BUS 401: Teaching Net Present Value (NPV) & Future Value (FV)

4.1

 The Time Value of Money

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 1 year. You may reasonably demand a higher future payment, but how much more should you receive? The situation is illustrated here using the timeline shown in Figure 4.1.

Figure 4.1

In this diagram "now," the present time, is assigned t = 0, or time zero. One year from now is assigned t = 1. The present value of the cash payment is $100 and is denoted PV0 (and read as "present value at time zero"). Its future value at t = 1 is denoted as FV1 (and read as "future value 1 year from now"). To find the amount that you could demand for deferring receipt of the money by 1 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 3% interest annually. However, you know from Chapter 1 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 additional risk that you think is inherent in the loan to your friend.

Suppose 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 arithmetic calculations, and so

(4.1)

FV1 = $100 + ($100)(0.10)

Factoring $100 from the right-hand side of Equation (4.1) gives

	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.

Let's say that your friend agrees to this interest rate but asks to delay payment for 2 years.

Figure 4.2

Now we must find FV2, the future value of the payment 2 years from today. This situation is illustrated by the timeline in Figure 4.2. 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. Our equation is then

(4.2)	FV2	= FV1 + FV1(0.10)
		= $110 + ($110)(0.10)
		= $110(1 + 0.10)
		= $110(1.10)
		= $121

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

The time value of money and the mathematics associated with it provide important tools for comparing the relative values of cash flows received at different times. Just as a hammer may be the most useful item in a carpenter's toolbox, time value of money mathematics is indispensable to a financial manager.

For example, recall from Chapter 1 that to increase shareholder wealth, managers must make investments that have greater value than their costs. Often, such investments require an immediate cash outlay, like buying a new delivery truck. The investment (the truck) then produces cash flows for the corporation in the future (delivery fee income, increased sales, lower delivery costs, etc.). To determine whether the future cash flows have greater value than the initial cost of the truck, managers must be able to calculate the present value of the future stream of cash flows produced by this investment.

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4.2

 Compound and Simple Interest

The preceding section showed that, at a 10% annual interest rate, $100 today is equivalent to $110 a year from now and $121 in 2 years.

This result may be generalized using the following formulas:

(4.3)

FV 1 =PV 0 (1+r)

(4.4)

FV 2 =PV 0 (1+r) 2

where FV1 and FV2 are, respectively, future values 1 year and 2 years from now, PV0 is the present value at time zero, respectively, and r is the interest rate.

Now, let's expand Equation (4.4):

(4.5)	FV2	= PV0 = (1 + r)(1 + r)
		= PV0(1 + 2r + r2)
		= PV0(1 + 2r) + PV0(r2)

The last line of Equation (4.5) 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 information we have used in our example. The second term, PV0(r2), yields $1. The value $120 equals your original principal ($100) plus the amount of interest earned ($20) if your friend paid simple interest. 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 2-year period with interest compounded annually. In this example we assumed annual compounding since nearly all transactions are now based on compound rather than simple interest. Not all compounding is done on an annual basis, however. Sometimes interest is added to an account every 6 months (semiannual compounding). Other contracts call for quarterly, monthly, or daily compounding. As you will see, the frequency of compounding can make a big difference when the time value of money is calculated.

4.3

 The Time Value of a Single Cash Flow

Continuing our example, let us suppose that your friend who wishes to delay paying you agrees to a 10% annual rate of interest over the 2-year period and will allow you to compound interest semiannually. What will you be paid in 2 years given this agreement? Semiannual compounding means that interest will be credited to you every 6 months, based on half of the annual rate. In effect you will be earning a 5% semiannual rate of interest over four 6-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 time periods (n = 1 through 4), each period being 1/2-year long. The new situation is illustrated in Figure 4.3.

Figure 4.3

Here, FV1 is the future value of the $100 at the end of period 1 (the first 6 months). As before, FV1 equals the $100 beginning principal plus interest earned over the 6 months at the 5% semiannual interest rate. Therefore we set this up using the following equation:

(4.6)	FV1	= $100 + $100(0.05)
		= $100(1.05)
		= $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:

(4.7)	FV2	= $105 + $105(0.05)
		= $105(1.05)
		= $110.25

Note that we could substitute [$100(1.05)] for $105 in the second line of Equation (4.7). By doing so, FV2 could be expressed as follows:

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

By following this pattern, finding FV3 and FV4 is straightforward. For the future value at the end of the third period, we have

(4.9)

FV 3 =$100(1.05) 3          =$115.76

and that at the end of the fourth period is

(4.10)

FV 4 =$100(1.05) 4          =$121.55

Equation (4.10) gives the answer we seek. The future value at the end of four 6-month periods is $121.55. Changing from annual compounding to semiannual compounding has increased the future value of your friend's obligation to you by $0.55. The additional interest earned from semiannual compounding, $0.55, doesn't seem like much, but imagine a firm borrowing $100 million; then the compounding period—annual, semiannual, quarterly—can turn into tens of thousands of dollars.

The Future Value of a Single Cash Flow

The pattern established here may be generalized into the formula for the future value of a single cash flow using compound interest:

(4.11)

FVn = PV0(1 + r)n

where

FVn = the future value at the end of n time periods PV0 = the present value of the cash flow   r = the periodic interest rate   n = the number of compounding periods until maturity, or(number of years until maturity)(compounding periods per year)

The periodic interest rate equals the annual nominal rate divided by the number of compounding periods per year,

r=annual nominal rate number of periods per year   

It is critical 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 after 6 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 times 12 months per year = 72 months). Students often adjust the interest rate and then forget to adjust the number of periods (or vice versa)! The answer to this problem is

FV 6×12 = (1+0.18 12  )  6×12    FV 72 = $100(1.015) 72            = $292.12   

Try It: Calculator Key Strokes and Excel Functions—Future Value

TI Business Analyst

Future Value of Single Cash Flow: If you put $400 in the bank today at 12% per year, and leave it there for 5 years, what will be the balance at the end of the time period?

400	[PV]	The PV key is used to input the present value of the deposit, $400.
5	[N]	The funds are invested for 5 years, so 5 is entered using the N key.
0	[PMT]	PMT is the key used to input a constant periodic payment or deposit, but in this problem there are no such cash flows, so 0 is entered using the PMT key.
12	[I/Y]	I/Y is the key used for entering the periodic interest rate, in this case 12% per year, so 12 is entered.
[CPT]	[FV]	CPT is the key that tells the calculator to calculate a value; in this case you are asked to find the future value of the deposit, so the calculator is told to compute the FV:
= $704.9366.

Note: These may be input in any order so long as the [FV] and [CPT] are at the end. Also, the calculator register will show the answer as a negative 704.9366, since you entered 400 as a positive number. Think of it like this: 400 is cash going one way (you are giving it to the bank), and the 704 is going the opposite direction (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 positive 704.9366. It does not matter which way you do this.

Excel

Use the FV function. The inputs for this function are RATE, NPER, PMT, PV, and TYPE, where

  RATE is the interest rate per period as a percentage,

  NPER is the number of compounding periods,

  PMT is any periodic payment (for the FV of a single cash flow this would be zero),

  PV is the present value,

and

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

  If you put $400 in the bank today at 12% per year, and leave it there for 5 years, what will be the balance at the end of the time period?

  Using the FV function in Excel gives

FV(12%,5,0,–400,0)

= 704.94

Note: Financial functions in Excel require that cash inflows and cash outflows have different arithmetic signs. We signed the PV (the amount you put in the bank today) negative because it is flowing away from you and into the bank. The result ($704.94) is positive because that is a cash flow to you. The inputs are separated by commas, so you cannot enter numbers with commas separating thousands (e.g., $1,000). Nor can you include dollar signs ($).

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

(4.12)

FV s n =PV 0 +(n)(PV 0 )(r)=PV 0 (1+nr)

where

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

  n = the number of periods until 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

FV= 6 s $100[1+ (6)(0.18)] = $208.00 

Thus, monthly compounding yielded a future value after 6 years of $292.12, or $84.12 more than simple interest in this example. Table 4.1 illustrates the future value of $100, bearing 18% annual interest, with different compounding assumptions. Be sure that you can replicate the solutions illustrated here using your calculator. 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 calculations. You should practice with your calculator until your answers match those given in Table 4.1. A graph of these results is shown in Figure 4.4.

Table 4.1: The future value of $100
Compounding assumption	n	r	FVn
Annual compounding	6	0.18	$269.96
Semiannual compounding	12	0.09	$281.27
Quarterly compounding	24	0.045	$287.60
Monthly compounding	72	0.015	$292.12
Weekly compounding	312	0.00346	$293.92
Daily compounding	2,190	0.000493	$294.39

Figure 4.4

The Present Value of a Cash Flow

We have solved for the future value of a current cash flow. Often, we must solve for the present value of a future cash flow, solving for PV rather than FV.

Suppose, for example, you are going to receive a bonus of $1,000 in 1 year. 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 so that you will still be able to pay off the loan in 1 year 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 (1 year, or 12 monthly compounding periods). Figure 4.5 shows a timeline illustrating the problem.

Figure 4.5

Try It: Calculator Key Strokes and Excel Functions—Present Value

Present Value of Single Cash Flow: How much money would you have to put in the bank today at 12% per year, to have $10,000 in exactly 4 years?

TI Business Analyst

   1000 [FV]

   3     [N]

   0     [PMT]

   12    [I/Y]

   [CPT] [PV]

    = $711.78

Note that the answer that your calculator produces will be negative if you follow these keystrokes. The future value was entered as a positive number (like a cash inflow) so the present value is negative (like a cash outflow).

Excel

Use the PV function with the format: PV(RATE,NPER,PMT,FV,TYPE).

The inputs for this example would be:

    = PV(12%,3,0,1000,0)

    = –$711.78

In this case n = 12, r = 1%, and FV12 = is known, whereas PV0 is unknown. We may still use Equation (4.11),

(4.11)

FV n =PV 0 (1+r) n

Substituting in the known quantities gives

$1,000=PV 0 (1.01) 12

and using some algebra we have

(4.13)

PV 0 =1,000(1.01) −12 =$1,0001 1.01 12                                          =$887.45

You could borrow $887.45 today and fully pay off the loan, given the bank's terms, in 1 year using your $1,000 bonus. Equation (4.13) may be generalized 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 discounting. In fact, compounding and discounting are two sides of the same coin. Compounding is used to express a value at a future date given a rate of interest. Discounting involves expressing a future value as an equivalent amount at an earlier date.

This formula is also called the discounting formula for a single future cash flow:

(4.14)

PV 0 =FV n (1+r) −n =FV n 1 (1+r) n

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; they are just solved for different unknowns.

Table 4.2: The present value of $1,000
Compounding assumption	n	r	PV0
Annual compounding	1	0.12	$892.86
Semiannual compounding	2	0.06	$890.00
Quarterly compounding	4	0.03	$888.49
Monthly compounding	12	0.01	$887.45
Weekly compounding	52	0.00231	$887.04
Daily compounding	365	0.000329	$886.94

Table 4.2 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 solutions on your calculator. A graph of these results is shown in Figure 4.6.

Figure 4.6

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. Here, you are solving for the number of time periods. The timeline is shown in Figure 4.7.

Figure 4.7

You may substitute the known quantities PV0 = $500, FVn = $1,000, r = 0.08 into either formula and solve for n. Let's use

(4.14)

PV0 = FVn(1 + r)–n

We can rearrange this equation into

PV0/FVn = (1 + r)–n

or

(1 + r)n = FVn/PV0

Taking the logarithm of both sides gives us

n log(1 + r) = log(FVn/PV0)

Finally, solving for n gives

n = log(FVn/PV0)/log(1 + r)

Plugging in our numbers gives

	n = log($1000/$500)/log(1 + 0.08)
	= 9

Therefore in 9 years the balance in your account will double.

Suppose the account earned 8% per year compounded monthly. To find the time until the account's balance doubled, you would convert the interest rate to reflect monthly compounding r = 0.08/12 = 0.00667 and solve for the number of compounding periods. Starting again with

(4.14)

PV0 = FVn(1 + r)–n

we substitute in numbers to get

$500 = $1,000(1.00667)–n

or

(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 time 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 savers (or 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.

Another type of problem is solving for the interest rate. This time let's suppose that an investment costing $200 will make a single payment of $275 in 5 years. What is the interest rate such an investment will yield? The timeline is shown in Figure 4.8.

Figure 4.8

Starting again with the formula

PV0 = FVn(1 + r)–n

we have for n = 5

PV0 = FV5(1 + r)–5

We want to solve for the interest rate r. Rearranging terms we get

(1 + r) 5  = FV 5  PV 0

so

1 + r = (FV5/PV0)0.20

or

r = (FV5/PV0)0.20 – 1

Substituting in PV0 = $200 and FV5 = $275, we get

r = ($275/$200)0.20 – 1

or

r = 0.06576

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.

David Sipress/The New Yorker Collection/www.cartoonbank.com

Effective Annual Percentage Rate

As you have seen, the frequency of compounding is important. Truth-in-lending laws now require that financial institutions reveal the effective annual percentage rate (EAR) to customers so that the true cost of borrowing is explicitly stated. Before this legislation, 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 effective annual rate, you must 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, and, finally, subtract 1 from this result.

The general formula for the effective annual percentage rate is

(4.15)

EAR=(1+APR CP  ) CP −1

For our example,

EAR A =(1+0.12 365  ) 365 −1=0.1275=12.75%  EAR B =(1+0.121 2  ) 2 −1=0.1247=12.47%

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.

A Closer Look: The Rule of 72

The rule of 72, which is very useful for making estimates when dealing with the time value of money, says that if the periodic rate times the number of compounding periods equals 72, then the future value will equal approximately twice the present value for a lump sum. Stated differently, if the rate times the periods equals 72, then your original deposit will double. Use the rule of 72 to solve the following problems.

1. You deposit $500 in an account that paid 8% interest per year, compounded annually. If you leave the money in the account for 9 years, what would be your approximate balance at the end of the 9 years?

2. If you deposit $400 in an account that bears 12% interest per year (compounded annually) and leave it there for 12 years, what would be the approximate balance in your account at the end of that time?

3. Gas in 1969 cost about $0.40 per gallon. If inflation has averaged about 4.5% per year since then, use the rule of 72 to estimate whether gas is more expensive, less expensive, or equally expensive now compared to what it was then.

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