FINANCE

Chapter 3

Time Is Money

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Learning Objectives

A�er studying this chapter, you should be able to:

Express the �me value of money and related mathema�cs, including present and future values, principal, and interest. Explain the significance of compounding frequency in rela�on to future and present cash flows and effec�ve annual percentage rates. Iden�fy the values of common cash flow streams, including perpetui�es, annui�es, and amor�zed loans.

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Ch. 3 Introduction

The saying “�me is money” could not be more true than it is in finance. People ra�onally prefer to collect money earlier rather than later. By delaying the receipt of cash, individuals forgo the opportunity to purchase desired goods or invest the funds to increase their wealth. The foregone interest, which could be earned if cash were received immediately, is called the opportunity cost of delaying its receipt. Individuals require compensa�on to reimburse them for the opportunity cost of not having the funds available for immediate investment purposes. This chapter describes how such opportunity costs are calculated. Because many business ac�vi�es require compu�ng a value today for a series of future cash flows, the techniques presented in this chapter apply not only to finance but also to marke�ng, manufacturing, and management. Here are examples of ques�ons that the tools introduced in this chapter can help answer:

How much should we spend on an adver�sing campaign today if it will increase sales by 5% in the future? Is it worth buying a new computerized lathe for $120,000 if the lathe reduces material waste by 15%? Which strategy should we employ, given their respec�ve costs and es�mated contribu�ons to future earnings? What types of health insurance and re�rement plans are best for our employees, given the amount of money we have available?

Being able to value the cash to be received in the future—whether dividends from a share of stock, interest from a bond, or profits from a new product—is one of the primary skills needed to run a successful business. This chapter provides you with an introduc�on to that skill.

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How do we determine the future value of $100 today?

How do we determine the future value of $100 today at a 10% interest rate?

3.1 The Time Value of Money

The �me value of money and the mathema�cs associated with it provide important tools for comparing the rela�ve values of cash flows received at different �mes. Just as a hammer may be the most useful item in a carpenter’s toolbox, �me value of money mathema�cs is indispensable to a financial manager.

Recall from Chapter 1 that to increase shareholder wealth managers must make investments that have greater value than their costs. O�en, such investments require an immediate cash outlay, like buying a new delivery truck. The investment (the truck) then produces cash flows for the corpora�on in the future (delivery fee income, increased sales, lower delivery costs, etc.). To determine whether the future cash flows have greater value than the ini�al cost of the truck, managers must be able to calculate the present value of the future stream of cash flows produced by this investment. Let’s take a closer look at the �me value of money in ac�on.

Present and Future Value

Suppose a friend owes you $100 and the payment is due today. You receive a phone call from this friend, who says she would like to delay paying you for one year. You may reasonably demand a higher future payment, but how much more should you receive? The situa�on is illustrated in Figure 3.1 as a �meline.

Figure 3.1: Determining future value

In this diagram “now,” the present �me, is assigned t = 0, or �me zero. One year from now is assigned t = 1. The present value of the cash payment is $100 and is denoted PV0. Its future value at t = 1 is denoted as FV1. To find the amount that you could demand for deferring receipt of the money by one year, you must solve for FV1, the future

value of $100 one year from now. The FV1 value will depend on the opportunity cost of forgoing immediate receipt of $100. You know, for instance, that if you had the

money today you could deposit the $100 in a bank account earning 5% interest annually. However, you know from Chapter 2 that value depends on risk. In your judgment, your friend is less likely to pay you next year than is the bank. Therefore, you will increase the rate of interest to reflect the addi�onal risk that you think is inherent in the loan to your friend.

Suppose that you decide that a 10% annual rate of interest is appropriate. The amount of the future payment, FV1, will be the original principal plus the interest that could

be earned at the 10% annual rate. Algebraically, you can solve for FV1, being careful always to convert percentages to decimals when doing arithme�c calcula�ons,

FV1 = $100 + $100 (0.10)

Factoring $100 from the right-hand side of the equa�on, we have

FV1 = $100(1 + 0.10) = $100(1.10) = $110

You may demand a $110 payment at t = 1 in lieu of an immediate $100 payment because these two amounts have equivalent value. No�ce that if you had deposited the $100 in the bank, you would have only $105 a�er one year. The higher the interest rate, the faster the amount will grow.

Let’s say that your friend agrees to this interest rate but asks to delay payment for two years. The new scenario is illustrated in Figure 3.2.

Figure 3.2: Determining the future value of $100 at %10 interest

Now we must find FV2, the future value of the payment 2 years from today. Since we know FV1 = $110 and we know the interest rate is 10%, we can solve for FV2 by

recognizing that FV2 will equal FV1 plus the interest that could be earned on FV1 during the second year.

FV2 = FV1 + FV1(0.10) = $110 + ($110)(0.10) = $110(1 + 0.10) = $121

You may demand a $121 payment at t = 2 because its �me value is equivalent to either $110 at t = 1 or $100 at t = 0, given the 10% interest rate.

Simple and Compound Interest

We just showed that, at a 10% annual interest rate, $100 today is equivalent to $110 a year from now and $121 in 2 years. Now, we look at how compound and simple interest affect the �me value of money. Look at the �meline shown in Figure 3.3.Processing math: 0%

How does 10% compound interest impact the value of $100 a�er two years?

Even when lending money to a friend, it’s important to iron out the details, agree upon the terms of repayment, and figure out the �me value of the money.

The Agency Collec�on/Ge�y Images

Figure 3.3: The future value of $100 at 10% compound interest

This result may be generalized using the following formulas,

(3.1) FV1 = PV0(1 + r)

(3.2) FV2 = PV0(1 + r) 2

where r is the interest rate.

Equa�on (3.2) can be restated as

(3.3) FV2 = PV0(1 + 2r) + PV0(r 2)

Equa�on (3.3) is broken down in a special way. The first term on the right side of the equal sign, PV0 (1 + 2r), would yield $120 given the informa�on we have used in our

example. The second term, PV0 (), yields $1. The value $120 equals your original principal ($100) plus the amount of interest earned ($20) if your friend paid simple interest.

Simple interest means that the same dollar amount of interest is received every period. For example, if you withdraw interest earned during each year at the end of that year, you would earn simple interest. In this case, you would receive $10 interest payments at the end of years 1 and 2, totaling $20. If, on the other hand, your friend credited (but did not pay) interest to you every year, then you would earn interest during year 2 on the interest credited to you at the end of year 1. Earning interest on previously earned interest is known as compounding. Thus, you would earn an extra dollar, a total of $121, over the two-year period with interest compounded annually. The example assumed annual compounding since nearly all transac�ons are now based on compound rather than simple interest. This problem is demonstrated in the Applying Finance: Annual Compounding feature. See Appendix A (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/appendix#appendix) for informa�on on se�ng up your calculator, and addi�onal financial applica�on problems.

Applying Finance: Annual Compounding

Future Value With Annual Compounding: To solve the problem we just looked at with a financial calculator or Excel is straigh�orward.

To Solve Using TI Business Analyst

A�er clearing the calculator, use the following inputs:

100 [+/−] [PV]

2 [N]

0 [PMT]

10 [I/Y]

[CPT] [FV]Processing math: 0%https://content.ashford.edu/books/AUBUS650.13.1/sections/appendix#appendix

Many of the contracts on the bills we pay involve compound interest, such as with our credit cards. Can you think of other examples of where compound interest is u�lized?

Rob Lewine/Ge�y Images

= $121.00

Note: These may be input in any order so long as the FV and Compute are at the end. We entered 100 as a nega�ve number. Think of it as $100 going away (you are giving it to the bank) and the $121 is being received (the bank is giving it back to you), so the two cash flows will have opposite signs. If you enter 100 [+/–] [PV] in this problem, then your answer will be a posi�ve 121. If you entered the PV of $100 as a posi�ve number, then the FV displayed would be signed nega�ve.

To Solve Using Excel

Use the FV func�on. The inputs for this func�on are: FV(RATE,NPER,PMT,PV,TYPE)

RATE: Interest rate per period as a %

NPER: Number of compounding periods

PMT: Any periodic payment (for the FV of a single cash flow this would be zero)

PV: Present value

TYPE: 0 if payments are made at the end of the period (most common) and 1 if payments are made at the beginning of the period

= FV(10%,2,0,-100,0)

= $121.00

Note: Financial func�ons in Excel require that cash inflows and cash ou�lows have different arithme�c signs. We signed the PV (the amount you put in the bank today) nega�ve because it is flowing away from you and into the bank. The result ($121.00) is posi�ve because that is a cash flow to you. Commas separate the inputs, so you cannot enter numbers with commas separa�ng thousands (e.g., $1,000). Nor can you include dollar signs ($).

Let’s extend this example to 20 years to be�er show the difference between simple and compound interest. At a 10% interest rate using simple interest our original deposit of $100 would grow into $300 over 20 years. This growth is based on receiving $10 of interest each year.

$100 + (20 × $10) = $100 + $200 = $300

Compare this to the result from compound interest.

FV20 = PV0(1 + r) 20 = 100(1 + 0.10)20 = $672.75

The difference between simple and compound interest is $372.75 over 20 years!

Not all compounding is done on an annual basis, however. Some�mes interest is added to an account every six months (semiannual compounding). Other contracts call for quarterly, monthly, or daily compounding. As you will see in the next sec�on, the frequency of compounding can make a big difference when the �me value of money is calculated.

Understanding Compound Interest

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Compound interest allows earned money to earn more money. Interest can be compounded daily, monthly, or annually. Why does a financial manager need to understand the difference in compounding periods?

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How does semiannual interest impact the value of $100 a�er two years?

3.2 Valuing a Single Cash Flow

As men�oned in the previous sec�on, how o�en a loan’s interest is compounded changes how we determine the �me value of money. There are different compounding periods: mortgage or car loans use monthly compounding; corporate bonds that pay interest semiannually use semiannual compounding; some cer�ficates of deposit use con�nuous compounding; and many credit cards use daily compounding. In this sec�on, we will show how different compounding periods affect the �me value of money formula used.

Con�nuing our example from Sec�on 3.1, let us suppose that your friend who wishes to delay paying you agrees to a 10% annual rate of interest over the two-year period and will allow you to compound interest semiannually. What will you be paid in two years given this agreement? Semiannual compounding means that interest will be credited to you every six months, based on half of the annual rate. In effect you will be earning a 5% semiannual rate of interest over four six-month periods. In other words, the periodic interest rate will be half the annual rate because you are using semiannual compounding and you will be earning interest for four �me periods (n = 1 through 4), each period being one-half year long. The new situa�on is illustrated in Figure 3.4.

Figure 3.4: Semiannual compounding

Here, FV1 is the future value of the $100 at the end of period 1 (the first six months). As before, FV1 equals the $100 beginning principal plus interest earned over the six

months at the 5% semiannual interest rate.

FV1 = $100 + $100(0.05) = $100(1 + 0.05) = $105

Therefore, at the end of period 1 (at n = 1), the principal balance you are owed will be $105. FV2 will be equal to the principal at the beginning of period 2 plus interest

earned during period 2.

FV2 = $105 + $105(0.05) = $110.25

Note that we could subs�tute [$100(1.05)] for $105 in the previous equa�on. Doing so, FV2 could be expressed as follows:

FV2 = $105(1.05) = [$100(1.05)](1.05) = $100(1.05) 2

Following this pa�ern, finding FV3 and FV4 is straigh�orward.

FV3 = $100(1.05) 3 = $115.76

FV4 = $100(1.05) 4 = $121.55

The final equa�on gives the answer we seek. The future value at the end of four six-month periods is $121.55. Changing from annual compounding to semiannual compounding has increased the future value of your friend’s obliga�on to you by $0.55. The addi�onal interest earned from semiannual compounding, $0.55, doesn’t seem like much but imagine a firm borrowing $100 million; then the amount earned from compounding—the interest earned on previous interest—can turn into tens of thousands of dollars.

The Future Value of a Single Cash Flow

The pa�ern established here may be generalized into the formula for the future value of a single cash flow using compound interest.

(3.4) FVn = PV0(1 + r) n

where

FVn = the future value at the end of n �me periods

PV0 = the present value of the cash flow

r = the periodic interest rate

which equals the annual nominal rate divided by the number of compounding periods per year,

n = the number of compounding periods un�l maturity, or

n = (number of years un�l maturity)(compounding periods per year)

It is cri�cal when using this formula to be certain that r and n agree with each other. If, for example, you are finding the future value of $100 a�er six years and the annual rate is 18%, compounded monthly, then the appropriate r is 1.5% per month (18%/12 = 1.5%) and n is 72 months (6 years �mes 12 months per year = 72 months). Students

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o�en adjust the interest rate and then forget to adjust the number of periods (or vice versa)! The answer to this problem is

FV72 = $100(1.015) 72 = $292.12

For quarterly compounding, you would divide the annual rate by 4 and mul�ply the number of years by 4. So $100 a�er six years and an annual rate of 18% with quarterly compounding would be found using a periodic interest rate of 4.5% (18%/4) and 24 periods (6 years �mes 4 periods per year).

FV24 = $100(1.045) 24 = $287.60

For simple interest, without compounding, the future value is simply equal to the annual interest earned, �mes the number of years, plus the original principal. The formula for the future value of a single cash flow using simple interest is

= PV0 + (n)(PV0)(r) = PV0(1 + nr)

where

= the future value at the end of n periods using simple interest

n = the number of periods un�l maturity (Generally n simply equals the number of years because there is no adjustment for compounding periods.)

r = the periodic rate (which also usually equals the annual rate because there is no adjustment for compounding periods)

For the previous example, the future value of $100 invested for 6 years in an account paying 18% per year using simple interest is

= $100[1 + (6)(0.18)] = $208.00

The adjustment process we just discussed works for all compounding periods except one: con�nuous compounding. We won’t go into the details of the math; we will just show the result.

(3.6) FVn = PV0(e rn)

The le�er e is one of those special numbers in mathema�cs that is assigned its own name. (Another one is π, which you may remember is approximately equal to 3.14.) The number e is approximately equal to 2.72 (more precisely 2.71828183). The exponent rn in Equa�on (3.6) doesn’t need to be adjusted. In our example of 18% for six years, rn will be the same with or without any of those adjustments we discussed. We always do math with decimals rather than percentages, so the rn exponent for 18% would be 0.18 × 6 = 1.08 as is 1.5% (0.015 × 72) or 4.5% (0.045 × 24). Most calculators have an key, which makes compu�ng con�nuous compounding fairly easy.

Monthly compounding yielded a future value a�er six years of $292.12, or $84.12 more than simple interest in this example. Table 3.1 illustrates the future value of $100, bearing 18% annual interest, with different compounding assump�ons. Use your calculator to replicate the solu�ons illustrated below. Be sure your n and r agree (e.g., both are monthly, yearly, etc.), and always be sure you express percentages as decimals before doing any calcula�ons. You should prac�ce with your calculator un�l your answers match those given. For more applica�ons, refer to the Applying Finance: Future Value feature box.

Table 3.1: The future value of $100 a�er six years at 18% annual interest, various compounding periods

Compounding assump�on n r FVn

Annual 6 0.18 $269.96

Semiannual 12 0.09 $281.27

Quarterly 24 0.045 $287.60

Monthly 72 0.015 $292.12

Weekly 312 0.00346 $293.92

Daily 2,190 0.000493 $294.39

Con�nuous ∞ $294.47

Applying Finance: Future Value

Future Value of Single Cash Flow: If you put $400 in the bank today at 12% per year, leave it there for five years, what will be the balance at the end of the �me period?

To Solve Using TI Business Analyst

400 [PV]

5 [N]

0 [PMT]

12 [I/Y]

[CPT] [FV]

= $704.9366Processing math: 0%

How much can you borrow today at 12% compounded monthly for 36 months, if you plan to pay it back with your $1000 bonus at the end of the loan?

Note: These may be input in any order so long as the FV and Compute are at the end. Also, the calculator register will show the answer as a nega�ve 704.9366 since you entered 400 as a posi�ve number. Think of it as 400 is cash going one way (you are giving it to the bank) and the 704 is going the opposite direc�on (the bank is giving it back to you), so the two cash flows will have opposite signs. If you enter 400 [+/–] [PV] in this problem, then your answer will be a posi�ve 704.9366. It does not ma�er which way you do this.

To Solve Using Excel

Use the FV func�on. The inputs for this func�on are: FV(RATE,NPER,PMT,PV,TYPE)

RATE: Interest rate per period as a %

NPER: Number of compounding periods

PMT: Any periodic payment (for the FV of a single cash flow this would be zero)

PV: Present value

TYPE: 0 if payments are made at the end of the period (most common) and 1 if payments are made at the beginning of the period

If you put $400 in the bank today at 12% per year, leave it there for five years, what will be the balance at the end of the �me period?

=FV(12%,5,0,-400,0)

=$704.94

Note: Financial func�ons in Excel require that cash inflows and cash ou�lows have different arithme�c signs. We signed the PV (the amount you put in the bank today nega�ve because it is flowing away from you and into the bank. The result ($704.94) is posi�ve because that is a cash flow to you. The inputs are separated by commas, so you cannot enter numbers with commas separa�ng thousands (e.g., $1,000). Nor can you include dollar signs ($).

The Present Value of a Single Cash Flow

We have solved for the future value of a current cash flow. O�en, we must solve for the present value of a future cash flow, solving for PV rather than FV. You can think of the present value as the amount that you have to put in the bank today to have some specific amount in the future. A higher interest rate causes a deposit to grow faster, so the higher the interest rate the smaller the amount of money that has to be deposited today to achieve a desired future amount. Similarly, the longer the �me un�l a future cash flow is collected, the smaller the amount deposited has to be. This is because the ini�al deposit has more �me to grow.

Suppose, for example, you are going to receive a bonus of $1,000 in three years. You could really use some cash today and are able to borrow from a bank that would charge you an annual interest rate of 12%, compounded monthly. You decide to borrow as much as you can now such that you will s�ll be able to pay off the loan in three years using the $1,000 bonus. In essence, you wish to solve for the present value of a $1,000 future value, knowing the interest rate (12% per year, compounded monthly) and the term of the loan (3 years, or 36 monthly compounding periods). Figure 3.5 is a �meline illustra�ng the problem. This problem is also prac�ced in the Applying Finance: Present Value feature.

Figure 3.5: Determining present value of $1000 in the future

Applying Finance: Present Value

Present Value of Single Cash Flow: How much money would you have to put in the bank today at 12% per year, with monthly compounding, to have $1,000 in exactly three years?

To Solve Using TI Business Analyst

Clear TVM worksheet

2nd [CLR TVM]

2nd [Quit]

Clear CF worksheetProcessing math: 0%

2nd [CLR WORK]

2nd [Quit]

Set the compounding period to monthly

2nd [P/Y] 12 [enter]

2nd [Quit]

1000 [FV]

36 [N]

0 [PMT]

12 [I/Y]

[CPT] [PV]

= $698.92

To Solve Using Excel

Use the PV func�on with the format: PV(RATE,NPER,PMT,FV,TYPE).

The inputs for this example would be:

=PV(1%,36,0,1000,0)

= –$698.92

In this case n = 36, r = 1%, and is known, whereas PV0 is unknown. We may s�ll use Equa�on (3.4), subs�tu�ng in the known quan��es and using some algebra.

(3.4) FVn = PV0(1 + r) n

$1,000 = PV0(1.01) 36

You could borrow $698.92 today and fully pay off the loan, given the bank’s terms, in three years using your $1,000 bonus. We can generalize the last expression into the formula for the present value of a single cash flow with compound interest. Solving for the present value of a future cash flow is also known as discoun�ng. In fact, compounding and discoun�ng are flip sides of the same coin. Compounding is used to express a value at a future date given a rate of interest. Discoun�ng involves expressing a future value as an equivalent amount at an earlier date.

This formula is also called the discoun�ng formula for a single future cash flow.

(3.7)

The variables PV0, FVn, n, and r are defined exactly as they are in the future value formula because both formulas are really the same, just solved for different unknowns.

To find the present value with con�nuous compounding, we would use

(3.8)

Table 3.2 solves for the present, or discounted, value of a $1,000 cash flow to be received in 1 year at a 12% per year discount rate using different compounding periods. You should be able to replicate these solu�ons on your calculator.

Table 3.2: The present value of $1,000 to be received in 1 year discounted at 12% annual interest, various compounding periods

Compounding assump�on N R PVN

Annual 6 0.12 $892.86

Semiannual 12 0.06 $890.00

Quarterly 24 0.03 $888.49

Monthly 72 0.01 $887.45

Weekly 312 0.00231 $887.04

Daily 2,190 0.000329 $886.94

Con�nuous ∞ $886.92

Present and future value formulas are very useful because they may be used to solve a variety of problems. Suppose you make a $500 deposit in a bank today and you want to know how long it will take your account to double in value, assuming that the bank pays 8% interest per year, compounded annually (shown in Figure 3.6). Here, you are solving for the number of �me periods. You may subs�tute the known quan��es PV0 = $500, FVn = $1,000, r = 0.08 into either formula and solve for n:Processing math: 0%

How many �me periods must pass to double your investment of $500 at 8% interest compounded annually?

What is the interest rate on an investment of $200 that results in a single payment of $275 in five years?

Figure 3.6: Determining number of �me periods

(3.7) PV0 = FVn(1 + r) −n

$500 = $1,000(1.08)−n

(1.08)n = $1,000/$500

(1.08)n = 2

At this point, without using logarithms, you must use trial and error to solve for n. Suppose you try n = 10 as your first guess for n:

(1.08)10 = 2.1589

This value yields a number higher than our objec�ve of 2. Therefore, try n = 9 because a lower value of n will yield a lower answer:

(1.08)9 = 1.999

which is close enough. In nine years, the balance in your account will double.

Suppose the account earned 8% per year compounded monthly. To find the �me un�l the account’s balance doubled, you would convert the interest rate below to reflect monthly compounding,

and then solve for the number of compounding periods.

(3.7) PV0 = FVn(1 + r) −n

$500 = $1,000(1.00667)−n

(1.00667)n = 2

Using trial and error, you get the answer n = 105. This should be interpreted as 105 months because you are dealing with monthly compounding periods. Thus, in 8.75 years the account will double in value when using monthly rather than annual compounding.

This example illustrates an important lesson. It takes less �me to achieve a desired amount of wealth with more frequent compounding at a given nominal interest rate. It is no surprise that borrowers prefer less frequent compounding, while lenders prefer compounding as frequently as possible. The difference between compounding frequencies offered at various banks makes shopping around worthwhile whether you are a borrower or a saver.

Finding Interest Rates

Another type of problem is solving for the interest rate. This �me let’s suppose that an investment cos�ng $200 will make a single payment of $275 in 5 years. What is the interest rate such an investment will yield? Subs�tute n = 5 into the formula and solve for r.

Figure 3.7 shows a �meline for finding the interest rate that equates a $200 deposit to a future value of $275 in 5 years.

Figure 3.7: Determining interest rate

We can use Equa�on (3.7) to find r.

(3.7) PV0 = FVn(1 + r) −n

$200 = $275(1 + r)−n

(1 + r)5 = 1.375

1 + r = (1.375)⅕

1 + r = 1.3750.20

r = (1.375)0.20 − 1

r = 0.06576Processing math: 0%

The answer, r = 0.06576, is based on an annual compound rate because we assumed n = 5 years. It is also expressed as a decimal and could be re-expressed as a percentage, 6.576% per year compounded annually. See the Applying Finance: Finding Interest Rates feature for prac�ce with this problem.

Applying Finance: Finding Interest Rates

Annual Compound Interest Rate: If a $200 deposit grows into $275 in five years, what is the annual compound interest rate?

To Solve Using TI Business Analyst

(Make sure to set P/Y to 1).

275 [FV]

5 [N]

0 [PMT]

200 [+/–] [PV]

[CPT] [I/Y]

= 6.58

To Solve Using Excel

Use the Rate func�on with the format: PV(NPER,PMT,PV,FV,TYPE,GUESS).

The inputs for this example would be:

= RATE(5,0,-200,275,,)

= 6.5763%

Effective Annual Percentage Rate

As you have seen, the frequency of compounding is important. Truth-in-lending laws now require that financial ins�tu�ons reveal the effec�ve annual percentage rate (EAR) to customers so that the true cost of borrowing is explicitly stated. Before this legisla�on, banks could quote customers annual interest rates without revealing the compounding period. Such a lack of disclosure can be costly to borrowers. For example, borrowing at a 12% yearly rate from bank A may be more costly than borrowing from bank B, which charges 12.1% yearly, if bank A compounds interest daily and bank B compounds semiannually. Both 12% and 12.1% are nominal rates—they reveal the rate “in name only” but not in terms of the true economic cost. To find the effec�ve annual rate, divide the nominal annual percentage rate (APR) by the number of compounding periods per year and add 1; then raise this sum to an exponent equal to the number of compounding periods per year. Finally, subtract 1 from this result.

(3.9)

For our example,

Thus, if you are a borrower, you would prefer to borrow from bank B despite its higher APR. The lower EAR translates into a lower cost over the life of the loan. The disclosure of EARs makes comparison shopping for rates much easier.

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The charity should bid no more than $6,153.07 for the hot dog stand, given the stream of expected cash flows.

In this case, interest is deferred un�l a�er the first �me period. This is not always true of future values of cash flow streams.

3.3 Valuing Multiple Cash Flows

Many problems in finance involve finding the �me value of mul�ple cash flows. Consider the following problem. A charity has the opportunity to purchase a used mobile hot dog stand being sold at an auc�on. The charity would use the hot dog stand to raise money at special events held in the summer each year (at the county fair, baseball and soccer games, etc.). The old hot dog stands will only last two years and then will be worthless. The charity es�mates that, a�er all opera�ng expenses, the stand will produce cash flows of $1,000 in both June and July in each of the next two years and cash flows of $1,500 in each of the next two Augusts. The auc�on takes place January 1, and the charity requires that its fundraising projects return 12% on their invested funds. How much should the charity bid for the hot dog stand? The strategy for solving this problem is shown in Figure 3.8.

Figure 3.8: Determining the present value of mul�ple cash flows

The present value of the stream of cash flows the stand is expected to produce is found by applying Equa�on (3.7) to each of the six future cash flows. Note that 1% is used as the periodic rate (12% per year/12 months) because cash flows are spaced in monthly intervals. The charity should bid a maximum of $6,153.07 for the hot dog stand. Given the level of expected cash flows, paying more than this amount would result in the charity earning a lower return than its 12% objec�ve.

The hot dog stand example illustrates the general formula for finding the present value of any cash flow stream,

(3.10)

where

n = the number of compounding periods from �me 0 CFn = the cash flow to be received exactly n compounding periods from �me 0 (e.g., CF1 is the cash flow received at the end of period 1, etc.)

r = the periodic interest rate N = the number of periods un�l the last cash flow

The future value formula for a cash flow stream is also found by finding the future value of each individual cash flow and summing. Terms in the formula are defined as in the present value formula.

(3.11) FVN = CF1(1 + r) N − 1 + CF2(1 + r)

N − 2 + … + CFN

You may ques�on why in Equa�on (3.11) the first term is raised to the exponent N – 1 and why the last term is not mul�plied by an interest factor. This situa�on may be clarified by using a �meline, as shown in Figure 3.9. The last cash flow (CFn) occurs at the end of the last �me period and therefore earns no interest.

Figure 3.9: Determining the future value of cash flow streams

As the �meline shows, CF1 will earn interest for N – 1 period, but CFn earns no interest and is simply added to the other sums to find the total future value. By conven�on,

we assume that the cash flows from investments do not start immediately but are deferred un�l the end of the first period. This is not always the case, however. Prac��oners must carefully analyze any problem to be certain exactly when cash flows will occur. A �meline is a useful aid in modeling when the cash flows from a project will occur.

PerpetuitiesProcessing math: 0%

What is the present value of a perpetuity that pays $50 a year forever, with a 10% return on investment?

What is the present value of an annuity that pays $50 semiannually for two years at 10% annual interest?

Some special pa�erns of cash flows are frequently encountered in finance. The nature of these pa�erns allows the general formulas to be simplified to a more concise form. The first special case is that of perpetui�es. These are cash flow streams where equal cash flow amounts are uniformly spaced in �me (every year, every month, etc.). Perpetuity means that these payments con�nue forever. To illustrate, suppose an investment is expected to pay $50 every year forever. Investors require a return of 10% on this investment. What should be its current price? Recognizing that today’s price should equal the present value of the investment’s future cash flows, the problem is illustrated using a �meline in Figure 3.10.

Figure 3.10: Determining the present value of a perpetuity

The arrow indicates that these cash flows con�nue into the future indefinitely. This poses a problem: If there are an infinite number of cash flows, how can we find all of their present values? Let’s consider the algebraic expression of this problem.

Summing this geometric series and using some algebra yields the following formula for the present value of a perpetuity:

(3.12)

Note that there is no subscript a�ached to CF because all the cash flows are the same.

Therefore, there is no need to dis�nguish CF1 from CF2, and so on. Let’s apply the formula to the example. CF = $50, r = 0.10, and

Annuities

Of all the special pa�erns of cash flow streams, annui�es are the most common. As we shall see, millions of fixed-rate home mortgages are annui�es. Re�rement payments, bond interest payments, automobile loan payments, and lo�ery jackpot payoffs all o�en fit the annuity pa�ern.

An annuity is a stream of equally sized cash flows, equally spaced in �me, which end a�er a fixed number of payments. Thus, annui�es are like perpetui�es, except they do not go on forever. The present value of an annuity can be found by summing the present values of all the individual cash flows.

(3.10)

Here N is the number of cash flows being paid and CF is the uniform amount of each cash flow. Solving for CF0 using Equa�on (3.10) would be a �me-consuming problem if

n were large. However, because the right-hand side of the equa�on is yet another geometric series, it can be simplified to yield the formula for finding the present value of an annuity.

(3.13)

To convince you that Equa�ons (3.10) and (3.13) are equivalent, let’s work an example using both approaches. Suppose you wished to know the present value of a stream of $50 payments made semiannually over the next two years. The first payment is scheduled to begin six months from today, and the annual rate of interest is 10%. The problem is illustrated with a �meline in Figure 3.11.

Figure 3.11: Determining the present value of an annuity

Using Equa�on (3.10), and recognizing that r = 5% = 0.05 semiannually, this problem may be solved as follows:

Alterna�vely, Equa�on (3.13) could be used to solve the same problem.

It may appear that using Equa�on (3.10) is just as �me-consuming as using Equa�on (3.13), but consider the work involved had there been 300 payments rather than 4.Processing math: 0%

Payments occur one period sooner in an annuity due, as opposed to an ordinary annuity.

What is the future value of an ordinary annuity that pays $100 a month for two years at 12% annual interest?

What is the future value of an annuity due that pays $100 a month for two years at 12% annual interest?

The problem just solved is an example of an ordinary annuity because cash flows commence at the end of the first period. Most loans require interest payments at the end of each period. Rent, on the other hand, is usually payable in advance. Annui�es in which cash flows are made at the beginning of each period are called annui�es due. Leases are usually structured as an annuity due; you make a payment before you get use of the asset. But loans are structured as regular annui�es because some interest has to build up before a payment is made. Let’s change the example we just worked slightly to require that the cash flows be made at the beginning of each period.

The �meline in Figure 3.12 shows that in a four-payment annuity due, each payment occurs one period sooner than in an otherwise similar ordinary annuity. Because of this characteris�c, each cash flow is discounted for one less period when finding the PV of an annuity due.

Figure 3.12: An ordinary annuity versus an annuity due

The formula for finding the present value of an annuity due is

(3.14)

This is simply the formula for an ordinary annuity �mes 1 + r, which adjusts for one less discoun�ng period. Thus, it is usually easier to find the PV of an ordinary annuity and mul�ply �mes 1 + r when solving for the PV of an annuity due.

(3.15)

Now suppose you save $100 each month for two years in an account paying 12% interest annually, compounded monthly. What will be the balance in the account at the end of two years if you make your first deposit at the end of this month? Figure 3.13 illustrates this problem with a �meline.

Figure 3.13: Determining the future value of an ordinary annuity

In this case we are trying to solve for the future value of an ordinary annuity.

FV24 = $100(1.01) 23 + 100(1.01)22 + . . . + $100

Solving our problem in this manner would take considerable �me. Fortunately, the future value of an annuity is also a geometric series, which can be simplified.

The formula for the future value of an ordinary annuity is

(3.16)

Subs�tu�ng the values for our example into Equa�on (3.16) yields the solu�on

If the first deposit were made immediately, our problem would be one of finding the future value of an annuity due. Figure 3.14 illustrates this problem using a �meline.

Figure 3.14: Determining the future value of an annuity due

Each cash flow in an annuity due earns one addi�onal period’s interest compared to the future value of an ordinary annuity. Thus, the future value of an annuity due is equal to the future value of an ordinary annuity �mes 1 + r.

(3.17) Processing math: 0%

In an amor�zed loan—such as a typical home loan—the interest por�on of the loan payment gets smaller over �me, so more principal is repaid with each payment, and the amount paid each month remains the same: $804.62. A�er 30 years, the $100,000 loan will be paid off, plus the 9% annual interest.

Home mortgages are an important example of an amor�zed loan. Can you think of any other examples of this type of loan?

Ge�y Images News/Ge�y Images

The future value of the deposits would therefore increase to $2,724.32 if they were made at the beginning of each period. No�ce the adjustment from an ordinary annuity to an annuity due is the same whether you are solving for PV or FV [compare Equa�ons (3.15) and (3.17)]. Note that both the present value and the future value of an annuity due are always larger than an otherwise similar ordinary annuity.

Application: Loan Amortization

Many loans, such as home mortgages, require a series of equal payments made to the lender. Each payment is for an amount large enough to cover both the interest owed for the period as well as some principal. In the early stages of the loan, most of each payment covers interest owed by the borrower and very li�le is used to reduce the loan balance. Later in the loan’s life, the small principal reduc�ons have added up to a sum that has significantly reduced the amount owed. Thus, as �me passes, less of each payment is applied toward interest and increasing amounts are paid on the principal. This type of loan is called an amor�zed loan. The final payment just covers both the remaining principal balance and the interest owed on that principal. An amor�zed loan is a direct applica�on of the present value of an annuity. The original amount borrowed is the present value of the annuity (PV0), while loan payments are the annuity’s cash flows (CFs).

If you borrow $100,000 to buy a house, what will your monthly payments be on a 30-year mortgage if the interest rate is 9% per year? For this problem, the formula for finding the present value of an annuity is used [Equa�on (3.13)]. The present value is the loan amount (PV0 = $100,000), there are 360 payments (N = 360),

and the monthly interest rate is 0.75% (9%/12 months). The payment amount (CF) is determined as follows:

(3.13)

A stream of 360 monthly payments of $804.62 will cover the interest owed each month and will pay off the en�re $100,000 loan as well. Figure 3.15 illustrates how the amount of each payment applied toward principal increases over �me, with a corresponding decrease in interest expense. As shown in Figure 3.15, $750.00 of the first payment is used to pay the interest owed the lender for the use of $100,000 during the first month at the 0.75% monthly rate. $54.62 of the first payment will be applied toward the principal. Thus, for the second month of the loan only $99,945.38 is owed. This reduces the amount of interest owed during the second month and increases the second month’s principal reduc�on. This pa�ern con�nues un�l the last payment when, as seen in Figure 3.15, only a $798.63 principal balance is remaining. The last month’s interest on this balance is $5.99. Therefore, the last $804.62 payment will just pay off the loan and pay the last month’s interest, too. Note that the ending balance (or the FV) of the loan equals zero a�er the last payment is made, so the loan is completely paid off with the last payment.

Figure 3.15: Components of an amor�zed loan over �me

Table 3.3 is an amor�za�on table showing principal and interest payments on a five-year, $10,000 loan, amor�zed using a 10% rate compounded annually. An amor�za�on table is useful because it can be used to find the unpaid balance owed on a loan a�er some payments have been made. Using Table 3.3, a borrower would know, for example, that $4,578.32 would be necessary to pay off the loan a�er the third annual payment is made.

Table 3.3: Loan amor�za�on table for $10,000 borrowed at 10% interest annually compounded for five years

Year Beginning principal balance Total payment Interest Principal reduc�on Ending principal balance Processing math: 0%

Crea�ng an amor�za�on table helps you keep track of monthly balance, principal payment, and interest payments. The video uses a TI Inspire to create the table. What are some other tools you could use to create an amor�za�on table?

1 $10,000 $2,637.97 $1,000 $1,637.97 $8,362.03

2 $8,362.03 $2,637.97 $836.20 $1,803.77 $6,560.26

3 $6,560.26 $2,637.97 $656.03 $1,981.94 $4,578.32

4 $4,578.32 $2,637.97 $457.83 $2,180.14 $2,398.18

5 $2,398.18 $2,637.97 $239.79 $2,398.18 $0

Note: Each year’s beginning balance equals the previous year’s ending balance. Each year’s interest equals the rate mul�plied by the total loan amount. Each year’s principal reduc�on equals the total payment minus the amount applied toward the interest. Each year’s ending principal balance equals the beginning balance minus the principal reduc�on.

Amor�za�on Table for Month to Month Payments

Field Trip: Loan Amor�za�on

Bankrate.com provides a mortgage amor�za�on schedule calculator that allows users to build an amor�za�on table using their own data.

Visit: h�p://www.bankrate.com/calculators/mortgages/amor�za�on-calculator.aspx (h�p://www.bankrate.com/calculators/mortgages/amor�za�on-calculator.aspx)

Experiment with different mortgage amounts, terms, and interest rates to see how they affect your monthly payments. You can even experiment with adding addi�onal payments to change the pay-off date of the loan.

Processing math: 0%http://www.bankrate.com/calculators/mortgages/amortization-calculator.aspx

Ch. 3 Conclusion

Chapter 3 has covered much of the topic of the �me value of money. Next, the concepts and techniques introduced here will be applied to finding the value of stocks, bonds, and other securi�es. Before that, however, it is best to prac�ce the newly acquired skills. The authors cannot overemphasize the importance of mastering �me value mathema�cs. Therefore, as you do your homework, make sure you feel confident in your ability. If you are not, now is a good �me to ask your instructor for assistance.

Processing math: 0%

Ch. 3 Learning Resources

Key Ideas

The foregone interest, which could be earned if cash were received immediately, is called the opportunity cost of delaying its receipt. The �me value of money and the mathema�cs associated with it provide important tools for comparing the rela�ve values of cash flows received at different �mes. You can think of the present value as the amount that you have to put in the bank today to have some specific amount in the future. How frequently a loan’s interest is compounded changes how we determine the �me value of money. An amor�zed loan is a direct applica�on of the present value of an annuity.

Key Equa�ons

Cri�cal Thinking Ques�ons

1. Suppose you own some land, purchased by your father 20 years ago for $5,000. You are able to trade this land for a brand new Corve�e sports car. What economic opportunity might you forego if you proceed with the trade? How would you es�mate the opportunity cost of proceeding with the trade?

2. The Corve�e dealership from Ques�on 1 is also willing to trade the car for an IOU you own that promises to pay you $2,000 at the end of each year for the next 10 years and $20,000 when it matures at the end of the 10-year period. Investors are currently valuing such IOUs using a 6% discount rate. What economic opportunity might you lose if you make the trade? How would you calculate the opportunity cost of the trade?

3. If the market for new automobiles and the real estate and bond markets are all efficient, what do you think you would discover about the opportunity costs of the trade in Ques�ons 1 and 2?

4. Usually we compute present values using a constant interest rate. But we know that interest rates vary over �me, and it is impossible to know what the interest rate will be in 10 or 20 years. Why is using the current interest rate a good approach? Or would we be be�er off to simply ignore cash flows arriving beyond the period for which we have reasonable interest rate es�mates? Explain your answer.

5. We discussed the EAR (effec�ve annual percentage rate). Private student loans o�en are structured so no payments are necessary while a student is in school (and for 6 months a�er). However, interest does accrue during this period. This interest is then added to the principal amount of the loan once the grace period ends. For example, you borrow $10,000 at 6% when you start a two-year graduate program. The interest is $50 per month. You complete your degree and take advantage of some of the postgraduate grace period and then begin making payments 25 months a�er the loan began. The new principal is now $10,000 plus the capitalized interest of $1,250 or $11,250. Lenders don’t state an effec�ve annual rate because of the uncertainty associated with the amount of capitalized interest. Does this seem fair? Can you think of a way that this could be expressed so student borrowers understand what they are commi�ng to when they get a private student loan with a capitalized interest feature?

Key Terms

Click on each key term to see the defini�on.

amor�zed loan (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

A loan that is paid off in equal periodic payments. Automobile loans and home mortgages are o�en amor�zed loans.

annui�es due (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

A finite stream of cash flows of a fixed amount, equally spaced in �me where payments are made at the beginning of each period.

annuity (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

A finite stream of cash flows of a fixed amount, equally spaced in �me.

compounding (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

discoun�ng (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

Solving for the present value of a future cash flow.

discount rate (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The interest rate used to find the present value of a future payment or series of payments. For many investments, investors’ required return is the discount rate used to find the present value.

effec�ve annual percentage rate (EAR) (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The annualized compound rate of interest.

future value (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

A cash flow, or stream of cash flows, re-expressed as an equivalent amount at some future date.

interest (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The amount of money paid by a borrower to a tender for the use of the borrowed principal. The rate is expressed as a percentage of the principal owed.

nominal annual percentage rate (APR) (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The stated interest rate per year without considering the effect of compounding.

nominal rates (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The stated rate or yield that reflects expecta�ons about infla�on.

opportunity cost (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The amount of the highest valued forgone alterna�ve.

ordinary annuity (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

A finite stream of cash flows of a fixed amount, equally spaced in �me, where payment are made at the end of each period.

periodic interest rate (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The rate of interest expressed per period, e.g., per month (12 periods per year); quarterly (4 periods per year); semi-annually (twice per year); weekly (52 periods per year); bi-annually (once every two years), etc.

perpetui�es (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

An infinite stream of equal cash flows, each equally spaced in �me.

present value (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

A future cash flow, or stream of cash flows, re-expressed as an equivalent current amount of money.

principal (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The amount of money borrowed.

simple interest (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The number of years mul�plied by the interest rate mul�plied by the amount originally invested.

�me value of money (h�p://content.thuzelearning.com/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/front_ma�er/books/AUBUS650.13.1/sec�ons/fro

The idea that, holding all else constant, people prefer to receive a given amount of money today rather than in the future.

The importance of the �me value of money concept is discussed here: h�p://www.qfinance.com/cash-flow-management-calcula�ons/�me-value-of-money (h�p://www.qfinance.com/cash-flow-management-calcula�ons/�me-value-of-money)

The effec�ve interest rate concept can be applied to mortgage interest rates, too. The Motley Fool website shows how to compute an effec�ve a�er-tax interest rate here: h�p://wiki.fool.com/How_to_Calculate_an_Effec�ve_Mortgage_Rate_With_a_Tax_ Writeoff (h�p://wiki.fool.com/How_to_Calculate_an_Effec�ve_Mortgage_Rate_With_a_Tax_ Writeoff)

Mortgage agreements have a stated note rate, which determines the interest component of each payment, and an annual percentage rate (APR). The APR is almost always higher than the note rate because it includes other costs associated with acquiring a mortgage for a home: origina�on fee, points, prepaid interest, and insurance. Here is a descrip�on of the APR: h�p://www.americanloansearch.com/info-apr.htm (h�p://www.americanloansearch.com/info-apr.htm)

For informa�on on mortgage rates, mortgage calculators, and historic rate informa�on, visit: h�p://www.mortgagenewsdaily.com/mortgage_rates/ (h�p://www.mortgagenewsdaily.com/mortgage_rates/)

Processing math: 0%http://www.qfinance.com/cash-flow-management-calculations/time-value-of-money http://wiki.fool.com/How_to_Calculate_an_Effective_Mortgage_Rate_With_a_Tax_%20Writeoff http://www.americanloansearch.com/info-apr.htm http://www.mortgagenewsdaily.com/mortgage_rates/

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