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time value of money 2:
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:
the Math Coach on...
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.
the Math Coach on...
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.
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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.
Log in to your Connect course to watch instructional videos on using spreadsheets.
Table 5.3 shows the effect of various compounding frequencies. We'd like to draw your attention to two important points in the table. First, the higher the compound frequency, the higher the future value will be. Second, the relative increase in value from increasing compounding frequency seems to diminish with increasing frequencies. For example, increasing frequency from annual to semiannual increased the future value by 36 cents. However, increasing frequency from daily to hourly compounding increases the future value by only 0.1 cent.3
EXAMPLE Car Loan Debt LGS-4
S-S
Now
you
would
like
to
buy
a
car.
You
have
reviewed
your
budget
and
determined
that
you
can
afford
to
pay
$500
per
month
as
a
car
payment.
How much
can
you
borrow if interest
rates
are 9
percent and
you
pay
the
loan
over
four
years?
How
much
could
you
borrow
if
you
agree
to
pay
for
six
years
instead?
SOLUTION:
The loan amount is the present value of the 48-month, $500 annuity. Note that the loan term will be 48 (= 4 x 12) months and the interest rate is 0.75 (= 9
""7
12) percent. Using equation 5-4, you discover that you can borrow up to $20,092 to buy a
car:
If
you
are willing to borrow
money
for six
years
instead of four, the small
change
to the
equation
results in
your
ability to borrow
$27,738. Although
this
would allow you
to
buy
a more
expensive car,
it
would
also require
two
more
years
of
$500
payments
(an
additional $12,000
of
payments!).
Similar to Problems 525, 526,
selftest
problem
4
annual percentage rate (APR) The interest rate per period times the number of periods in a year. effective annual rate (EAR) An interest rate that reflects annualizing with compounding figured in.
When we work with annuity cash flows, the compound frequency used is the same as the timing of the cash flows. When annuity cash flows are paid monthly, then interest is also compounded monthly, as seen in Examples 5-5 and 5-6.
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EARs and APRs If you borrowed $100 at a 12 percent interest rate, you would expect to pay
$112 in one year. If the loan compounded monthly, then you would owe $112.68 at the end of the year, as Table 5.3 shows. So a 12 percent loan compounded monthly means that you really pay more
than 12 percent. In fact, you would pay 12.68 percent. In this example, the 12 percent rate is called the annual percentage rate (APR). The higher rate, 12.68 percent, is called the effective annual rate (EAR)-a more accurate measurement of what you will actuallypay.
Lenders are legally required to show potential borrowers the APR on any loan offered. While the difference in APR and EAR is not that large in this example, it's interesting that the law requires only the less accurate (and lower) one to be shown. Since the EAR is a more accurate measure of what you will pay, it's useful to know how to convert a stated APR to an EAR. Equation 5-8 shows this conversion with a compounding frequency of m times per year:
Table 5.4 shows various EAR conversions. If compounding occurs annually, you will see that the EAR and the APR will be the same. If compounding happens more than once a year, then the EAR will be higher than the APR. The table also demonstrates that the compound frequency effect grows substantially for higher interest rates or longer term loans. Compounded quarterly, the EAR is hardly different at all from a 5 percent APR: 5.09 percent versus 5 percent. The difference is larger when the APR is 12 percent. Compounded quarterly, the EAR is higher at 12.55 percent.
TABLE S.3 Future Value in One Year and Compounding Frequency of $100 at 12 percent
EXAMPLE S-6
Making Monthly Pension Contributions LGS-7
Reexamine your original plan to contribute to your company retirement plan. Instead of a total contribution of $4,500 per year for 40 years, you are able to contribute monthly. Given your expected 7 percent per year
investment return, how much money can you expect in your retirement account?
SOLUTION:
Now your total monthly contribution will be $375 (= $4,500 ""7 12), which will continue for 480 months and earn a 0.58333 (= 7 ""7 12) percent monthly return. The results of equation 5-2 show that the future value of this annuity is:
When you made contributions annually, the future value was $898,358 (Example 5-2). By changing to monthly contributions, your retirement nest egg increased by nearly $86,000 to $984,305!
Similar to Problems 551, 552, selftest problem 3
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Common Mistakes
As you figure present and future values of annuity cash flows, check that all terms are consistent: the number of payments, interest rate, and payment size all need to use common terms. If your payments are monthly, then the number of payments must reflect
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the number of months; the interest rate must be stated as a per-month rate, and the payment register must reflect that monthly
time out!
S-7 Why is EAR a more accurate measure of the rate actually paid than APR?
S-8 What would have a smaller present value: a future sum discounted annually or one discounted monthly?
TABLE S.4 The EAR Is Higher than the APR
Note: This compound frequency effect grows substantially for higher interest rates or longer term loans.
EXAMPLE
S-7 Evaluating Credit Card Offers LGS-7
As a college student, you probably receive many credit card offers in the mail. Consider these two offers. The first card charges a 16 percent APR. An examination of the footnotes reveals that this card compounds monthly. The second credit card charges 15.5 percent APR and compounds weekly. Which card has a lower effective annual rate?
SOLUTION:
Compute the EAR of each card to compare them in common (and realistic) terms. The first card has an EAR of:
The EAR of the second card is:
You should pick the second credit card because it has a lower effective annual rate. But note also that you will always be better off if you pay your credit card balance whenever the bill comes due.
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Similar to Problems 515, 516, selftest problem 3
ANNUITY LOANS
In this chapter, we've focused on computing the future and present value of annuities. But in many situations, these values are already known and what we really need to compare are the payments or implied interest rate- usually, the highest interest rate offered.
What Is the Interest Rate? LGS-8
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Many business and personal applications already state the cost of an investment, as well as the annuity cash flows and time period. We need, then, to solve for the implied interest rate of this investment. Unfortunately, we have no general, easy equation to solve for the interest rate. Even financial calculators use an iterating process, which causes them to "think" a little longer before displaying the estimated interest rate result.
Consider the plight of a manager of a small doctor's office who has the opportunity to buy a piece of imaging equipment for $100,000. The equipment will allow the office to generate $25,000 in profits for six years, at which time the equipment will be worn out and without value in the United States.4 What rate of return does this purchase offer the doctor's office? The time line for this problem appears as:
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More Common Mistakes
As we noted in Chapter 4, when computing the interest rate, make sure that the present value and the
annuity payments are of different signs (positive versus negative). Otherwise, the calculator will show an error.
For the financial calculator solution, input N = 6, PV = -100000, PMT = 25000, and FV = 0. The interest rate result is then 12.98 percent. So, if this is a high enough return relative to other uses of the
$100,000, the doctor's office should seriously consider purchasing the imaging machine.
Finding Payments on an Amortized Loan LGS-9
Many consumers and small business owners already know how much money they want to borrow and the level of current interest rates. Usually, they need to translate this information into the actual payments to determine if they can really afford the purchase. A loan structured for annuity payments that completely pay off the debt is called an amortized loan. To compute the annuity cash flow of an amortized loan, rearrange the present value of an annuity formula, equation 5-4, to solve for the payment:
EXAMPLE S-8
Computing Interest Rate Needed LGS- 8
After saving diligently throughout your entire career, you and your spouse are finally ready to retire with a nest egg of $800,000. You need to invest this money in a mix of stocks and bonds that will allow you to withdraw $6,000 per month for 30 years. What interest rate do you need to earn?
SOLUTION:
Use a financial calculator and input N = 360, PV = -800000, PMT = 6000, and FV = 0. The interest rate result is 0.6860 percent. But remember, since the periods and payments are in months, the interest rate is too. It is customary to report this as an APR: 8.23 percent (= 0.6860 percent x 12). However, the EAR more accurately reflects the true interest rate, 8.55 percent (= 1.0068612 - 1). In order for your money to last for 30 years while funding a $6,000 per month income, you must earn at least an 8.23 APR per year return.
If you have uneven cash flows, use the calculator CF worksheet and then solve with the IRR function. Similar to Problems 533, 534, 535, 536
Most car loans require monthly payments for three to five years. Assume that you need a $10,000 loan to buy a car. The loan is for four years and interest rates are 9 percent per year APR. To implement equation 5-9, use an interest rate of 0.75 percent (= 9
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percent/12) and 48 periods (= 4 x 12)as:
So, when interest rates are 9 percent, it takes monthly payments of $248.85 to pay off a $10,000 loan in four years.
Interest rate levels and loan length strongly affect how large your payments will be. Table 5.5 shows the monthly payments needed to pay off a mortgage debt at various interest rates and lengths of time. (Try computing the payments yourself!) Note that as the interest rate declines, the monthly payment also declines. This is why people rush to refinance their mortgages after interest rates fall. A decline of 1 or 2 percent can save a homeowner hundreds of dollars every month. You will also see from the table that paying off a mortgage in only 15 years requires larger payments, but generally saves thousands in interest.
Amortized Loan Schedules When you pay a car loan or home mortgage, you will often find it useful to know how much of the debt, or loan principal, you still owe. For example, consider a case wherein you bought a car two years ago using a 4-year loan. In order to sell the car now, the loan balance will have to be paid off. Being able to compute this principal balance may influence your chances of selling the car.
An interest-only loan allows the borrower to make payments that consist totally of interest payments, so none of the debt is reduced. A $10,000 interest-only loan with a 9 percent APR paid monthly will cost $75 per month (= $10,000 x 0.09 ""7 12). Amortizing this loan over four years requires monthly payments of $248.85 (see earlier car loan problem). The difference in the first month's payment on the two loans is $173.85 (= $248.85 - $75) and represents the amount of the regular amortized loan's payment that goes to reducing the principal balance. So after the first month's payment, the amortized loan's balance has fallen to $9,826.15, while the interest-only loan still has a balance of $10,000.
In the second month, the interest incurred on the regular amortized loan is $73.70 (= $9,826.15 x 0.09 ""7 12), so the $248.85 second-month payment represents principal payment of $175.15. These numbers are shown in the amortization schedule of Table 5.6. The table will show you that the early payments on a car loan go mostly to paying the interest rather than reducing the principal. That interest component declines over time, and then the principal balance declines.
The amortization schedule shows that if you wish to sell the car after two years, you will have to pay the loan company a car loan (principal) debt of $5,447.13. Of course, if you had an interest-only loan, you would still owe the full principal of $10,000 after two years. Amortization schedules are also useful for determining other things, like the total amount of interest that you will pay over the life of the loan. In this case, if you take a regular loan in which you pay both principal and interest, you pay
$10,000 in principal and nearly $1,945 in interest during the four years of the loan. The interest component is an even larger component of longer-term loans, like 30-year mortgages. Depending on the interest rate charged, the first payment in a mortgage consists of 75 percent to 95 percent interest. The home mortgage principal balance falls very slowly in the first years of the loan.
TABLE S.S Monthly Payments on a $225,000 Loan
amortized loan A loan in which the borrower pays interest and principal over time. loan principal The balance yet to be paid on a loan.
amortization schedule A table detailing the periodic loan payment, interest payment, and debt balance over the life of the loan.
We construct amortization schedules by showing the loan's principal balance at the beginning of the month. This is the same as the balance at the end of the previous month (except for the very first payment). Then we compute the interest owed on that balance
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for the month. After paying that interest, what's left of the monthly payment reduces the loan balance for the next month. Because of these repetitive computations, spreadsheets make amortization schedules easy to construct.
Compute the Time Period LGS-10 You might also find it useful to know how long it will take to pay off a loan with specific annuity payments. To find the number of periods, you can solve equation 5-9 for N-the number of payments-but the equation becomes quite complicated.6 Many people just use a financial calculator or spreadsheet. We can check to see if the $248.85 monthly payment would indeed pay off the $10,000, 9 percent car loan in four years. Finding the solution with a financial calculator entails entering I = 0.75, PV = 10000, PMT = - 248.85, and FV = 0. The answer is 48 months.
Depending on the interest rate charged, the first payment in a mortgage consists of 75 percent to 95 percent interest.
TABLE S.6 Amortization Schedule over Four Years (9 percent APR)
EXAMPLE
S-9 Monthly Mortgage Payments LGS-9
Say that you have your heart set on purchasing a beautiful, old Tudor-style house for $250,000. A mortgage
broker says that you can qualify for a mortgage for 80 percent (or $200,000) of the price. If you get a 15-year mortgage, the interest rate will be 6.1 percent APR. A 30-year mortgage costs 6.4 percent. One of the factors that will help you decide which mortgage to take is the magnitude of the monthly payments. What will they be? 5
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SOLUTION:
To pay off the mortgage in only 15 years, the payments would have to be larger than for the 30-year mortgage. The higher payment will be eased somewhat because the interest rate is lower on the 15-year mortgage. The payment for the 15-year mortgage is:
The payment for the 30-year mortgage would be:
So, the payments on the 15-year mortgage are nearly $450 more each month than the 30-year mortgage payments. You must decide whether the cost of paying the extra $450 each month is worth it to own the house with no debt 15 years sooner. The decision would depend on your financial budget and the strength of your desire to be debt free.
Similar to Problems 539, 540, selftest problems 1 and 4
time out!
S-9 How might credit card companies keep their cardholders in debt for a long time? What payment do the credit card companies expect your friend to make so that he never pays down the debt?
S-10 Can you find the interest rate if you know the annuity payments and a future value? Under what circumstances might you want to solve this kind of problem? Which equation would you use?
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Using the AMORT Function in TVM Calculators
TVM calculators have preprogrammed functions to compute the amount of principal paid part way through a mortgage. For example, if you took out a 30-year, $200,000 mortgage at a 6 percent APR, how much principal have you paid after five years? How much do you still owe? How much interest have you paid?
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To answer these questions, first enter the mortgage information to compute the monthly payments. Then use the AMORT function. This example uses the Texas Instruments BA II + as an example. The Hewlett- Packard and other TVM calculators have similar functions. The AMORT function allows you to compute the loan balance at any time during the mortgage period. It also computes the amount of principal and interest that has been paid during any time period. To answer the questions above:
1. Press 2nd AMORT and P1 = 1 appears.s (This refers to the first payment of the mortgage.)
2. Press the down arrow L, P2 = appears. (This refers to the last payment made.)
3. The question refers to 5 years of payments, which is 60 months. Enter 60 and press ENTER.
The calculator has now computed the loan balance after the 60th payment and the amount of principal and interest that have been paid between the first and the 60th payment.
4. Press the down arrow L, displayed is BAL = 186,108.71, which is the loan balance.
5. Press the down arrow L, displayed is PRN = 13,891.29, which is the principal paid in the first 5 years.
6. Press the down arrow L, displayed is INT = -58,054.78, which is the interest paid in the first 5 years. Note that in the beginning of a mortgage, far more interest is paid than principal.
EXAMPLE Time to Pay Off a Credit Card
S-10 Balance LGS-10
Through poor financial management, your friend has racked up $5,000 in debt on his credit card. The card charges a 19 percent APR and compounds monthly. His latest bill shows that he must pay a minimum of
$150 this month. At this rate, how long will it take your friend to pay off his credit card debt?
SOLUTION:
Using the financial calculator, input I = 1.58333 (= 19/12), PV = 5000, PMT = -150, FV = 0. The answer is 48 months, or 4 years. If the friend pays the minimum payment, then it will be a long time before he will be out of debt. The credit card company is very content to continue to earn the high return for many years- essentially, the interest on the loan and a very small portion of the principal. Your friend should pay more than the minimum charge to reduce his debt quicker.
Similar to Problems 541, 542, selftest problem 4
add-on interest A calculation of the amount of interest determined at the beginning of the loan and then added to the principal.
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Add-On Interest One method of calculating payments of a loan that is popular in payday lending is called add-on interest. This method computes the amount of the interest payable at the beginning of the loan, which is then added to the principal of the loan. This total is then divided into the number of payments to be made. Consider a loan of $1,000 to be paid with 9 percent add-on interest and repaid in six monthly payments.
The total interest for this loan is computed as 9 percent of $1,000 for 6 months, or $45 (= 0.09 x
$1,000 x 1/). This is added to the principal for a total of $1,045. Each of the six monthly payments is then $1,045 ""7 6 = $174.17. Be alert that the add-on interest method seriously understates the real interest rate that is being paid! If you borrow $1,000 and repay a $174.17 monthly annuity for six months, the monthly interest rate is 1.27 percent. This is a 15.27 percent APR (= 1.27% x 12)and a
16.39 percent EAR (= 1.012712 - 1)-both much higher than the advertised 9 percent interest rate of this loan.
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Your Turn . . .
Questions
1. How can you add a cash flow in year 2 and a cash flow in year 4? In year 7? (LG51)
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2. People can become millionaires in their retirement years quite easily if they start saving early in employer 401(k) or 403(b) programs (or even if their employers don't offer such programs). Demonstrate the growth of a $250 monthly contribution for 40 years earning 9 percent APR. (LG52)
3. When you discount multiple cash flows, how does the future period that a cash flow is paid affect its present value and its contribution to the value of all the cash flows? (LG53)
4. How can you use the present value of an annuity concept to determine the price of a house you can afford? (LG54)
S. Since perpetuity payments continue forever, how can a present value be computed? Why isn't the present value infinite? (LG55)
6. Explain why you use the same adjustment factor, (1 + i), when you adjust annuity due payments for both future value and present value. (LG56)
7. Use the idea of compound interest to explain why EAR is larger than APR. (LG57)
8. Would you rather pay $10,000 for a 5-year, $2,500 annuity or a 10-year, $1,250 annuity? Why? (LG58)
9. The interest on your home mortgage is tax deductible. Why are the early years of the mortgage more helpful in reducing taxes than in the later years? (LG59)
10. How can you use the concepts illustrated in computing the number of payments in an annuity to figure how to pay off a credit card balance? How does the magnitude of the payment impact the number of months? (LG510)
Problems
BASIC PROBLEMS
S-1 FutureValue Compute the future value in year 9 of a $2,000 deposit in year 1 and another $1,500 deposit at the end of year 3 using a 10 percent interest rate. (LG51)
S-2 FutureValue Compute the future value in year 7 of a $2,000 deposit in year 1 and another $2,500 deposit at the end of year 4 using an 8 percent interest rate. (LG51)
S-3 Future Value of an Annuity What is the future value of a $900 annuity payment over five years if interest rates are 8 percent? (LG52)
S-4 Future Value of an Annuity What is the future value of a $700 annuity payment over six years if interest rates are 10 percent? (LG52)
S-S Present Value Compute the present value of a $2,000 deposit in year 1 and another $1,500 deposit at the end of year 3 if interest rates are 10 percent. (LG53)
S-6 Present Value Compute the present value of a $2,000 deposit in year 1 and another $2,500 deposit at the end of year 4 using an 8 percent interest rate. (LG53)
S-7 Present Value of an Annuity What's the present value of a $900 annuity payment over five years if interest rates are 8 percent? (LG54)
S-8 Present Value of an Annuity What's the present value of a $700 annuity payment over six years if interest rates are 10 percent? (LG54)
S-9 Present Value of a Perpetuity What's the present value, when interest rates are 7.5 percent, of a $50 payment made every year forever? (LG55)
S-10 Present Value of a Perpetuity What's the present value, when interest rates are 8.5 percent, of a $75 payment made every year forever? (LG55)
S-11 Present Value of an Annuity Due If the present value of an ordinary, 7-year annuity is $6,500 and interest rates are 7.5 percent, what's the present value of the same annuity due? (LG56)
S-12 Present Value of an Annuity Due If the present value of an ordinary, 6-year annuity is $8,500 and interest rates are 9.5 percent, what's the present value of the same annuity due? (LG56)
S-13 Future Value of an Annuity Due If the future value of an ordinary, 7-year annuity is
$6,500 and interest rates are 7.5 percent, what is the future value of the same annuity due? (LG56)
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S-14 FutureValue of an Annuity Due If the future value of an ordinary, 6-year annuity is $8,500 and interest rates are 9.5 percent, what's the future value of the same annuity due? (LG56)
S-1S Effective Annual Rate A loan is offered with monthly payments and a 10 percent APR. What's the loan's effective annual rate (EAR)? (LG57)
S-16 Effective Annual Rate A loan is offered with monthly payments and a 13 percent APR. What's the loan's effective annual rate (EAR)? (LG57)
S-17 Future Value Given a 4 percent interest rate, compute the year 6 future value of deposits made in years 1, 2, 3, and 4 of $1,100, $1,200, $1,200, and $1,500. (LG51)
INTERMEDIATE PROBLEMS
S-18 Future Value Given a 5 percent interest rate, compute the year 6 future value of deposits made in years 1, 2, 3, and 4 of $1,000, $1,300, $1,300, and $1,400. (LG51)
S-19 Future Value of Multiple Annuities Assume that you contribute $200 per month to a retirement plan for 20 years. Then you are able to increase the contribution to $300 per month for another 30 years. Given a 7 percent interest rate, what is the value of your retirement plan after the 50 years? (LG52)
S-20 Future Value of Multiple Annuities Assume that you contribute $150 per month to a retirement plan for 15 years. Then you are able to increase the contribution to $350 per month for the next 25 years. Given an 8 percent interest rate, what is the value of your retirement plan after the 40 years? (LG52)
S-21 Present Value Given a 6 percent interest rate, compute the present value of payments made in years 1, 2, 3, and 4 of $1,000, $1,200, $1,200, and $1,500. (LG53)
S-22 Present Value Given a 7 percent interest rate, compute the present value of payments made in years 1, 2, 3, and 4 of $1,000, $1,300, $1,300, and $1,400. (LG53)
S-23 Present Value of Multiple Annuities A small business owner visits her bank to ask for a loan. The owner states that she can repay a loan at $1,000 per month for the next three years and then $2,000 per month for two years after that. If the bank is charging customers 7.5 percent APR, how much would it be willing to lend the business owner? (LG54)
S-24 Present Value of Multiple Annuities A small business owner visits his bank to ask for a loan. The owner states that he can repay a loan at $1,500 per month for the next three years and then $500 per month for two years after that. If the bank is charging customers 8.5 percent APR, how much would it be willing to lend the business owner? (LG54)
S-2S Present Value You are looking to buy a car. You can afford $450 in monthly payments for four years. In addition to the loan, you can make a $1,000 down payment. If interest rates are 5 percent APR, what price of car can you afford? (LG54)
S-26 Present Value You are looking to buy a car. You can afford $650 in monthly payments for five years. In addition to the loan, you can make a $750 down payment. If interest rates are 8 percent APR, what price of car can you afford? (LG54)
S-27 Present Value of a Perpetuity A perpetuity pays $100 per year and interest rates are 7.5 percent. How much would its value change if interest rates increased to 9 percent? Did the value increase or decrease? (LG55)
S-28 Present Value of a Perpetuity A perpetuity pays $50 per year and interest rates are 9 percent. How much would its value change if interest rates decreased to 7.5 percent? Did the value increase or decrease? (LG55)
S-29 Future and Present Value of an Annuity Due If you start making $50 monthly contributions today and continue them for five years, what's their future value if the compounding rate is 10 percent APR? What is the present value of this annuity? (LG56)
S-30 Future and Present Value of an Annuity Due If you start making $75 monthly contributions today and continue them for four years, what is their future value if the compounding rate is 12 percent APR? What is the present value of this annuity? (LG56)
S-31 Compound Frequency Payday loans are very short-term loans that charge very high
interest rates. You can borrow $225 today and repay $300 in two weeks. What is the compounded annual rate implied by this 33.33 percent rate charged for only two weeks? (LG57)
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S-32 Compound Frequency Payday loans are very short-term loans that charge very high interest rates. You can borrow $500 today and repay $590 in two weeks. What is the compounded annual rate implied by this 18 percent rate charged for only two weeks? (LG57)
S-33 Annuity Interest Rate What's the interest rate of a 6-year, annual $5,000 annuity with present value of
$20,000? (LG58)
S-34 Annuity Interest Rate What's the interest rate of a 7-year, annual $4,000 annuity with present value of
$20,000? (LG58)
S-3S Annuity Interest Rate What annual interest rate would you need to earn if you wanted a $1,000 per month contribution to grow to $75,000 in six years? (LG58)
S-36 Annuity Interest Rate What annual interest rate would you need to earn if you wanted a $600 per month contribution to grow to $45,000 in six years? (LG58)
S-37 Add-On Interest Payments To borrow $500, you are offered an add-on interest loan at 8 percent. Two loan payments are to be made, one at six months and the other at the end of the year. Compute the two equal payments. (LG58)
S-38 Add-On Interest Payments To borrow $800, you are offered an add-on interest loan at 7 percent. Three loan payments are to be made, one at four months, another at eight months, and the last one at the end of the year. Compute the three equal payments. (LG58)
S-39 Loan Payments You wish to buy a $25,000 car. The dealer offers you a 4-year loan with a 9 percent APR. What are the monthly payments? How would the payment differ if you paid interest only? What would the consequences of such a decision be? (LG59)
S-40 Loan Payments You wish to buy a $10,000 dining room set. The furniture store offers you a 3-year loan with an 11 percent APR. What are the monthly payments? How would the payment differ if you paid interest only? What would the consequences of such a decision be? (LG59)
S-41 Number of Annuity Payments Joey realizes that he has charged too much on his credit card and has racked up $5,000 in debt. If he can pay $150 each month and the card charges 17 percent APR (compounded monthly), how long will it take him to pay off the debt? (LG510)
S-42 Number of Annuity Payments Phoebe realizes that she has charged too much on her credit card and has racked up $6,000 in debt. If she can pay $200 each month and the card charges 18 percent APR (compounded monthly), how long will it take her to pay off the debt? (LG510)
S-43 Future Value Given an 8 percent interest rate, compute the year 7 future value if deposits of $1,000 and
$2,000 are made in years 1 and 3, respectively, and a withdrawal of $700 is made in year 4. (LG510)
ADVANCED PROBLEMS
S-44 Future Value Given a 9 percent interest rate, compute the year 6 future value if deposits of $1,500 and
$2,500 are made in years 2 and 3, respectively, and a withdrawal of $600 is made in year 5. (LG51)
S-4S EAR of Add-On Interest Loan To borrow $2,000, you are offered an add-on interest loan at 10 percent with 12 monthly payments. First compute the 12 equal payments and then compute the EAR of the loan. (LG57, LG58)
S-46 EAR of Add-On Interest Loan To borrow $700, you are offered an add-on interest loan at 9 percent with 12 monthly payments. First compute the 12 equal payments and then compute the EAR of the loan. (LG57, LG58)
S-47 Low Financing or Cash Back? A car company is offering a choice of deals. You can receive $500 cash back on the purchase or a 3 percent APR, 4-year loan. The price of the car is $15,000 and
you could obtain a 4-year loan from your credit union, at 6 percent APR. Which deal is cheaper? (LG54, LG59)
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S-48 Low Financing or Cash Back? A car company is offering a choice of deals. You can receive $1,000 cash back on the purchase, or a 2 percent APR, 5-year loan. The price of the car is $20,000 and you could obtain a 5-year loan from your credit union, at 7 percent APR. Which deal is cheaper? (LG54, LG59)
S-49 Amortization Schedule Create the amortization schedule for a loan of $15,000, paid monthly over three years using a 9 percent APR. (LG59)
S-S0 Amortization Schedule Create the amortization schedule for a loan of $5,000, paid monthly over two years using an 8 percent APR. (LG59)
S-S1 Investing for Retirement Monica has decided that she wants to build enough retirement wealth that, if invested at 8 percent per year, will provide her with $3,500 of monthly income for 20 years. To date, she has saved nothing, but she still has 30 years until she retires. How much money does she need to contribute per month to reach her goal? (LG54, LG59)
S-S2 Investing for Retirement Ross has decided that he wants to build enough retirement wealth that, if invested at 7 percent per year, will provide him with $3,000 of monthly income for 30 years. To date, he has saved nothing, but he still has 20 years until he retires. How much money does he need to contribute per month to reach his goal? (LG54, LG59)
S-S3 Loan Balance Rachel purchased a $15,000 car three years ago using an 8 percent, 4-year loan. She has decided that she would sell the car now, if she could get a price that would pay off the balance of her loan. What is the minimum price Rachel would need to receive for her car? (LG59)
S-S4 Loan Balance Hank purchased a $20,000 car two years ago using a 9 percent, 5-year loan. He has decided that he would sell the car now, if he could get a price that would pay off the balance of his loan. What's the minimum price Hank would need to receive for his car? (LG59)
S-SS Teaser Rate Mortgage A mortgage broker is offering a $183,900 30-year mortgage with a teaser rate. In the first two years of the mortgage, the borrower makes monthly payments on only a 4 percent APR interest rate. After the second year, the mortgage interest rate charged increases to 7 percent APR. What are the monthly payments in the first two years? What are the monthly payments after the second year? (LG59)
S-S6 Teaser Rate Mortgage A mortgage broker is offering a $279,000 30-year mortgage with a teaser rate. In the first two years of the mortgage, the borrower makes monthly payments on only a 4.5 percent APR interest rate. After the second year, the mortgage interest rate charged increases to 7.5 percent APR. What are the monthly payments in the first two years? What are the monthly payments after the second year? (LG59)
S-S7 Spreadsheet Problem Consider a person who begins contributing to a retirement plan at age 25 and contributes for 40 years until retirement at age 65. For the first ten years, she contributes $3,000 per year. She increases the contribution rate to $5,000 per year in years 11 through 20. This is followed by increases to $10,000 per year in years 21 through 30 and to $15,000 per year for the last ten years. This money earns a 9 percent return. First compute the value of the retirement plan when she turns age 65. Then compute the annual payment she would receive over the next 40 years if the wealth was converted to an annuity payment at 8 percent. (LG52, LG59)
Combined Chapter 4 and Chapter 5 Problems
4&S-1 Future Value Consider that you are 35 years old and have just changed to a new job. You have $80,000 in the retirement plan from your former employer. You can roll that money into the retirement plan of the new employer. You will also contribute $3,600 each year into your new employer's plan. If the rolled-over money and the new contributions both earn a 7 percent return, how much should you expect to have when you retire in 30 years?
4&S-2 Future Value Consider that you are 45 years old and have just changed to a new job.
You have $150,000 in the retirement plan from your former employer. You can roll that money into the retirement plan of the new employer. You will also contribute $7,200 each
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year into your new employer's plan. If the rolled-over money and the new contributions both earn an 8 percent return, how much should you expect to have when you retire in 20 years?
4&S-3 Future Value and Number of Annuity Payments Your client has been given a trust fund valued at $1 million. He cannot access the money until he turns 65 years old, which is in 25 years. At that time, he can withdraw $25,000 per month. If the trust fund is invested at a 5.5 percent rate, how many months will it last your client once he starts to withdraw the money?
4&S-4 Future Value and Number of Annuity Payments Your client has been given a trust fund valued at
$1.5 million. She cannot access the money until she turns 65 years old, which is in 15 years. At that time, she can withdraw $20,000 per month. If the trust fund is invested at a 5 percent rate, how many months will it last your client once she starts to withdraw the money?
4&S-S Present Value and Annuity Payments A local furniture store is advertising a deal in which you buy a
$3,000 dining room set and do not need to pay for two years (no interest cost is incurred). How much money would you have to deposit now in a savings account earning 5 percent APR, compounded monthly, to pay the $3,000 bill in two years? Alternatively, how much would you have to deposit in the savings account each month to be able to pay the bill?
4&S-6 Present Value and Annuity Payments A local furniture store is advertising a deal in which you buy a
$5,000 living room set with three years before you need to make any payments (no interest cost is incurred). How much money would you have to deposit now in a savings account earning 4 percent APR, compounded
monthly, to pay the $5,000 bill in three years? Alternatively, how much would you have to deposit in the savings account each month to be able to pay the bill?
4&S-7 House Appreciation and Mortgage Payments Say that you purchase a house for $200,000 by getting a mortgage for $180,000 and paying a $20,000 down payment. If you get a 30-year mortgage with a 7 percent interest rate, what are the monthly payments? What would the loan balance be in ten years? If the house appreciates at 3 percent per year, what will be the value of the house in ten years? How much of this value is your equity?
4&S-8 House Appreciation and Mortgage Payments Say that you purchase a house for $150,000 by getting a mortgage for $135,000 and paying a $15,000 down payment. If you get a 15-year mortgage with a 7 percent interest rate, what are the monthly payments? What would the loan balance be in five years? If the house appreciates at 4 percent per year, what will be the value of the house in five years? How much of this value is your equity?
4&S-9 Construction Loan You have secured a loan from your bank for two years to build your home. The terms of the loan are that you will borrow $200,000 now and an additional $100,000 in one year. Interest of 10 percent APR will be charged on the balance monthly. Since no payments will be made during the 2-year loan, the balance will grow at the 10 percent compounded rate. At the end of the two years, the balance will be converted to a traditional 30-year mortgage at a 6 percent interest rate. What will you be paying as monthly mortgage payments (principal and interest only)?
4&S-10 Construction Loan You have secured a loan from your bank for two years to build your home. The terms of the loan are that you will borrow $100,000 now and an additional $50,000 in one year. Interest of 9 percent APR will be charged on the balance monthly. Since no payments will be made during the 2-year loan, the balance will grow. At the end of the two years, the balance will be converted to a traditional 15- year mortgage at a 7 percent interest rate. What will you pay as monthly mortgage payments (principal and interest only)?
1 The contract actually contains some complications like incentives to play well and salary deferral. We ignore those complicating factors here.
2 Rodriguez opted out of the contract after the 2007 season and then re-signed with the New York Yankees with a new contract.
3It is also possible to continuously compound. The future value of a continuously compounded deposit is FVN = PV x e(i x N), where e has a value of 2.7183.
4 Some charities are now gathering "obsolete" U.S. medical equipment and sending the materials to less developed countries-a situation in which everybody wins.
5Most homeowners are actually most interested in their total payment, which will include hazard insurance for the home and property taxes. Such payments are referred to as PITI-principal, interest, taxes, and insurance. For simplicity, we use only PI payments here-principal and interest.
6The equation for solving for the number of periods in an annuity is:
