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Present Value of Annuity
The present value of an annuity is the current value of a series of equal cash flows to be received or paid at regular intervals over a specified period, discounted at a certain interest rate. It is used to determine the current worth of a stream of future payments or receipts, helping in decision-making related to investments, loans, and other financial matters.
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An Introduction
- Herbert Mayo(Author)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
The present value of an annuity is simply the sum of the present value of each indi -vidual cash flow. Each cash inflow is discounted back to the present at the appropriate discount factor and the amounts are summed. Suppose you expect to receive $100 at the end of each year for three years and want to know how much this series of pay -ments is worth if you can earn 8 percent in an alternative investment. To answer the question, you discount each payment at 8 percent: Payment Year Interest Factor Present Value $100 1 0.926 $92.60 100 2 0.857 85.70 100 3 0.794 79.40 $257.70 The process determines the present value to be $257.70. That is, if you invest $257.70 now and earn 8 percent annually, you can withdraw $100 at the end of each year for the next three years. This process is expressed in more general terms by Equation 3.5. The present value (PV) of the annual payments (PMT) is then found by discounting these payments at the appropriate interest rate ( i ) for n time periods. P V 5 P M T 1 1 1 i 2 1 1 c 1 P M T 1 1 1 i 2 n 3.5 5 a n t 5 1 P M T 1 1 1 i 2 t . present value of an annuity The present worth of a series of equal payments. Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 C H A P T E R T H R E E The Time Value of Money 69 When the values from the previous example are inserted into the equation, it reads P V 5 $ 1 0 0 1 1 1 0 . 0 8 2 1 $ 1 0 0 1 1 1 0 . 0 8 2 2 1 $ 1 0 0 1 1 1 0 . 0 8 2 3 5 $ 1 0 0 1 . 0 8 0 1 $ 1 0 0 1 . 1 6 6 1 $ 1 0 0 1 . 2 6 0 5 $ 2 5 7 . 7 0 . Since the payments are equal and made annually, this example is an annuity, and the present value is simply the product of the payment and the interest factor. Interest tables have been developed for the interest factors for the present value of an annuity (see the fourth table in Appendix A). Selected interest rates are read horizontally along the top, and the number of periods is read vertically at the left.
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.
eBook - ePub- (Author)
- 2020(Publication Date)
- Wiley(Publisher)
6. THE PRESENT VALUE OF A SERIES OF CASH FLOWS
Many applications in investment management involve assets that offer a series of cash flows over time. The cash flows may be highly uneven, relatively even, or equal. They may occur over relatively short periods of time, longer periods of time, or even stretch on indefinitely. In this section, we discuss how to find the present value of a series of cash flows.6.1. The Present Value of a Series of Equal Cash Flows
We begin with an ordinary annuity. Recall that an ordinary annuity has equal annuity payments, with the first payment starting one period into the future. In total, the annuity makes N payments, with the first payment at t = 1 and the last at t = N. We can express the present value of an ordinary annuity as the sum of the present values of each individual annuity payment, as follows:where- A = the annuity amount
- r = the interest rate per period corresponding to the frequency of annuity payments (for example, annual, quarterly, or monthly)
- N = the number of annuity payments
Because the annuity payment (A) is a constant in this equation, it can be factored out as a common term. Thus the sum of the interest factors has a shortcut expression:In much the same way that we computed the future value of an ordinary annuity, we find the present value by multiplying the annuity amount by a present value annuity factor (the term in brackets in Equation 11 ).EXAMPLE 11
The Present Value of an Ordinary AnnuitySuppose you are considering purchasing a financial asset that promises to pay €1,000 per year for five years, with the first payment one year from now. The required rate of return is 12 percent per year. How much should you pay for this asset?
eBook - PDF- Donald E. Kieso, Jerry J. Weygandt, Terry D. Warfield(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
6-20 CHAPTER 6 Accounting and the Time Value of Money Annuities (Present Value) LEARNING OBJECTIVE 4 Solve present value of ordinary and annuity due problems. The present value of an annuity is the single sum that, if invested at compound interest now, would provide for an annuity (a series of withdrawals) for a certain number of future periods. Present Value of an Ordinary Annuity The present value of an ordinary annuity is the present value of a series of equal rents, to be withdrawn at equal intervals at the end of the period. One approach to finding the present value of an annuity determines the present value of each of the rents in the series and then totals their individual present values. For example, we may view an annuity of $1, to be received at the end of each of 5 periods, as separate amounts. We then compute each present value using the table of present values (see Table 6.2), assuming an interest rate of 5%. Illustration 6.28 shows this approach. Present Value at Beg. of Year 1 1 2 3 4 5 $0.95238 $1.00 .90703 $1.00 .86384 $1.00 .82270 $1.00 .78353 $1.00 $4.32948 Total (present value of an ordinary annuity of $1.00 for five periods at 5%) End of Period in Which $1.00 Is to Be Received ILLUSTRATION 6.28 Solving for the Present Value of an Ordinary Annuity This computation tells us that if we invest the single sum of $4.33 today at 5% interest for 5 periods, we will be able to withdraw $1 at the end of each period for 5 periods. We can summarize this cumbersome procedure by the following formula. PVF-OA n,i = 1 − 1 (1 + i) n i The expression PVF-OA n,i refers to the present value of an ordinary annuity of 1 factor for n peri- ods at i interest. Ordinary annuity tables base present values on this formula. Illustration 6.29 shows an excerpt from such a table.
eBook - PDF- Donald E. Kieso, Jerry J. Weygandt, Terry D. Warfield(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
Present Value of an Ordinary Annuity The present value of an ordinary annuity is the present value of a series of equal rents, to be withdrawn at equal intervals at the end of the period. One approach to finding the present value of an annuity determines the present value of each of the rents in the series and then totals their individual present values. Illustration 5.8 shows this approach. We may view an annuity of $1, to be received at the end of each of 5 periods, as separate amounts. We then compute each present value using Table 5.2, assuming an interest rate of 5%. 5-22 CHAPTER 5 Accounting and the Time Value of Money ILLUSTRATION 5.8 Solving for the Present Value of an Ordinary Annuity End of Period in Which $1.00 Is to Be Received Present Value at Beg. of Year 1 1 2 3 4 5 $0.95238 $1.00 .90703 $1.00 .86384 $1.00 .82270 $1.00 .78353 $1.00 $4.32948 Total (present value of an ordinary annuity of $1.00 for five periods at 5%) ILLUSTRATION 5.9 Excerpt from Table 5.4 Present Value of an Ordinary Annuity of 1 (Excerpt from Table 5.4) Period 4% 5% 6% 1 .96154 .95238 .94340 2 1.88609 1.85941 1.83339 3 2.77509 2.72325 2.67301 4 3.62990 3.54595 3.46511 5 4.45182 4.32948* 4.21236 *Note that this annuity table factor is equal to the sum of the present value of 1 factors shown in Illus- tration 5.8. The general formula for the present value of any ordinary annuity is as follows. Present value of an ordinary annuity = R (PVF-OA n,i ) where: R = periodic rent (ordinary annuity) PVF-OA n,i = present value of an ordinary annuity of 1 for n periods at i interest This computation tells us that if we invest the single sum of $4.33 today at 5% interest for five periods, we will be able to withdraw $1 at the end of each period for five periods. We can summarize this cumbersome procedure by the following formula. PVF-OA n ,i = 1− 1 _____ (1+ i) n ________ i The expression PVF-OA n,i refers to the present value of an ordinary annuity of 1 factor for n periods at i interest.
eBook - PDF- Donald E. Kieso, Jerry J. Weygandt, Terry D. Warfield(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
6-22 CHAPTER 6 Accounting and the Time Value of Money Present Value of an Annuity Due In our discussion of the present value of an ordinary annuity, we discounted the final rent based on the number of rent periods. In determining the present value of an annuity due, there is always one fewer discount period. Illustration 6.31 shows this distinction. What Do the Numbers Mean? Up in Smoke Time value of money concepts also can be relevant to public policy debates. For example, many governments must evalu- ate the financial cost-benefit of selling to a private operator the future cash flows associated with government-run services, such as toll roads and bridges. In these cases, the policymaker must estimate the present value of the future cash flows in de- termining the price for selling the rights. In another example, some governmental entities had to determine how to receive the payments from tobacco companies as settlement for a national lawsuit against the companies for the healthcare costs of smoking. In one situation, a governmental entity was due to collect 25 years of payments totaling $5.6 billion. The government could wait to collect the payments, or it could sell the payments to an in- vestment bank (a process called securitization). If it were to sell the payments, it would receive a lump-sum payment today of $1.26 billion. Is this a good deal for this governmental entity? Assuming a discount rate of 8% and that the payments will be received in equal amounts (e.g., an annuity), the present value of the tobacco payment is: $5.6 billion ÷ 25 = $224 million $224 million × 10.67478 * = $2.39 billion * PV-OA (i = 8%, n = 25) Why would the government be willing to take just $1.26 bil- lion today for an annuity whose present value is almost twice that amount? One reason is that the governmental entity was facing a hole in its budget that could be plugged in part by the lump-sum payment.
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 - PDF- Donald E. Kieso, Jerry J. Weygandt, Terry D. Warfield(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
In addition, GAAP may require the use of expected cash flows in determining present value. In this section, we examine: 1. Deferred annuities. 2. Bond problems. 3. Present value measurement. Deferred Annuities A deferred annuity is an annuity in which the rents begin after a specified number of periods. A deferred annuity does not begin to produce rents until two or more periods have expired. For example, “an ordinary annuity of six annual rents deferred 4 years” means that no rents will occur during the first 4 years and that the first of the six rents will occur at the end of the fifth year. “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 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, Sut- ton 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 (page 292) shows a time diagram of this situation. R = ? ? ? ? ? ? ? ? n = 8 i = 2% 0 1 2 3 4 5 6 7 8 Present Value PV-OA = $36,000 ? ? ? ? ? ? ? R = ? ILLUSTRATION 6-36 Time Diagram for Ordinary Annuity for a College Fund LEARNING OBJECTIVE 5 Solve present value prob- lems related to deferred annuities, bonds, and expected cash flows.
eBook - ePub- Brümmer LM, Hall JH, Du Toit E(Authors)
- 2017(Publication Date)
- Van Schaik Publishers(Publisher)
Diagram 1.3Diagram 1.3This time we consult Table A-4 , which provides the present value factors for an annuity, or PVFA. You will find that the PVFA over three years at 10% is 2.487, which, multiplied by the R100, gives us a present value of R248.70.Using a financial calculator to find the present value of an annuity
Similar to calculating the future value of an annuity, one can use a financial calculator to calculate the present value of an ordinary annuity.158Using a spreadsheet to find the present value of an annuity
A spreadsheet can be used as follows to calculate the present value of an ordinary annuity.A B 1 PRESENT VALUE OF AN ORDINARY ANNUITY 2 Annual payment 100.00 3 Interest rate, percentage per year, compounded annually 10% 4 Number of years 3 5 Present value 248.68 The entry in B5 is = PV(B3,B4,–B2) We write a minus sign in front of cell B2 because the present value is an outflow (money taken out of your account and paid into an ordinary annuity). Finding the present value of unequal annual amounts
When we have annual cash inflows of unequal amounts, we follow the same procedure as the one set out in the example where the future value of a mixed stream of cash inflows was calculated. This will look as in Diagram 1.4 .Diagram 1.4In this case, we consult Table A-3 in order to make use of the present value of a single amount. Supposing we had future cash inflows of R100 in the first year, R150 in the second and R200 in the third, all to be discounted at 10%. The calculation to find the total present value of these unequal amounts would be as follows:- Year 1:
R100.00 × 0.909 = R90.90 - Year 2:
R150.00 × 0.826 = R123.90 - Year 3:
R200.00 × 0.751 = R150.20 - Total:
R450.00 R365.00 Thus a total cash inflow of R450 over three years, discounted at a cost of capital of 10%, gives a total present value of R365.00. As you will note, whenever we have a mixed stream of inflows, each annual cash inflow must be discounted separately from the others. In this example, you will also note that the individual annual deposits do not need the special time arrangement required in our example of 159
- Robert Brechner, Geroge Bergeman(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
Present value (ordinary annuity) = Ordinary annuity table factor × Annuity payment iStock.com/Nikada EXAMPLE 4 CalCulating the present value of an ordinary annuity How much must be deposited now at 9% compounded annually to yield an annuity payment of $5,000 at the end of each year for 10 years? SOLUTION STRATEGY Step 1. The rate per period is 9% (9% ÷ 1 period per year) . Step 2. The number of periods is 10 (10 years × 1 period per year) . Step 3. From Table 12-2, the table factor for 9% , 10 periods is 6.41766 . Step 4. Present value = Ordinary annuity table factor × Annuity payment Present value = 6.41766 × 5,000 = $32,088.30 TRY IT EXERCISE 4 The Broadway Movieplex needs $20,000 at the end of each 6-month movie season for renovations and new projection equipment. How much must be deposited now at 8% compounded semiannu-ally to yield this annuity payment for the next 6 years? C H E C K Y O U R A N S W E R W I T H T H E S O L U T I O N O N P A G E 4 1 2 . Copyright 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. C ALCULATING THE P RESENT V ALUE OF AN A NNUITY D UE BY U SING T ABLES The present value of an annuity due is calculated by using the same table as ordinary annui-ties, with some modifications in the steps. 12-5 STEPS FOR CALCULATING PRESENT VALUE OF AN ANNUITY DUE step 1. Calculate the number of periods of the annuity (years × periods per year) and subtract one period from the total. step 2. Calculate the interest rate per period (nominal rate ÷ periods per year) .
eBook - PDF- Scott Besley, Eugene Brigham, Scott Besley(Authors)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
In other words, on a time value of money basis, if your opportunity rate is 10 percent, you should be indifferent if you had to choose whether to take $700 today, $770 in one year, $847 in two years, or $931.70 in three years. These values differ only by the amount of interest they contain, which is interest that can be earned by everyone who invests for the specified period of time. In fact, if we take out all the interest in each of the future amounts, the current (present) value of each is $700. Let’s apply the logic presented here to the ordi- nary annuity we discussed earlier in the chapter, where we showed that if, beginning in one year, Alice deposits $400 per year in a savings account that pays 5 percent interest annually, her account will have a balance of $1,261 at the end of three years. We also discovered that Alice would need to deposit $1,089.30 today in a savings account that pays 5 percent interest annually if she wants to pay herself $400 per year for the next three years, as- suming that she withdraws the first $400 from the ac- count at the end of this year. Suppose you win a raffle that allows you to choose from one of three prizes: Prize 1 is $1,089.30 that will be paid to you today; Prize 2 is $1,261 that will be paid 7 For the sale to occur quickly and for you to receive $700, we assume no commissions, taxes, or other costs are associated with the sale of the prize and there are many people who would want to buy such a prize; that is, there are many people who want to invest $700 today at 10 percent so they will have $931.70 to spend in three years.
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