Calculating apr on ba ii plus

analyzing annuity cash flows

We explained basic time value computations in the previous chapter. Those TVM equations covered moving a single cash flow from one point in time to another. While this circumstance does describe some problems that businesses and individuals face, most debt

and investment applications of time value of money feature multiple cash flows. In fact, most situations require many equal payments over time. Since these situations require a bit more complicated analysis, this chapter continues the TVM topic for applications that require many equal payments over time. For example, car loans and home mortgage loans require the borrower to make the same monthly payment for many months or years. People save for the future through monthly contributions to their pension portfolios. People in retirement must convert their savings into monthly income. Companies also make regular payments. Johnson & Johnson (ticker: JNJ) will pay level semiannual interest payments through 2033 on money it borrowed. General Motors (ticker: GM) paid a $0.50 per share quarterly dividend to stockholders for six straight years until 2006, when it switched to a $0.25 dividend. These examples require payments (and compounding) over different time intervals (monthly for car loans and semiannually for company debt). How are we to value these payments into common or comparable terms? In this chapter, we illustrate how to value multiple cash flows over time, including many equal payments, and how to incorporate different compounding frequencies.

LEARNING GOALS

LGS-1 Compound multiple cash flows to the future.

LGS-2 Compute the future value of frequent, level cash flows. LGS-3 Discount multiple cash flows to the present.

LGS-4 Compute the present value of an annuity.

LGS-S Figure cash flows and present value of a perpetuity.

LGS-6 Adjust values for beginning-of-period annuity payments.

LGS-7 Explain the impact of compound frequency and the difference between the annual percentage rate and the effective annual rate.

LGS-8 Compute the interest rate of annuity payments.

LGS-9 Compute payments and amortization schedules for car and mortgage loans.

LGS- 10
Calculate the number of payments on a loan.

FUTURE VALUE OF MULTIPLE CASH FLOWS
Chapter 4 illustrated how to take single payments and compound them into the future. To save enough money for a down payment on a house or for retirement, people typically make many contributions over time to their savings accounts. We can add the

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future value of each contribution together to see what the total will be worth at some future point in time-such as age 65 for retirement or in two years for a down payment on a house.

Finding the Future Value of Several Cash Flows LGS-1

Consider the following contributions to a savings account over time. You make a $100 deposit today, followed by a $125 deposit next year, and a $150 deposit at the end of the second year. If interest rates are 7 percent, what's the future value of your deposits at the end of the third year? The time line for this problem is illustrated as:

Note that the first deposit will compound for three years. That is, the future value in year 3 of a cash flow in year 0 will compound 3 (= 3 - 0) times. The deposit at the end of the first year will compound twice (= 3 - 1). In general, a deposit in year m will compound N- m times for a future value in year N. We can find the total amount at the end of three years by computing the future value of each deposit and then adding them together. Using the future value equation from Chapter 4, the future value of today's deposit is $100 x (1 + 0.07)3 = $122.50. Similarly, the future value of the next two deposits are $125 x (1 + 0.07)2 = $143.11 and $150 x (1 + 0.07)1 = $160.50, respectively.

Putting these three individual future value equations together would yield:

FV3 = $100 x (1 + 0.07)3 + $125 x (1 + 0.07)2 + $150 x (1 + 0.07)1 = $426.11

The general equation for computing the future value of multiple and varying cash flows (or payments) is:

In this equation, the letters m, n, and p denote when the cash flows occur in time. Each deposit can be different from the others.

view points

business APPLICATION

Walkabout Music, Inc., issued $20 million in debt ten years ago to finance its factory construction. The debt allows Walkabout to make interest-only payments at a 7 percent coupon rate, paid semiannually for 30 years. Debt issued today would carry only 6 percent interest. The company's CFO is considering whether or not to issue new debt (for 20 years) to pay off the old debt. To pay off the old debt early, Walkabout would have to pay a special "call premium" totaling $1.4 million to its debt holders. To issue new debt, the firm would have to pay investment bankers a fee of $1.2 million. Should the CFO replace the old debt with new debt?

Future Value of Level Cash Flows LGS-2
Now suppose that each cash flow is the same and occurs every year. Level sets of frequent cash flows are common in finance-we call them annuities. The first cash flow of an annuity occurs at the end of the first year (or other time period) and continues every year to the last year. We derive the equation for the future value of an annuity from the general equation for future value of multiple cash flows, equation 5-1. Since each cash flow is the same, and the cash flows are every period, the equation appears as:

FVAN = Future value of first payment

x Future value of second payment + ··· + Last payment

= PMT x (1 + i)N-1 + PMT

x (1 + i)N-2 + PMT x (1 + i)N-3 + ··· + PMT(1 + i)0

The term FVA is used to denote that this is the future value of an annuity. Factoring out the common level cash flow, PMT, we can summarize and reduce the equation as:

annuity A stream of level and frequent cash flows paid at the end of each time period-often referred to as an ordinary annuity.

personal APPLICATION

Say that you obtained a mortgage for $150,000 three years ago when you purchased your home. You've been paying monthly payments on the 30-year mortgage with a fixed 8 percent interest rate and have $145,920.10 of principal left to pay. Recently, your mortgage broker called to

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mention that interest rates on new mortgages have declined to 7 percent. He suggested that you could save money every month if you refinanced your mortgage. You could find a 27-year mortgage at the new interest rate for a

$1,000 fee. Should you refinance your mortgage?

But what if you want to move in the next few years? Is it

still a good idea? Scan the QR code for an extended look. Turn to the back of the book for solutions to these applications.
Suppose that $100 deposits are made at the end of each year for five years. If interest rates are 8 percent per year, the future value of this annuity stream is computed using equation 5-2 as:

Say that beginning with your freshman year in college, you will be working as a house painter for each of the next three summers. You intend to set aside some money from each summer's paycheck to buy a car for your senior year. If you can deposit $2,000 from the first summer, $2,500 in the second summer, and $3,000 in the last summer, how much money will you have to buy a car if interest rates are 5 percent? SOLUTION:We can show these deposits and future value on a time line as:

Saving for a Car LGS-1Five deposits of $100 each were made. So, the $586.66 future value represents $86.66 of interest paid. As with almost any TVM problem, the length of time of the annuity and the interest rate for compounding are very important factors in accumulating wealth within the annuity. Consider the examples in Table 5.1. A $50 deposit made every year for 20 years will grow to $1,839.28 with a 6 percent interest rate. Doubling the annual deposits to $100 also doubles the future value to $3,678.56. However, making $100 deposits for twice the amount of time, 40 years, more than quadruples the future value to $15,476.20! Longer time periods lead to more total compounding and much more wealth. Interest rates also have this effect. Doubling the interest rate from 6 to 12 percent on the 40- year annuity results in nearly a five-fold increase in the future value to $76,709.14.

EXAMPLE S-1

The time line for the forecast is:

The first cash flow, which occurs at the end of the first year, will compound for two years. The second cash flow will be invested for only one year. The last contribution will not have any time to grow before the purchase of the car. Using equation 5-1, the solution is

FV3 = [$2,000 x (1 + 0.05)3-1] + [$2,500 x (1 + 0.05)3-2] + [$3,000 x (1 + 0.05)3-3]

= ($2,000 x 1.1025) + ($2,500 x 1.05) + ($3,000 x 1) = $7,830

You will have $7,830 in cash to purchase a car for your senior year. Similar to Problems 5­1, 5­2, 5­17, 5­18, 5­43, 5­44

TABLE S.1 Magnitude of Periodic Payments, Number of Years Invested, and Interest Rate on FV of Annuity

Think about it: Depositing only $100 per year (about 25 lattes per year) can generate some serious money over time. See Figure 5.1. How much would $2,000 annual deposits generate?

Future Value of Multiple Annuities
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At times, multiple annuities can occur in both business and personal life. For example, you may find that you can increase the amount of money you save each year because of a promotion or a new and better job. As an illustration, reconsider the annual $100 deposits made for five years at 8 percent per year. This time, the deposit can be increased to $150 for the fourth and fifth years. How can we use the annuity equation to compute the future value when we have two levels of cash flows? In this case, the cash flow can be categorized as two annuities. The first annuity is a $100 cash flow for five years. The second annuity is a $50 cash flow for two years. We demonstrate this as:

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Annuities and the Financial Calculator

In the previous chapter, the level payment button (PMT) in the financial calculator was always set to zero because no constant payments were made every period. We use the PMT button to input the annuity amount. For calculators, the present value is of the opposite sign (positive versus negative) from the future value. This is also the case with annuities. The level cash flow will be of the opposite sign as the future value, as the previous time line shows.

You would use the financial calculator to solve the problem of depositing $100 for five years via the following inputs: N = 5, I = 8, PV = 0, PMT = - 100. In this case, the input for present value is zero because no deposit is made today. The result of computing the future value is

EXAMPLE S-2
Saving in the Company Pension Plan LGS-2

You started your first job after graduating from college. Your company offers a retirement plan for which the company contributes 50 percent of what you contribute each year. So, if you contribute $3,000 per year from your salary, the company adds another $1,500. You get to decide how to invest the total annual contribution from several portfolio choices that the plan administrator provides. Suppose that you pick a

mixture of stocks and bonds that is expected to earn 7 percent per year. If you plan to retire in 40 years, how big will you expect that retirement account to be? If you could earn 8 percent per year, how much money would be available?

SOLUTION:

Every year, you and your employer will set aside a total of $4,500 for your retirement. Using equation 5-2 shows that the future value of this annuity is:

Note that you can build a substantial amount of wealth ($898,358) through your pension plan at work. If you can earn just 1 percent more each year, 8 percent total, you could be a millionaire!

Similar to Problems 5­3, 5­4, self­test problem 1

FIGURE S.1 Future Value of a $100 Annuity at 6 Percent

EXAMPLE Growing Retirement

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Contributions LGS-2 In the previous example, you are investing a total of $4,500 per year for 40 years in your employer's retirement program. You believe that with raises and promotions , you will eventually be able to contribute more money each year. Consider that halfway through your career, you are able to increase your investment in the retir ement program to $6,000 per year (your contribution plus the company match). What would be the future value of your retirement wealth from this program if investments are compounded at 7 percent? SOLUTION: You can compute the future value using two annuities. The first annuity is one with payments of $4,500 that lasts 40 years. The second is a $1,500 (= $6,000 - $4,500) annuity that lasts only 20 years. We already computed the future value of the first annuity in the previous example: $898,358. The future value of the second annuity is:

S-3

So , your retirement wealth from this program would be $959,851 (= $898,358 + $61,493). Similar to Problems 5­19, 5­20, self­test problem 1the Math Coach on...

Solving Multiple Annuities
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The trick to solving multiple annuity problems is to disentangle cash flows into groups of level payments ending in the future value year that we've

To determine the future value of these two annuities, compute the future value of each one separately, and then simply add them together. The future value of the $100 annuity is the same as computed before, $586.66. The future value of the $50 annuity, using the TVM equation for the future value of a cash stream, is:

time out!

S-1 Describe how compounding affects the future value computation of an annuity.

S-2 Reconsider your original retirement plan example to invest $4,500 per year for 40 years. Now consider the result if you don't contribute anything for four years (years 19 to 22) while your child goes to college. How many annuity equations will you need to find the future value of your 401(k) in this situation?

So, the future value of both of the annuities is $690.66 (= $586.66 + $104). In the same way, we could easily compute the future value if the last two cash flows are $50 lower ($50 each), instead of $50 higher ($150 each). To solve this alternative version, we would simply subtract the $104 future value instead of adding it.

PRESENT VALUE OF MULTIPLE CASH FLOWS
The future value concept is very useful to understand how to build wealth for the future. The present value concept will help you most particularly for personal applications such as evaluating loans (like car and mortgage loans) and business applications (like determining the value of business opportunities).

Finding the Present Value of Several Cash Flows LGS-3
Consider the cash flows that we showed at the very beginning of the chapter: You deposit $100 today, followed by a $125 deposit next year, and a $150 deposit at the end of the second year. In the previous situation, we sought the future value when interest rates are 7 percent. Instead of future value, we compute the present value of these three cash flows. The time line for this problem appears as:

The first cash flow is already in year zero, so its value will not change. We will discount the second cash flow one year and the third cash flow two years. Using the present value equation from the previous chapter, the present value of today's payment is simply $100 ""7 (1 + 0.07)0 = $100. Similarly, the present value of the next two cash flows are $125 ""7 (1 + 0.07)1 = $116.82 and $150 ""7 (1 + 0.07)2 =

$131.02, respectively. Therefore, the present value of these cash flows is $347.84 (= $100 + $116.82 +

$131.02).

financeat at work //:

personal

Who Will Save for Their Future?
Though it seems way too early for you to think about planning for your "golden years,"

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financially wise people realize that it's never too early to start. Unfortunately, most people save little for their retirement years. Seventy percent of workers surveyed report that they have saved less than $50,000. How far does that get you? Using a 6 percent investment return, $50,000 can generate a monthly income of only

$299.78 for 30 years, at which time it is used up. That is less than $3,600 per year! The average Social Security monthly benefit is just over $1,230 per month, or about $13,400 per year. While seventy percent of workers have saved less than $50,000, only 10 percent have saved over $250,000.

This chapter illustrates that much higher amounts of wealth can be accumulated if you start early! One easy way to do this is through a retirement plan at work. Most company and government employers offer employees defined contribution plans. (The corporate version is called a 401(k) plan; a nonbusiness plan is usually referred to as a 403(b) plan-both named after the legislation that created the plans.) These plans place all of the responsibility on employees to provide for their retirement. Employees contribute from their own paychecks and decide how to invest. Employees' decisions about how much to contribute and how early to start contributing have a dramatic impact on retirement wealth. Consider employees who earn $50,000 annually for 40 years and then retire. Note that if the employees contribute for 40 years, they must start by age 25 or so- starting early is vitally important! Contributing 5 percent of their salaries ($2,500) to the 401(k) plan every year and having it earn a 4 percent return will generate $237,564 for retirement. A 10 percent contribution ($5,000) would create $475,128 for retirement. Finally, investment decisions that yield an 8 percent return would yield $1.3 million with a 10 percent contribution. This is quite a range of retirement wealth generated from just three important decisions each employee must make-how much to contribute, how to invest the funds, and when to start! Unfortunately, too many people make poor decisions. The average 401(k) account value for people in their 60s is only $136,400-often because people start 401(k) contributions too late to allow the funds to compound much.

Saving and investing money through a defined contribution plan is a good way to build wealth for retirement. But you must follow these rules: Start Early, Save Much, and Don't Touch!

Want to know more?

Key Words to Search for Updates: Employee Benefit Research Institute (go to www.ebri.org ), retirement income

Sources: "Preparing for Retirement in America," 2012 Retirement Confidence Survey Fact Sheet #3. http://www.ebri.org/pdf/surveys/rcs/2012/fs-03-rcs-12-fs3-saving.pdf.

Putting these three individual present value equations together would yield:

PV = [$100 ""7 (1 + 0.07)0] + [$125 ""7 (1 + 0.07)1] + [$150 ""7 (1 + 0.07)2] = $347.84

The general equation for discounting multiple and varying cash flows is:

In this equation, the letters m, n, and p denote when the cash flows occur in time. Each deposit can differ from the others in terms of size and timing.

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Using a Financial Calculator-Part 2

The five TVM buttons/functions in financial calculators have been fine, so far, for the types of TVM problems we've been solving. Sometimes we had to use them two or three times for a single problem, but that was usually because we needed an intermediate calculation to input into another TVM equation.

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Luckily, most financial calculators also have built-in worksheets specifically designed for computing TVM in problems with multiple nonconstant cash flows.

To make calculator worksheets as flexible as possible, they are usually divided into two parts: one for input, which we'll refer to as the CF (cash flow) worksheet, and one or more for showing the calculator solutions. We'll go over the conventions concerning the CF worksheet here, and we'll discuss the output solutions in Chapter 13.

The CF worksheet is usually designed to handle inputting sets of multiple cash flows as quickly as possible. As a result, it normally consists of two sets of variables or cells-one for the cash flows and one to hold a set of frequency counts for the cash flows, so that we can tell it we have seven $1,500 cash flows in a row instead of having to enter $1,500 seven times.

Using the frequency counts to reduce the number of inputs is handy, but you must take care. Frequency counts are only good for embedded annuities of identical cash flows. You have to ensure that you don't mistake another kind of cash flow for an annuity.

Also, using frequency counts will usually affect the way that the calculator counts time periods. As an example, let's talk about how we would put the set of cash flows shown here into a CF worksheet:

To designate which particular value we'll place into each particular cash flow cell in this worksheet, we'll note the value and the cell identifier, such as CF0, CF1, and so forth. We'll do the same for the frequency cells, using F1, F2, etc., to identify which CF cell the frequency cell goes with. (Note that, in most calculators, CF0 is treated as a unique value with an unalterable frequency of 1; we're going to make the same assumption here so you'll never see a listing for F0.) For this sample timeline, our inputs would be:

To compute the present value of these cash flows, use the NPV calculator function. The NPV function computes the present value of all the future cash flows and then adds the year 0 cash flow. Then, on the NPV worksheet, you would simply need to enter the interest rate and solve for the NPV:

Note a few important things about this example:

1. We had to manually enter a value of $0 for CF3. If we hadn't, the calculator wouldn't have known about it and would have implicitly assumed that CF4 came one period after CF2.

2. Once we use a frequency cell for one cash flow, all numbering on any subsequent cash flows that we enter into the calculator is going to be messed up, at least from our point of view. For instance, the first $75 isn't what we would call "CF5," is it? We'd call it "CF7" because it comes at time period 7; but calculators usually treat CF5 as "the fifth set of cash flows," so we'll just have to try to do the same to be consistent.

3. If we really don't need to use frequency cells, we will usually just leave them out of the guidance instructions in this chapter to save space.

Present Value of Level Cash Flows LGS-4
You will find that this present value of an annuity concept will have many business and personal applications throughout your life. Most loans are set up so that the amount borrowed (the present value) is repaid through level payments made every period (the

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annuity). Lenders will examine borrowers' budgets and determine how much each borrower can afford as a payment. The maximum loan offered will be the present value of that annuity payment. The equation for the present value of an annuity can be derived from the general equation for the present value of multiple cash flows, equation 5-3. Since each cash flow is the same, and the borrower pays the cash flows every period, the present value of an annuity, PVA, can be written as:

Suppose that someone makes $100 payments at the end of each year for five years. If interest rates are 8 percent per year, the present value of this annuity stream is computed using equation 5-4 as:

TABLE S.2 Magnitude of the Annuity, Number of Years Invested, and Interest Rate on PV

The time line for these payments and present value appears as:

Notice that although five payments of $100 each were made, $500 total, the present value is only

$399.27. As we've noted previously, the span of time over which the borrower pays the annuity and the interest rate for discounting strongly affect present value computations. When you borrow money from the bank, the bank views the amount it lends as the present value of the annuity it receives over time from the borrower. Consider the examples in Table 5.2.

A $50 deposit made every year for 20 years is discounted to $573.50 with a 6 percent discount rate. Doubling the annual cash flow to $100 also doubles the present value to $1,146.99. But extending the time period does not impact the present value as much as you might expect. Making $100 payments for twice the amount of time-40 years-does not double the present value. As you can see in Table 5.2, the present value increases less than 50 percent to only $1,504.63! If the discount rate increases from 6 percent to 12 percent on the 40-year annuity, the present value will shrink to $824.38.

EXAMPLE

S-4 Value of Payments LGS-4

Your firm needs to buy additional physical therapy equipment that costs $20,000. The equipment manufacturer will give you the equipment now if you will pay $6,000 per year for the next four years. If your firm can borrow money at a 9 percent interest rate, should you pay the manufacturer the $20,000 now or accept the 4-year annuity offer of $6,000?

SOLUTION:

We can find the cost of the 4-year, $6,000 annuity in present value terms using equation 5-4:

The cost of paying for the equipment over time is $19,438.32. This is less, in present value terms, than paying $20,000 cash. The firm should take the annuity payment plan.

Similar to Problems 5­7, 5­8, self­test problem 2

The present value of a cash flow made far into the future is not very valuable today, as Figure 5.2 illustrates. That's why doubling the number of years in the table from 20 to 40 only increased the present value by approximately 30 percent. Notice how the

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present value of $100 annuity payments declines for the cash flows made later in time, especially at higher discount rates. The $100 cash flow in year 20 is worth less than $15 today if we use a 10 percent discount rate; they're worth more than double, at nearly $38 today, if we use a discount rate of 5 percent. The figure also shows how quickly present value declines with a higher discount rate relative to a lower rate. As we showed above, the present values of the annuities in the figure are the sums of the present values shown. Since the present values for the 10 percent discount rate are smaller, the present value of an annuity is smaller as interest rates rise.

Present Value of Multiple Annuities

Just as we can combine annuities to solve various future value problems, we can also combine annuities to solve some present value problems with changing cash flows. Consider Alex Rodriguez's (A-Rod's) baseball contract in 2000 with the Texas Rangers. This contract made A-Rod into the "$252 million man." The contract was structured so that the Rangers paid A-Rod a $10 million signing bonus, $21 million per year in 2001 through 2004, $25 million per year in 2005 and 2006, and $27 million per year in 2007 through 2010.1 Notethatadding the signing bonus to the annual salary equals the $252 million figure. However, Rodriguez would receive the salary in the future. Using an 8 percent discount rate, what is the present value of A-Rod's contract?

The reported values for many sports contracts may be misleading in present value terms.

FIGURE S.2 Present Value of Each Annuity Cash Flow

https://phoenix.vitalsource.com/#/books/1259827178/cfi/6/24!/4/2@0:0 29 /45

We begin by showing the salary cash flows with the time line:

First create a $27 million, 10-year annuity. Here are the associated cash flows:

Now create a -$2 million, six-year annuity:

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Notice that creating the -$2 million annuity also resulted in the third annuity of -$4 million for four years. This time line shows three annuities. If you add the cash flows in any year, the sum is A-Rod's salary for that year. Now we can find the present value of each annuity using equation 5-4 three times.

Adding the value of the three annuities reveals that the present value of A-Rod's salary was $158.67 million (= $181.17m - $9.25m - $13.25m). Adding in the $10 million signing bonus produces a contract value of $168.67 million. So, the present value of A-Rod's contract turns out to be quite considerable, but you might not call him the $252 million man!2

Perpetuity-A Special Annuity LGS-S
perpetuity An annuity with cash flows that continue forever. consols Investment assets structured as perpetuities.

A perpetuity is a special type of annuity with a stream of level cash flows that are paid forever. These arrangements are called perpetuities because payments are perpetual. Assets that offer investors perpetual payments are preferred stocks and British 21/% Consolidated Stock, a debt referred to as

consols.
The value of an investment like this is the present value of all future annuity payments. As the cash flow continues indefinitely, we can't use equation 5-4. Luckily, mathematicians have figured out that when the number of periods, N, in equation 5-4 goes to infinity, the equation reduces to a very simple one:

Present value of a perpetuity = Payment ""7 Interest rate

For example, the present value of an annual $100 perpetuity discounted at 10 percent is $1,000 (=

$100 ""7 0.10). Compare this to the present value of a $100 annuity of 40 years as shown in Table 5.2. The 40-year annuity's value is $977.91. You'll see that extending the payments from 40 years to an infinite number of years adds only $22.09 (= $1,000 - $977.91) of value. This demonstrates once again how little value today is placed on cash flows paid many years into the future.

time out!

S-3 How important is the magnitude of the discount rate in present value computations? Do significantly higher interest rates lead to significantly higher present values?

S-4 Reconsider the physical therapy equipment example. If interest rates are only 7 percent, should you pay the up-front fee or the annuity?

ORDINARY ANNUITIES VERSUS ANNUITIES DUE LGS-6
So far, we've assumed that every cash flow comes in at the end of every period. But in many instances, cash flows come in at the beginning of each period. An annuity in which the cash flows occur at the beginning of each period is called an annuity due.

annuity due An annuity in which cash flows are paid at the beginning of each time period.

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Consider the 5-year $100 annuity due. The cash flow in the beginning of year 1 looks like it's actually a cash flow today.

Note that these five annuity-due cash flows are essentially the same as a payment today and a four- year ordinary annuity.

Future Value of an Annuity Due So, how do we calculate the future value of the 5-year

annuity due shown in the time line? The first cash flow of an ordinary 5-year annuity can compound for four years. The last cash flow does not compound at all. From the time line, you can see that the first cash flow of the annuity due essentially occurs in year zero, or today. So the first cash flow compounds for five years. The last cash flow of an annuity due compounds one year. The main difference between an annuity due and an ordinary annuity is that all the cash flows of the annuity due compound one more year than the ordinary annuity. The future value of the annuity due will simply be the future value of the ordinary annuity multiplied by (1 + i):

Earlier in the chapter, the future value of this ordinary annuity was shown to be $586.66. Therefore, the future value of the annuity due is $633.59 (= $586.66 x 1.08).

Present Value of an Annuity Due What is a five-year annuity due, shown previously, worth today? Remember that we discount the first cash flow of an ordinary five-year annuity one year. We discount the last cash flow for the full five years. But since the first cash flow of the annuity due is already paid today, we don't discount it at all. We discount the last cash flow of an annuity due only four years. Indeed, we discount all the cash flows of the annuity due one year less than we would discount the ordinary annuity. Therefore, the present value of the annuity due is simply the present value of the ordinary annuity multiplied by (1 + i):

Earlier in the chapter, we discovered that the present value of this ordinary annuity was $399.27. So the present value of the annuity due is $431.21 (= $399.27 x 1.08).

Interestingly, we make the same adjustment, (1 + i), to both the ordinary annuity present value and future value to compute the annuity due value.

THE MAIN DIFFERENCE BETWEEN AN ANNUITY DUE AND AN ORDINARY ANNUITY IS THAT ALL THE CASH FLOWS OF THE ANNUITY DUE COMPOUND ONE MORE YEAR THAN THE ORDINARY

ANNUITY.

the Math Coach on...

Setting Financial Calculators for Annuity Due

Financial calculators can be set for beginning-of-period payments. Once set, you compute future and present values of annuities due just as you would the ordinary annuity. To set the HP calculator, press the color button followed by the BEG/END button. To set the TI calculator for an annuity due, push the 2ND button, followed by the BGN button, followed by the 2ND button again, followed by the SET button, and followed by the 2ND button a third time, finally the QUIT button. To set the HP and TI calculators back to end-of-period cash flows, repeat these procedures.

finance at work //: behavioral

Take Your Lottery Winnings Now or Later?
On March 31, 2012, three lottery players co-won a record Powerball jackpot of $640 million.

The winners had two choices for payment: They could split a much-discounted lump sum cash payment immediately or take 30 annuity payments (one immediately and then one every year

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for 29 years, which is a 30-year annuity due). The annuity payment to be split would be $24.61 million (=

$640 million ""7 26)for 26 years. The immediate lump sum offered to be split was $462.3 million. One way to decide between the two alternatives would be to use the time value of money concepts. The winners might have computed the present value of the annuity and compared it to the lump sum cash payment.

At the time, long-term interest rates were 3.2 percent. The present value of the annuity offered was $443.85 million. (Compute this yourself.) Notice that winning $640 million does not deliver $640 million of value! If the decision was made from this perspective, the group should take the lump sum choice because it has more value than the annuity alternative, and that is what the three winners did. In fact, most winners do. Financial advisors tend to recommend that lottery winners take the lump sum because they believe that the money can earn a higher return than the 3.2 percent interest rate.

Good reasons arise for taking the annuity, however. To earn the higher return on the lump sum, the advisor (and the group of owners) would have to take risks. In addition, most of the lump sum would have to be invested. But most people who choose the lump sum end up spending much of it in the first couple of years. Stories abound about lottery winners who declare bankruptcy a few years after receiving their money. Choosing the annuity helps instill financial discipline, since the winners can't waste money today that they won't receive for years.

Want to know more?

Key Words to Search for Updates: Powerball winners (go to www.powerball.com )

Source: Ronald D. Orol, "3 Winning Tickets Sold in $640 Million Lottery," Market Watch, March 31, 2012, http://articles.marketwatch.com/2012-03-31/general/31265196_1_million-lottery-kansas-lottery-ticket.

time out!

S-S In what situations might you need to use annuity due analysis instead of an ordinary annuity analysis?

S-6 Reconsider your retirement plan earlier in this chapter. What would your retirement wealth grow to be if you started contributing today?

COMPOUNDING FREQUENCY
So far, all of our examples and illustrations have used annual payments and annual compounding or discounting periods. But many situations that use cash flow time-value-of-money analysis require more frequent or less frequent time periods than simple yearly entries. Bonds make semiannual interest payments; stocks pay quarterly dividends. Most consumer loans require monthly payments. Monthly payments require monthly compounding. In this section, we'll discuss the implications of compounding more than once a year.

Effect of Compounding Frequency
Consider a $100 deposit made today with a 12 percent annual interest rate. What's the future value of this deposit in one year? Equation 4-2 from the previous chapter shows that the answer is $112. What would happen if the bank compounded the interest every six months instead of at the end of the year? Halfway through the year, the bank would compute that the deposit has grown 6 percent (half the annual 12 percent rate) to $106. At the end of the year, the bank would compute another 6 percent interest payment. However, this 6 percent is earned on $106, not the original $100 deposit. The end-of- year value is therefore $112.36 (= $106 x 1.06). By compounding twice per year instead of just once, the future value is $0.36 higher. Though this amount may seem negligible, you might be surprised to see how quickly the difference becomes significant.

Instead of compounding annually or semiannually, what might happen if compounding were quarterly? Since each year contains four quarters, the interest rate per quarter would be 3 percent (= 12 percent ""7 4 quarters). The future value in one year, compounded quarterly, is $112.55 (= $100 x

1.034). Again, the compounding frequency increased and so did the future value.

the Math Coach on...

Annuity Computations in Spreadsheets

The TVM functions in a spreadsheet handle annuity payments similar to financial calculators. The spreadsheet Math Coach in Chapter 4 shows the functions. The functions have an annuity input.

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The example illustrated earlier in this chapter asks for the FV of annual $100 deposits earning 8 percent. The spreadsheet solution is the same as the equation and calculator solutions. If you want the FV of an annuity due, just change the type from 0 to 1.