Corporate Finance Fundamentals [FN1]


 Barrie Parker
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1 Page 1 of 32 Foundation review Introduction Throughout FN1, you encounter important techniques and concepts that you learned in previous courses in the CGA program of professional studies. The purpose of this foundation review is to give you an opportunity to review and practise some of these techniques and concepts before you apply them in FN1. This foundation review is optional. Each topic has review questions and solutions so that you can assess your strengths and weaknesses. Try the problems and questions. If you are strong in these topics, just skim the material and proceed with the course. Avoid referring to the solutions until you are finished. If you are weak in a particular topic, work on the refresher reading. The first topic in the foundation review will look at the various forms of organizations. How the types of organizations affect their financing needs will be addressed in Module 1. The topics of present value and future value are also fundamental to FN1, and thus are covered in the second topic. Module 2 takes the principles and applies them to financial instruments. It is essential that you understand the calculation and application of present and future values to succeed in this course. Learning objectives Understand and assess the needs of a business based on its legal form. Apply present value and future value calculations to basic cash flow patterns encountered in finance. FR.1 Legal forms of organization Accountants provide services to all types of business entities. It is important, therefore, that you be familiar with the characteristics of the different forms of business organization. The three legal forms of business ownership are summarized as follows: A single or sole proprietorship is a business that is owned by one individual, but is not established as a separate entity under the law. A partnership differs from a single proprietorship only in that it has more than one owner. The owner or owners of proprietorships and partnerships are personally liable for the debts of the business. A corporation is established under the law as a separate entity; hence, its owners (shareholders) are not liable for the debts of the corporation. These concepts were covered in both FA2 and LW1. As it may have been a while since you covered this material, review questions are provided to allow you to assess your understanding of the essentials of the various organizational forms. Refresher reading If you need additional help, use the following refresher reading to review the legal forms an organization can take.
2 Page 2 of 32 Major legal forms of organization The three most widely used legal forms of enterprise in Canada are sole proprietorship, partnership, and corporation. The sole proprietorship and partnership forms of organization are distinguished from the corporate form in a fundamental way: in the sole proprietorship and partnership forms, the business and the owners are not distinguishable in law, while in the corporate form, the owners are separate from the business. Because of this structure, owners of sole proprietorship and partnership firms include business income and expenses on their personal tax returns, since there is no distinction between the business and the owner. In addition, if the sole proprietorship or partnership business incurs debt, all the assets of the owners, whether related to the business or not, are accessible by the lenders to recover any debt payments due. On the other hand, owners of the corporation are separate from the business; that is, the corporation is a separate legal entity. The corporate business entity files its own tax return, totally independent from those of its owners. And, if the corporation incurs debt, only the assets of the corporation are exposed to recoupment by the lenders. The corporate form is used for large business enterprises; there are many sole proprietorships, but corporations are responsible for the greatest dollar volume of business. One reason for this dominance of the corporation is that sole proprietorships and partnerships directly link the enterprise to the owner. For example, when the owner of a sole proprietorship dies, so does the business. Likewise, income from a proprietorship is included with the owner's other sources of income for tax purposes; that is, the proprietorship is not viewed as a separate entity by Canada Revenue Agency. Key differences between forms of organization To further describe the differences outlined above, you can categorize four ways in which the corporation differs from the partnership or sole proprietorship: 1. The corporation treats the taxation of the business separately from that of the owner. 2. The corporation provides its owners with limited liability. 3. The longevity of the corporation is independent of the lifespan of its owners. 4. The corporation is better suited for raising large amounts of funds for investment in capital assets. Example FR.11 highlights these differences. Example FR.11: Comparing investment in a partnership with investment in a corporation Mrs. B. Peal won $5,000 in a lottery. She will use this money to either (1) buy equity shares of BCE Inc., a corporation, or (2) start a babysitting partnership with her friend, Mrs. Smith. Suppose Mrs. Peal enters the partnership and, a year later, she and Mrs. Smith realize that they cannot repay a bank loan payment of $8,000 out of the partnership's cash flow. Because partners in a partnership are not
3 Page 3 of 32 automatically protected by limited liability, the bank can attempt to take possession of both women's personal wealth. Suppose Mrs. Peal now discovers that Mrs. Smith is broke! The bank may go after Mrs. Peal for the entire $8,000. Alternatively, Mrs. Peal could have bought the BCE shares. If BCE subsequently defaulted on its bank loans, the limited liability feature of the corporate form would apply: The banks that lent funds to BCE cannot take Mrs. Peal's personal wealth. At worst, Mrs. Peal's shares in BCE will become worthless. Limited liability in the corporate form means that Mrs. Peal does not have to investigate the personal wealth of all BCE's shareholders, just in case BCE defaults. No matter what happens to BCE, Mrs. Peal's loss is limited to her original investment, and any loss (for example, through declining share prices) is shared equally among all shareholders. The facts of limited liability and liquidity of ownership shares (that is, ease of resale to other investors) enable the corporate form to accommodate many owners. This is a necessary requirement to raise the large amounts of capital required for an efficient organization of real assets. FR.2 Future and present value Note: As a refresher, three methods for solving time value of money problems are illustrated in this foundation review: the table method, the financial calculator method, and the spreadsheet method. FN1 emphasizes the use of financial calculators. Therefore, for the purposes of online quizzes, assignments, and the final examination, you should use a financial calculator wherever applicable. In FN1, you are required to manipulate the present values of a lump sum, an annuity, and an annuity due. Present values are a critical valuation method for longterm liabilities and bonds. Present values are also used in evaluating capital budgeting projects. Be sure that you can do the following: Calculate the present value of an annuity and a lump sum. Calculate a debt payment that includes both principal and interest (a blended payment). Determine an interest rate implicit in a repayment pattern. Divide blended payments into principal and interest components. You should be able to calculate present values using lookup tables, a financial calculator, and a spreadsheet program. During the examination, you will only be allowed to use lookup tables and a financial calculator. In the answers to the review questions, solutions are presented based on lookup tables because this method universally illustrates the logic of the calculations. A set of lookup tables is provided in Appendix FR1 (To
4 Page 4 of 32 view the content from this link go to end of document.). Refresher reading For additional help, review the following refresher reading. Future value and present value of single payments The following two formulas calculate the present and future value factors. The following examples will outline how these factors are applied to situations in order to determine the present and future value amounts: Future value (FV n ) = Present value (PV o ) (1 + k) n Present value (PV o ) = Future value (FV n ) 1 (1 + k) n where k = stated interest rate per period n = number of periods The future value equation above is referred to as the basic compounding equation, and the last term [(1 + k) n ] is referred to as the future value interest factor (FVIF) or the compound value interest factor (CVIF). This compound value interest factor gives the future value of $1 at various interest rates and periods. For example, at an interest rate of 5%, for 6 periods, the CVIF is 1.340: (1.05) 6 = 1.340). Similarly, the second equation above can be used to calculate the present value of a future amount. This equation states that you can divide the future value by the CVIF to get the present value. This is the basic discounting equation, and the last term [1 (1 + k) n ] is referred to as the discount factor or present value interest factor (PVIF). For example, at an interest rate of 5%, for six periods, the PVIF is 0.746: 1 (1.05) 6 = If the future amount is $500, the present value is $ = $373. Lookup tables Note that in this foundation review, the calculations use the figures provided by the tables rounded to three decimal places. Table 3 can be used to calculate the future value of a present amount. This table shows the compound amount (future value interest factor) of $1 at various interest rates and periods. The table contains three types of information: the interest rate (across the top) the number of interest periods (down the side) the future value interest factor (at the intersection of a column and row) For example, at an interest rate of 5%, for 6 periods, the future value interest factor is To find the future value of a specific amount for a given interest rate and number of periods, look up the future value interest factor from Table 3, then multiply the specific amount by the factor from the table. For example, the future value of $500 at an interest rate of 5% for six periods is $ = $670. Table 1 can be used to calculate the present value of a future amount. For example, at an interest rate of 5%,
5 Page 5 of 32 for six periods, the present value interest factor is If the future amount is $500, the present value is $ = $373. Thus, the amount of $500 invested at 5% per year for six years has a future value of $ A similar investment that is valued six years from now at $500 has a present value of $373. Financial calculators Most financial calculators provide special keys to calculate future values and present values. It is important that you become very familiar with the way to use your calculator for these problems. Use your calculator now to confirm the preceding value computations. Note: In FN1, the financial calculator results are those obtained with TI BA II Plus. In general, for many calculations, the calculator is set at the "end" mode. The calculator is set at the "begin" mode when the present value of an annuity due is involved. Formula used: Future value of a lump sum FV n = PV 0 (1 + k) n PMT = 0; PV = 500; I/Y = 5; N = 6; CPT FV. FV = $ Formula used: present value of a lump sum PV o = FV n [1 (1 + k) n ] PMT = 0; FV = 500; I/Y = 5; N = 6; CPT PV. PV = $ Spreadsheet formulas Excel provides builtin functions to calculate future value and present value. FV(interest rate, term,, present value) PV(interest rate, term,, future value) Notice that you need to specify the present value or future value in the Excel functions as negative values. For details on Excel, see CT2. Examples FR.21 to FR.24 review three different methods for calculating future values and present values. Example FR.21: Future value of a present amount
6 Page 6 of 32 $1,000 is deposited in a savings account on January 1, 20X1. Interest of 12% is compounded annually. What is the balance in the account at the end of the fifth year (December 31, 20X5)? Principal (present value at January 1, 20X1) $1,000 Annual interest rate 12% Number of years 5 You need to calculate the future value on December 31, 20X5. You can calculate this value by looking up a table, using a calculator, or using a spreadsheet program. Lookup table method From Table 3, under the 12% column, looking across period 5, the future value interest factor is Thus, the future value of $1,000 is $1, = $1,762. Financial calculator method Enter the following on the calculator: Number of years: 5 Annual interest rate: 12% Present value: 1000 PMT = 0; PV = 1000; I/Y = 12; N = 5; CPT FV. FV = $1, Spreadsheet method Start Excel. Open file FN1FR2E from the data folder and click the FRE2.1 sheet tab. If you use the spreadsheet formula provided above, the spreadsheet should provide a final future value of $1, Given that cell B3 contains the present value ($1,000), cell B4 contains the annual interest rate, and cell B5 contains the number of years, then the formula is =FV(B4,B5,, B3) Your spreadsheet should look as follows: Example FR.22: Calculating savings account balances The savings account has
7 Page 7 of 32 Present value: $15,000 Annual interest rate: 8% To calculate the account balance at the end of 10, 15, and 25 years, you need to calculate the future value of the present amount of $15,000, with an annual interest rate of 8%. Lookup table method Look up the future value interest factor in Table 3 for 10, 15, and 25 years under the 8% column, and multiply the factors by $15,000 to obtain the future values: Number of years Annual interest rate Future value interest factor Future value 10 8% $ 32, % $ 47, % $ 102,720 Financial calculator method Number of years (a) PMT = 0; PV = 15000; I/Y = 8; N = 10; CPT FV (b) PMT = 0; PV = 15000; I/Y = 8; N = 15; CPT FV (c) PMT = 0; PV = 15000; I/Y = 8; N = 25; CPT FV Notice the differences between the calculator and lookup table results; the table values use a lower standard of precision. Spreadsheet method Future value 10 $ 32, (a) 15 $ 47, (b) 25 $ 102, (c) Using the worksheet used in Example FR21, enter the appropriate values for present value, annual interest rate, and number of years. The worksheet will calculate the future values. You should obtain the following results: Number of years Future value 10 $ 32, $ 47, $ 102, Notice that the future values calculated using Excel are slightly different from those obtained using the financial calculator owing to differences in precision. The spreadsheet programs carry more decimal places and yield the most accurate result of the three methods.
8 Page 8 of 32 Example FR.23: Present value of a future amount What is the present value of a single payment of $10,000, which is to be received three years from now using a discount rate of 10%? Future value: $10,000 Annual interest rate: 10% Number of years: 3 You need to calculate the present value. Lookup table method From Table 1, under the 10% column, looking across period 3, the present value interest factor is Thus, the present value of $10,000 three years from now is $10, = $7, Financial calculator method Number of years: 3 Annual interest rate: 10% Future value: PMT = 0; FV = 10000; I/Y = 10; N = 3; CPT PV. PV = $7, Notice that the calculator carries more decimal places than the table and thus yields a more accurate result. Spreadsheet method Using the same worksheet as previously, set up this model to calculate the present value. Here is how your completed, formatted worksheet should look: The formula for the present value amount in cell B11 should be =PV(B9,B10,, B8) Save this worksheet. Example FR.24: Calculating present values of a bond Assume that you have a legal contract, such as a bond, that specifies that you will receive $200,000 cash in
9 Page 9 of 32 the future. Assuming a 9% interest rate, how much must you pay now if the amount of $200,000 is to be received in 10, 15, or 25 years? Future value: $200,000 Annual interest rate: 9% You want to calculate the present value of bonds that will be paid in 10, 15, or 25 years. Lookup table method The present values calculated using the present value interest factors from Table 1 are Number of years Annual interest rate Present value interest factor Present value 10 9% $ 84, % $ 55, % $ 23, Financial calculator method Use the financial functions of your calculator to compute the present values. You should obtain the following results: (a) PMT = 0; FV = ; I/Y = 9; N = 10; CPT PV (b) PMT = 0; FV = ; I/Y = 9; N = 15; CPT PV (c) PMT = 0; FV = ; I/Y = 9; N = 25; CPT PV Notice again the difference in precision between the lookup table method and the calculator method. Spreadsheet method Number of years Present value 10 $ 84, (a) 15 $ 54, (b) 25 $ 23, (c) Using your worksheet from the previous examples, enter the appropriate values for future value, annual interest rate, and number of years. The worksheet will calculate the present values. You should obtain the following results: Number of years Present value 10 $ 84, $ 54, $ 23,193.57
10 Page 10 of 32 Determining an unknown interest rate You may encounter a situation where the present and future values and the number of periods are known, but the interest rate is not known. To analyze and compare the yield of one investment to another, you must be able to determine what interest rate each investment will earn. Example FR.25 demonstrates three methods of determining the interest rate. Example FR.25: Three methods of determining interest rate A $100,000 investment will yield $146,933 in five periods. What is the implicit interest rate earned on the investment? You need to find the annual interest rate that will result in a future value of $146,933 in five years, with an initial investment (present value) of $100,000. Lookup table method The future value interest factor is $146,933 $100,000 = rounded to In Table 3, look across period 5 for a future value interest factor of This value is located in the 8% column. Thus, the annual interest rate is 8%. Financial calculator method Number of years: 5 Future value: Present value: PMT = 0; FV = ; PV = ; N = 5; CPT I/Y. I/Y = 8 percent Spreadsheet method Using the same worksheet, enter the following formula for the annual interest rate in cell B16: =RATE(B15,, B14,B13)
11 Page 11 of 32 Ordinary annuities and annuities due Many financial transactions require a series of identical payments, received or paid at equally spaced intervals. When compound interest is also involved, this type of transaction is known as an annuity. An annuity is a series of identical payments, at identical periods or intervals, over a specified term, at a constant interest rate. The most common example of an annuity is a series of mortgage payments. Automobile leasing, which requires regular fixed monthly payments, is another example of an annuity. There are several types of annuities. An annuity paid or received at the end of the interest compounding period is an ordinary annuity. An annuity paid or received at the beginning of the interest compounding period is an annuity due. Canadian mortgage payments, where monthly payments are due at the end of each month, are examples of an ordinary annuity. A rental agreement where the monthly rents are paid at the beginning of each month is an example of an annuity due. Another type of annuity is a deferred annuity, where payments do not start immediately; an example is the purchase of equipment for which payments do not start until three months after delivery. The following notes are designed to assist you in performing annuity calculations by using lookup tables, financial calculator, and spreadsheet methods. The future value of an annuity is the sum of the future values of each of the payments. Similarly, it is possible to calculate the present value of an ordinary annuity or an annuity due by calculating the present value of each of the payments, then adding up these present value amounts. Looking up a table, using a calculator, or using a spreadsheet program such as Excel greatly facilitates these tasks. Calculations for annuities involve the following four items: payment amount per period interest rate per period number of interest periods future value or present value of the annuity You can calculate any one of the four items if you know the remaining three. For example, if you know the payment amount per period, the interest rate per period, and the number of interest periods, you can calculate the future or present value of the annuity. Remember the importance of stating interest rates and periods in the same units of time. People often make errors in computation, perhaps because of the word "annuity." While the word originally signified annual payments, usage changed and the payment periods and terms of annuities are now frequently stated in other time periods. Notice also the use of the word "payment." In annuity calculations, this word may mean either an amount paid or received, depending on the context of the case. Lookup tables Table 4 shows the future value of an ordinary annuity of $1 at various interest rates and for various time periods. Similarly, Table 2 shows the present value of an ordinary annuity of $1. The same tables are used for calculating an annuity due of $1. The methodology of changing the values in Table 2 and Table 4 to that of an annuity due will be covered in the section entitled Future value of annuity due and Present value of annuity due.
12 Page 12 of 32 The values in Tables 4 are determined using the following equation, which can be used to estimate the future value of an ordinary annuity of $1 at various interest rates and for various time periods. This formula is usually referred to as the compound value annuity formula (CVAF) to distinguish it from the single sum CVIF. below:, which is then multiplied by the payment amount to determine the FV, as shown Similarly, the values in Table 2 are determined using the formula for determining the PV of ordinary annuities, which is given below: This formula is usually referred to as the present value annuity formula (PVAF) to distinguish it from the PVIF used for valuing single sum problems. Spreadsheet functions Most spreadsheet programs provide a number of builtin functions that can be used to calculate annuities. Excel provides the =FV, =PV, =PMT and =NPER functions (the Excel functions are explained in detail in CT2). These functions are summarized in Exhibit FR.21. Exhibit FR.21: Summary of financial functions in Excel Function =FV(rate, nper, pmt, pv, type) =PV(rate, nper, pmt, fv, type) =PMT(rate, nper, pv, fv, type) =NPER(rate, pmt, pv, fv, type) Purpose Calculates the future value of an annuity or a present amount Calculates the present value of an annuity or a future amount Calculates the payment per period for an annuity Calculates the number of interest periods for an annuity Examples FR.26 and Fr.27 detail the three methods of calculating for annuities.
13 Page 13 of 32 Example FR.26: Future value of an ordinary annuity It is January 1, 20X1. You have agreed to invest $5,000 per year, over six years, into an investment account. The payments are to be made at the end of each year on December 31. The interest rate is 8%. What is the investment account balance on December 31, 20X6 after the last payment? Annual payment amount: $ 5,000 Annual interest rate: 8% Number of payments: 6 You need to find the future value of the ordinary annuity. Lookup table method From Table 4, looking across period 6 under the 8% column, the future value factor of an ordinary annuity is Thus, the future value is $5, = $36, Financial calculator method Number of years: 6 Payment amount: 5000 Interest rate: 8% Make sure that your calculator is set at the "end" mode. PMT = 5000; N = 6; PV = 0; I/Y = 8; CPT FV. FV = 36, (FV = $36,679.64) The financial calculator displays the future value as a negative number. Ignore the negative sign. If you obtained an incorrect answer, you probably did not clear the mode registers. Check your calculator manual on how to do this, and repeat the calculation. Spreadsheet method Continue with the previous worksheet. Add this model to the worksheet to calculate the future value of an ordinary annuity: The formula for the future value in cell B21 should be =FV(B19,B20, B18) Save the worksheet.
14 Page 14 of 32 Example FR.27: Present value of an ordinary annuity Suppose the situation is similar to that in Example FR.26, but you wish to determine the present value of the investment. Suppose also that the payments are semiannual and the payment amount is $2,500. In this case, the annuity is an ordinary annuity. You need to find out the present value of the ordinary annuity with a periodic interest rate of 4% (8% 2) and a total number of 12 payments (6 years 2 semiannual payments per year). Lookup table method From Table 2, looking across period 12 under the 4% column, the present value factor of an ordinary annuity is Thus, the present value is $2, = $23, Financial calculator method Number of periods: 12 Payment amount: 2500 Interest rate: 4% Set your calculator at the "end" mode. PMT = 2500; N = 12; FV = 0; I/Y = 4; CPT PV. PV = $23, (Ignore the negative sign.) Spreadsheet method Continue with the Excel worksheet. Add the following section to the worksheet to calculate the present value of an ordinary annuity: The formula for the present value in cell B26 should be =PV(B24,B25, B23) Compare your result with that shown. If you have obtained different results, click on the FRE2.1S tab and locate the area that has caused problems. Find and compare the information in the cell that is different from what you expected in the solution sheet to your sheet to determine where the error is. Using time lines and tables to calculate annuity due The difference between an ordinary annuity and an annuity due is the timing of the payment. For an ordinary annuity, the payment comes at the end of each interest period, whereas for an annuity due, the payment
15 Page 15 of 32 comes at the beginning of each interest period. You can see this difference by comparing the time line of an ordinary annuity with three annual payments to the time line of an annuity due with three annual payments, as shown in Exhibit FR.22. Exhibit FR.22: Comparison of an ordinary annuity and an annuity due In the time lines, you can see that the cash flow for an ordinary annuity is made up of three payments starting one period from the initial loan or investment date. In the case of an annuity due, the payments start one period ahead of the ordinary annuity, beginning with the first payment at the initial loan or investment date. In fact, the future value of an annuity due is equal to the future value of an ordinary annuity, compounded for one more interest period. Similarly, the present value of an annuity due is equal to the present value of an ordinary annuity, discounted for one less interest period. This understanding makes it possible for you to use an ordinary annuity table to perform calculations for annuity due. Note that it is important for you to learn this technique because on the CGA examinations you will not have annuity due tables available. Future value of annuity due Since we have shown that the future value of an annuity due equals the future value of an ordinary annuity, compounded one additional period at (1 + k), we can estimate the future value of an annuity due as the FV of an ordinary annuity times (1 + k), or When using the tables, if the future value of an annuity due table is not available, but you have the future value of an ordinary annuity table, you can still calculate the future value of an annuity due. Suppose the interest rate is 8%, with three payments. To find the future value of an annuity due of three periods and 8% from the ordinary annuity table, look up the row for period 4 under 8% in Table 4, which yields This value includes four ordinary annuity payments, with the last payment made at the end of the fourth period. If you subtract the last payment from this amount, the result is exactly the same as the future value of an
16 Page 16 of 32 annuity due with three payments at 8%. Thus, using the ordinary annuity table, you determine that the future value of the annuity due should be ( ) or Remember that the general method is to look up the future value using an ordinary annuity table, add one more interest period, then subtract 1 from the result to obtain the future value of the annuity due. Present value of annuity due Recall that a three year annuity due is equivalent to a three year ordinary annuity that receives one less period of discounting at 1 (1 + k). Therefore, the PV of an annuity due is simply the PV of an ordinary annuity times (1 + k), or A similar technique can be employed to determine the present value of an annuity due from the present value table of an ordinary annuity. The difference is that you are discounting rather than compounding. Using the same example of three payments and 8%, look up the present value of an ordinary annuity table (Table 2). Look across period 2 in the 8% column, which yields This amount did not include the first payment; thus, you need to add back the first payment. You determine that the present value of the annuity due should be ( ) = Again, remember that the general method is to look up the present value using an ordinary annuity table, subtract one interest period, then add 1 to the result to obtain the present value of the annuity due. Examples FR.28 through FR.210 follow, illustrating three typical annuity problems. Work through each example using tables, calculator, and spreadsheet. Before completing this foundation review, you should also be competent with the use of time lines. Example FR.28: Future value of an annuity due For the same investment covered in Example FR26, suppose that instead of making the annual payment on December 31 of each year, the investments are to be made at the beginning of each year, on January 1, with the first investment to be made on January 1, 20X1. In this case, the annuity is an annuity due and you wish to calculate its future value at maturity. Lookup table method From Table 4, looking across year 7 under the 8% column, the future value factor of an annuity due is Then subtract 1 from this value to obtain Thus, the future value is $5, = $39,615.00
17 Page 17 of 32 Financial calculator method Number of years: 6 Payment amount: 5000 Interest rate: 8% Set your calculator at the "begin" mode: 2 nd BEG 2 nd SET PMT = 5000; N = 6; PV = 0; I/Y = 8; CPT FV. FV = $39, (Ignore the negative sign.) Spreadsheet method Continue with the worksheet. Add the following model to the worksheet to calculate the future value of an annuity due. Pay particular attention to the formula in this model: The formula in cell B31 should be =FV(B29,B30, B28,,1) Example FR.29: Present value of an annuity due Suppose that for the same investment as in Example FR28, you wish to calculate the present value. Once again, observe that the investment is an annuity due. Lookup table method Using the ordinary annuity table (Table 2), look across year 5 under the 8% column to obtain Then add 1 to this value to obtain Thus, the present value is $5, = $24, Financial calculator method Number of years: 6 Payment amount: 5000 Interest rate: 8% Set your calculator at the "begin" mode: 2 nd BEG 2 nd SET PMT = 5000; N = 6; FV = 0; I/Y = 8; CPT PV. PV = $24, (Ignore the negative sign.) Spreadsheet method Continue with the worksheet. Add the following model to the worksheet to calculate the present value of the annuity due:
18 Page 18 of 32 The formula in cell B36 should be =PV(B34,B35, B33,,1) Example FR.210: Annuity payment per period On May 1, 20X2, Job Company obtains a $100,000 loan from the bank and promises to repay the loan in three equal annual payments. The payments are to be made each April 30, with the first payment due on April 30, 20X3. For this type of loan, the bank charges 12% interest per annum. Is this an ordinary annuity or an annuity due? Compute the annual payment amounts, using lookup tables, calculator, and Excel. Lookup table method This is an ordinary annuity. From Table 2, looking across period 3 under the 12% column, you obtain This is the present value of an ordinary annuity of $1 over three years at 12%. Thus, the annuity payment when the present value (principal) is $100,000 is 100,000 divided by 2.402, yielding $41, Remember that you must divide the principal (present value) by the interest factor to obtain the payment. Financial calculator method Number of years: 3 Annual interest rate: 12% Present value: If you are using a TI BAIIPlus, then the data entry will be as follows: PV = ; I/Y = 12; N = 3; FV = 0; CPT PMT. PMT = $41, Spreadsheet method Continue with the worksheet. Add the following model to the worksheet to calculate the annuity payment: The formula in cell B41 should be =PMT(B39,B40, B38)
19 Page 19 of 32 Foundation review Review questions Topic FR.1 1. What is a sole proprietorship? 2. What are some of the consequences that flow from a sole proprietorship s lack of a separate legal identity from its owner? 3. Do partnerships have a separate legal identity? 4. Describe the implications of separate legal existence for a corporation and its shareholders. 5. Outline the advantages associated with the corporate method of carrying on business. Solutions Topic FR.2 1. Compute the following amounts. Each case is independent. Round to the nearest dollar. a. On January 1, 20X1, Dardon Corporation signs a contract agreeing to pay $40,000 on December 31, 20X3. Assuming the following factors, what is the present value of the payment? 1. Annual compounding, 8% annual interest 2. Semiannual compounding, 8% annual interest 3. Quarterly compounding, 8% annual interest b. On January 1, 20X2, Dardon Corporation agrees to pay Servicon Corporation $4,000 per year for five years in exchange for the right to use a patented process. Assuming the following factors, what is the present value of the payment stream? 1. Payments in advance each January 1, 12% annual interest, annual compounding 2. Payments each January 1, 12% annual interest, semiannual compounding 3. Payments each December 31, 12% annual interest, annual compounding 4. Payments each December 31, 12% annual interest, semiannual compounding c. On January 1, 20X3, Dardon Corporation agrees to pay Canadian Finance Co. as follows: December 31, 20X3 $ 6,000 December 31, 20X4 $ 6,000 December 31, 20X5 $ 6,000 December 31, 20X6 $ 6,000
20 Page 20 of 32 December 31, 20X7 $ 106,000 Canadian Finance Co. advances the present value of this payment stream to Dardon Corporation on January 1, 20X3; the present value of the payment stream is the principal amount of the loan, while the rest is interest. Complete the following table: d. On January 1, 20X1, Dardon Corporation agrees to lease a machine, with the following terms required by the lease contract: What is the present value of the payment stream, assuming 1. 6% annual interest, annual compounding 2. 16% annual interest, annual compounding 2. Compute the following amounts. Each case is independent. Round to the nearest dollar or percentage. a. On January 1, 20X0, Marcon Corporation borrowed $120,000 from The Canadian Bank. Repayment is to be in six equal annual instalments, including both principal and interest. Compounding is annual. Calculate the annual payment for 1. December 31 payment, 10% annual interest 2. December 31 payment, 6% annual interest 3. January 1 payment, 10% annual interest 4. January 1 payment, 6% annual interest b. On January 1, 20X2, Marcon Corporation borrowed $40,000 from The Canadian Bank. Repayment is to be made in equal annual instalments, including both principal and interest. Compounding is annual. Calculate the implicit interest rate associated with 1. December 31 payment of $10,856, 6 payments 2. January 1 payment of $5,323, 10 payments Calculate the number of payments needed for Principal 1. 6% annual interest, annual compounding $ $ 2. 8% annual interest, annual compounding $ $ 3. 4% annual interest, annual compounding $ $ 4. 6% annual interest, semiannual compounding $ $ 5. 8% annual interest, semiannual compounding $ $ Interest December 31, 20X120X5, per year $ 40,000 December 31, 20X620Y0, per year $ 20,000 December 31, 20Y1 $ 10,000 December 31, 20Y2 $ 5,000
21 Page 21 of December 31 payment of $4,074, 8% 4. January 1 payment of $4,936, 10% 3. It is January 1, 20X7, and Terry Corporation is about to borrow $100,000 from The Canadian Bank. The loan will be repaid in five equal instalments, including both principal and compound interest at 10%; interest is compounded annually. Required a. Compute the annual loan payment that would be made if (1) the first payment is made on December 31, 20X7, or (2) the first payment is made on January 1, 20X7. b. Prepare a debt amortization schedule for each alternative as follows: Instalment payment Date Beginning principal Interest Principal Ending principal Solutions FR.3 Foundation review  Review questions  Content Links Solution FR.1 1. A sole proprietorship is perhaps the most common and simplest form of business organization. The business has no separate legal identity from the owner of the business, since the business has not been incorporated. A sole proprietorship is a person who is carrying on business for herself or himself, although this person may have employees working for him or her. 2. First, the lack of separate legal identity of a sole proprietorship means that the liability of sole proprietors is unlimited. They are personally responsible for all the debts of the business. If the business does not have sufficient assets to meet the sums owed to creditors, the personal assets of the sole proprietor (such as savings accounts, RRSPs, automobile, house, boat and cottage) are available to satisfy the debts of the sole proprietorship. Second, the owner owns all the assets and is entitled to all the profits. Consequently, the sole proprietorship does not file a separate income tax return since the profits of the business of the sole proprietorship are taxable in the hands of the proprietor, and the expenses and losses deductible in his or her hands. Third, a person engaged in business as a sole proprietorship may be able to claim certain tax advantages that would not be available to employees. However, the profits of the sole proprietorship are taxed in the hands of the owner at progressive personal tax rates as opposed to corporate tax rates.
22 Page 22 of 32 The sole proprietorship may end up paying more taxes than would a corporation with similar income. Fourth, an owner is free of outside interference. The owner does not have to report to shareholders or have audited statements prepared (unless required by a lender). The proprietorship will still have to keep adequate books and records, and obtain licences as necessary. 3. Partnerships do not have a legal identity that is separate from the partners; the partners are the partnership and are therefore personally responsible for the debts and the actions of the partnership. 4. The shareholders of a corporation enjoy the benefit of limited liability. They are only liable for the debts of the corporation to the extent of their capital contribution. The corporation is solely responsible and liable for the payment of its debts. Its debts are not the debts of its owners, the shareholders. The liability of a corporation for its debts is limited to its assets; a corporation's creditors have no claim on the personal assets of its shareholders for the payment of the corporation's debts. There is no need for the approval of other shareholders before purchasing a corporation's shares. Lacking any rules to the contrary, such as those found in a shareholder agreement, no restrictions exist on the sale and purchase or transfer of ownership of a corporation's shares. This absence of restrictions also makes it easier to raise capital to fund the corporation's operations, which is one of the reasons why people choose to incorporate. A corporation has a continuous existence. Shareholders do not have a duty of good faith towards the corporation, nor must they act in the corporation's best interests. The shareholders, the owners, are not necessarily the managers. The corporate vehicle allows for the separation of ownership and management. It is the officers of the corporation and not its shareholders who manage the corporation and who may bind the corporation by forming contracts in the name of the corporation. 5. Major advantages of carrying on business through a corporation are the limited liability of the members, the separation of ownership and management, the ability to raise capital by share issues, a separate existence for tax purposes, the ease of transfer of ownership, particularly in publicly traded companies, and the fact that a corporation has a continuous existence. Solution FR.2 1. Note: these solutions are based first on the lookup tables and then using financial calculators. Solutions obtained with a financial calculator will be slightly more accurate because rounding errors are eliminated. PV$1 means the present value of $1 (lump sum), PVOA stands for the present value of an ordinary annuity, PVAD is the present value of an annuity due, and "n" represents the number of compounding periods. a. 1. $31,760 [$40,000 (PV$1, 3 = n, 8%)]; ($40, ) 2. $31,600 [$40,000 (PV$1, 6 = n, 4%)]; ($40, ) 3. $31,520 [$40,000 (PV$1, 12 = n, 2%)]; ($40, ) b. 1. $16,148 [$4,000 (PVAD, 5 = n, 12%)]; ($4, ) 2. $16,056 * 3. $14,420 [$4,000 (PVOA, 5 = n, 12%)]; ($4, ) 4. $14,288 *
23 Page 23 of 32 * These are annual payments but semiannual compounding; annuity tables can only be used if the payment and compounding points are identical. Therefore, the lump sum tables must be used. Payment PV$1 factor Total PV$1 factor Total 1. $4,000 0 = n, 6% 1.0 $ 4,000 2 = n, 6% ,000 2 = n, 6% ,560 4 = n, 6% ,000 4 = n, 6% ,168 6 = n, 6% ,000 6 = n, 6% ,820 8 = n, 6% ,000 8 = n, 6% , = n, 6% $16,056 Financial calculator method PV o = FV n [1 (1 + k) n ] Solving for PV 0 (Ignore the negative signs.) a. 1. $31,754 b. PMT = 0; N = 3; I/Y = 8; FV = 40000; CPT PV. PV = 31, $31,613 PMT = 0; N = 3 x 2 = 6; I/Y = 8/2 = 4; FV = 40000; CPT PV. PV = 31, $31,540 PMT = 0; N = 3 x 4 = 12; I/Y = 8/4 = 2; FV = 40000; CPT PV. PV = 31, $16,149 Formula used: Present value of an annuity due
24 Page 24 of 32 Set your calculator at "begin" mode. PMT = 4000; N = 5; I/Y = 12; FV = 0; CPT PV. PV = 16, $16,058 This looks like a problem with the present value of an annuity due. The formula of the present value of an annuity due assumes that the point of payment and the point of compounding are identical. Since the payments are made annually but the compounding is semiannual, the present value of a lump sum is applied to each payment along with the appropriate compounding periods. Formula used: Present value of a lump sum PV o = FV n [1 (1 + k) n ] 1 st payment (i): 0 compounding period 2 nd payment (ii): 2 compounding periods 3 rd payment (iii): 4 compounding periods 4 th payment (iv): 6 compounding periods 5 th payment (v): 8 compounding periods (i) 4, (ii) PMT = 0; FV = 4000; I/Y = 12/2 = 6; N = 2; CPT PV. PV = 3, (iii) (iv) (v) PMT = 0; FV = 4000; I/Y = 12/2 = 6; N = 4; CPT PV. PV = 3, PMT = 0; FV = 4000; I/Y = 12/2 = 6; N = 6; CPT PV. PV = 2, PMT = 0; FV = 4000; I/Y = 12/2 = 6; N = 8; CPT PV. PV = 2, , $14,419 Formula used: Present value of an ordinary annuity PMT = 4000; I/Y = 12; N = 5; FV = 0. CPT PV. PV = 14, $14,291 This looks like a problem with the present value of an ordinary annuity. The formula of the
25 Page 25 of 32 present value of an ordinary annuity assumes that the point of payment and the point of compounding are identical. Since the payments are made annually but the compounding is semiannual, the present value of a lump sum is applied to each payment along with the appropriate compounding periods. Formula used: Present value of a lump sum PV o = FV n [1 (1 + k) n ] 1 st payment (i): 2 compounding periods 2 nd payment (ii): 4 compounding periods 3 rd payment (iii): 6 compounding periods 4 th payment (iv): 8 compounding periods 5 th payment (v): 10 compounding periods (i) (ii) (iii) (iv) (v) PMT = 0; FV = 4000; I/Y = 12/2 = 6; N = 2; CPT PV. PV = 3, PMT = 0; FV = 4000; I/Y = 12/2 = 6; N = 4; CPT PV. PV = 3, PMT = 0; FV = 4000; I/Y = 12/2 = 6; N = 6; CPT PV. PV = 2, PMT = 0; FV = 4000; I/Y = 12/2 = 6; N = 8; CPT PV. PV = 2, PMT = 0; FV = 4000; I/Y = 12/2 = 6; N = 10; CPT PV. PV = 2, c. (1) (2) (3) Principal Interest Total 1. $ 99,972 1 $ 30,028 $ 130, , , , , , , , , , , , ,000 14, [$6,000 (PVOA, 4 = n, 6%)] + [$106,000 (PV$1, 5 = n, 6%)]; ($6, ) + ($106, ) 2 [$6,000 (PVOA, 4 = n, 8%)] + [$106,000 (PV$1, 5 = n, 8%)]; ($6, ) + ($106, ) 3 [$6,000 (PVOA, 4 = n, 4%)] + [$106,000 (PV$1, 5 = n, 4%)] ($6, ) + ($106, ) 4 There are annual payments but semiannual compounding; annuity tables can only be used if the payment and compounding points are identical. Therefore, the lump sum tables must be used.
26 Page 26 of 32 Payment PV$1 factor Total PV$1 factor Total 1. $ 6,000 2 = n, 3% $ 5,658 2 = n, 4% $ 5, ,000 4 = n, 3% ,328 4 = n, 4% , ,000 6 = n, 3% ,022 6 = n, 4% , ,000 8 = n, 3% ,734 8 = n, 4% , , = n, 3% , = n, 4% ,656 5 ($6,000 4) + $106,000 = $130,000 Financial calculator method Note: Ignore the negative signs in the following results. $99,606 $91,462 (1) (2) (3) Principal Interest Total 1. $ 100,000 1 $ 30,000 $ 130, , , , , , , , , , , , ,000 Formulas used: Present value of an ordinary annuity + Present value of a lump sum + FV n [1 (1 + k) n ] 1 PMT = 6000; N= 4; I/Y= 6; FV= 0; CPT PV. PV= 20, PMT = 0; FV = ; N = 5; I/Y = 6. CPT PV. PV = 79, , PMT = 6000; N= 4; I/Y= 8; FV= 0; CPT PV. PV= 19, PMT = 0; FV = ; N = 5; I/Y = 8. CPT PV. PV = 72, , PMT = 6000; N= 4; I/Y= 4; FV= 0; CPT PV. PV= 21, PMT = 0; FV = ; N = 5; I/Y = 4. CPT PV. PV = 87, , ,6 This looks like a problem with the present value of an ordinary annuity. The formula of the present value of an ordinary annuity assumes that the point of payment and the point of compounding are identical. Since the payments are made annually but the compounding is semiannual, the present value of a lump sum is applied to each payment along with the appropriate compounding periods.
27 Page 27 of 32 Formula used: Present value of a lump sum PV o = FV n [1 (1 + k) n ], along with different compounding periods 1 st payment (i): 2 compounding periods 2 nd payment (ii): 4 compounding periods 3 rd payment (iii): 6 compounding periods 4 th payment (iv): 8 compounding periods 5 th payment (v): 10 compounding periods 4 PMT = 0 ; FV = 6000; I/Y = 6/2 = 3; N = 2; CPT PV. PV = 5, PMT = 0; FV = 6000; I/Y = 6/2 = 3; N = 4; CPT PV. PV = 5, PMT = 0; FV = 6000; I/Y = 6/2 = 3; N = 6; CPT PV. PV = 5, PMT = 0; FV = 6000; I/Y = 6/2 = 3; N = 8; CPT PV. PV = 4, PMT = 0; FV = ; I/Y = 6/2 = 3; N = 10; CPT PV. PV = 78, , ($6,000 4) + 106,000 = $130,000 6 PMT = 0; FV = 6000; I/Y = 8/2 = 4; N = 2; CPT PV. PV = 5, PMT = 0; FV = 6000; I/Y = 8/2 = 4; N = 4; CPT PV. PV = 5, PMT = 0; FV = 6000; I/Y = 8/2 = 4; N = 6; CPT PV. PV = 4, PMT = 0; FV = 6000; I/Y = 8/2 = 4; N = 8; CPT PV. PV = 4, PMT = 0; FV = ; I/Y = 8/2 = 4; N = 10; CPT PV. PV = 71, , d. 1. $239, $164,918 Payment PV factor Amount PV factor Amount 1. $ 40,000 PVOA, 5 = n, 6% $168,480 PVOA, 5 = n, 16% ,000 PVOA, 5 = n, 6% PVOA, 5 = n, 16% PV$1, 5 = n, 6% ,927 PV$1, 5 = n, 16% ,000 PV$1, 11 = n, 6% ,270 PV$1, 11 = n, 16% ,000 PV$1, 12 = n, 6% ,485 PV$1, 12 = n, 16% $239,162 For complex examples, you may find it helpful to draw a timeline and count the compounding periods. Note that application of the PVOA factor to the $20,000 annuity (five periods) provides its present value as at January 1, 20X6. You then apply the lump sum PV$1 factor (five periods) to move it back to January 1, 20X1. Financial calculator method d. 1. $239,202
28 Page 28 of $164,947 I/Y = 6 I/Y = 16 N Payment amount amount $ 40,000 5 $168,495 1 $130, , , , , , , , , , 2, 5, 6 Formula used: Present value of an ordinary annuity 239, ,947 3, 4, 7, 8 Formula used: Present value of a lump sum PV o = FV n [1 (1 + k) n ] 1 PMT = 40000; N = 5; I/Y = 6; FV = 0;CPT PV. PV = 168, PMT = 20000; N = 10; I/Y = 6; FV = 0;CPT PV. PV = 147, PMT = 20,000; N = 5; I/Y = 6; FV = 0;CPT PV. PV= 84, , PMT = 0; N = 11; I/Y = 6; FV = 10,000;CPT PV. PV = 5, PMT = 0; N = 12; I/Y = 6; FV = 5,000;CPT PV. PV = 2, , PMT = 40000; N = 5; I/Y = 16; 6 2. a. FV = 0;CPT PV. PV = 130, PMT = 20000; N = 10; I/Y = 16; FV = 0;CPT PV. PV = 96, PMT = 20000; N = 5; I/Y = 16; FV = 0;CPT PV. PV = 65, , PMT = 0; N = 11; I/Y = 16; FV = 10,000;CPT PV. PV = 1, PMT = 0; N = 12; I/Y = 16; FV = 5,000;CPT PV. PV = , $27,554 [$120,000/(PVOA, 6 = n, 10%)]; ($120,000/4.355) 2. $24,405 [$120,000/(PVOA, 6 = n, 6%)]; ($120,000/4.917) 3. $25,047 [$120,000/(PVAD, 6 = n, 10%)]; ($120,000/4.791) 4. $23,024 [$120,000/(PVAD, 6 = n, 6%)]; ($120,000/5.212) b % ($40,000/$10,856) ; 6 period line in PVOA table 2. 7% ($40,000/$5,323) ; 10 period line in PVAD table
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