# Time value of money. appendix B NATURE OF INTEREST

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1 appendix B Time value of money LEARNING OBJECTIVES After studying this appendix, you should be able to: Distinguish between simple and compound interest. Solve for future value of a single amount. Solve for future value of an annuity. 4 Identify the variables fundamental to solving present value problems. Solve for present value of a single amount. 6 Solve for present value of an annuity. 7 Calculate the present values in capital budgeting situations. Would you rather receive \$000 today or a year from now? You should prefer to receive the \$000 today because you can invest the \$000 and earn interest on it. As a result, you will have more than \$000 a year from now. What this example illustrates is the concept of the time value of money. Everyone prefers to receive money today rather than in the future because of the interest factor. LEARNING OBJECTIVE Distinguish between simple and compound interest. NATURE OF INTEREST Interest is payment for the use of another person s money. It is the difference between the amount borrowed or invested (called 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:. Principal (p): the original amount borrowed or invested.. Interest rate (i): an annual percentage of the principal.. Time (n): the number of years 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 figure B.. Interest = Principal p Rate i Time n Figure B. Interest calculation For example, if you borrowed \$000 for years at a simple interest rate of % annually, you would pay \$00 in total interest calculated as follows: Interest = p i n = \$ = \$00 0 Appendix B

2 COMPOUND INTEREST Compound interest is calculated on principal and on any interest earned that has not been paid or withdrawn. It 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. To illustrate the difference between simple and compound interest, assume that you deposit \$000 in Bank One, where it will earn simple interest of 9% per year, and you deposit another \$000 in CityCorp, where it will earn compound interest of 9% per year compounded annually. Also assume that in both cases you will not withdraw any interest until years from the date of deposit. The calculation of interest to be received and the accumulated year-end balances are indicated in figure B.. Figure B. Simple vs. compound interest Bank One City Corp Simple interest calculation Simple interest Accumulated year-end balance Compound interest calculation Compound interest Accumulated year-end balance Year \$ % \$ \$ Year \$ % \$ \$ Year \$ % \$80.00 Year \$ % 98.0 \$88.0 Year \$ % \$ \$70.00 \$.0 Difference Year \$88.0 9% 06.9 \$ 9.0 \$9.0 Note in figure B. that simple interest uses the initial principal of \$000 to calculate the interest in all years. Compound interest uses the accumulated balance (principal plus interest to date) at each year-end to calculate interest in the succeeding year which explains why your compound interest account is larger. Obviously, if you had a choice between investing your money at simple interest or at compound interest, you would choose compound interest, all other things especially risk being equal. In the example, compounding provides \$.0 of additional interest revenue. For practical purposes, compounding assumes that unpaid interest earned becomes a part of the principal, and the accumulated balance at the end of each year becomes the new principal on which interest is earned during the next year. As can be seen in figure B., you should invest your money at CityCorp, which compounds interest annually. Compound interest is used in most business situations. Simple interest is generally applicable only to short-term situations of year or less. FUTURE VALUE CONCEPTS FUTURE VALUE OF A SINGLE AMOUNT The future value of a single amount is the value at a future date of a given amount invested assuming compound interest. For example, in figure B., \$9.0 is the future value of the \$000 at the end of years. The \$9.0 could be determined more easily by using the following formula: where: FV = p ( + i ) n FV = future value of a single amount p = principal (or present value) i = interest rate for one period n = number of periods LEARNING OBJECTIVE Solve for future value of a single amount. Appendix B 0

3 Figure B. Time diagram The \$9.0 is calculated as follows: FV = p ( + i ) n = \$000 ( + i ) = \$ = \$9.0 The.90 is calculated by multiplying ( ). The amounts in this example can be depicted in the time diagram in figure B.. Present value (p) i = 9% Future value 0 \$000 \$9.0 n = years TABLE Future value of Another method that may be used to calculate the future value of a single amount involves the use of a compound interest table, which shows the future value of for n periods. Table, shown below, is such a table. (n) Periods 4% % 6% 8% 9% 0% % % % In table, n is the number of compounding periods, the percentages are the periodic interest rates, and the 6-digit decimal numbers in the respective columns are the future value of factors. In using table, the principal amount is multiplied by the future value factor for the specified number of periods and interest rate. For example, the future value factor for periods at 9% is.880. Multiplying this factor by \$000 equals 04 Appendix B

4 \$88.0, which is the accumulated balance at the end of year in the CityCorp example in figure B.. The \$9.0 accumulated balance at the end of the third year can be calculated from table by multiplying the future value factor for periods (.90) by the \$000. The demonstration problem in figure B.4 illustrates how to use table. Figure B.4 Demonstration problem using table for FV of John and Mary Rich invested \$0 000 in a savings account paying 6% interest at the time their son, Matt, was born. On his 8th birthday, Matt withdraws the money from his savings account. How much did Matt withdraw from his account? Present value (p) i = 6% Future value =? 0 \$ n = 8 years Answer: The future value factor from table is.844 (8 periods at 6%). The future value of \$0 000 earning 6% per year for 8 years is \$ (\$ ). FUTURE VALUE OF AN ANNUITY The preceding discussion involved the accumulation of only a single principal sum. Individuals and entities often encounter situations in which a series of equal dollar amounts are to be paid or received periodically, such as loans or lease (rental) contracts. Such payments or receipts of equal dollar amounts are referred to as annuities. The future value of an annuity is the sum of all the payments (receipts) plus the accumulated compound interest on them. In calculating the future value of an annuity, it is necessary to know () the interest rate, () the number of compounding periods, and () the amount of the periodic payments or receipts. To illustrate the calculation of the future value of an annuity, assume that you invest \$000 at the end of each year for years at % interest compounded annually. This situation is depicted in the time diagram in figure B.. LEARNING OBJECTIVE Solve for future value of an annuity. Figure B. Time diagram for a -year annuity i = % Future Present value \$000 \$000 value =? \$000 0 n = years As can be seen in figure B., the \$000 invested at the end of year will earn interest for years (years and ) and the \$000 invested at the end of year will earn interest Appendix B 0

5 for year (year ). However, the last \$000 investment (made at the end of year ) will not earn any interest. The future value of these periodic payments could be calculated using the future value factors from table as shown in figure B.6. Figure B.6 Future value of periodic payments Year invested Amount invested \$000 \$000 \$000 Future value of factor at % Future value.00 \$ \$60 TABLE Future value of an annuity of The first \$000 investment is multiplied by the future value factor for periods (.0) because years interest will accumulate on it (in years and ). The second \$000 investment will earn only year s interest (in year ) and therefore is multiplied by the future value factor for year (.000). The final \$000 investment is made at the end of the third year and will not earn any interest. Consequently, the future value of the last \$000 invested is only \$000 since it does not accumulate any interest. This method of calculation is required when the periodic payments or receipts are not equal in each period. However, when the periodic payments (receipts) are the same in each period, the future value can be calculated by using a future value of an annuity of table. Table, shown below, is such a table. (n) Periods 4% % 6% 8% 9% 0% % % % Table shows the future value of to be received periodically for a given number of periods. From table it can be seen that the future value of an annuity of factor for periods at % is.0. The future value factor is the total of the three individual future value factors as shown in figure B.6. Multiplying this amount by the annual investment of \$000 produces a future value of \$ Appendix B

6 The demonstration problem in figure B.7 illustrates how to use table. Henning Printing knows that in 4 years it must replace one of its existing printing presses with a new one. To ensure that some funds are available to replace the machine in 4 years, the business is depositing \$ 000 in a savings account at the end of each of the next 4 years (four deposits in total). The savings account will earn 6% interest compounded annually. How much will be in the savings account at the end of 4 years when the new printing press is to be purchased? i = 6% Future Present value \$ 000 \$ 000 \$ 000 value =? \$ n = 4 years Answer: The future value factor from table is (four periods at 6%). The future value of \$ 000 invested at the end of each year for 4 years at 6% interest is \$ (\$ ). PRESENT VALUE CONCEPTS PRESENT VALUE VARIABLES The present value, like the future value, is based on three variables: () the dollar amount to be received (future amount), () the length of time until the amount is received (number of periods), and () the interest rate (the discount rate). The process of determining the present value is referred to as discounting the future amount. In this book, present value calculations are used in measuring several items. For example, capital budgeting and other investment proposals are evaluated using present value calculations. All rate of return and internal rate of return calculations involve present value techniques. PRESENT VALUE OF A SINGLE AMOUNT To illustrate present value concepts, assume that you are willing to invest a sum of money that will yield \$000 at the end of year. In other words, what amount would you need to invest today to have \$000 in a year from now? If you want a 0% rate of return, the investment or present value is \$ (\$000.0). The calculation of this amount is shown in figure B.8. Figure B.7 Demonstration problem using table for FV of an annuity of LEARNING OBJECTIVE Identify the variables fundamental to solving present value problems. 4 LEARNING OBJECTIVE Solve for present value of a single amount. Present value PV PV PV = = = = Future value ( + i ) n FV ( + 0%) \$000.0 \$ Figure B.8 Present value calculation \$000 discounted at 0% for year The future amount (\$000), the discount rate (0%), and the number of periods () are known. The variables in this situation can be depicted in the time diagram in figure B.9. Appendix B 07

7 Present value (?) i = 0% Future value Figure B.9 Finding present value if discounted for one period \$ n = year \$000 If the single amount of \$000 is to be received in years and discounted at 0% [PV = \$000 ( + 0%) ], its present value is \$86.4 [\$000 (.0.0)], as depicted in figure B.0. Present value (?) i = 0% Future value Figure B.0 Finding present value if discounted for two periods 0 \$86.4 n = years \$000 TABLE Present value of The present value of may also be determined through tables that show the present value of for n periods. In table, n is the number of discounting periods involved. The percentages are the periodic interest rates or discount rates, and the 6-digit decimal numbers in the respective columns are the present value of factors. (n) Periods 4% % 6% 8% 9% 0% % % % Appendix B

8 When table is used, the future value is multiplied by the present value factor specified at the intersection of the number of periods and the discount rate. For example, the present value factor for period at a discount rate of 0% is , which equals the \$ (\$ ) calculated in figure B.8. For periods at a discount rate of 0%, the present value factor is 0.864, which equals the \$86.4 (\$ ) calculated previously. Note that a higher discount rate produces a smaller present value. For example, using a % discount rate, the present value of \$000 due year from now is \$869.7 versus \$ at 0%. It should also be recognised that the further removed from the present the future value is, the smaller the present value. For example, using the same discount rate of 0%, the present value of \$000 due in years is \$60.9 versus \$000 due in year is \$ The following demonstration problem (figure B.) illustrates how to use table. Determine the amount Metal Fabricators Ltd must deposit now in its savings account, paying 9% interest, in order to accumulate \$ in 4 years time for an addition to its factory. PV =? i = 9% \$ Now n = 4 4 years Answer: The present value factor from table is (4 periods at 9%). Thus, Metal Fabricators Ltd would need to deposit \$ 4 0 (\$ ) at 9% now in order to receive \$ in 4 years time. PRESENT VALUE OF AN ANNUITY The preceding discussion involved the discounting of only a single future amount. Entities and individuals often 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, 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 () the discount rate, () the number of discount periods, and () the amount of the periodic receipts or payments. To illustrate the calculation of the present value of an annuity, assume that you will receive \$000 cash annually for years at a time when the discount rate is 0%. This situation is depicted in the time diagram in figure B.. Figure B. Demonstration problem using table for PV of LEARNING OBJECTIVE Solve for present value of an annuity. 6 PV =? \$000 \$000 \$000 i = 0% Now n = years Figure B. Time diagram for a -year annuity Appendix B 09

9 The present value in this situation may be calculated as follows: Figure B. Present value of a series of future amounts calculation Future amount \$000 (one year away) 000 (two years away) 000 (three years away) Present value of factor at 0% = Present value \$ \$ This method of calculation is required when the periodic cash flows are not uniform in each period. However, when the future receipts are the same in each period, there are two other ways to calculate present value. First, the annual cash flow can be multiplied by the sum of the three present value factors. In the previous example, \$ equals \$ Second, annuity tables may be used. As illustrated in table 4 below, these tables show the present value of to be received periodically for a given number of periods. TABLE 4 Present value of an annuity of (n) Periods 4% % 6% 8% 9% 0% % % % From table 4 it can be seen that the present value of an annuity of factor for three periods at 0% is.4868.* This present value factor is the total of the three individual present value factors as shown in figure B.. Applying this amount to the annual cash flow of \$000 produces a present value of \$ The demonstration problem in figure B.4 illustrates how to use table 4. * The difference of between and.4868 is due to rounding. 060 Appendix B

10 Steel Products Ltd has just signed an agreement to purchase equipment for instalment payments of \$6000 each, to be paid at the end of each of the next years. In setting the amount of the payments, the seller used a discount rate of %. What is the present value of the instalment payments i.e. how much is Steel Products Ltd paying for the equipment and how much is it paying in total interest over the term of the instalment contract? PV =? Now \$6000 \$6000 \$6000 \$6000 i = % n = 4 \$6000 years Answer: The present value factor from table 4 is ( periods at %). The present value of payments of \$6000 each discounted at % is \$ (\$ ). Therefore, the cost of the equipment to Steel Products Ltd is \$ and the interest is \$87. [(\$6000 ) \$ 68.68]. Figure B.4 Demonstration problem using table 4 for PV of an annuity of TIME PERIODS AND DISCOUNTING In the preceding calculations, the discounting has been done on an annual basis using an annual interest rate. Discounting may also be done over shorter periods of time such as monthly, quarterly or half-yearly. When the time frame is less than year, it is necessary to convert the annual interest rate to the applicable time frame. Assume, for example, that the investor in figure B. received \$00 every 6 months for years instead of \$000 annually. In this case, the number of periods becomes 6 ( ), the discount rate is % (0% ), the present value factor from table 4 is.0769, and the present value of the future cash flows is \$7.8 (.0769 \$00). This amount is slightly higher than the \$ calculated in figure B. because interest is calculated twice during the same year; therefore interest is earned on the first 6 months interest. CALCULATING THE PRESENT VALUES IN A CAPITAL BUDGETING DECISION The decision to make long-term capital investments is best evaluated using discounting techniques that recognise the time value of money, i.e the present value of the cash flows involved in a capital investment. The evaluation must reduce all cash inflows and outflows to a common comparable amount. That can be accomplished by either future valuing to some future date all the cash flows, or present valuing (discounting) to the present date all cash flows. Although both are useful for evaluating the investment, the present value (discounting) technique is more appealing and universally used. Nagel Ltd is considering adding another truck to its fleet because of a purchasing opportunity. Navistar Ltd, Nagel Ltd s main supplier of trucks, is overstocked and offers to sell its biggest truck for \$4 000 cash payable on delivery. Nagel Ltd knows that the truck will produce a net cash flow per year of \$ for years (received at the end of each year), at which time it will be sold for an estimated residual value of \$ 000. Nagel Ltd s discount rate in evaluating capital expenditures is 0%. Should Nagel Ltd commit to the purchase of this truck? The cash flows that must be discounted to present value by Nagel Ltd are as follows: Cash payable on delivery (now): \$ Net cash flow from operating the rig: \$ for years (at the end of each year). Cash received from sale of truck at the end of years: \$ 000. The time diagrams for the latter two cash flows are shown in figure B.. LEARNING OBJECTIVE Calculate the present values in capital budgeting situations. 7 Appendix B 06

11 Diagram for net operating cash flows Present value? Now i = 0% Net operating cash flows \$ \$ \$ \$ \$ n = 4 year Diagram for residual value Present value? Now i = 0% n = 4 Cash from sale \$ 000 year Figure B. Time diagrams for Nagel Ltd Figure B.6 Present value calculations at 0% Figure B.7 Present value calculations at % Note from the diagrams that calculating the present value of the net operating cash flows (\$ at the end of each year) is discounting an annuity (table 4) whereas calculating the present value of the \$ 000 residual value is discounting a single sum (table ). The calculation of these present values is shown in figure B.6. Present values using a 0% discount rate Present value of net operating cash flows received annually over years: \$ PV of received annually for years at 0% \$ \$ 6.60 Present value of residual value (cash) to be received in years: \$ 000 PV of received in years at 0% \$ Present value of cash inflows Present value of cash outflows (purchase price due now at 0%): \$4 000 PV of due now \$ Net present value \$ Because the present value of the cash receipts (inflows) of \$ (\$ \$ 7.0) exceeds the present value of the cash payments (outflows) of \$ , the net present value of \$ is positive and the decision to invest should be accepted. Now assume that Nagel Ltd uses a discount rate of % not 0% because it wants a greater return on its investments in capital assets. The cash receipts and cash payments by Nagel Ltd are the same. The present values of these receipts and cash payments discounted at % are shown in figure B.7. Present values using a 0% discount rate Present value of net operating cash flows received annually over years at %: \$ \$ Present value of residual value (cash) to be received in years at %: \$ Present value of cash inflows Present value of cash outflows (purchase price due now at %): \$ Net present value \$ (.0) 06 Appendix B

12 Because the present value of the cash payments (outflows) of \$4 000 exceeds the present value of the cash receipts (inflows) of \$ (\$ \$7 40.0), the net present value of \$.0 is negative and the investment should be rejected. The above discussion relied on present value tables in solving present value problems. Electronic handheld calculators may also be used to calculate present values without the use of these tables. Some calculators have present value (PV) functions that allow you to calculate present values by merely identifying the proper amount, discount rate and periods, then pressing the PV key. SUMMARY OF LEARNING OBJECTIVES Distinguish between simple and compound interest. Simple interest is calculated on the principal only whereas compound interest is calculated on the principal and any interest earned that has not been withdrawn. Solve for future value of a single amount. Prepare a time diagram of the problem. Identify the principal amount, the number of compounding periods, and the interest rate. Using the future value of table, multiply the principal amount by the future value factor specified at the intersection of the number of periods and the interest rate. Solve for future value of an annuity. Prepare a time diagram of the problem. Identify the amount of the periodic payments, the number of compounding periods, and the interest rate. Using the future value of an annuity of table, multiply the amount of the payments by the future value factor specified at the intersection of the number of periods and the interest rate. 4 Identify the variables fundamental to solving present value problems. The following three variables are fundamental to solving present value problems: () the future amount, () the number of periods, and () the interest rate (the discount rate). Solve for present value of a single amount. Prepare a time diagram of the problem. Identify the future amount, the number of discounting periods, and the discount (interest) rate. Using the present value of table, multiply the future amount by the present value factor specified at the intersection of the number of periods and the discount rate. 6 Solve for present value of an annuity. Prepare a time diagram of the problem. Identify the future amounts (annuities), the number of discounting periods, and the discount (interest) rate. Using the present value of an annuity of table, multiply the amount of the annuity by the present value factor specified at the intersection of the number of periods and the interest rate. 7 Calculate the present values in capital budgeting situations. Calculate the present values of all cash inflows and all cash outflows related to the capital budgeting proposal (an investment-type decision). If the net present value is positive, accept the proposal (make the investment). If the net present value is negative, reject the proposal (do not make the investment). GLOSSARY Annuity A series of equal dollar amounts to be paid or received periodically (p. 0). Compound interest The interest calculated on the principal and any interest earned that has not been paid or received (p. 0). Discounting the future amount(s) The process of determining present value (p. 07). Future value of a single amount The value at a future date of a given amount invested assuming compound interest (p. 0). Future value of an annuity The sum of all the payments or receipts plus the accumulated compound interest on them (p. 0). Interest Payment for the use of another s money (p. 0). Present value The value now of a given amount to be invested or received in the future assuming compound interest (p. 07). Present value of an annuity A series of future receipts or payments discounted to their value now assuming compound interest (p. 09). Principal The amount borrowed or invested (p. 0). Simple interest The interest calculated on the principal only (p. 0). Appendix B 06

13 BRIEF EXERCISES (USE TABLES TO SOLVE EXERCISES) Calculate the future value of a single amount. (LO) Using future value tables. (LO,) BEB. Don Smith invested \$000 at 6% annual interest, and left the money invested without withdrawing any of the interest for 0 years. At the end of the 0 years, Don withdrew the accumulated amount of money. (a) What amount did Don withdraw assuming the investment earns simple interest? (b) What amount did Don withdraw assuming the investment earns interest compounded annually? BEB. For each of the following cases, indicate (a) to what interest rate columns and (b) to what number of periods you would refer in looking up the future value factor.. In table (future value of ): (a) (b) Annual rate 6% % Number of years invested. In table (future value of an annuity of ): (a) (b) Annual rate % 4% Number of years invested 0 6 Compounded Annually Half-yearly Compounded Annually Half-yearly Calculate the future value of a single amount. (LO) Calculate the future value of a single amount and of an annuity. (LO,) Calculate the future value of a single amount. (LO) Using present value tables. (LO,6) BEB. Porter Ltd signed a lease for an office building for a period of 0 years. Under the lease agreement, a security deposit of \$0 000 is made. The deposit will be returned at the expiration of the lease with interest compounded at % per year. What amount will Porter receive at the time the lease expires? BEB.4 David and Kathy Hatcher invested \$000 in a savings account paying 6% annual interest when their daughter, Sue, was born. They also deposited \$000 on each of her birthdays until she was 8 (including her 8th birthday). How much will be in the savings account on her 8th birthday (after the last deposit)? BEB. Ron Watson borrowed \$0 000 on July 000. This amount plus accrued interest at 6% compounded annually is to be repaid on July 00. How much will Ron have to repay on July 00? BEB.6 For each of the following cases, indicate (a) to what interest rate columns and (b) to what number of periods you would refer in looking up the discount rate.. In table (present value of ): (a) (b) (c) Annual rate % 0% 8% Number of years involved 6 8 Discounts per year Annually Annually Half-yearly. In table 4 (present value of an annuity of ): (a) (b) (c) Annual rate % 0% 8% Number of years involved 0 4 Number of payments involved 0 8 Frequency of payments Annually Annually Half-yearly Determining present values. (LO,6) BEB.7 (a) What is the present value of \$0 000 due 8 periods from now, discounted at 8%? (b) What is the present value of \$0 000 to be received at the end of each of 6 periods, discounted at 9%? 064 Appendix B

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