Present Value Concepts

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 are more widely applied than under traditional Canadian GAAP. This appendix will explain the basics that you must be aware of to understand related topics in this text. INTEREST RATES Interest is payment for the use of money. It is the difference between the amount borrowed or invested (the principal) and the amount repaid or collected. The amount of interest to be paid or collected is usually stated as a rate over a specific period of time. The rate of interest is generally stated as an annual rate. The amount of interest involved in any financing transaction is based on three elements: 1. Principal (p): The original amount borrowed or invested 2. Interest rate (i): An annual percentage of the principal 3. Number of periods (n): The time period that the principal is borrowed or invested Simple Interest Simple interest is calculated on the principal amount only. It is the return on the principal for one period. Simple interest is usually expressed as shown in Illustration PV-1. Interest Principal Interest Rate Number of Periods (p) (i) (n) Illustration PV-1 Simple interest formula For example, if you borrowed $1,000 for 3 years at a simple interest rate of 9% annually, you would pay $270 in total interest, calculated as follows: Interest p i n Year 1 Year 2 Year 3 $1,000 9% 3 $90 $90 $90 $270 $270

2 Present Value Concepts Compound Interest Compound interest is the return on (or growth of) the principal for two or more time periods. Compounding calculates interest not only on the principal but also on the interest earned to date on that principal, assuming the interest is left on deposit (i.e., added to the original principal amount). To illustrate the difference between simple and compound interest, assume that you deposit $1,000 in the Last Canadian Bank, where it will earn simple interest of 9% per year, and you deposit another $1,000 in the First Canadian Bank, where it will earn interest of 9% per year compounded annually. Also assume that in both cases you will not withdraw any interest until three years from the date of deposit. The calculation of interest to be received and the accumulated year-end balances are given in Illustration PV-2. Last Canadian Bank Accumulated Simple Interest Simple Year-End Calculation Interest Balance Year 1 $1,000.00 9% $ 90.00 $1,090.00 Year 2 $1,000.00 9% 90.00 $1,180.00 First Canadian Bank Accumulated Compound Interest Compound Year-End Calculation Interest Balance Year 1 $1,000.00 9% $ 90.00 $1,090.00 Year 2 $1,090.00 9% 98.10 $1,188.10 Year 3 $1,000.00 9% 90.00 $1,270.00 Year 3 $1,188.10 9% 106.93 $1,295.03 $270.00 $25.03 $295.03 Difference Illustration PV-2 Simple versus compound interest Note in Illustration PV-2 that simple interest uses the initial principal of $1,000 to calculate the interest in all three years. Compound interest uses the accumulated balance (principal plus interest to date) at each year end to calculate interest in the following year. This explains why your compound interest account is larger: you are earning interest on interest. For practical purposes, compounding assumes that unpaid interest earned becomes a part of the principal. The accumulated balance at the end of each year becomes the new principal on which interest is earned during the next year. Assuming all else is equal (especially risk), if you had a choice between investing your money at simple interest or at compound interest, you would choose compound interest. In the example, compounding provides $25.03 of additional interest income. Compound interest is used in most business situations. Simple interest is generally applicable only to short-term situations of one year or less. Present value concepts use compound interest. CALCULATING PRESENT VALUES In the previous section on compound and simple interest, the initial principal was given. It was used to calculate the interest earned and the value of the investment at the end of three years. The initial principal, invested at the beginning of year one, is the present value of the investment. The value of the investment at the end of three years is the future value of the investment. You are probably more accustomed to being given the present value and then calculating the future value. But in business, there are many situations in which the future value is given, and it is necessary to calculate the present value. For example, we determine the market price of a bond by calculating the present value of the principal and interest payments. Calculating the amount to be reported for fixed and intangible assets, notes payable, pensions, and finance lease liabilities can also involve present value calculations. Present value calculations are always based on three variables:

Calculating Present Values 3 1. The dollar amount to be received (the future amount or future value) 2. The length of time until the amount is received (the number of periods) 3. The interest rate (the discount rate) per period The process of determining the present value is often referred to as discounting the future cash flows. The word discount has many meanings in accounting, each of which varies with the context in which it is being used. Be careful not to confuse the use of this term. Present Value of a Single Future Amount In the following section, we will show three methods of calculating the present value of a single future amount: present value formula, present value tables, and financial calculators. Present Value Formula To illustrate present value concepts, assume that you want to invest a sum of money at 5% in order to have $1,000 at the end of one year. The amount that you would need to invest today is called the present value of $1,000 discounted for one year at 5%. The variables in this example are shown in the time diagram in Illustration PV-3. Present Value = $952.38 Future Amount = $1,000 i 5% n 1 year Now 1 Year Illustration PV-3 Time diagram for the present value of $1,000 discounted for one period at 5% The formula used to determine the present value for any interest (discount) rate (i), number of periods (n), and future amount (FV) is shown in Illustration PV-4. Future value (FV) Present value (PV) (l i) n FV (1 i) n Illustration PV-4 Present value of a single future amount formula In applying this formula to calculate the present value (PV) for the above example, the future value (FV) of $1,000, the interest (discount) rate (i) of 5%, and the number of periods (n) of one are used as follows: PV $1,000 (1 5%) 1 $1,000 1.05 $952.38 If the single future cash flow of $1,000 is to be received in two years and discounted at 5%, its present value is calculated as follows: PV $1,000 (1 5%) 2 $1,000 1.05 2 or [($1,000 1.05) 1.05] $907.03 The time diagram in Illustration PV-5 shows the variables used to calculate the present value when cash is received in two years.

4 Present Value Concepts Illustration PV-5 Time diagram for present value of $1,000 discounted for two periods at 5% Present Value = $907.03 Future Amount = $1,000 Now i 5% n 2 years 1 2 Years Present Value Tables The present value may also be determined through tables that show the present value of 1 for n periods for different periodic interest rates or discount rates. In Table PV-1, the rows represent the number of discounting periods and the columns the periodic interest or discount rates. The five-digit decimal numbers in the respective rows and columns are the factors for the present value of 1. When present value tables are used, the present value is calculated by multiplying the future cash amount by the present value factor specified at the intersection of the number of periods and the discount rate. For example, if the discount rate is 5% and the number of periods is 1, Table PV-1 shows that the present value factor is 0.95238. Then the present value of $1,000 discounted at 5% for one period is calculated as follows: PV $1,000 0.95238 $952.38 For two periods at a discount rate of 5%, the present value factor is 0.90703. The present value of $1,000 discounted at 5% for two periods is calculated as follows: PV $1,000 0.90703 $907.03 Note that the present values in these two examples are identical to the amounts determined previously when using the present value formula. This is because the factors in a present value table have been calculated using the present value formula. The benefit of using a present value table is that it can be quicker than using the formula. If you are using a simple calculator (not a financial calculator) or doing the calculations by hand, there are more calculations involved as the number of periods increase, making it more tedious than using the present value tables. Table PV-1 can also be used if you know the present value and wish to determine the future cash flow. The present value amount is divided by the present value factor specified at the intersection of the number of periods and the discount rate in Table PV-1. For example, it can easily be determined that an initial investment of $907.03 will grow to yield a future amount of $1,000 in two periods, at an annual discount rate of 5% ($1,000 $907.03 0.90703).

Calculating Present Values 5 Table PV-1 PRESENT VALUE OF 1 PV 1 (1 i) n (n) Periods 2% 2 1 /2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 15% 1 0.98039 0.97561 0.97087 0.96154 0.95238 0.94340 0.93458 0.92593 0.91743 0.90909 0.90090 0.89286 0.86957 2 0.96117 0.95181 0.94260 0.92456 0.90703 0.89000 0.87344 0.85734 0.84168 0.82645 0.81162 0.79719 0.75614 3 0.94232 0.92860 0.91514 0.88900 0.86384 0.83962 0.81630 0.79383 0.77218 0.75131 0.73119 0.71178 0.65752 4 0.92385 0.90595 0.88849 0.85480 0.82270 0.79209 0.76290 0.73503 0.70843 0.68301 0.65873 0.63552 0.57175 5 0.90573 0.88385 0.86261 0.82193 0.78353 0.74726 0.71299 0.68058 0.64993 0.62092 0.59345 0.56743 0.49718 6 0.88797 0.86230 0.83748 0.79031 0.74622 0.70496 0.66634 0.63017 0.59627 0.56447 0.53464 0.50663 0.43233 7 0.87056 0.84127 0.81309 0.75992 0.71068 0.66506 0.62275 0.58349 0.54703 0.51316 0.48166 0.45235 0.37594 8 0.85349 0.82075 0.78941 0.73069 0.67684 0.62741 0.58201 0.54027 0.50187 0.46651 0.43393 0.40388 0.32690 9 0.83676 0.80073 0.76642 0.70259 0.64461 0.59190 0.54393 0.50025 0.46043 0.42410 0.39092 0.36061 0.28426 10 0.82035 0.78120 0.74409 0.67556 0.61391 0.55839 0.50835 0.46319 0.42241 0.38554 0.35218 0.32197 0.24718 11 0.80426 0.76214 0.72242 0.64958 0.58468 0.52679 0.47509 0.42888 0.38753 0.35049 0.31728 0.28748 0.21494 12 0.78849 0.74356 0.70138 0.62460 0.55684 0.49697 0.44401 0.39711 0.35553 0.31863 0.28584 0.25668 0.18691 13 0.77303 0.72542 0.68095 0.60057 0.53032 0.46884 0.41496 0.36770 0.32618 0.28966 0.25751 0.22917 0.16253 14 0.75788 0.70773 0.66112 0.57748 0.50507 0.44230 0.38782 0.34046 0.29925 0.26333 0.23199 0.20462 0.14133 15 0.74301 0.69047 0.64186 0.55526 0.48102 0.41727 0.36245 0.31524 0.27454 0.23939 0.20900 0.18270 0.12289 16 0.72845 0.67362 0.62317 0.53391 0.45811 0.39365 0.33873 0.29189 0.25187 0.21763 0.18829 0.16312 0.10686 17 0.71416 0.65720 0.60502 0.51337 0.43630 0.37136 0.31657 0.27027 0.23107 0.19784 0.16963 0.14564 0.09293 18 0.70016 0.64117 0.58739 0.49363 0.41552 0.35034 0.29586 0.25025 0.21199 0.17986 0.15282 0.13004 0.08081 19 0.68643 0.62553 0.57029 0.47464 0.39573 0.33051 0.27651 0.23171 0.19449 0.16351 0.13768 0.11611 0.07027 20 0.67297 0.61027 0.55368 0.45639 0.37689 0.31180 0.25842 0.21455 0.17843 0.14864 0.12403 0.10367 0.06110 Financial Calculators Present values can also be calculated using financial calculators. A financial calculator will perform the same calculation as a simple calculator, but it requires fewer steps and less input. Basically, with a financial calculator, you input the future value, the discount rate, and the number of periods, and then tell it to calculate the present value. We will not illustrate how to use a financial calculator because the details vary with different financial calculators. But you should know that the present value amounts calculated with a financial calculator can be slightly different than those calculated with present value tables. That is because the numbers in a present value table are rounded. For example, in Table PV-1 the factors are rounded to five digits. In a financial calculator, only the final answer is rounded to the number of digits you have specified. A major benefit of using a financial calculator is that you are not restricted to the interest rates or numbers of periods on a present value table. In Table PV-1, present value factors have been calculated for 13 interest rates (there are 13 columns in that table) and the maximum number of periods is 20. With a financial calculator you could, for example, calculate the present value of a future amount to be received in 25 periods using 5.75% as the discount rate. Later in this appendix, we will illustrate how to partly overcome this limitation of present value tables through a method called interpolation. Regardless of the method used in calculating present values, a higher discount rate produces a smaller present value. For example, using an 8% discount rate, the present value of $1,000 due one year from now is $925.93 versus $952.38 at 5%. It should also be recognized that the further away from the present the future cash flow is, the smaller the present value. For example, using the same discount rate of 5%, the present value of $1,000 due in five years is $783.53. The present value of $1,000 due in one year is $952.38.

6 Present Value Concepts BEFORE YOU GO ON Review It 1. Distinguish between simple and compound interest. 2. Distinguish between present value and future value. 3. What are the three variables used in calculating present value? 4. When calculating present value, what are the benefits of using present value tables and of using financial calculators? Do It Suppose you have a winning lottery ticket and the lottery commission gives you the option of taking $10,000 three years from now, or taking the present value of $10,000 now. If an 8% discount rate is used, what is the present value of your winnings if you take the option of receiving $10,000 three years from now? Show the calculation using either (a) the present value formula or (b) Table PV-1. Action Plan Note that $10,000 is the future value, the number of periods is 3, and the discount rate is 8%. Recall that the present value of $10,000 to be received in 3 years is less than $10,000 because interest can be earned on an amount invested today (i.e., the present value) over the next three years. Understand that the discount rate used to calculate the present value is the compound interest rate that would be used to earn $10,000 in 3 years if the present value is invested today. Draw a time diagram showing when the future value will be received, the discount rate, and the number of periods. PV (?) $10,000 i 8% n 3 Now 1 2 3 Years Solution (a) Using the present value formula: PV $10,000 (1 8%) 3 $10,000 1.08 3 or {[($10,000 1.08) 1.08] 1.08} $7,938.30 (b) Using the present value factor from Table PV-1: The present value factor for three periods at 8% is 0.79383. PV $10,000 0.79383 $7,938.30 Do It Again Determine the amount you must deposit now in your savings account, paying 3% interest, in order to accumulate $5,000 for a down payment on a hybrid electric car four years from now, when you graduate. Show the calculation using either (a) present value formula, or (b) Table PV-1. Action Plan Note that $5,000 is the future value, n 4, and i 3%. Draw a time diagram showing when the future value will be received, the discount rate, and the number of periods. PV (?) $5,000 i 3% n 4 2 Now 1 3 4 Years

Present Value of a Series of Future Cash Flows (Annuities) 7 Solution The amount you must deposit now in your savings account is the present value calculated as follows: (a) Using the present value formula: PV $5,000 (1 3%) 4 $5,000 1.03 4 or ({[($5,000 1.03) 1.03] 1.03} 1.03) $4,442.43 (b) Using the present value factor from Table PV-1: The present value factor for four periods at 3% is 0.88849. PV $5,000 0.88849 PV $4,442.45 Note: The difference in the two answers is due to rounding the factors in Table PV-1. Related exercise material: BEPV-1, BEPV-2, BEPV-3, BEPV-4, and BEPV-5. PRESENT VALUE OF A SERIES OF FUTURE CASH FLOWS (ANNUITIES) The preceding discussion was for the discounting of only a single future amount. Businesses and individuals frequently engage in transactions in which a series of equal dollar amounts are to be received or paid periodically. Examples of a series of periodic receipts or payments are loan agreements, instalment sales, mortgage notes, lease (rental) contracts, and pension obligations. These series of periodic receipts or payments are called annuities. In calculating the present value of an annuity, it is necessary to know (1) the discount rate (i), (2) the number of discount periods (n), and (3) the amount of the periodic receipts or payments (FV). To illustrate the calculation of the present value of an annuity, assume that you will receive $1,000 cash annually for three years, and that the discount rate is 4%. This situation is shown in the time diagram in Illustration B-6. PV (?) $1,000 $1,000 $1,000 i 4% n 3 Now 1 2 3 Years Illustration PV-6 Time diagram for a three-year annuity One method of calculating the present value of this annuity is to use the present value formula to determine the present value of each of the three $1,000 payments and then add those amounts as follows: PV [$1,000 (1 4%) 1 ] [$1,000 (1 4%) 2 ] [$1,000 (1 4%) 3 ] $961.54 $924.56 $889.00 $2,775.10 The same result is achieved by using present value factors from Table PV-1, as shown in Illustration PV-7. Present Value of 1 Future Value Factor at 4% Present Value $1,000 (one year away) 0.96154 $ 961.54 1,000 (two years away) 0.92456 924.56 1,000 (three years away) 0.88900 889.00 2.77510 $2,775.10 Illustration PV-7 Present value of a series of future cash flows

8 Present Value Concepts Determining the present value of each single future cash flow, and then adding the present values, is required when the periodic cash flows are not the same in each period. But when the future receipts are the same in each period, there are three other ways to calculate present value. The first is to use the present value of an ordinary annuity formula, as shown in Illustration PV-8. Illustration PV-8 Present value of an ordinary annuity of 1 formula Present value (PV ) Future value (FV ) 1 1 (1 i) n $1,000 [(1 (1 (1 + 4%) 3 )) 4%] $1,000 [(1 (1 (1.04) 3 )) 0.04] $1,000 [(1 (1 1.124864)) 0.04] $1,000 [(1 0.888996358) 0.04] $1,000 2.77509 $2,775.09 i Table PV-2 PRESENT VALUE OF AN ANNUITY OF 1 PV 1 1 (1 i) n (n) Periods 2% 2 1 /2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 15% 1 0.98039 0.97561 0.97087 0.96154 0.95238 0.94340 0.93458 0.92593 0.91743 0.90909 0.90090 0.89286 0.86957 2 1.94156 1.92742 1.91347 1.88609 1.85941 1.83339 1.80802 1.78326 1.75911 1.73554 1.71252 1.69005 1.62571 3 2.88388 2.85602 2.82861 2.77509 2.72325 2.67301 2.62432 2.57710 2.53129 2.48685 2.44371 2.40183 2.28323 4 3.80773 3.76197 3.71710 3.62990 3.54595 3.46511 3.38721 3.31213 3.23972 3.16987 3.10245 3.03735 2.85498 5 4.71346 4.64583 4.57971 4.45182 4.32948 4.21236 4.10020 3.99271 3.88965 3.79079 3.69590 3.60478 3.35216 6 5.60143 5.50813 5.41719 5.24214 5.07569 4.91732 4.76654 4.62288 4.48592 4.35526 4.23054 4.11141 3.78448 7 6.47199 6.34939 6.23028 6.00205 5.78637 5.58238 5.38929 5.20637 5.03295 4.86842 4.71220 4.56376 4.16042 8 7.32548 7.17014 7.01969 6.73274 6.46321 6.20979 5.97130 5.74664 5.53482 5.33493 5.14612 4.96764 4.48732 9 8.16224 7.97087 7.78611 7.43533 7.10782 6.80169 6.51523 6.24689 5.99525 5.75902 5.53705 5.32825 4.77158 10 8.98259 8.75206 8.53020 8.11090 7.72173 7.36009 7.02358 6.71008 6.41766 6.14457 5.88923 5.65022 5.01877 11 9.78685 9.51421 9.25262 8.76048 8.30641 7.88687 7.49867 7.13896 6.80519 6.49506 6.20652 5.93770 5.23371 12 10.57534 10.25776 9.95400 9.38507 8.86325 8.38384 7.94269 7.53608 7.16073 6.81369 6.49236 6.19437 5.42062 13 11.34837 10.98319 10.63496 9.98565 9.39357 8.85268 8.35765 7.90378 7.48690 7.10336 6.74987 6.42355 5.58315 14 12.10625 11.69091 11.29607 10.56312 9.89864 9.29498 8.74547 8.24424 7.78615 7.36669 6.98187 6.62817 5.72448 15 12.84926 12.38138 11.93794 11.11839 10.37966 9.71225 9.10791 8.55948 8.06069 7.60608 7.19087 6.81086 5.84737 16 13.57771 13.05500 12.56110 11.65230 10.83777 10.10590 9.44665 8.85137 8.31256 7.82371 7.37916 6.97399 5.95423 17 14.29187 13.71220 13.16612 12.16567 11.27407 10.47726 9.76322 9.12164 8.54363 8.02155 7.54879 7.11963 6.04716 18 14.99203 14.35336 13.75351 12.65930 11.68959 10.82760 10.05909 9.37189 8.75563 8.20141 7.70162 7.24967 6.12797 19 15.67846 14.97889 14.32380 13.13394 12.08532 11.15812 10.33560 9.60360 8.95011 8.36492 7.83929 7.36578 6.19823 20 16.35143 15.58916 14.87747 13.59033 12.46221 11.46992 10.59401 9.81815 9.12855 8.51356 7.96333 7.46944 6.25933 i The second is to use a present value of an annuity table. As illustrated in Table PV-2, these tables show the present value of 1 to be received periodically for a given number of periods. From Table PV-2, it can be seen that the present value factor of an annuity of 1 for three periods at 4% is 2.77509. This present value factor is the total of the three individual present value factors, as shown in Illustration PV-7.1 Applying this present value factor to the annual cash flow of $1,000 produces a present value of $2,775.09 ($1,000 2.77509).

Present Value of a Series of Future Cash Flows (Annuities) 9 The third method is to use a financial calculator and input the number of periods, the interest rate, and the amount of the annual payment. When using a financial calculator to calculate the present value of an annuity, it is also necessary to specify if the annual cash flow is an ordinary annuity or an annuity in arrears. In an ordinary annuity, the first payment is at the end of the first period, as in our example. In an annuity in arrears, the first payment starts immediately, at the beginning of the first period. In this appendix and textbook, all of the annuity examples are ordinary annuities. Interest Rates and Time Periods In the preceding calculations, the discounting has been done on an annual basis using an annual interest rate. There are situations where adjustments may be required to the interest rate, the time period, or both. Using Time Periods of Less Than One Year Discounting may be done over shorter periods of time than one year, such as monthly, quarterly, or semi-annually. When the time frame is less than one year, it is necessary to convert the annual interest rate to the applicable time frame. Assume, for example, that the investor in Illustration PV-6 received $500 semi-annually for three years instead of $1,000 annually. In this case, the number of periods (n) becomes six (three annual periods 2), and the discount rate (i) is 2% (4% 6 /12 months). If present value tables are used to determine the present value, the appropriate present value factor from Table PV-2 is 5.60143. The present value of the future cash flows is $2,800.72 (5.60143 $500). This amount is slightly higher than the $2,775.09 calculated in Illustration PV-8 because interest is calculated twice during the same year. Thus, interest is compounded on the first half-year s interest. Interpolation As previously discussed, one of the limitations of the present value tables is that the tables contain a limited number of interest rates. But in certain situations where the factor for a certain interest rate is not available, it can be deduced or interpolated from the tables. For example, if you wished to find the factor to determine the present value of 1 for a 3 1 /2% interest rate and five periods, it would be difficult to use Table PV-1 to do so. Table PV-1 (or Table PV-2 for that matter) does not have a 3 1 /2% interest rate column. We can, however, average the factors for the 3% and 4% interest rates to approximate the factor for a 3 1 /2% interest rate. Factor, n 5, i 3% 0.86261 Factor, n 5, i 4% 0.82193 Sum of two factors 1.68454 Average two factors 2 Factor, n 5, i 31 2% 0.84227 Illustration PV-9 Interpolation of 3 1 /2% interest rate factor 1 The difference of 0.00001 between 2.77509 and 2.77510 is due to rounding. As previously mentioned, interpolation is only necessary when using present value tables. If a financial calculator is used, any interest rate can be used in the calculations.

10 Present Value Concepts BEFORE YOU GO ON Review It 1. What is an annuity? 2. How is Table PV-2 used to calculate the present value of an annuity? 3. If an annuity is paid every three months for five years and 8% is the annual discount rate, what are the number of periods (n) and the interest rate (i) used in the present value calculation? 4. When is it necessary to interpolate an interest rate? Do It Corkum Company has just signed a capital lease contract for equipment that requires rental payments of $6,000 each, to be paid at the end of each of the next five years. The appropriate discount rate is 6%. What is the present value of the rental payments; that is, the amount used to capitalize the leased equipment? Show the calculation using either (a) the present value of an annuity formula or (b) Table PV-2. Action Plan Draw a time diagram showing when the future value will be received, the discount rate, and the number of periods. PV (?) $6,000 $6,000 $6,000 $6,000 $6,000 i 6% n 5 Now 1 2 3 4 5 Years Note that each of the future payments is the same amount paid at even intervals; therefore, use present value of an annuity calculations to determine the present value (i 6% and n 5). Solution The present value of lease rental payments of $6,000 paid at the end of each year for five years, discounted at 6%, is calculated as follows: (a) Using the present value of an annuity formula: PV $6,000 [(1 (1 (1 6%) 5 ) 6%] $6,000 [(1 (1 (1.06) 5 ) 0.06] $6,000 [(1 (1 1.33823) 0.06] $6,000 [(1.747256) 0.06] $6,000 4.21236 $25,274.16 (b) Using the present value factor from Table PV-2: The present value factor from Table PV-2 is 4.21236 (five periods at 6%). PV $6,000 4.21236 PV $25,274.16 Related exercise material: BEPV-6, BEPV-7, and BEPV-8. APPLYING PRESENT VALUE CONCEPTS Calculating the Market Price or Present Value of a Bond The present value (or market price) of a bond is a function of three variables: (1) the payment amounts, (2) the length of time until the amounts are paid, and (3) the market interest rate, also known as the discount rate. The first variable (dollars to be paid) is made up of two elements: (1) the principal amount (a single sum), and (2) a series of interest payments (an annuity). To calculate the present value of the bond, both the principal amount and the interest payments must be discounted, which requires two different calculations. The present value of a bond can be calculated using present value formulas, factors from the two present value tables, or a financial calculator. We will illustrate only present value tables.

It is important to note that the interest rate used to determine the annual or semi-annual interest payments is fixed over the life of a bond. This is the bond s contractual interest rate. The company issuing the bond chooses the specific interest rate before issuing the bonds. But investors may use a different interest rate when determining how much they are willing to pay (the present value) for the bonds. Investors are influenced by both general economic conditions and their assessment of the company issuing the bonds. Investors use the market interest rate to determine the present value of the bonds. In the following sections, we will illustrate how to calculate the present value of the bonds using a market interest rate that is equal to, greater than, or less than the contractual interest rate. Market Interest Rate Equals the Contractual Interest Rate Applying Present Value Concepts 11 When the investor s market interest rate is equal to the bond s contractual interest rate, the bonds present value will equal their face value. To illustrate, assume there is a bond issue of five-year, 6% bonds with a face value of $100,000. Interest is payable semi-annually on January 1 and July 1. In this case, the investor will receive (1) $100,000 at maturity, and (2) a series of 10 $3,000 interest payments [($100,000 6%) 6 /12 months] over the term of the bonds. The length of time (n) is the total number of interest periods (10 periods 5 years 2 payments per year), and the discount rate (i) is the rate per semi-annual interest period (3% 6% 6 /12 months). The time diagram in Illustration PV-10 shows the variables involved in this discounting situation. DIAGRAM FOR PRINCIPAL Principal Present Amount Value (?) $100,000 i 3% n 10 Now 1 2 3 4 5 6 7 8 9 10 Periods Illustration PV-10 Time diagram for the present value of a five-year, 6% bond paying interest semi-annually DIAGRAM FOR INTEREST Interest Present Payments Value (?) $3,000 $3,000 $3,000 $3,000 $3,000 $3,000 $3,000 $3,000 $3,000 $3,000 i 3% n 10 Now 1 2 3 4 5 6 7 8 9 10 Periods The calculation of the present value of these bonds using factors from the appropriate present value tables is shown in Illustration PV-11. 6% Contractual Rate and 6% Market Rate Present value of principal to be received at maturity $100,000 PV of 1 due in 10 periods (n) at 3% (i) $100,000 0.74409 (Table PV-1) $ 74,409 Present value of interest to be received periodically over the term of the bonds $3,000 PV of 1 due periodically for 10 periods (n) at 3% (i) $3,000 8.53020 (Table PV-2) 25,591* Present value of bonds $100,000 Illustration PV-11 Present value of principal and interest (face value) *Rounded

12 Present Value Concepts Thus, when the market rate is the same as the contractual rate, the bonds will sell at face value. Market Interest Rate Is Greater Than the Contractual Interest Rate Now assume that the investor s market rate of return is 8%, not 6%. The future cash flows are again $100,000 and $3,000, respectively. These cash flows are based on the bond contract and do not vary with the investor s rate of return. But the investor s rate of return can vary, depending on available rates in the marketplace. If the market interest rate is 8%, then the present value is calculated using this rate. In this case, 4% (8% 6 /12 months) will be used because the bonds pay interest semi-annually. The present value of the bonds is $91,889, as calculated in Illustration PV-12. Illustration PV-12 Present value of principal and interest (discount) 6% Contractual Rate and 8% Market Rate Present value of principal to be received at maturity $100,000 PV of 1 due in 10 periods (n) at 4% (i) $100,000 0.67556 (Table PV-1) $67,556 Present value of interest to be received periodically over the term of the bonds $3,000 PV of 1 due periodically for 10 periods (n) at 4% (i) $3,000 8.11090 (Table PV-2) 24,333* Present value of bonds $91,889 *Rounded In this situation, the bonds will sell for $91,889, at a discount of $8,111. If the market interest rate is greater than the contract interest rate, the bonds will always sell at a discount. If investors determine that the bond s contract interest rate is too low, they will compensate by paying less for the bonds. Note that they will still collect the full $100,000 at the maturity date. Market Interest Rate Is Less Than the Contractual Interest Rate On the other hand, the market rate might be lower than the contract interest rate. In this case, the interest paid on the bonds is higher than what investors expected to earn. As a result, they will compensate by paying more for the bonds. If the market interest rate is 5%, the present value will be calculated using 2.5% (5% 6 /12 months) as the discount rate. The cash payments and number of periods remain the same. In this case, the present value of the bonds is $104,376, calculated as in Illustration PV-13. Illustration PV-13 Present value of principal and interest (premium) 6% Contractual Rate and 5% Market Rate Present value of principal to be received at maturity $100,000 PV of 1 due in 10 periods (n) at 2.5% (i) $100,000 0.78120 (Table PV-1) $ 78,120 Present value of interest to be received periodically over the term of the bonds $3,000 PV of 1 due periodically for 10 periods (n) at 2.5% (i) $3,000 8.75206 (Table PV-2) 26,256* Present value of bonds $104,376 *Rounded

Applying Present Value Concepts 13 These bonds will sell at a $4,376 premium for $104,376. If the market interest rate is less than the contract interest rate, the bonds will always sell at a premium. The above discussion relied on present value tables to solve present value problems. As previously discussed, financial calculators may also be used to calculate present values without the use of these tables. In addition, most computer spreadsheets and programs also have built-in formulas to perform the discounting functions. BEFORE YOU GO ON Review It 1. Explain how the present value of a bond is determined. 2. Distinguish between the market rate of interest and the contractual rate of interest. How is each of these used in determining the present value of a bond? 3. If the bonds have interest payable semi-annually, how is the discount rate (i) determined? How is the number of periods (n) determined? 4. If the market rate is greater than the contract rate, will the bonds sell at a premium or a discount? Explain. Do It Forest Lake Enterprises issued $1 million of six-year, 4.5% bonds that pay interest semi-annually. The market rate of interest for the bonds at the issue date is 4%. What cash proceeds did Forest Lake Enterprises receive from the issue of the bonds? Action Plan Note that Forest Lake will be able to sell these bonds at a premium because the bonds pay higher interest (4.5%) than the current market interest rate (4%). Recall that the contractual interest rate is used to determine the interest payment; the market rate is used to determine the present value. Adjust the interest rates and number of periods for the effect of the semi-annual periods. Use Table PV-1 to determine the present value of the principal. Use Table PV-2 to determine the present value of the interest payments. Solution 1. Amount to be received at maturity is the face value of the bonds, $1,000,000 2. Semi-annual interest payment $22,500 ($1,000,000 4.5% 6 /12 months) 3. Number of periods n 12 (6 years 2 payments a year) 4. Discount rate i 2% (4% 2 payments a year) The cash proceeds that Forest Lake will receive from issuing the bonds is the present value of principal to be received at maturity plus the present value of the interest received periodically, calculated as follows: Present value of principal to be received at maturity: $1,000,000 0.78849 (PV of $1 due in 12 periods at 2% from Table PV-1) $788,490 Present value of interest to be received periodically over the term of the bonds: $22,500 10.57534 (PV of $1 due each period for 12 periods at 2% from Table PV-2) 237,945 Present value of bonds $1,026,435 Related exercise material: BEPV-9 and BEPV-10. Calculating the Present Value of Notes Payable Long-term notes payable are normally repayable in a series of periodic payments. These payments may be fixed principal payments plus interest, or blended principal and interest payments. The accounting treatment of notes is similar to that for bonds. The present value of a note is a function of the same three variables: (1) the payment amounts, (2) the length of time until the amounts are paid (time to maturity date), and (3) the market interest rate. Examples of long-term notes payable include unsecured notes, mortgages (which are secured notes on real property, such as a house), and loans (e.g., student or car).

14 Present Value Concepts Note Payable: Fixed Principal Payments Plus Interest Let s assume Heathcote Company obtains a five-year note payable, with an 8% interest rate, to purchase a piece of equipment costing $25,000. If we first assume that repayment is to be in fixed principal payments plus interest, paid annually, then the payment amount would be $5,000 ($25,000 5 years) plus 8% interest on the outstanding balance. Illustration PV-14 details the total cost of this loan over the five-year period. Illustration PV-14 Cost of equipment loan fixed principal payments plus interest Year 1 Year 2 Year 3 Year 4 Year 5 Total Principal $5,000 $5,000 $5,000 $5,000 $5,000 $25,000 Interest 1 2,000 1,600 1,200 800 400 6,000 Total 7,000 6,600 6,200 5,800 5,400 $31,000 Present value factor 2 0.92593 0.85734 0.79383 0.73503 0.68058 Present value $6,482 $5,658 $4,922 $4,263 $3,675 $25,000 1 Interest is calculated on the outstanding principal balance at the beginning of the year. In Year 1, it is $25,000 8% $2,000. In Year 2, it is $20,000 ($25,000 $5,000) 8% $1,600, and so on. 2 These factors are taken from Table PV-1, for i 8% and the appropriate n, or number of periods. Note Payable: Blended Principal Plus Interest Payments In the case of blended principal and interest payments, present value concepts are used to calculate the amount of the annual payment. If we divide the total loan amount of $25,000 by the present value factor for an annuity from Table PV-2 for i 8% and n 5, then we can determine that the annual payment is $6,261 ($25,000 3.99271). Illustration PV-15 details the total cost of this loan over the five-year period. Illustration PV-15 Cost of equipment loan blended principal and interest payments Year 1 Year 2 Year 3 Year 4 Year 5 Total Payment $6,261 $6,261 $6,261 $6,261 $6,261 $31,305 Present value factor 1 0.92593 0.85734 0.79383 0.73503 0.68058 Present value 2 $5,797 $5,368 $4,970 $4,602 $4,261 $25,000 1 These factors are taken from Table PV-1, for i 8% and the appropriate n, or number of periods. 2 The sum of the annual present values is not exactly equal to $25,000 because of rounding. If full digits were used, the total would be $25,000. Comparison of Notes Payable with Fixed Principal versus Blended Payments Both options result in a present value cost of $25,000, but the blended principal and interest payment option results in a higher total cash outflow, $31,305, compared with $31,000 for the fixed principal payments plus interest option. Note the difference also in the annual cash flows required under each option. For example, the fixed principal and interest payment option results in a higher cash outflow in the first two years and lower cash outflow in the last three years. These examples illustrate the effect of the timing and amount of cash flows on the present value. Basically, payments made further away from the present are worth less in present value terms, and payments made closer to the present are worth more.

Notes Payable: Stated Interest Rate below Market Interest Rate In the previous two examples, the stated (contract) interest rate and the market (discount) interest rate were both 8%. As a result of these rates being equal, the present value and the principal amount of the loan were both $25,000. We found the same result in the previous section on bonds payable. Businesses sometimes offer financing with stated interest rates below market interest rates to stimulate sales. Examples of notes with stated interest rates below market are seen in advertisements offering no payment for two years. These zero-interest-bearing notes do not specify an interest rate to be applied, but the face value is greater than the present value. The present value of these notes is determined by discounting the repayment stream at the market interest rate. As an example, assume that a furniture retailer is offering No payment for two years, or $150 off on items with a sticker price of $2,000. The implicit interest cost over the two years is $150, and the present value of the asset is then $1,850 ($2,000 $150). From Table PV-1, the effective interest rate can be determined: PV FV discount factor $1,850 $2,000 0.925 Applying Present Value Concepts 15 Looking at the n 2 row, this would represent an interest rate of approximately 4%. Most notes have an interest cost, whether explicitly stated or not. If a financial calculator is used, the exact interest rate can be determined by inputting $1,850 as the present value, $2,000 as the future value, 2 as the number of periods, and then instructing the calculator to compute the interest rate. The result is an interest rate of 3.975%, which is very close to the 4% estimated by using the present value tables. Also note that the purchaser should record the furniture at a cost of $1,850 even if the purchaser chose the option of no payment for two years. In this case, the purchaser will recognize interest expense of $74 ($1,850 4%) in the first year and $76 [($1,850 $74) 4%] in the second year (numbers have been rounded so total interest expense over two years is equal to $150). Canada Student Loans The Canadian federal government, in an effort to encourage post-secondary education, offers loans to eligible students with no interest accruing while they maintain their student status. When schooling is complete, the entire loan must be repaid within 10 years. Repayment of the loan can be at a fixed rate of prime 5%, or a floating rate of prime 2.5%. Let s assume that you attend school for four years and borrow $2,500 at the end of each year. Upon graduation, you have a total debt of $10,000. If you had not been eligible for the student loan, and had borrowed the money as needed through a conventional lender who agreed to let the interest accrue at 9% over the four years, with no payments required, how much would you owe upon graduation? Compound interest would accrue as shown in Illustration PV-16. Year 1 Year 2 Year 3 Year 4 Balance, beginning of period $ 0 $2,500 $5,225 $ 8.195 Interest (@ 9%) 0 225 470 738 0 2,725 5,695 8,933 Increase in borrowings 2,500 2,500 2,500 2,500 Balance, end of period $2,500 $5,225 $8,195 $11,433 Illustration PV-16 Cost of student loan with a conventional lender

16 Present Value Concepts The loan would total $11,433 at the end of Year 4. Note that this is the future value of the loan at an interest rate of 9%. This amount can also be determined using present value concepts as shown in Illustration PV-17. Illustration PV-17 Cost of student loan with a conventional lender using PV calculations Year 1 Year 2 Year 3 Year 4 Balance, beginning of period (PV) $ 0 $2,500 $5,225 $ 8.195 From Table PV-1; FV PV/0.91743 1 0 2,725 5,695 8,933 Increase in borrowings 2,500 2,500 2,500 2,500 Balance, end of period $2,500 $5,225 $8,195 $11,433 1 n 1, i 9%; calculation done for each year separately, based on the balance at the beginning of each period. A third way of calculating this future value is to first calculate the PV of an annuity of $2,500 over four years at 9%, and to then calculate the FV of this figure, as follows: From Table PV-2, the present value of an annuity of $2,500 per year for 4 years at 9% is: $2,500 3.23972 $8,099.30. From Table PV-1, the future value (4 years at 9%) of $8,099.30 is: $8,099.30 0.70843 $11,432.75. The benefit of having a Canada Student Loan, as opposed a loan with a conventional lender, is shown in Illustration PV-18 by comparing the cost of repaying these two types of loans. Illustration PV-18 Calculation of annuity required to repay two loans Canada Student Loan Conventional Loan Value of loan at the end of 4 years $10,000 $11,433 PV factor from Table PV-2: n 10; i 9% 6.41766 6.41766 Annual annuity required to pay the loan $1,558.20 $1,781.49 Under the Canada Student Loan, there is a nominal savings of $2,232.90 [10 ($1,781.49 $1,558.20)] over the life of the loan. This example uses present value calculations to illustrate the effect of not incurring interest while you are in school. BEFORE YOU GO ON Review It 1. Explain how calculating the present value of a note is similar to, or different from, calculating the present value of a bond. 2. Differentiate between the two kinds of payment options on a note payable. 3. What are the benefits of having a Canada Student Loan instead of a conventional bank loan? Do It You are about to purchase your first car. You decide to finance the car at an annual interest rate of 7% over a period of 48 months. If your monthly payment is $574.71, how much is the car worth today? Action Plan The present value factor of an annuity for n 48 and i 0.58% (7% 12 months) is 41.76019. Solution The day the car is purchased, it is worth the present value of the future cash payments. The present value of $574.71 to be paid monthly for four years, discounted at 7%, is $24,000 ($574.71 41.76019). Related exercise material: BEPV-11, BEPV-12, BEPV-13, BEPV-14, BEPV-15, BEPV-16, and BEPV-17.

Assets Estimating Value in Use Using Future Cash Flows As we have learned in a previous section, a bond can be valued based on future cash flows. So too can an asset. In Chapter 9, you learned that companies are required to regularly determine whether the value of property, plant, and equipment has been impaired. Recall that an asset is impaired if the carrying amount reported on the balance sheet is greater than its recoverable amount. The recoverable amount is either the asset s fair value or its value in use. Determining the value in use requires the application of present value concepts. The computation of value in use is a two-step process: (1) future cash flows are estimated, and (2) the present value of these cash flows is calculated. For example, assume JB Company owns a specialized piece of equipment used in its manufacturing process. JB needs to determine the asset s value in use to test for impairment. As the first step in determining value in use, JB s management estimates that the equipment will last for another five years and that it will generate the following future cash flows at the end of each year: Year 1 Year 2 Year 3 Year 4 Year 5 $9,000 $10,000 $13,000 $10,000 $7,000 Applying Present Value Concepts 17 In the second step of determining value in use, JB calculates the present value of each of these future cash flows. Using a discount rate of 8%, the present value of each future cash flow is shown in Illustration PV-19. Year 1 Year 2 Year 3 Year 4 Year 5 Future cash flows $9,000 $10,000 $13,000 $10,000 $7,000 Present value factor 1 0.92593 0.85734 0.79383 0.73503 0.68058 Present value amount $8,333 $ 8,573 $10,320 $ 7,350 $4,764 Illustration PV-19 Present value of estimated future cash flows of specialized equipment 1 The appropriate interest rate to be used is based on current market rates; however, adjustments for uncertainties related to the specific asset may be made. Further discussion on this topic is covered in more advanced texts. The value in use of JB s specialized equipment is the sum of the present value of each year s cash flow, $39,340 ($8,333 $8,573 $10,320 $7,350 $4,764). If this amount is less than the asset s carrying amount, JB will be required to record an impairment as shown in Chapter 9. The present value method of estimating the value in use of an asset can also be used for intangible assets. For example, assume JB purchases a licence from Redo Industries for the right to manufacture and sell products using Redo s processes and technologies. JB estimates it will earn $6,000 per year from this licence over the next 10 years. What is the value in use to JB of this licence? Since JB expects to earn the same amount each year, Table PV-2 is used to find the present value factor of the annuity after determining the appropriate discount rate. As pointed out in the previous example, JB should choose a rate based on current market rates; however, adjustments for uncertainties related to the specific asset may be required. Assuming JB uses 8% as the discount rate, the present value factor from Table PV-2 for 10 periods is 6.71008. The value in use of the licence is $40,260 ($6,000 6.71008).

18 Present Value Concepts BEFORE YOU GO ON Review It 1. When is it necessary to calculate the value in use of an asset? 2. Explain the two steps in calculating the value in use of an asset. Do It You are attempting to estimate the value in use of your company s production equipment, which you estimate will be used in operations for another eight years. You estimate that the equipment will generate annual cash flows of $16,000, at the end of each year, for the remainder of its productive life, and that 9% is the appropriate discount rate. What is the value in use of this equipment? Action Plan Identify future cash flows. Use Table PV-2 to determine the present value of an annuity factor for n 8 and i 9% and calculate the present value. Solution The value in use is equal to the present value of the estimated annual future cash flows for the remaining life of the asset discounted at an appropriate discount rate. Future annual cash flows: $16,000 Number of periods: 8 Discount rate: 9% Present value annuity factor (n 8, i 9%): 5.53482 Present value: $88,557 $16,000 5.53482 Related exercise material: BEPV-18, BEPV-19, and BEPV-20. Brief Exercises Calculate simple and compound interest. Calculate present value of a single-sum investment. Calculate future value of a single-sum investment and demonstrate the effect of compounding. Calculate number of periods of a single investment sum. Calculate interest rate on single sum. Calculate present value of an annuity investment. Determine number of periods and discount rate. BEPV 1 Determine the amount of interest that will be earned on each of the following investments: (i) (n) Investment Interest Rate Number of Periods Type of Interest (a) $100 5% 1 Simple (b) $500 6% 2 Simple (c) $500 6% 2 Compound BEPV 2 Smolinski Company is considering an investment that will return a lump sum of $500,000 five years from now. What amount should Smolinski Company pay for this investment in order to earn a 4% return? BEPV 3 Mehmet s parents invest $8,000 in a 10-year guaranteed investment certificate (GIC) in his name. The investment pays 4% annually. How much will the GIC yield when it matures? Compare the interest earned in the first and second five-year periods, and provide an explanation for the difference. BEPV 4 Janet Bryden has been offered the opportunity to invest $44,401 now. The investment will earn 7% per year, and at the end of that time will return Janet $100,000. How many years must Janet wait to receive $100,000? BEPV 5 If Kerry Dahl invests $3,152 now, she will receive $10,000 at the end of 15 years. What annual rate of interest will Kerry earn on her investment? Round your answer to the nearest whole number. BEPV 6 Kilarny Company is considering investing in an annuity contract that will return $25,000 at the end of each year for 15 years. What amount should Kilarny Company pay for this investment if it earns a 6% return? BEPV 7 For each of the following cases, indicate in the chart below the appropriate discount rate (i) and the appropriate number of periods (n) to be used in present value calculations. Show calculations. The first one has been completed as an example.

Brief Exercises 19 Annual Number Frequency (n) Number (i) Discount Interest Rate of Years of Payments of periods Rate 1. 8% 3 Quarterly 3 4 12 8% 4 2% 2. 5% 4 Semi-annually 3. 7% 5 Annually 4. 4% 3 Quarterly 5. 6% 6 Semi-annually 6. 6% 15 Monthly BEPV 8 Insert the appropriate discount factor into the table below for each of the situations given. These factors have to be interpolated from Tables PV-1 and PV-2. Interpolate discount factors. (a) n 4, i 4½% (b) n 6, i 6½% PV of 1 PV of an Annuity of 1 (Table PV-1) (Table PV-2) BEPV 9 Cross Country Railroad Co. is about to issue $100,000 of 10-year bonds that pay a 5.5% annual interest rate, with interest payable semi-annually. The market interest rate is 5%. How much can Cross Country expect to receive for the sale of these bonds? BEPV 10 Assume the same information as BEPV-9, except that the market interest rate is 6% instead of 5%. In this case, how much can Cross Country expect to receive from the sale of these bonds? BEPV 11 Caledonian Company receives a six-year, $50,000 note that bears interest at 8% (paid annually) from a customer at a time when the market interest rate is 7%. What is the present value of the note received by Caledonian? BEPV 12 Hung-Chao Yu Company issues a six-year, 8% mortgage note on January 1, 2011, to obtain financing for new equipment. The terms provide for semi-annual instalment payments of $112,825. What were the cash proceeds received from the issue of the note? BEPV 13 You are told that a note has repayment terms of $4,000 per year for five years, with a stated interest rate of 4%. How much of the total payment is for principal, and how much is for interest? BEPV 14 You would like to purchase a car with a list price of $30,000, and the dealer offers financing over a five-year period at 8%. If repayments are to be made annually, what would your annual payments be? BEPV 15 Assume the same information as in BEPV-14, except that you can only afford to make annual payments of $6,000. If you decide to trade in your present car to help reduce the amount of financing required, what trade-in value would you need to negotiate to ensure your annual payment is $6,000? BEPV 16 As CFO of a small manufacturing firm, you have been asked to determine the best financing for the purchase of a new piece of equipment. If the vendor is offering repayment options of $10,000 per year for five years, or no payment for two years followed by one payment of $46,000, which option would you recommend? The current market rate of interest is 8%. Calculate present value of bonds. Calculate present value of bonds. Calculate present value of note. Calculate present value of note. Determine interest and principal payments. Calculate annual payments. Calculate trade-in value of car. Compare financing options. BEPV 17 If the market rate of interest in BEPV-16 is 10%, would you choose the same option? Compare financing options. BEPV 18 Barney Googal owns a garage and is contemplating purchasing a tire retreading machine for $16,100. After estimating costs and revenues, Barney projects a net cash flow from the retreading machine of $2,690 annually for eight years. Barney hopes to earn a return of 11% on such investments. What is the present value of the retreading operation? Should Barney Googal purchase the retreading machine? BEPV 19 Chang Company must perform an impairment test on its equipment. The equipment will produce the following cash flows: Year 1, $35,000; Year 2, $45,000; Year 3, $55,000. Chang requires a minimum rate of return of 10%. What is the value in use for this equipment? Calculate value of machine for purchase decision. Calculate value in use for a machine.

20 Present Value Concepts Calculate value in use of a patent. BEPV 20 ABC Company signs a contract to sell the use of its patented manufacturing technology to DEF Corp for 15 years. The contract for this transaction stipulates that DEF Corp pays ABC $18,000 at the end of each year for the use of this technology. Using a discount rate of 9%, what is the value in use of the patented manufacturing technology? Solutions