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Future Value of Annuity
The future value of an annuity refers to the total value of a series of equal payments or cash flows at a specified future date, assuming a certain interest rate. It is a key concept in finance for evaluating the growth of investments or the cost of borrowing. Calculating the future value of an annuity helps in making informed financial decisions and planning for the future.
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12 Key excerpts on "Future Value of Annuity"
- Robert Brechner, Geroge Bergeman(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
Section i • Future Value oF an annuity: ordinary and annuity due 381 F UTURE V ALUE OF AN A NNUITY : O RDINARY AND A NNUITY D UE The concepts relating to compound interest in Chapter 11 were mainly concerned with lump sum investments or payments. Frequently in business, situations involve a series of equal periodic payments or receipts rather than lump sums. These are known as annuities. An annuity is the payment or receipt of equal cash amounts per period for a specified amount of time. Some common applications are insurance and retirement plan premiums and payouts; loan payments; and savings plans for future events such as starting a business, going to col-lege, or purchasing expensive items (e.g., real estate or business equipment). In this chapter, you learn to calculate the future value of an annuity, the amount accu-mulated at compound interest from a series of equal periodic payments. You also learn to calculate the present value of an annuity, the amount that must be deposited now at compound interest to yield a series of equal periodic payments. Exhibit 12-1 graphically shows the dif-ference between the future value of an annuity and the present value of an annuity. All the exercises in this chapter are of the type known as simple annuities . This means that the number of compounding periods per year coincides with the number of annuity pay-ments per year. For example, if the annuity payments are monthly, the interest is compounded monthly; if the annuity payments are made every six months, the interest is compounded semiannually. Complex annuities are those in which the annuity payments and compounding periods do not coincide. As with compound interest, annuities can be calculated manually, by tables, and by for-mulas. Manual computation is useful for illustrative purposes; however, it is too tedious because it requires a calculation for each period.
eBook - PDF- (Author)
- 2015(Publication Date)
- Wiley(Publisher)
Future Value of a Lump Sum Future value Present value * 1 = + ( ) ( ) i n (11.3) Where i denotes the interest rate and n the number of time periods. Future Value of an Ordinary Annuity FV * Ordinary Annuity = + ( ) - C i i n 1 1 (11.4) Where C = cash flow per period; i = interest rate; n = number of payments. Ordinary Annuities and Annuities Due The financial planner should understand the differences between the two types of annuities: ordinary annuities and annuities due. An ordinary annuity refers to a series of payments made at the end of each period over a definite amount of time (e.g., interest payments from bond issuers usually paid semiannually or quarterly dividends from a company). Equation (11.2) and equation (11.4) demonstrate how to calculate the present value and future value of an ordinary annuity. Alternatively, an annuity due differs from an ordinary annuity in that the payments are made immediately or at the beginning of each period (e.g., lease and rental payments). The formulas for computing present value and future value of an annuity due are provided in equation (11.5) and equation (11.6). Present Value of an Annuity Due PV * Annuity Due = - + ( ) + ( ) C i i i n 1 1 1 * (11.5) Where C = cash flow per period; i= interest rate; n = number of payments. Future Value of an Annuity Due FV * Annuity Due = + ( ) - + ( ) C i i i n 1 1 1 * (11.6) Where C = cash flow per period; i = interest rate; n = number of payments. 100 TIME VALUE OF MONEY Even and Uneven Cash Flows We have assumed thus far that the cash flows of time value of money cal- culations remain constant over the evaluation period (e.g., bonds and an- nuities). This broad category in TVM is called even cash flows calculation. However, often in practice the cash flows of investments or projects are not equal during every evaluation period. This type of TVM calculation is re- ferred to as uneven cash flows.
eBook - PDF- R. Charles Moyer, James McGuigan, Ramesh Rao, , R. Charles Moyer, James McGuigan, Ramesh Rao(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
3. Present value calculations determine the value today (present value) of some amount to be received in the future. 4. An annuity is a series of equal periodic payments. a. Ordinary annuity payments are made at the end of each period. b. Annuity due payments are made at the beginning of each period. 5. The future value of an annuity calculation determines the future value of an annuity stream of payments. 6. The present value of an annuity calculation determines the present value of an annuity stream of payments. 7. The net present value rule is the primary decision-making rule used throughout the practice of financial management. a. The net present value of an investment is equal to the present value of the future cash flows minus the initial outlay. b. The net present value of an investment made by a firm represents the contribution of that investment to the value of the firm and, accordingly, to the wealth of shareholders. 8. Other important topics include: a. Compounding frequency b. Determining the present value of perpetuities c. Determining the present value of uneven cash flow streams d. Determining the present value of deferred annuities 144 Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 FINANCIAL CHALLENGE Pension Obligations and the Discount Rate Most larger companies, as well as federal, state, and local governments and not-for-profit organiza- tions, offer their employees pension benefits. Two basic types of plans are offered—defined contri- bution plans, for example 401(k)s, and defined benefit plans. With a defined contribution plan, through contributions from the employer and/or employee, the plan is fully funded on the day the employee retires, and the employer has no further financial liability to the pension plan.
- Donald E. Kieso, Jerry J. Weygandt, Terry D. Warfield(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
Present Value The concept of present value is described as the amount that must be invested now to produce a known future value. This is the opposite of the compound interest discussion in which the present value was known and the future value was determined. An example of the type of question addressed by the present value method is: What amount must be invested today at 6% interest compounded annually to accumulate $5,000 at the end of 10 years? In this question the present value method is used to determine the initial dollar amount to be invested. The present value method can also be used to determine the number of years or the interest rate when the other facts are known. LO 3: Solve future value of ordinary and annuity due problems. Future Value of an Annuity An annuity is a series of equal periodic payments or receipts called rents. An annuity requires that the rents be paid or received at equal time intervals, and that compound interest be applied. The future value of an annuity is the sum (future value) of all the rents (payments or receipts) plus the accumulated compound interest on them. If the rents occur at the end of each time period, the annuity is known as an ordinary annuity. If rents occur at the beginning of each time period, it is an annuity due. Thus, in determining the amount of an annuity for a given set of facts, there will be one less interest period for an ordinary annuity than for an annuity due. LO 4: Solve present value of ordinary and annuity due problems. Present Value of an Annuity The present value of an annuity is a sum of money invested today at compound interest that will provide for a series of equal withdrawals for a specified number of future periods. If the annuity is an ordinary annuity, the initial sum of money is invested at the beginning of the first period and withdrawals are made at the end of each period.
eBook - PDF- Donald E. Kieso, Jerry J. Weygandt, Terry D. Warfield(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
280 Chapter 6 Accounting and the Time Value of Money For example, a life insurance contract involves a series of equal payments made at equal intervals of time. Such a process of periodic payment represents the accumulation of a sum of money through an annuity. An annuity, by definition, requires the follow- ing: (1) periodic payments or receipts (called rents) of the same amount, (2) the same- length interval between such rents, and (3) compounding of interest once each interval. The future value of an annuity is the sum of all the rents plus the accumulated com- pound interest on them. Note that the rents may occur at either the beginning or the end of the periods. If the rents occur at the end of each period, an annuity is classified as an ordinary annuity. If the rents occur at the beginning of each period, an annuity is classified as an annuity due. Future Value of an Ordinary Annuity One approach to determining the future value of an annuity computes the value to which each of the rents in the series will accumulate, and then totals their individual future values. For example, assume that $1 is deposited at the end of each of 5 years (an ordinary annuity) and earns 5% interest compounded annually. Illustration 6-17 shows the com- putation of the future value, using the “future value of 1” table (Table 6-1 on pages 314–315) for each of the five $1 rents. ILLUSTRATION 6-17 Solving for the Future Value of an Ordinary Annuity END OF PERIOD IN WHICH $1.00 IS TO BE INVESTED Value at End Present 1 2 3 4 5 of Year 5 $1.00 $1.21551 $1.00 1.15762 $1.00 1.10250 $1.00 1.05000 $1.00 1.00000 Total (future value of an ordinary annuity of $1.00 for 5 periods at 5%) $5.52563 Because an ordinary annuity consists of rents deposited at the end of the period, those rents earn no interest during the period in which they are deposited. For example, the third rent earns interest for only two periods (periods four and five).
eBook - PDF- Donald E. Kieso, Jerry J. Weygandt, Terry D. Warfield(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Such a process of periodic payment represents the accumulation of a sum of money through an annuity. An annuity , by definition, requires the following: (1) periodic pay- ments or receipts (called rents) of the same amount, (2) the same length interval between such rents, and (3) compounding of interest once each interval. The future value of an annuity is the sum of all the rents plus the accumulated compound interest on them. Note that the rents may occur at either the beginning or the end of the periods. If the rents occur at the end of each period, an annuity is classified as an ordinary annuity . If the rents occur at the beginning of each period, an annuity is classified as an annuity due. Future Value of an Ordinary Annuity One approach to determining the future value of an annuity computes the value to which each of the rents in the series will accumulate, and then totals their individual future values. PV = €800,000 5 FV = €1,070,584 4 3 2 1 0 n = 5 i = ? i = ? ILLUSTRATION 6.15 Time Diagram to Solve for Unknown Interest Rate Using the future value factor of 1.33823, refer to Table 6.1 and read across the 5-period row to find that factor in the 6% column. Thus, the company must invest the €800,000 at 6% to accumulate to €1,070,584 in five years. Or, using the present value factor of .74726 and Table 6.2, again find that factor at the juncture of the 5-period row and the 6% column. Annuities (Future Value) 6-15 ILLUSTRATION 6.17 Solving for the Future Value of an Ordinary Annuity End of Period in Which $1.00 Is to Be Invested Value at End Present 1 2 3 4 5 of Year 5 $1.00 $1.21551 $1.00 1.15762 $1.00 1.10250 $1.00 1.05000 $1.00 1.00000 Total (future value of an ordinary annuity of $1.00 for 5 periods at 5%) $5.52563 For example, assume that $1 is deposited at the end of each of five years (an ordinary annuity) and earns 5% interest compounded annually.
eBook - PDF- Donald E. Kieso, Jerry J. Weygandt, Terry D. Warfield(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
Future Value of a Deferred Annuity Computing the future value of a deferred annuity is relatively straightforward. Because there is no accumulation or investment on which interest may accrue, the future value of a deferred annuity is the same as the future value of an annuity not deferred. That is, computing the future value simply ignores the deferred period. To illustrate, assume that Sutton Corporation plans to purchase a land site in 6 years for the construction of its new corporate headquarters. Because of cash flow problems, Sutton budgets deposits of $80,000 on which it expects to earn 5% annually, only at the end of the fourth, fifth, and sixth periods. What future value will Sutton have accumulated at the end of the sixth year? Illustration 6.37 shows a time diagram of this situation. i = 5% R = $80,000 $80,000 n = 3 (first 3 periods are ignored) 0 1 2 3 4 5 6 FV - OA = ? Future Value FV-OA = ? $80,000 ILLUSTRATION 6.37 Time Diagram for Future Value of Deferred Annuity Sutton determines the value accumulated by using the standard formula for the future value of an ordinary annuity: Future value of an ordinary annuity = R (FVF-OA n,i ) = $80,000 (FVF-OA 3,5% ) = $80,000 (3.15250) = $252,200 Present Value of a Deferred Annuity Computing the present value of a deferred annuity must recognize the interest that accrues on the original investment during the deferral period. To compute the present value of a deferred annuity, we compute the present value of an ordinary annuity of 1 as if the rents had occurred for the entire period. We then subtract the present value of rents that were not received during the deferral period. We are left with the present value of the rents actually received subsequent to the deferral period. To illustrate, Bob Boyd has developed and copyrighted tutorial software for students in advanced accounting. He agrees to sell the copyright to Campus Learning Systems for 6 annual payments of $5,000 each.
eBook - PDF- Laurence Booth, Ian Rakita(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
We could solve these problems to find the ending value using “brute force,” finding the future value of each payment the twins make individually. However, we will solve them after we develop more formally the concept of an annuity. Ordinary Annuities So far, we have dealt with PV and FV concepts as they apply to only two cash flows—one today (i.e., the PV) and one in the future (i.e., the FV). In practice, we will often need to compare different series of receipts or payments that occur through time. An annuity is a series of payments or receipts, which we will simply call cash flows, that are for the same amount and paid at the same interval—that is, for example, they are paid annually, monthly, or annuity regular payments on an investment that are for the same amount and are paid at the same interval of time cash flows the actual cash generated from an investment 5.4 Annuities and Perpetuities 5-13 weekly—over a given period. Annuities are common in finance; the ones you may be familiar with are car loans or mortgage payments. These involve an identical payment made at regular intervals for a loan based on a single interest rate. Ordinary annuities involve end‐of‐period payments. Example 5.8 demonstrates how to determine the FV and PV of an ordinary annuity. We have the same values as in our earlier discussion: FV, PV, n, and k. However, now we have another term, PMT, for the regular annu- ity payment or receipt. ordinary annuities equal payments that are made at the end of each period of time EXAMPLE 5.8 FV and PV of an Ordinary Annuity In 2018–19, the average NHL player earned $2.78 million per year, while the average career was approximately six years. Assuming a 41-percent tax rate, this would generate $1,640,200 in annual “disposable” (i.e., after‐tax) income.
- Robert Parrino, Hue Hwa Au Yong, Nigel Morkel-Kingsbury, Jennifer James, Paul Mazzola, James Murray, Lee Smales, Xiaoting Wei(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
6.1 Describe how to calculate the present value and the future value of an ordinary annuity, and how an ordinary annuity differs from an annuity due. An ordinary annuity is a series of equally spaced, level cash flows over time. The cash flows for an ordinary annuity are assumed to take place at the end of each period. To find the present value of an ordinary annuity, we multiply the present value of an annuity factor, which is equal to (1 − Present value factor)/i, by the amount of the constant cash flow. An annuity due is an annuity in which the cash flows occur at the beginning of each period. A lease is an example of an annuity due. In this case, we are effectively prepaying for the service. To calculate the value of an annuity due, we calculate the present value (or future value) as though the cash flows were an ordinary annuity. We then multiply the ordinary annuity value times (1 + i). Section 6.2 discusses the calculation of present value of an ordinary annuity and annuity due. 6.1 Explain what perpetuities are and where we see them in business, and calculate the present values A perpetuity is like an annuity except that the cash flows are perpetual — they never end. British Treasury bonds, called consols, were the first widely used securities of this kind. The most common example of a perpetuity today is preference shares. The issuer of preference shares promises to pay fixed-rate dividends forever. The cash flows from companies can also look like perpetuities. To calculate the present value of a perpetuity, we simply divide the promised constant payment (CF) by the interest rate (i). 6.1 Discuss why the effective annual interest rate (EAR) is the appropriate way to annualise interest rates and calculate the EAR. The EAR is the annual growth rate that takes compounding into account. Thus, the EAR is the true cost of borrowing or lending money.
eBook - PDF- Robert Parrino, David S. Kidwell, Thomas Bates(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
DISCOUNTED CASH FLOWS AND VALUATION 186 three years total €110 000, versus €100 000 for A. However, to make the decision on the basis of the undiscounted cash flows ignores the time value of money. By discounting the cash flows, we eliminate the time value of money effect by converting all cash flows to current money. The present value of business A is €85 270 and that of B is €83 810. Thus, you should acquire business A. Before You Go On 1. Explain how to calculate the future value of a stream of cash flows. 2. Explain how to calculate the present value of a stream of cash flows. 3. Why is it important to adjust all cash flows to a common date? 6.2 Level Cash Flows: Annuities and Perpetuities In finance, we commonly encounter contracts that call for the payment of equal amounts of cash over several time periods. For example, most business term loans and insurance policies require the holder to make a series of equal payments, usually monthly. Similarly, nearly all consumer finance, such as motor, personal and home mortgage loans, call for equal monthly payments. Any financial contract that calls for equally spaced and level cash flows over a finite number of periods is called an annuity . If the cash flow payments continue forever, the contract is called a perpetuity . Most annuities are structured so that cash payments are received at the end of each period. Because this is the most common structure, these annuities are often called ordinary annuities . LEARNING OBJECTIVE 2 annuity A series of equally spaced and level cash flows extending over a finite number of periods. perpetuity A series of level cash flows that continue forever. ordinary annuity An annuity in which payments are made at the ends of the periods. WEB Visit the following webform that provides an online annuity calculator http://www.feike.biz/annuity.php Present Value of an Annuity We frequently need to find the present value of an annuity (PVA) .
eBook - ePub- Brümmer LM, Hall JH, Du Toit E(Authors)
- 2017(Publication Date)
- Van Schaik Publishers(Publisher)
future value factors for an annuity, or FVFAs. (Note that these tables are for “ordinary annuities” as opposed to “annuities due” – these expressions will be explained shortly.)An annuity may be defined as a stream of equal and constant cash inflows over periods of more than one year (or parts of a year, such as a half year or even a month) into the future. Diagram 1.1 illustrates what an annuity does:For the purposes of this book and for the sake of simplicity, we shall keep to a year as our standard investment period, rather than a shorter period.Diagram 1.1The FVFA formula
Before we discuss the calculation of the future value of an annuity, we must note that there are several kinds of annuity. We shall, however, concern ourselves with 149 only the two most common kinds: an “ordinary annuity” and an “annuity due”. The basic difference between the two is to be found in the time at which the annual amount is invested. For the sake of simplicity, we shall use amounts of R100, which will be invested annually.In the case of an ordinary annuity, the R100 annual deposits are invested at the end of each of the investment years. This means that where the term of the annuity is three years, the first R100 earns interest for only two years, the second R100 for one year, while the last sum earns no interest at all. The third R100 earns no interest because each of the investments is made at the end of the year. In other words, an ordinary annuity is similar to a payment in arrears (made at the end of the period), as opposed to an annuity due, which is similar to a payment in advance (made at the beginning of the period).You may ask: “Why do we have such a strange method of investment? Surely when a deposit is made, the date of investment should mark the beginning, and not the end, of an investment period?” Ordinary annuities are generally used only in the payment of certain debts and other obligations, such as mortgage bonds and insurance premiums. The first payment on a mortgage bond, for example, is usually the down payment consisting of, say, 10% of the total debt. If the debt is R100 000, the down payment will be R10 000. The next payment due will be the first of a whole series of fixed equal amounts payable at the end of the year (i.e. in advance of the next year), as it will continue to be for all succeeding years until the debt is repaid.
eBook - PDF- Donald E. Kieso, Jerry J. Weygandt, Terry D. Warfield(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
• “An annuity due of six annual rents deferred 4 years” means that no rents will occur during the first 4 years and that the first of six rents will occur at the beginning of the fifth year. Future Value of a Deferred Annuity Computing the future value of a deferred annuity is relatively straightforward. Because there is no accumulation or investment on which interest may accrue, the future value of a deferred annuity is the same as the future value of an annuity that is not deferred. Computing the future value simply ignores the deferred period. Example 5.21 Future Value of a Deferred Annuity FACTS Sutton Corporation plans to purchase a land site in 6 years for the construction of its new corporate headquarters. Because of cash flow problems, Sutton budgets deposits of $80,000 on which it expects to earn 5% annually, only at the end of the fourth, fifth, and sixth periods. QUESTION What future value will Sutton have accumulated at the end of the sixth year? SOLUTION The following shows a time diagram of this situation. 0 2 3 4 1 n = 3 (first 3 periods are ignored) FV-OA = ? i = 5% 6 5 R = $80,000 $80,000 $80,000 Sutton determines the value accumulated by using the standard formula for the future value of an ordinary annuity and Table 5.3 as follows. Future value of an ordinary annuity = R (FVF-OA n,i ) = $80,000 (FVF-OA 3,5% ) = $80,000 (3.15250) = $252,200 Present Value of a Deferred Annuity Computing the present value of a deferred annuity must recognize the interest that accrues on the original investment during the deferral period. To compute the present value of a deferred annuity: Compute the present value of an ordinary annuity of 1 as if the rents had occurred for the entire period. 1 2 3 Subtract the present value of rents that were not received during the deferral period. The difference is the present value of the rents actually received subsequent to the deferral period.
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