound interest?
CHAPTER 6 Accounting and the Time Value of Money ASSIGNMENT CLASSIFICATION TABLE (BY TOPIC) Topics Questions Brief Exercises Exercises Problems 1. Present value concepts. 1, 2, 3, 4, 5, 9, 17, 19 2. Use
Present Value Concepts Present value concepts are widely used by accountants in the preparation of financial statements. In fact, under International Financial Reporting Standards (IFRS), these concepts
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
Appendix B Time Value of Learning Objectives CAP Conceptual C1 Describe the earning of interest and the concepts of present and future values. (p. B-1) Procedural P1 P2 P3 P4 Apply present value concepts
EYK 15-1 Present Value (PV) Tutorial The concepts of present value are described and applied in Chapter 15. This supplement provides added explanations, illustrations, calculations, present value tables,
Time Value of Money 15.511 Corporate Accounting Summer 2004 Professor S. P. Kothari Sloan School of Management Massachusetts Institute of Technology July 2, 2004 1 LIABILITIES: Current Liabilities Obligations
Chapter The Time Value of Money PPT 9-2 Chapter 9 - Outline Time Value of Money Future Value and Present Value Annuities Time-Value-of-Money Formulas Adjusting for Non-Annual Compounding Compound Interest
TIME VALUE OF MONEY (TVM) INTEREST Rate of Return When we know the Present Value (amount today), Future Value (amount to which the investment will grow), and Number of Periods, we can calculate the rate
MATH 1003 Calculus and Linear Algebra (Lecture 3) Future Value of an Annuity Definition An annuity is a sequence of equal periodic payments. We call it an ordinary annuity if the payments are made at the
Basic Concept of Time Value of Money CHAPTER 1 1.1 INTRODUCTION Money has time value. A rupee today is more valuable than a year hence. It is on this concept the time value of money is based. The recognition
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
6-1 Chapter 6 Time Value of Money Concepts 6-2 Time Value of Money Interest is the rent paid for the use of money over time. That s right! A dollar today is more valuable than a dollar to be received in
Introduction to Real Estate Investment Appraisal Maths of Finance Present and Future Values Pat McAllister INVESTMENT APPRAISAL: INTEREST Interest is a reward or rent paid to a lender or investor who has
CHAPTER 6 Accounting and the Time Value of Money ASSIGNMENT CLASSIFICATION TABLE (BY TOPIC) Topics Questions Brief Exercises Exercises Problems 1. Present value concepts. 1, 2, 3, 4, 5, 9, 17 2. Use of
2 Time-Value-of-Money and Amortization Worksheets The Time-Value-of-Money and Amortization worksheets are useful in applications where the cash flows are equal, evenly spaced, and either all inflows or
LO.a: Interpret interest rates as required rates of return, discount rates, or opportunity costs. 1. The minimum rate of return that an investor must receive in order to invest in a project is most likely
CHAPTER 6 Accounting and the Time Value of Money 6-1 LECTURE OUTLINE This chapter can be covered in two to three class sessions. Most students have had previous exposure to single sum problems and ordinary
Chapter 6 Learning Objectives Principles Used in This Chapter 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams 1. Distinguish between an ordinary annuity and an annuity due, and calculate present
1 Appendix Time Value of Money After studying Appendix 1, you should be able to: 1 Explain how compound interest works. 2 Use future value and present value tables to apply compound interest to accounting
CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 1. The simple interest per year is: $5,000.08 = $400 So after 10 years you will have: $400 10 = $4,000 in interest. The total balance will be
CHAPTER 6 DISCOUNTED CASH FLOW VALUATION Answers to Concepts Review and Critical Thinking Questions 1. The four pieces are the present value (PV), the periodic cash flow (C), the discount rate (r), and
D. Dimov Most financial decisions involve costs and benefits that are spread out over time Time value of money allows comparison of cash flows from different periods Question: You have to choose one of
Integrated Case 5-42 First National Bank Time Value of Money Analysis You have applied for a job with a local bank. As part of its evaluation process, you must take an examination on time value of money
2 Determinants of Valuation Part Two 4 Time Value of Money 5 Fixed-Income Securities: Characteristics and Valuation 6 Common Shares: Characteristics and Valuation 7 Analysis of Risk and Return The primary
FIN 301 Homework Solution Ch4 Chapter 4: Time Value of Money 1. a. 10,000/(1.10) 10 = 3,855.43 b. 10,000/(1.10) 20 = 1,486.44 c. 10,000/(1.05) 10 = 6,139.13 d. 10,000/(1.05) 20 = 3,768.89 2. a. $100 (1.10)
Part 5 Statistical Models for Forecasting and Planning Chapter 16 Financial Calculations: Interest, Annuities and NPV chapter 16 Financial Calculations: Interest, Annuities and NPV Outcomes Financial information
BASIC ANNUITIES There are many accounting transactions that require the payment of a specific amount each period. A payment for a auto loan or a mortgage payment are examples of this type of transaction.
Chapter 4 The Time Value of Money 1 Learning Outcomes Chapter 4 Identify various types of cash flow patterns Compute the future value and the present value of different cash flow streams Compute the return
Chapter 6 The Time Value of Money: Annuities and Other Topics Chapter 6 Contents Learning Objectives 1. Distinguish between an ordinary annuity and an annuity due, and calculate present and future values
Slide 1 Interest Due Slide 2 Fin 5413 CHAPTER FOUR FIXED RATE MORTGAGE LOANS Interest Due is the mirror image of interest earned In previous finance course you learned that interest earned is: Interest
CALCULATOR TUTORIAL INTRODUCTION Because most students that use Understanding Healthcare Financial Management will be conducting time value analyses on spreadsheets, most of the text discussion focuses
CHAPTER 4 The Time Value of Money Chapter Synopsis Many financial problems require the valuation of cash flows occurring at different times. However, money received in the future is worth less than money
Chapter Two THE TIME VALUE OF MONEY Conventions & Definitions Introduction Now, we are going to learn one of the most important topics in finance, that is, the time value of money. Note that almost every
CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY Answers to Concepts Review and Critical Thinking Questions 1. The four parts are the present value (PV), the future value (FV), the discount
The Concept of Present Value If you could have $100 today or $100 next week which would you choose? Of course you would choose the $100 today. Why? Hopefully you said because you could invest it and make
Time Value of Money 1 This topic introduces you to the analysis of trade-offs over time. Financial decisions involve costs and benefits that are spread over time. Financial decision makers in households
The Time Value of Money C H A P T E R N I N E Figure 9-1 Relationship of present value and future value PPT 9-1 $1,000 present value $ 10% interest $1,464.10 future value 0 1 2 3 4 Number of periods Figure
Compound Interest Invest 500 that earns 10% interest each year for 3 years, where each interest payment is reinvested at the same rate: Table 1 Development of Nominal Payments and the Terminal Value, S.
Time Value of Money Work book Section I True, False type questions State whether the following statements are true (T) or False (F) 1.1 Money has time value because you forgo something certain today for
1 Simple interest 2 5. Time value of money With simple interest, the amount earned each period is always the same: i = rp o We will review some tools for discounting cash flows. where i = interest earned
QUANTITATIVE METHODS THE TIME VALUE OF MONEY Reading 5 http://proschool.imsindia.com/ 1 Learning Objective Statements (LOS) a. Interest Rates as Required rate of return, Discount Rate and Opportunity Cost
Sample Examination Questions CHAPTER 6 ACCOUNTING AND THE TIME VALUE OF MONEY MULTIPLE CHOICE Conceptual Answer No. Description d 1. Definition of present value. c 2. Understanding compound interest tables.
Reading 5 The Time Value of Money Money has a time value because a unit of money received today is worth more than a unit of money to be received tomorrow. Interest rates can be interpreted in three ways.
P a g e 1 Study Questions for Actuarial Exam 2/FM By: Aaron Hardiek June 2010 P a g e 2 Background The purpose of my senior project is to prepare myself, as well as other students who may read my senior
Compound Interest Interest is the amount you receive for lending money (making an investment) or the fee you pay for borrowing money. Compound interest is interest that is calculated using both the principle
The time value of money: Part II A reading prepared by Pamela Peterson Drake O U T L I E 1. Introduction 2. Annuities 3. Determining the unknown interest rate 4. Determining the number of compounding periods
Chapter 4 Time Value of Money ANSWERS TO END-OF-CHAPTER QUESTIONS 4-1 a. PV (present value) is the value today of a future payment, or stream of payments, discounted at the appropriate rate of interest.
Bond valuation A reading prepared by Pamela Peterson Drake O U T L I N E 1. Valuation of long-term debt securities 2. Issues 3. Summary 1. Valuation of long-term debt securities Debt securities are obligations
4.1 a. Future Value = C 0 (1+r) T Chapter 4: Net Present Value = $1,000 (1.05) 10 = $1,628.89 b. Future Value = $1,000 (1.07) 10 = $1,967.15 c. Future Value = $1,000 (1.05) 20 = $2,653.30 d. Because interest
MAT116 Project 2 Chapters 8 & 9 1 8-1: The Project In Project 1 we made a loan workout decision based only on data from three banks that had merged into one. We did not consider issues like: What was the
6 Formulas Discounted Cash Flow Valuation McGraw-Hill/Irwin Copyright 2008 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter Outline Future and Present Values of Multiple Cash Flows Valuing
380.760: Corporate Finance Lecture 2: Time Value of Money and Net Present Value Gordon Bodnar, 2009 Professor Gordon Bodnar 2009 Financial Decision Making Finance decision making is about evaluating costs
1 2 TIME VALUE OF MONEY APPENDIX 3 The simplest tools in finance are often the most powerful. Present value is a concept that is intuitively appealing, simple to compute, and has a wide range of applications.
Part 2 Important Financial Concepts Chapter 4 Time Value of Money Chapter 5 Risk and Return Chapter 6 Interest Rates and Bond Valuation Chapter 7 Stock Valuation 130 LG1 LG2 LG3 LG4 LG5 LG6 Chapter 4 Time
CHAPTER 3 CONCEPT REVIEW QUESTIONS 1. Will a deposit made into an account paying compound interest (assuming compounding occurs once per year) yield a higher future value after one period than an equal-sized
KEATMX01_p001-008.qxd 11/4/05 4:47 PM Page 1 Calculations for Time Value of Money In this appendix, a brief explanation of the computation of the time value of money is given for readers not familiar with
and Loans Mathematics Rome, University of Tor Vergata and Loans Future Value for Simple Interest Present Value for Simple Interest You deposit E. 1,000, called the principal or present value, into a savings
Chapter 6 Key Concepts and Skills Be able to compute: the future value of multiple cash flows the present value of multiple cash flows the future and present value of annuities Discounted Cash Flow Valuation
CHAPTER 4 DISCOUNTED CASH FLOW VALUATION Solutions to Questions and Problems NOTE: All-end-of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability
CHAPTER 8 INTEREST RATES AND BOND VALUATION Solutions to Questions and Problems 1. The price of a pure discount (zero coupon) bond is the present value of the par value. Remember, even though there are
Management 3 Quantitative Methods The Time Value of Money Part 2B Present Value of Annuities Revised 2/18/15 New Scenario We can trade a single sum of money today, a (PV) in return for a series of periodic
Chapter 3 Mathematics of Finance Section 3 Future Value of an Annuity; Sinking Funds Learning Objectives for Section 3.3 Future Value of an Annuity; Sinking Funds The student will be able to compute the
Foundation review: Introduction Foundation review Introduction Throughout FN1, you will be expected to apply techniques and concepts that you learned in prerequisite courses. The purpose of this foundation
Introduction to the Hewlett-Packard (HP) 10BII Calculator and Review of Mortgage Finance Calculations Real Estate Division Sauder School of Business University of British Columbia Introduction to the Hewlett-Packard
2 The Mathematics of Finance Copyright Cengage Learning. All rights reserved. 2.3 Annuities, Loans, and Bonds Copyright Cengage Learning. All rights reserved. Annuities, Loans, and Bonds A typical defined-contribution
TIME VALUE OF MONEY PROBLEM #7: MORTGAGE AMORTIZATION Professor Peter Harris Mathematics by Sharon Petrushka Introduction This problem will focus on calculating mortgage payments. Knowledge of Time Value
Applying Time Value Concepts C H A P T E R 3 based on the value of two packs of cigarettes per day and a modest rate of return? Let s assume that Lou will save an amount equivalent to the cost of two packs
Lecture 8: Asset Markets c 2009 Je rey A. Miron Outline:. Present and Future Value 2. Bonds 3. Taxes 4. Applications Present and Future Value In the discussion of the two-period model with borrowing and
Mathematics of Finance Interest is the rental fee charged by a lender to a business or individual for the use of money. charged is determined by Principle, rate and time Interest Formula I = Prt $100 At
What is an annuity? The basics Part 1 of 8 You may be considering an annuity, and, if that s the case, it is important that know what an annuity is and isn t. Basically, an annuity is a contract with an
CHAPTER Risk Analysis in Engineering and Economics ENGINEERING ECONOMICS AND FINANCE A. J. Clark School of Engineering Department of Civil and Environmental Engineering 6a CHAPMAN HALL/CRC Risk Analysis
How Does Money Grow Over Time? Suggested Grade & Mastery Level High School all levels Suggested Time 45-50 minutes Teacher Background Interest refers to the amount you earn on the money you put to work
Ordinary Annuities (Young: 6.2) In this Lecture: 1. More Terminology 2. Future Value of an Ordinary Annuity 3. The Ordinary Annuity Formula (Optional) 4. Present Value of an Ordinary Annuity More Terminology
Chapter 2 - Sample Problems 1. If you wish to accumulate $140,000 in 13 years, how much must you deposit today in an account that pays an annual interest rate of 14%? 2. What will $247,000 grow to be in
PRESENT VALUE ANALYSIS Time value of money equal dollar amounts have different values at different points in time. Present value analysis tool to convert CFs at different points in time to comparable values
appendix B Time value of money LEARNING OBJECTIVES After studying this appendix, you should be able to: Distinguish between simple and compound interest. Solve for future value of a single amount. Solve
Vilnius University Faculty of Mathematics and Informatics Gintautas Bareikis CONTENT Chapter 1. SIMPLE AND COMPOUND INTEREST 1.1 Simple interest......................................................................
C H A P T E R6 The Time Value of Money When plumbers or carpenters tackle a job, they begin by opening their toolboxes, which hold a variety of specialized tools to help them perform their jobs. The financial
Chapter 2 Applying Time Value Concepts Chapter Overview Albert Einstein, the renowned physicist whose theories of relativity formed the theoretical base for the utilization of atomic energy, called the