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The Time Value of Money (TVM)
A fundamental skill that any Chief Financial Officer should have is using the time value of money
to help solve problems and develop information to make decisions.
Money has a time value, not because of inflation but because, just like most other things, it can
be rented. The rent is called “interest.” Because the option of renting it out is always available, it
becomes a fundamental alternative that must be considered and measured in any decision where
the usage of money is being weighed.
Let’s take an example. Suppose a salesman approaches your NPO on a new lighting system,
and claims that it can save you money because it is fully programmable and therefore does not
need a person to attend it during performances. The salesman claims that the savings you’ll get
by laying that person off will more than pay your NPO back for the cost of the system. Sounds
good, but…there’s another factor that must be addressed, and that is the opportunity cost of
using money to buy the system rather than putting that same money in an interest-bearing
savings account (which is always an alternative available to us). So, if the system is to pass the
more complete test, it must not only pay you back for its cost, but also for the interest on that
cost, which you are giving up to buy the system. Using time value of money techniques, we can
measure whether the new system has a chance of meeting this fundamental requirement.
The Types of Time Value of Money Problems
There are 4 basic categories of TVM problems.
1. The present value of an annuity (PVA1)
2. The present value of a single amount (PV1)
3. The future value of an annuity (FVA1)
4. The future value of a single amount (FV1)
The subscript “1” means one dollar, which is merely the profession’s way of saying that all
problems can be solved using $1 as the common amount and then multiplying that answer by the
number of dollars involved in your particular situation. The word “annuity” describes any series of
periodic payments (if you’re on the paying side) or receipts (if you’re on the receiving side). Most
problems in real life deal with this kind of situation (#1 or #3 above). The annuity can be regular
(weekly, monthly, quarterly, annual, etc.), or irregular; and it can be of the same dollar amount
every time, or a different dollar amount each time. For this course, and for most situations, we
usually assume a regular periodicity and the same dollar amount because it greatly simplifies the
computation. Irregular timing and dollar amounts can be dealt with, but the computation takes
longer and is more complicated.
Know 3, Find the 4th
The most common TVM problems have 4 pieces of information. The first 3 you either have to
know or make a good guess at (read that: assume), and then the 4th one can be solved. Let’s
take the familiar example of figuring out the monthly payment on a loan. The 4 pieces of
information are:
1. The loan amount (its Present Value or PV)
2. The interest rate (I)
3. The number of months involved (N)
4. The payment (PMT)
We’ll either know or assume the first 3 pieces of information to solve for the 4 th. For example:
1. PV = $15,000 (known or assumed)
2. I = 9.99% (known or assumed)
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3. N = 48 (known or assumed)
4. PMT = $380.37 (solved)
This is good news! You only have to get 3 pieces of information for any TVM problem, and you
can always solve for the 4th.
Know, Assume,
or Estimate
Know, Assume,
or Estimate
Know, Assume,
or Estimate
Solve
Present Value of
Annuity
Problems
Payment
Rate
Periods
Present Value
Rate
Periods
Present Value
Payment
Periods
Present Value
Payment
Rate
Present Value
Payment
Rate
Periods
Future Value of
Annuity Problems
Payment
Rate
Periods
Future Value
Rate
Periods
Future Value
Payment
Periods
Future Value
Payment
Rate
Future Value
Payment
Rate
Periods
Present or Future
Value of A Single
Deposit Problems
Present Value
Rate
Periods
Future Value
Rate
Periods
Future Value
Present Value
Periods
Future Value
Present Value
Rate
Future Value
Present Value
Rate
Periods
Compounding and Its Magic
When we talk about compounding we’re talking about compounded interest…or interest on
interest. If we deposit $100 in a savings account that earns 10%, compounded annually, then at
the end of the first year we will have $110 in the account. At the end of the second year we will
have $121 in the account (our original $100 + $10 earned the first year + another $10 earned the
second year + $1 earned the second year on the $10 earned the first year). That last $1 is
compounding; it is interest earned on interest. This is about as close to magic as you can get. It’s
wonderful.
In most of the TVM problems that we’ll discuss here, we’ll assume monthly compounding. That
means that we’ll do our interest computations monthly rather than annually, and we’ll add that
interest to the “pot” at the end of each month. Of course, we won’t use the full annual rate (10% in
the example above), but one-twelfth of that rate (0.83%) since the time involved is a month
instead of a year. The interesting thing is that frequent compounding is better. For example, 10%
on $100 compounded annually yields $10.00 in interest, but 10% on $100 compounded monthly
yields $10.47 in interest. That little extra thrown in the “pot” each month makes the difference.
(We always presume that interest rates are stated annually if there is no additional language to
help us out. If we are given an interest rate of 10%, then we are to assume 10% per year unless
we are told otherwise. At a 10% rate, $100 would earn $10 for the whole year. If we left the $100
on deposit for only one month, then the interest would be $0.83, which is $10 divided by 12
months.)
Working The TI BA II Plus Calculator
The Texas Instruments Business Analyst II Plus (TI BA II Plus) calculator can perform all of the
TVM computations that we’ll be doing in this class. We need to understand the specific keys we’ll
be using.
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2nd
You’ll notice that this key is light blue. Whenever you press this key it will set the keyboard up
so that the blue functions just above the various keys are performed rather than what is
indicated on the key itself.
QUIT
This is the blue function above the CPT key. When you are in the 2nd mode and want to get
out of it, you press this key. If you are not in the 2nd mode, then the key functions as the CPT
(compute) key.
P/Y
This is the light blue function above the I/Y key. When you press the 2nd key followed by the
P/Y key, you will see how many compounding periods per year you have specified. Annual
compounding would be “1,” quarterly compounding would be “4,” monthly compounding
would be “12.” To set it to 12, type in 12 and then the SET key (which is normally the ENTER
key).
CLR TVM
This is the light blue function above the FV key. When you are in the 2nd mode you press
this key to clear all of the calculator’s registers (sort of like holding tanks for data). It is
important in most cases to make sure these registers are cleared out, so you will use this
function after solving each problem and before starting on the next one.
N
This is the leftmost key, third row from the top. It registers the number of periods (in all of the
problems in this class, months) for the problem you’re solving. You type in the number of
months, for example 36, and then press this key. Sometimes, though, you may be solving for
N, in which case you would have already entered your 3 known variables. Then you would
press the CPT key (for compute) before you press the N key.
I/Y
This key is just to the right of the N key. It registers the interest rate for the problem you’re
solving. You type in the annual interest rate (no need to convert it to monthly because you’re
setting the P/Y to 12 automatically does that for you), for example 10.75, and then press this
key. Sometimes, though, you may be solving for I/Y, in which case you would have already
entered your 3 known variables. Then you would press the CPT key (for compute) before you
press the I/Y key.
PV
This key is just to the right of the I/Y key. It registers the present value for the problem you’re
solving. You type in the dollar value and then press this key. Sometimes, though, you may be
solving for PV, in which case you would have already entered your 3 known variables. Then
you would press the CPT key (for compute) before you press the PV key.
PMT
This key is just to the right of the PV key. It registers the amount of the periodic payment for
the problem you’re solving. You type in the dollar value and then press this key. Sometimes,
though, you may be solving for PMT, in which case you would have already entered your 3
known variables. Then you would press the CPT key (for compute) before you press the PMT
key.
This key is just to the right of the PMT key. It is the rightmost key, third row from the top. It
registers the future value of the problem you’re solving. You type in the dollar value and then
press this key. Sometimes, though, you may be solving for FV, in which case you would have
already entered your 3 known variables. Then you would press the CPT key (for compute)
before you press the FV key.
This key is on the bottom row, just to the left of the = key. It changes the sign of a number
from positive to negative, or from negative to positive, like a toggle switch. Whenever two of
your three known variables in a TVM problem are dollar values (such as PMT and PV), one
of them must be negative and the other must be positive. You use the +/- key to change the
sign on one of them.
FV
+/-
Clearing Your Calculator
Before every TVM problem, you should clear your calculator of any existing TVM values in any of
the registers. You do this by the following keystrokes.
2nd then QUIT then 2nd then CLR TVM
It’s not a bad idea to repeat this series of keystrokes two or three times in a row before you begin
your next TVM problem, just to make sure the registers are emptied.
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Present Value of an Annuity of One (PVA1)
This is the most difficult TVM category to understand, which is why we’ll deal with it first. The best
way to approach it is to examine the fundamental question it answers:
“What single amount am I willing to invest now, at a given interest rate of I, in order to receive (or
alternatively avoid paying) a periodic payment of $X for the next N periods?”
Perhaps the better term should be:
“The Starting Out Amount.” Because you
will find situations in which you need to
compute “Present Value of an Annuity” but
that “answer” is actually in the future. Take
the “Retirement” problem, for example.
In this type of TVM problem, we’ll know or make a good guess at
1. PMT - The payment
2. I – The interest rate
3. N – The number of payments (months)
We punch these three “knowns” into our calculator, press CPT and then PV, and the calculator
will display our solution.
Example: What would we be willing to invest now in an account that pays 10% per year,
compounded monthly, in order to receive (or avoid paying) a monthly payment of $500 for the
next six months? We make the following keystrokes into our calculator:
1.
2.
3.
4.
$500 then PMT
10 then I
6 then N
CPT then PV
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The calculator displays the solution of $2,914.41. We could then construct a table like the one
below to prove the solution (of course, you don’t need to do this every time).
Balance at Beginning of Month
Interest Earned During Month
Month 1
Month 2
Month 3
Month 4
$2,914.41
$2,438.70
$1,959.02
$1,475.34
Month 5
$987.64
Month 6
$495.87
$24.29
$20.32
$16.33
$12.29
$8.23
$4.13
Subtotal
$2,938.70
$2,459.02
$1,975.34
$1,487.64
$995.87
$500.00
Less Payment Made/Received
($500.00)
($500.00)
($500.00)
($500.00) ($500.00) ($500.00)
Balance at End of Month
$2,438.70
$1,959.02
$1,475.34
$987.64
$495.87
$0.00
So, a single investment of $2,914.41 yields us a monthly cash flow of $500.00, or $3,000.00 over
the six-month period. The power of compounded interest makes up the difference between the
$2,914.41 and the $3,000.00.
What if we could find an account that pays 20% compounded monthly? Would our answer go up
or down? Let’s do the keystrokes.
1.
2.
3.
4.
$500 then PMT
20 then I
6 then N
CPT then PV
The calculator displays the solution of $2,832.50. Our answer goes down because the higher
interest rate works harder for us, allowing us to invest less to get the same result. The table below
demonstrates how the solution would roll out over the six months.
Balance at Beginning of Month
Interest Earned During Month
Subtotal
Month 1
Month 2
Month 3
Month 4
$2,832.50
$2,379.71
$1,919.37
$1,451.36
$975.55
$47.21
$39.66
$31.99
$24.19
$16.26
$8.20
$2,879.71
$2,419.37
$1,951.36
$1,475.55
$991.80
$500.00
($500.00) ($500.00) ($500.00)
Less Payment Made/Received
($500.00)
($500.00)
($500.00)
Balance at End of Month
$2,379.71
$1,919.37
$1,451.36
$975.55
Month 5
$491.80
Month 6
$491.80
($0.00)
By the way, in each of the tables above, you compute the interest earned during the month by
taking the beginning balance and multiplying it times the interest rate, and then dividing that
answer by 12 (because it’s just one month’s worth of interest). Try it and see if you arrive at the
same results.
We can change the problem wording slightly to ask the question, “What is the biggest amount we
would be willing to offer now to ‘buy ourselves’ out of an agreement in which we are obligated to
make payments of $X each for the next N months, assuming an interest rate of I?” A similar
variation is, “What is the biggest amount we would be willing to invest now to achieve a cost
savings (or perhaps a revenue enhancement) of $X per month for the next N months, assuming a
required rate of return of I?” It doesn’t matter whether we talk about the payments from the
incoming side or the outgoing side. We use the exact same keystrokes to solve the problem.
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Future Value of an Annuity of One (FVA1)
The fundamental question answered by this type of TVM problem is:
“If I invest $X per month for the next N months at a given interest rate (or rate of return) of I%, to how
much will that accumulate?”
In this type of TVM problem, as in the PVA1 problem, we’ll know or make a good guess at
1. PMT - The payment
2. I – The interest rate
3. N – The number of payments
(months)
We punch these three “knowns” into our calculator, press CPT and then FV, and the calculator
will display our solution.
Example: What if we invest $500 per month for the next 6 months in an account that pays 10%
compounded monthly. What amount would we have in the account at the end of the 6 months?
We make the following keystrokes into our calculator:
1.
2.
3.
4.
$500 then PMT
10 then I
6 then N
CPT then FV
The calculator displays the solution of $3,063.20. We could then construct a table like the one
below to prove the solution.
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Month 1
Balance at Beginning of Month
$0.00
Interest Earned During Month
$0.00
Subtotal
$0.00
Month 2
Month 3
Month 4
Month 5
Month 6
$500.00 $1,004.17 $1,512.53 $2,025.14 $2,542.02
$4.17
$8.37
$12.60
$16.88
$21.18
$504.17 $1,012.53 $1,525.14 $2,042.02 $2,563.20
Add Payment Made/Received
$500.00
Balance at End of Month
$500.00 $1,004.17 $1,512.53 $2,025.14 $2,542.02 $3,063.20
$500.00
$500.00
500.00
$500.00
$500.00
Importantly, the FVA1 problem is NOT a “mirror image” of the PVA1 problem, as is the case
with our other two TVM problems: FV1 and PV1.
Present Value of One (PV1)
The fundamental question answered by this type of TVM problem is:
“What single amount (deposit) must I invest now, at a given interest rate of I, if I want to have an amount
of $X at the end of the next N periods?”
In the figure below, the FINISH is known, and we’re solving for the START.
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We’ll know or make a good guess at
1. FV - The future value that is our objective
2. I – The interest rate
3. N – The number of months
We punch these three “knowns” into our calculator, press CPT and then PV, and the calculator
will display our solution.
Example: What amount do we need to deposit now into an account that pays 10% compounded
monthly in order to have $500 in that account 6 months from now? We make the following
keystrokes into our calculator:
1.
2.
3.
4.
$500 then FV
10 then I
6 then N
CPT then PV
The calculator displays the solution of $475.71. We could then construct a table like the one
below to prove the solution.
Month 1
Balance at Beginning of Month
Month 2
Month 3
Month 4
Month 5
Month 6
$0.00
$479.67
$483.67
$487.70
$491.77
Add Deposit
$475.71
$0.00
$0.00
$0.00
$0.00
$0.00
Subtotal
$475.71
$479.67
$483.67
$487.70
$491.77
$495.86
Interest Earned During Month
Balance at End of Month
$495.86
$3.96
$4.00
$4.03
$4.06
$4.10
$4.13
$479.67
$483.67
$487.70
$491.77
$495.86
$500.00
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Future Value of One (FV1)
The fundamental question answered by this type of TVM problem is:
“What amount will I have at the end of N months if I invest a single amount now of $X in account that
pays an interest rate of I?”
In the figure below, the START is known, and we’re solving for the FINISH.
We’ll know or make a good guess at
1. PV - The starting deposit
2. I – The interest rate
3. N – The number of months
We punch these three “knowns” into our calculator, press CPT and then FV, and the calculator
will display our solution.
Example: If I deposit $475.71 now in an account that pays 10% compounded monthly, what
amount will I have in that account at the end of 6 months? We make the following keystrokes into
our calculator:
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1.
2.
3.
4.
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$475.71 then PV
10 then I
6 then N
CPT then FV
The calculator displays the solution of $500.00. Our solution table will be exactly the same as the
one above, indicating that PV1 and FV1 problems are mirror images of each other.
Variations of the TVM Problems
It could be however that we know both dollar amounts and are trying to solve for I (the interest
rate), or N (the number of periods – months for purposes of this class).
In cases like this, one of the dollar amounts that you punch in to your TI BA II Plus
calculator must be negative and the other must be positive. As mentioned above, you
change the sign of the dollar amount before you input it into the problem by pressing the +/- key
(on the bottom row of your calculator). The +/- key acts as a toggle switch, changing the sign of
the amount you have punched in, to negative and then to positive, back and forth, as long as you
keep pressing it.
Example: Our landlord comes to us with a proposal for us to pay her $2,914.41 now in order to
satisfy our obligation to pay her $500.00 per month for the next 6 months. Since $2,914.41 is less
than $3,000.00 ($500.00 X 6), we know it’s worth considering. By figuring out what the implied
interest rate is on her offer, we can have better information. We make the following keystrokes
into our calculator:
1.
2.
3.
4.
$2,914.41 then +/- then PMT
6 then N
$500.00 the PMT
CPT then I/Y
The calculator displays the solution of 10.00% (compounded monthly). If we have the $2,914.41
available in an account which is only paying 6.00% compounded monthly, we will be putting our
money to better use by accepting the landlord’s offer. On the other hand, if our account is paying
14.00%, our money is put to better use by leaving it where it is, and we will either reject the
landlord’s offer, or give her a counter-proposal of a lower amount that beats our current 14.00%
rate of return.
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Doing TVM Problems In Excel
Once you become knowledgeable and skilled in using the electronic spreadsheet program,
Microsoft Excel, you can solve TVM problems even quicker and easier.
We learned that TVM problems are generally 4-variable problems and that by knowing (or
sometimes assuming or developing) any 3 variables we can solve the 4th. Here is what we know
(or assume) for your retirement question:
1. i = 4%, compounded monthly (0.33% each month [4% ÷ 12 ] )
2. n = 300 (85 – 60 = 25 years X 12 = 300 months)
3. pmt = $6,154 (your needed before-tax amount assuming a federal income tax rate of 28%
and a state income rate of 7% [$4,000 ÷ { 1 – 0.35} ]
Let’s open an Excel spreadsheet to work out the solution. First, let’s set up the three “knowns” in
the exhibit below.
Having done that, our cursor, in this example, now needs to be in cell D5, the solution cell. Let’s then click
on Excel’s Function Wizard (fx) to find our 4th variable, the Starting-Out Amount of this annuity situation
(remember, “annuity” is simply a periodic payment). This dialog box will appear.
Click on this “down arrow.”
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Click on “Financial”
Scroll down the list and
click on “PV”
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The next dialog box will appear as shown below
Click in the “Rate” white space in the box, and then click on the monthly rate cell (D2 in this example).
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Then click in the “Nper” white space in the box, followed by a click on the number-of-months cell (D3 in
this example).
Then click in the “Pmt” white space in the box, type in a minus sign, followed by a click on the monthlypayment-needed cell (D4 in this example). Finally, click on “OK.”
(Without the minus sign, Excel will still give us the correct answer, but with a minus sign in front of it. The
explanation for why it works that way is a little too long for our purposes here. Just think of it as a toggle
switch. Typing in the minus sign as instructed above keeps it out of the answer; not typing it in allows the
it to show up in front of the answer. Either way, it’s the absolute value of the answer we’re interested in,
and either way, that value will be the same.)
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In the solution cell (D5 in this example), Excel returns the value of $1,165,891. Notice also, that in the
formula area, Excel indicates the formula for that cell.
There’s your number! If you retire at age 60, die at age 85, can find an account during your retirement
that will pay an annual rate of 4%, compounded monthly, and need $4,000 a month – after taxes – to live
on…your Starting-Out Amount in that account (the Present Value…yep, that’s what it’s called even
though we’re talking about the future) will need to be $1,165,891. That Starting-Out-Amount – that
Present Value – will allow you to make a withdrawal of $6,154 each month. From that you should be able
to set aside what you’ll need for your federal and state income taxes and live comfortably on what’s left
($4,000). So, what’s the lesson here? Make sure you die on schedule…Just kidding.
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Buying a Business
Closely related to TVM problems is that of buying a business. There are many professionals who
make an excellent living in the “business valuation” field. At first glance, we may think that valuing a
business is difficult and complex. What is it we’re buying? How do we put a value on the business
activity that the current owner has built up? Surprisingly, the solution is fairly simple.
Fundamentally, when you buy (or sell) a business you are dealing with three components:

The assets of the business

The liabilities of the business

The cash flow generated by the business
The Assets. Putting a price on the assets is relatively straightforward. Typically, you will engage an
appraiser to help you assign a value to the assets that you’re about to acquire. Do you have to pay
what the appraiser estimates? No. The appraiser merely produces for you a reference point from
which you and the seller can start bargaining. Ultimately, the price you pay is decided only by you and
the seller.
The Liabilities. Also, it is not uncommon in some business purchases for the buyer to assume any
liabilities that the seller may owe at the time of sale. The total amount of liabilities you assume as the
buyer is a part of the sale price, and you should factor this into your bargaining strategy with the
seller. That is you will probably want to subtract the amount of the liabilities you’re assuming from the
asset valuation. Basically, the objective is for the seller to receive, and for the buyer to pay, a fair
estimation of seller’s equity (assets minus liabilities) in the assets of the business.
The Cash Flow. Putting a price on the cash flow is also fairly straightforward. Conceptually, investing
in a business is the same as investing in a savings account: giving up the use of money in order to
receive a stream of revenues in the future. Let’s take an example. When you invest $100,000 in a
savings account that pays 10% per year, you’re actually giving up the use of that money in order to
receive $10,000 per year in interest income. When putting a price on the cash flow portion of a
business purchase, you are estimating the value of an already-established pattern of income
(revenues minus expenses). In arriving at the $10,000 interest income above, we multiplied the
investment times the interest rate. ($100,000 X .10 = $10,000). By reversing the math, we can
compute the single amount we would be willing to invest in order to receive the annual income
($10,000 ÷ 0.10 = $100,000). Accordingly, in arriving at an estimate of our price for the cash flow, we
would begin by dividing the business’ average annual cash flow from operations by our required rate
of return. We would NOT want to include cash flow from investing activities or cash flow from
financing activities because these have nothing to do with the cash flow generated by the business’
customers and suppliers, which is what we’re interested in purchasing.
Example: An analysis of the annual cash flow statements of the business we want to buy indicates an
average annual cash flow from operations of $200,000. We’re not interested in investing in the
business unless we can get at least a rate of return of 15%. Therefore the maximum price we would
be willing to pay is computed as follows:
$200,000 ÷ 0.15 = $1,333,333.33
Our best offer to the seller for the cash flow portion of the business purchase would be
$1,333,333.33. As our required rate of return increases, the less we are willing to offer for the cash
flow. If we upped our requirement to 20%, the maximum price we would be willing to pay would be:
$200,000 ÷ 0.20 = $1,000,000.00
So, if we changed our minds during negotiations, we would be faced with reducing our offer to the
seller. The point is that the higher the rate of return required by the buyer for a given average annual
cash flow from operations, the lower the computed maximum offering price. This makes sense
because if a $1 million investment can produce the same $200,000 cash flow as a $1.3 million
investment, then the $1 million investment must be the harder working one. The harder work is
demonstrated by the 20% rate of return versus the 15% one.