Foundation review. Introduction. Learning objectives


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1 Foundation review: Introduction Foundation review Introduction Throughout FN1, you will be expected to apply techniques and concepts that you learned in prerequisite courses. The purpose of this foundation review is to give you an opportunity to review some of these techniques and concepts before you apply them in FN1. The first topic in this foundation review looks at the various legal forms of organizations. The second topic reviews some basic concepts of the time value of money, and illustrates how to solve for the various parameters using a preprogrammed financial calculator. Learning objectives Explain fundamental differences in taxation, legal liability, longevity, and sourcing funds based on the legal form of the business. Interpret and solve basic time value of money questions. Foundation Review 1: Legal forms of organization Foundation Review 2: Time value of money Print the Foundation Review Foundation review: Introduction
2 FR.1 Legal forms of organization FR.1 Legal forms of organization Accountants provide services to all types of business entities. It is important, therefore, that you are familiar with the characteristics of the different forms of business ownership. The three most common 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 legal entity. 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 legal entity; hence, its owners (shareholders) are not liable for the debt obligations of the corporation. These concepts were covered in both FA2 and LW1. As it may have been a while since you covered this material, the following material and review questions are provided to allow you to assess your understanding of the essentials of the various organizational forms. For taxation purposes, 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 usually accessible by the lenders to recover any defaulted debt obligations. 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 most large business enterprises; there are many sole proprietorships and partnerships, 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. For the purpose of this course, the major ways in which the corporation differs from the partnership or sole proprietorship are as follows: The corporation treats the taxation of the business separately from that of the owner. The corporation provides its owners with limited liability. The longevity of the corporation is independent of the lifespan of its owners. The corporation is generally better suited for raising large amounts of funds for investment in capital assets. Example FR.11: Comparing investment in a partnership with investment in a corporation Mrs. 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 debt of $8,000 now due out of the partnership's cash flow. Because partners in a partnership are not 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 take legal action against 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 loaned funds to BCE cannot litigate for 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 FR.1 Legal forms of organization
3 FR.1 Legal forms of organization 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.1 Legal forms of organization
4 FR.2 Time value of money FR.2 Time value of money Note: The preprogrammed financial calculator is used extensively throughout FN1. You are expected to be able to use a preprogrammed financial calculator for solving "time value of money" questions in online quizzes, assignments, and the final examination. Microsoft Excel is used throughout the course to prepare simple financial models for which you can easily change parameters and assess the resulting impact. However, during the final examination, you will only be allowed to use a financial calculator. 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. In addition to this short review, you may wish to consult the CGA Publication and Reference Handbook entitled Mathematics 4 Business, by T.B. Killip, or another basic business mathematics text. Future value and present value of single payments The following two formulas calculate the present value (PV) and future value (FV) of a single payment: FV n = PV o (1 + i) n PV o = FV n (1 + i) n where i = 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 + i) 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 (1.05) 6 or Similarly, you can use the second equation above 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 + i) 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 1 (1.05) 6 or If the future amount is $500, the present value is $ = $ 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. 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 FR.2 Time value of money
5 FR.2 Time value of money present value of an annuity due is involved. For example, to find the future value of $500 invested for 6 years at 5%: FV n = PV 0 (1 + i) n PMT = 0; PV = 500; I/Y = 5; N = 6; CPT FV FV = $ To find the present value of $500 invested for 6 years at 5%: PV o = FV n (1 + i) 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 Computer Tutorial 2. Examples FR.21 to FR.24 demonstrate how to calculate future values and present values. Example FR.21: Future value of a present amount $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)? Using a financial calculator: For this question, you are given: Number of years: 5 Annual interest rate: 12% Present value: $1,000 Enter the following on the calculator: PMT = 0; PV = 1,000; I/Y = 12; N = 5; CPT FV FV = $1, Using Excel: Open Excel file FN1FR2E from the data folder and click the FRE2.1 sheet tab. (Before you begin working on the data files in FN1, you must first download them and save them to your hard drive. Click the data files link in the Course Modules window, and follow the instructions for downloading and saving the files.) FR.2 Time value of money
6 FR.2 Time value of money Note that cell B3 contains the present value ($1,000), cell B4 contains the annual interest rate, and cell B5 contains the number of years. Enter the following in cell B6: =FV(B4,B5,, B3) The future value is $1, Example FR.22: Calculating savings account balances To calculate the account balance at the end of 10, 15, and 25 years, calculate the future value of the present amount of $15,000, with an annual interest rate of 8%. Using a financial calculator: Number of years Future value 10 $ 32, (a) 15 $ 47, (b) 25 $ 102, (c) (a) PMT = 0; PV = 15,000; I/Y = 8; N = 10; CPT FV (b) PMT = 0; PV = 15,000; I/Y = 8; N = 15; CPT FV (c) PMT = 0; PV = 15,000; I/Y = 8; N = 25; CPT FV Using Excel: Using the worksheet from Example FR21, enter the appropriate values for present value, annual interest rate, and number of years in cells B3 to B5. You should obtain the same results in cell B6. 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%? Using a financial calculator: Number of years: 3 Annual interest rate: 10% Future value: $10,000 FR.2 Time value of money
7 FR.2 Time value of money PMT = 0; FV = 10,000; I/Y = 10; N = 3; CPT PV PV = $7, Using Excel: Using the same worksheet, calculate the present value. The data for this question is in cells B8 through B10. Enter the following in cell B11: =PV(B9,B10,, B8) The present value is $7, 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 the future. Assuming a 9% annual 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. Using a financial calculator: Use the financial functions of your calculator to compute the present values. You should obtain the following results: Number of years Present value 10 $ 84, (a) 15 $ 54, (b) 25 $ 23, (c) (a) PMT = 0; FV = 200,000; I/Y = 9; N = 10; CPT PV (b) PMT = 0; FV = 200,000; I/Y = 9; N = 15; CPT PV (c) PMT = 0; FV = 200,000; I/Y = 9; N = 25; CPT PV Using Excel: Using your worksheet from the previous examples, enter the appropriate values for future value, annual interest rate, and number of years in cells B8 to B10. You should obtain the same results in cell B11. Determining the interest rate You may encounter a situation where the present and future values and the number of periods are known, but you do not FR.2 Time value of money
8 FR.2 Time value of money know the interest rate. 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 how to determine the interest rate. Example FR.25: Determining interest rate A $100,000 investment will yield $146,933 in five periods. What is the 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. Using a financial calculator: PMT = 0; FV = 146,933; PV = 100,000; N = 5; CPT I/Y I/Y = 8 percent Using Excel: Using the same worksheet, enter the following formula for the annual interest rate in cell B16: =RATE(B15,, B14,B13) Ordinary annuities and annuities due Many financial transactions require a series of identical payments, received or paid at equally spaced intervals. 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 leases, which require regular fixed monthly payments, are also 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 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. Calculations for annuities involve the following four items: payment amount per period interest rate per period number of interest periods FR.2 Time value of money
9 FR.2 Time value of money 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. Here is the formula to calculate the future value of an ordinary annuity: Here is the formula to calculate the PV of an ordinary annuity: 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 Excel provides the FV, PV, PMT and NPER functions (the Excel functions are explained in detail in Computer Tutorial 2). These functions are summarized in Exhibit FR.21. Exhibit FR.21: 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 show you how to calculate annuities. Example FR.26: Future value of an ordinary annuity FR.2 Time value of money
10 FR.2 Time value of money 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. Using a financial calculator: Make sure that your calculator is set to "end" mode. PMT = 5,000; N = 6; PV = 0; I/Y = 8; CPT FV FV = $36, Using Excel: Continue with the previous worksheet. Enter the formula for the future value in cell B21: =FV(B19,B20, B18) 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, the interest rate is 8% compounded semiannually, the payment amount is $2,500, and the payments occur over 6 years. In this case, the annuity is an ordinary annuity. You need to calculate 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). Using a financial calculator: PMT = 2,500; N = 12; FV = 0; I/Y = 4; CPT PV PV = $23, Using Excel: Continue with the Excel worksheet. Enter the formula for the present value in cell B26: =PV(B24,B25, B23) FR.2 Time value of money
11 FR.2 Time value of money Compare your result with that shown. If you obtained different results, click on the FRE2.1S tab. Find and compare the information in the cell that is different from what you expected in the solution sheet to your sheet to determine your error. Using time lines to calculate an annuity due The difference between an ordinary annuity and an annuity due is the timing of the payment. For an ordinary annuity, the payment comes at the end of each interest period, whereas for an annuity due, the payment comes at the beginning of each interest period. You can see this difference by comparing the time line of an ordinary annuity with three annual payments to the time line of an annuity due with three annual payments, as shown in Exhibit 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. Future value of annuity due Since the future value of an annuity due equals the future value of an ordinary annuity, compounded one additional period at (1 + i), you can calculate the future value of an annuity due as the FV of an ordinary annuity times (1 + i), or FR.2 Time value of money
12 FR.2 Time value of money Present value of annuity due Recall that a threeyear annuity due is equivalent to a threeyear ordinary annuity that receives one less period of discounting at 1 (1 + i). Therefore, the PV of an annuity due is simply the PV of an ordinary annuity times (1 + i), or Examples FR.28 through FR.210 illustrate three typical annuity problems. Work through each example, using your calculator and Excel. Example FR.28: Future value of an annuity due For the investment in Example FR.26, suppose that instead of making the annual payment on December 31 of each year, the investments are made at the beginning of each year, on January 1, with the first investment on January 1, 20X1. In this case, the annuity is an annuity due and you wish to calculate its future value at maturity. Using a financial calculator: Number of years: 6 Payment amount: 5,000 Interest rate: 8% Set your calculator at the "begin" mode: 2 nd BEG 2 nd SET PMT = 5,000; N = 6; PV = 0; I/Y = 8; CPT FV FV = $39, Using Excel: Continue with the worksheet. Enter the following formula in cell B31: =FV(B29,B30, B28,,1) Example FR.29: Present value of an annuity due Suppose that for the same investment in Example FR28, you wish to calculate the present value. Once again, observe that the investment is an annuity due. FR.2 Time value of money
13 FR.2 Time value of money Using a financial calculator: Number of years: 6 Payment amount: 5,000 Interest rate: 8% Set your calculator at the "begin" mode: 2 nd BEG 2 nd SET PMT = 5,000; N = 6; FV = 0; I/Y = 8; CPT PV PV = $24, Using Excel: Continue with the worksheet. Enter the following formula in cell B36: =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 a financial calculator: This is an ordinary annuity. Number of years: 3 Annual interest rate: 12% Present value: $100,000 Set your calculator to "end" mode: 2 nd BGN 2 nd SET PV = 100,000; I/Y = 12; N = 3; FV = 0; CPT PMT PMT = $41, Using Excel: Continue with the worksheet. Enter the following formula in cell B41: =PMT(B39,B40, B38) FR.2 Time value of money
14 FR.2 Time value of money FR.2 Time value of money
15 Foundation review Review questions 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. Describe the advantages associated with the corporate method of carrying on business. Solutions Topic FR.2 1. Compute the following amounts. Each case is independent. 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 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: Principal Interest 1. 6% annual interest, annual compounding $ $ 2. 8% annual interest, annual compounding $ $ 3. 4% annual interest, annual compounding $ $ Foundation review Review questions
16 Foundation review Review questions 4. 6% annual interest, semiannual compounding $ $ 5. 8% annual interest, semiannual compounding $ $ d. On January 1, 20X1, Dardon Corporation agrees to lease a machine, with the following terms required by the lease contract: December 31, 20X120X5, per year $ 40,000 December 31, 20X620Y0, per year $ 20,000 December 31, 20Y1 $ 10,000 December 31, 20Y2 $ 5,000 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. 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 with 6 payments 2. January 1 payment of $5,323 with 10 payments Calculate the number of payments needed for 3. December 31 payment of $4,074 at 8% 4. January 1 payment of $4,936 at 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. a. Calculate 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: Date Beginning principal Instalment payment Interest Principal Ending principal Solutions Foundation review Review questions
17 Solution FR.1 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 is not incorporated. A sole proprietorship is a person who is carrying on business for herself or himself, although this person may have employees. 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, 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 (income less expenses) of the sole proprietorship are taxable in the hands of the proprietor. 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. 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 still has 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.1
18 Solution FR.2 Solution FR.2 Notes: PV = present value of a lump sum PVOA = present value of an ordinary annuity PVAD = present value of an annuity due N = number of compounding periods Question 1a This is an ordinary annuity with the formula: PV o = FV n (1 + i) n 1. FV = $40,000; I/Y = 8; N = 3; PMT = 0; CPT PV PV = $31, FV = $40,000; I/Y = 8 2 = 4; N = 3 2 = 6; PMT = 0; CPT PV PV = $31, FV = $40,000; I/Y = 8 4 = 2; N = 3 4 = 12; PMT = 0; CPT PV PV = $31, Question 1b 1. This is the future value of an annuity due with the formula: Set your calculator to "begin" mode. PMT = 4,000; N = 5; I/Y = 12; FV = 0; CPT PV PV = $16, This is an annuity due. However, the payments are made annually but the compounding is semiannual. You need to calculate the present value of each payment using the formula for the present value of a lump sum: PV o = FV n (1 + i) n 1st payment (i): 0 compounding periods 2nd payment (ii): 2 compounding periods 3rd payment (iii): 4 compounding periods 4th payment (iv): 6 compounding periods Solution FR.2
19 Solution FR.2 5th payment (v): 8 compounding periods (i) 4, (ii) PMT = 0; FV = 4,000; I/Y = 12/2 = 6; N = 2; CPT PV 3, (iii) PMT = 0; FV = 4,000; I/Y = 12/2 = 6; N = 4; CPT PV 3, (iv) PMT = 0; FV = 4,000; I/Y = 12/2 = 6; N = 6; CPT PV 2, (v) PMT = 0; FV = 4,000; I/Y = 12/2 = 6; N = 8; CPT PV 2, $16, Alternatively, you can find the annual compounding rate that is equivalent to 12% compounded semiannually: ( /2) 2 = (1 + i) 1 i = (1.06) 2 1 i = or 12.36% A rate of 12.36% compounded annually is equivalent to 12% compounded semiannually. Use this rate to solve the question using a financial calculator: Set your calculator to "begin" mode. PMT = 4,000; N = 5; I/Y = 12.36; FV = 0; CPT PV PV = $16, This is the future value of an ordinary annuity. The formula is: Set your calculator to "end" mode. PMT = 4,000; I/Y = 12; N = 5; FV = 0; CPT PV PV = $14, This is an ordinary annuity. However, the payments are made annually but the compounding is semiannual. You need to calculate the present value of each payment using the formula for the present value of a lump sum: PV o = FV n (1 + i) n 1st payment (i): 2 compounding periods 2nd payment (ii): 4 compounding periods 3rd payment (iii): 6 compounding periods 4th payment (iv): 8 compounding periods 5th payment (v): 10 compounding periods (i) PMT = 0; FV = 4,000; I/Y = 12/2 = 6; N = 2; CPT PV 3, (ii) PMT = 0; FV = 4,000; I/Y = 12/2 = 6; N = 4; CPT PV 3, (iii) PMT = 0; FV = 4,000; I/Y = 12/2 = 6; N = 6; CPT PV 2, (iv) PMT = 0; FV = 4,000; I/Y = 12/2 = 6; N = 8; CPT PV 2, (v) PMT = 0; FV = 4,000; I/Y = 12/2 = 6; N = 10; CPT PV 2, $14, Alternatively, you can find the annual compounding rate that is equivalent to 12% compounded semiannually. In Solution FR.2
20 Solution FR.2 other words, ( /2) 2 = (1 + i) 1 i = (1.06) 2 1 i = or 12.36% A rate of 12.36% compounded annually is equivalent to 12% compounded semiannually. Use this rate to solve the question using a financial calculator: Set your calculator to "end" mode. PMT = 4,000; N = 5; I/Y = 12.36; FV = 0; CPT PV PV = $14, Question 1c This is the present value of an ordinary annuity + the present value of a lump sum: (1) (2) (3) Principal Interest Total 1. $ 100,0001 $ 30,000 $ 130, , , , , , , , , , , , ,000 1 PMT = 6,000; N= 4; I/Y= 6; FV= 0; CPT PV 20, PMT = 0; FV = 106,000; N = 5; I/Y = 6; CPT PV 79, , PMT = 6,000; N= 4; I/Y= 8; FV= 0; CPT PV 19, PMT = 0; FV = 106,000; N = 5; I/Y = 8; CPT PV 72, , PMT = 6,000; N= 4; I/Y= 4; FV= 0; CPT PV 21, PMT = 0; FV = 106,000; N = 5; I/Y = 4; CPT PV 87, , , 6 There are annual payments but semiannual compounding. You need to calculate the present value of each payment using the formula for the present value of a lump sum: PV o = FV n (1 + i) n 1st payment (i): 2 compounding periods 2nd payment (ii): 4 compounding periods 3rd payment (iii): 6 compounding periods 4th payment (iv): 8 compounding periods Solution FR.2
21 Solution FR.2 5th payment (v): 10 compounding periods 4 PMT = 0; FV = 6,000; I/Y = 6/2 = 3; N = 2; CPT PV 5, PMT = 0; FV = 6,000; I/Y = 6/2 = 3; N = 4; CPT PV 5, PMT = 0; FV = 6,000; I/Y = 6/2 = 3; N = 6; CPT PV 5, PMT = 0; FV = 6,000; I/Y = 6/2 = 3; N = 8; CPT PV 4, PMT = 0; FV = 106,000; I/Y = 6/2 = 3; N = 10; CPT PV 78, , (6,000 4) + 106,000 = 130,000 6 PMT = 0; FV = 6,000; I/Y = 8/2 = 4; N = 2; CPT PV 5, PMT = 0; FV = 6,000; I/Y = 8/2 = 4; N = 4; CPT PV 5, PMT = 0; FV = 6,000; I/Y = 8/2 = 4; N = 6; CPT PV 4, PMT = 0; FV = 6,000; I/Y = 8/2 = 4; N = 8; CPT PV 4, PMT = 0; FV = 106,000; I/Y = 8/2 = 4; N = 10; CPT PV 71, , Question 1d N I/Y = 6 I/Y = 16 Payment $ 40,000 5 Amount $168,4951 Amount $130, ,000 10, ,9542 5, ,1796 1,9547 5, , $239,202 $164,947 1, 2, 5, 6 Present value of an ordinary annuity: 3, 4, 7, 8 Present value of a lump sum: PV o = FV n (1 + i) n 1 PMT = 40,000; N = 5; I/Y = 6; FV = 0; CPT PV 168, PMT = 20,000; N = 10; I/Y = 6; FV = 0; CPT PV 147, PMT = 20,000; N = 5; I/Y = 6; FV = 0; CPT PV 84, , PMT = 0; N = 11; I/Y = 6; FV = 10,000; CPT PV PMT = 0; N = 12; I/Y = 6; FV = 5,000; CPT PV 5, , , PMT = 40,000; N = 5; I/Y = 16; FV = 0; CPT PV 130, PMT = 20,000; N = 10; I/Y = 16; FV = 0; CPT PV PMT = 20,000; N = 5; I/Y = 16; FV = 0; CPT PV 96, , , PMT = 0; N = 11; I/Y = 16; FV = 10,000; CPT PV 1, PMT = 0; N = 12; I/Y = 16; FV = 5,000; CPT PV , Question 2a For 1 and 2, use the formula for the present value of an ordinary annuity and solve for PMT: Solution FR.2
22 Solution FR.2 1. PV= 120,000; N = 6; I/Y = 10; FV = 0; CPT PMT PMT = $27, PV= 120,000; N = 6; I/Y= 6; FV = 0; CPT PMT PMT = $24,404 For 3 and 4, use the formula for the present value of an annuity due and solve for PMT: Set your calculator to "begin" mode. 3. PV = 120,000; N = 6; I/Y = 10; FV = 0; CPT PMT PMT = $25, PV = 120,000; N = 6; I/Y = 6; FV = 0; CPT PMT PMT = $23,022 Question 2b For 1 and 3, use the formula for the present value of an ordinary annuity: 1. PMT = 10,856; N = 6; FV = 0; PV = 40,000; CPT I/Y I/Y = 16.00% 3. PMT = 4,074; FV = 0; PV = 40,000; I/Y = 8; CPT N Solution FR.2
23 Solution FR.2 N = 20 payments For 2 and 4, use the formula for the present value of an annuity due: Set your calculator to "begin" mode. 2. PMT = 5,323; N = 10; FV = 0; PV = 40,000; CPT I/Y I/Y = 7.00% 4. PMT = 4,936; FV = 0; PV = 40,000; I/Y = 10; CPT N N = 14 payments Question 3a 1. Use the formula for the present value of an ordinary annuity and solve for PMT: PV = 100,000; N = 5; FV = 0; I/Y = 10; CPT PMT PMT = $26, Use the formula for the present value of an annuity due and solve for PMT: Set your calculator to "begin" mode. PV = 100,000; N = 5; FV = 0; I/Y = 10; CPT PMT PMT = 23, Question 3b 1. Payment rounded to $26,380. Solution FR.2
24 Solution FR.2 Instalment payment Date Beginning principal (1) Interest (2) Principal (3) Ending principal (4) Dec. 31, 20X7 $100,000 $10,000 $16,380 $83,620 Dec. 31, 20X8 83,620 8,362 18,018 65,602 Dec. 31, 20X9 65,602 6,560 19,820 45,782 Dec. 31, 20Y0 45,782 4,578 21,802 23,980 Dec. 31, 20Y1 23,980 2,400* 23,980 Calculations: (2) Interest = (1) 10% (3) Principal = $26,380 (2) (4) Ending principal = (1) (3) * Adjusted $2 to compensate for rounding the payment. 2. Payment rounded to $23,981 Instalment payment Date Beginning principal (1) Interest (2) Principal (3) Ending principal (4) Jan. 1, 20X7 $100,000 $23,981 $76,019 Jan. 1, 20X8 76,019 $ 7,602 16,380 59,639 Jan. 1, 20X9 59,639 5,964 18,018 41,621 Jan. 1, 20Y0 41,621 4,162 19,820 21,801 Jan. 1, 20Y1 21,801 2,180 21,801 Calculations: (2) Interest = (1) 10% (3) Principal = $23,981 (2) (4) Ending principal = (1) (3) Solution FR.2
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