TIME VALUE OF MONEY
In following we will introduce one of the most important and powerful
concepts you will learn in your study of finance;
the time value of money.
It is generally acknowledged that money has a time value. One
dollar today is worth more than one dollar tomorrow.
TIME VALUE OF MONEY
The reason why the value of money increases with time is because
of the interest rate, which represents the cost of borrowing the
money or the return obtainable by investing it.
Because of the cost of money, interest must be considered by
contractors when making decisions regarding their equipment. This
requires a cash flow analysis, which recognizes that money has a
different economic value depending on when it is received or paid.
TIME VALUE OF MONEY
In finance, we are often concerned with evaluating an investment
(asset) that will pay out certain cash flows at given times in the
future. In order to know what a reasonable price for such an
investment (asset) is, we need to ask the question,
„How much are the cash flows we will receive in the future
worth today?"
At other times, we might be trying to save for some future event,
such as retirement or college. In cases like these we may ask,
"How much will my savings be worth when I need to use
them?"
TIME VALUE OF MONEY
The answers we get to these questions will allow us to do things
like save for our retirement, purchase cars and homes, invest in
rare collectibles, and buy a business.
TIME VALUE OF MONEY
Objective 1:
Compute how much a sum deposited today will grow to in the
future.
This is also referred to as the future value of a lump sum. What if
your wealthy uncle had deposited $1,000 into an account when you
were born and the account earned a return of 8% per year. How
much would that account be worth on your 20th birthday? This is the
type of problem that future value of a lump sum techniques are
designed to solve.
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Objective 2:
Compute the value today of a sum you will not receive until
some future date.
This is also referred to as the present value of a lump sum, since
you are looking for the present value (the value today) of a sum of
money that is promised to you at some point in the future. Say that
as the beneficiary of a trust you were to receive a sum of $20,000 on
your 25th birthday. However, when you graduated from high school
at age 18 you could not afford to go to college. Rather than delay
your education until you turn 25. You might be able to sell the
payment you will receive from the trust. How much would it be
worth? This question is easily answered using the concepts of
present value of a lump sum.
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Objective 3:
Compute how much a series of equal payments will grow to
over time.
This is also referred to as the future value of an annuity. The most
common application of this concept is in retirement planning. Let‘s
say that you deposited $2,000 per year into an individual retirement
account, also called an IRA, and were able to earn 12% per year on
your investment. If you continue your annual investing from the age
of 25 until you retire at age 65, how large would your retirement nest
egg be? It is easy to see the important implications of this type of
calculation, and using the future value of annuity techniques will
make problems like this easy to solve.
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Objective 4:
Compute the value today of a series of equal payments.
This final time value of money concept is also called finding the
present value of an annuity. Say you won a million dollars in the
lottery. However, like most lotteries; this one pays out the money in
twenty equal installments of $50,000 each.
While the total dollar value received over the twenty-year period
equals one million dollars, it is obviously not the same as having a
million dollars in your hands today.
Many lotteries now have started giving winners the choice of the
traditional installment payments or a lump sum, which is paid out
immediately. In effect, you are trading twenty annual payments of
$50,000 for some dollar value received today. How much should you
get if you make the trade?
This is a question you can answer using present value of an annuity
techniques.
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Objective 5:
Apply the time value of money methods to answer a variety of
questions.
The beauty of the techniques you will learn is that they can be
applied to such a wide variety of problems and answer such a large
number of important questions.
Whether you are concerned with how much money you will have at
retirement.
Or whether you can afford the payments on a new sport utility
vehicle, the concepts encompassed in this chapter will allow you to
answer the question. If you learn these concepts well, you will be
amazed at how often you will use them throughout the rest of your
life.
TIME VALUE OF MONEY
Simple versus Compound Interest
Simple Interest is money earned on an investment, which is not
reinvested. It is easy to understand simple interest if you think of the
interest payments as cash that is placed in a safe after it is received.
Since the interest earned is sitting in a safe gathering dust, it cannot
be reinvested. Hence, it "simply" accumulates over time, but does
not grow in value once it is earned.
Compound Interest is money (or interest) earned on an investment
that is immediately reinvested, typically in the same investment.
In contrast to simple interest, compound interest allows the investor
to earn interest on interest. Over long periods of time, and this
interest on interest can generate large amounts of money, and is
often referred to as the power of compounding.
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Assume that you invested $100 for 3 years at an interest rate of 10%. If
the money was earning simple interest, at the end of the three years you
would have a total of $130. This is found in the following manner:
Year
Interest Earned
0
1
2
3
$100(.10)= $10.00 $110.00
$100(.10)= $10.00 $120.00
$100(.10)= $10.00 $130.00
Cumulative
Value
$100.00
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Now assume, however, that the investment was set up to
automatically reinvest the earnings at the same rate of return. In this
case, you would be earning interest on interest, or compound
interest. The total value of the investment at the end of three years
is then found as follows:
Year
Interest Earned
0
1
2
3
$100(.10)= $10.00 $110.00
$110(.10)= $11.00 $121.00
$121(.10)= $12.10 $132.00
Cumulative
Value
$100.00
Notice that the compound interest approach results in a final
investment value that is $2.10 larger than the simple interest
approach. This "extra" money earned represents what was earned
on the reinvested interest.
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The Future Value of a Lump Sum
We can use the example above to develop a formula that will allow
us to find the future value of any lump sum investment under a
compound interest scenario.
In order to general this formula, all we need to do is examine how
we arrived at the cumulative values in the table above. We found
that the value of a $100 investment at the end of one year was
$110.
This can also be written as:
$100(1+0.10)=$100(1+0.10)1 = $110.00
TIME VALUE OF MONEY
Similarly, the value at the end of the second year, $121.00, can be
written as:
$100(1+.10)(1+.10)=$100(1+.10)2= $121.00
Finally, the value at the end of the third year can be rewritten as:
$100(1+.10)(1+.10)(1+.10)= $100(1+.10)3 = $132.10
TIME VALUE OF MONEY
You can see the pattern. To find the future value of a lump sum, you
only need to multiply the original investment times (1+i)n
where i is the interest rate and n is the number of years for which
the investment will be compounded.
Now we can use this information to develop a generic equation that
we can use for any lump sum amount and any number of periods.
The formula is:
FV = PVo(1+i)n
Where FVn is the future value the lump sum will grow to n years
from today, PVo is the present value of the investment at time zero
(the lump sum you invested today), i is the interest rate and n is the
number of years.
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But what if your calculator doesn't have an exponent key?
Without one, it can be impractical to find something like (1+.10)25.
Fortunately, these values have already been computed for you and
put into an easy to use table.
To use the tables, we convert preliminary equation into time value of
money format.
FVn =PVo (FVIFi,n)
We know (1+i)n = (FVIFi.n). The values of FVIF – Future Value
Interest Factor you can find out in the tables or at
http://finance.thinkanddone.com/fvif.htm
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Example
Assume we want to place $1,000 into an account today, could that
the investment will earn an annual rate of return of 10% over the
next five years.
How much will we have at the end of the period?
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NONANNUAL COMPOUNDING PERIODS
In the previous examples, interest was paid once per year, which is
also called annual compounding.
Frequently, interest in accounts is paid more often than once per
year. Many bank accounts, for example, pay interest monthly.
Therefore, we must adjust our formula to allow for any interest
payment schedule that may arise.
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Equation shows how to adjust for any number of compounding
periods per year:
FVn = PV(1 + i/m)mn
where
FVn =
PV =
i
=
n
=
m
=
the future value of a deposit at the end of the
nth period
the initial deposit
the annual interest rate
the number of years the deposit is allowed to
compound
the number of times compouding occurs during
the year
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Similarly, we can adjust Equation to reflect nonannual compounding
periods, with the result being:
FVn =PVo (FVIFi/n,mxn)
and where all the variables are the same as defined above.
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Now we can apply either of these new formulas to the following
example
Example:
Assume we wanted to place $125 into a bank account earning 12%
interest, compounded quarterly. If we left the investment in place for
five years, what would be its value at the end of the period?
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Effective Interest Rates
If the bank increases the number of compounding periods, the
amount earned on a deposit increases. This implies that a higher
effective interest rate is being earned.
The effective interest rate is the amount you would need to earn
annually to be as well off as you are with multiple compounding
periods per year.
To determine the exact effective interest rate, we use the following
equation:
Effective Rate = (1+ i/m)m
Where m is the number of compounding periods per year.
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Accumulating a Future Balance
Another application of future value is computing the initial deposit
required to some future amount of money.
Example:
Suppose you are saving for your child's education, and want to have
$10,000 available 10 years from now. If you can earn 8% annually
on your investment, how much will you need to deposit today?
TIME VALUE OF MONEY
In other words, if you put $4,631.93 into an account earning 8%
interest, it would grow to exactly $10,000 in ten years.
This problem can also be solved in a similar manner using either the
tables or a financial calculator.
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Solving for Number of Periods and Interest Rates
Up until now we have used the future value equation to find out
either how much a deposit will grow to in the future, or how much we
need to deposit today to have a given amount at some future time.
But what if we want to determine the rate of return on an investment
we have already made, or want to find out how long it will take for an
investment to grow to a certain value?
Solving for these different variables is not difficult; we just need to
identity the future and present value amounts and at least one other
variable and plug them into the previous equations.
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Example:
1. Finding the interest rate i:
Assume your uncle invested $1,000 m a trust fund for you 20years
ago. Since then, the fund has grown in value to $6,727.
What is the average annual compound rate of return earned on the
account?
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This problem can be solved easily using either financial factor tables
or a financial calculator. The table solution starts by plugging the
known variables into the future value equation.
FVn
= PV (FVIFi,n)
FV20
= PV0(FVIFi,20 yrs.)
$6,727 = $1,000(FVIFi,20 yrs.)
6.727 = FVIF i,20 yrs. )
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Now go to the FVIF table and on the row corresponding to 20
periods, find the value closest to 6.727 and look at the top of that
column to find the interest rate. In this case, the interest rate is 10%.
If we wanted to use a financial calculator to solve the problem, we
would enter the following inputs:
FV = $6,727
PV = -$1,000
N = 20
PMT = 0
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Then compute the interest rate using the calculator key marked I. As
with the tables, the answer you receive should be 10%.
Note that the sign on the $1,000 initial deposit is negative. Some
calculators require this because the PV is assumed to be an
investment, and hence is really a cash outflow or negative cash flow.
Some calculators do not follow this convention, however, and you
will need to experiment to see which method gives you the correct
answer.
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2. Finding the Number of Periods n:
Now suppose you have $5,000 to invest and your investment
advisor tells you to expect a 12% rate of return.
How long will it take you to double your money? Using the tables,
and the FV formula, we have:
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FV = PV (FVIF i,n)
FVn= PVo (FVIF12%,n)
$10,000 = $5,000(FVIFi 12%,n )
2.000 =FVIF12%,n
The next step is to consult the FVIF table.
Run down the 12% interest rate column until you come as close as
possible to the value 2.000. The closest value you will find is 1.974,
which corresponds to 6 years. In other words, it will take slightly more
than 6 years to double your money if you can earn a 12% rate of
return.
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Future Value of an Annuity
An annuity is a stream of equal cash flows. The two basic types of
annuities are the ordinary annuity and the annuity due. With an
ordinary annuity, payments occur at the end of each period. In
contrast, an annuity due's payments occur at the beginning of each
period.
The future value of an annuity can be found by summing the future
values of each year's cash flows. It is important to remember that
with an ordinary annuity, payment is made at the end of the period,
so in the last period no interest will be received.
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To simplify the future value of an ordinary annuity calculation, a precalculated future value interest factor for an annuity is provided.
When interest is compounded annually at "i" percent for "n"
periods, this factor is multiplied by the amount to be deposited
annually at the end of each year, resulting in the future value of an
annuity.
FV annuity = PMT (FVIFAi,n)
Where:
PMT
FVIFA i,n
= the equal payment made at equal intervals, and
= the future value interest factor of an annuity from
table
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The payments for an annuity due are received at the beginning of
each period.
Because of this, the equation for finding the future value of an
annuity must be converted to work for the future value of an annuity
due.
Previous equation can be converted for use with an annuity due
simply by multiplying the right-hand side of the equation by 1+i. This
results in the following:
FVannuity due = PMT (FVIFAi,n)(1 + i)
This new equation recognizes the fact that the payments of an
annuity due are received one period sooner, and they therefore are
compounded for one additional period as well
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Suppose you planned to invest $2,000 per year into an Individual
Retirement Account (IRA) for the next 20 years. What would be the
value of your account if you could receive a 12% annual rate of
return on your investment? Assume first that you are dealing with an
ordinary annuity, and then solve assuming you have an annuity due.
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To solve the problem as an ordinary annuity, we apply following:
FV annuity = PMT (FVIFAi,n)
FV annuity = $2,000(FVIFA12%,20yrs.)
FV annuity = $2,000(72.052)
FV annuity = $144,104
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To solve the problem as an annuity due, we simply apply the annuity
due formula.
FVannuity due = PMT (FVIFAi,n)(1 + i)
FVannuity due = $2,000(FVIFA i,n12%,20yrs.)(1 + 0.12)
FVannuity due = $2,000(72.052)(1.12)
FVannuity due = $161,396
Note that depositing the annual payment at the beginning of the year
instead of the end of the year results in more man $17,000 in
additional retirement funds at the end of 20 years. This provides a
good illustration of the power of compounding.
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Present Value of a Lump Sum
We previously derived equation, which establishes the future value
of a deposit. Take another look at this equation:
FVn = PV0 (1 + i)n
To convert this equation to a present value equation, all that we
need is a little algebra. Rearrange the terms in the equation so that
PV is all by itself on the right-hand side, and we have the following:
PV0 = FVn / (1 + i)n
And when the PV interest factors from the tables in the book are used, the
equation becomes:
PV0 = FV(PVIFi,n)
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The present value of a single amount is found by discounting the
cash flow back to time zero. The process of discounting is simply the
inverse of compounding interest. The application of discounting is
concerned with answering the question:
"If l am earning i percent on my money, what will I be willing to pay
today (at time zero) for the opportunity to receive the designated
cash flow n periods from now?"
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To illustrate, let‘s work through the following example.
Assume you have the chance to buy a rare baseball card as on
investment. You believe that you could sell this card in 8 years for
$12,000 to help pay part of your daughter's college expenses as a
freshman at an exclusive private school. If you require a 10% rate of
return on your investments, how much should you be willing to pay
for the card today?
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PV = FVn / (1 + i)n
PV = $12,000 / (1 + 0.10)8
PV = $12,000 / 2.144
PV = $5,597.01
In other words, if you paid $5,597.01 for the card today, and sold
it 8 years later for $12,000, you would have earned a 10%
annual rate of return on your investment.
If you paid more than $5,597.01, you would not be able to earn
the required 10%.
Of course, if you paid less than $5,597.01, you would earn more
than 10%.
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The Present Value of Mixed Streams
Say you want to find the present value of an investment that pays a
series of unequal cash flows over a period of years.
To solve problems like this the technique is simply to find the
individual PV of each cash flow and sum the PV's together.
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Example:
Assume you were considering buying part ownership in a hot dog
stand. You expect business to grow over time, and you have
estimated the following annual cash flows from / your portion of the
stand‘s ownership.
Assuming an 8% required rate of return, how much should you pay
for your ownership share?
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PV = $500(PVIF8%,1) + $600(PVIF8%,2) + $800(PVIF8%,3) +
$950(PVIF8%,4)
PV = $500(0.926) + $600(0.857) + $800(0.794) + $950(0.735)
PV=$463.00+$514.20+$635.20+$698.25
PV= $2,310.65
The calculator solution is $2,310.71, which is slightly larger due to
rounding.
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Increasing the Compounding Period
As with future value, non-annual compounding periods require a bit
of adjustment before the PV can be found. The PV equation for
multiple compounding periods is:
PV=-FVn / (1 + i/m)mn
where
m = number of compounding periods per year
n = number of years
i = annual interest rate
We can also write this in TVM notation as follows:
PV=FVn (PVIPi/m,mn)
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Example:
Find the present value of $1,500 to be received 5 years from now, if
the appropriate rate of return is 12% compounded quarterly.
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PV= $1,500 PVIF12%/4,4(5))
PV=$1,500(PVIF3%,20periods)
PV=$1,500(0.554)
PV=$831.00
The calculator solution for this problem is $830.51.
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Present Value of an Annuity
If we want to find the PV of a stream of equal cash flows, we use a
concept called the present value of an annuity.
The formula is:
PVannuity = PMT (PVIFAi,n)
where
PMT = the $ amount of the equal payments.
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Example:
Assume you just won the Illinois lottery, and the prize was $50,000
per year for the next 20 years. Lottery officials claim you won a "one
million dollar" prize because the payments sum to $1,000,000.
However, what if you decided you needed all the cash today, and
wanted to sell your stream of payments? If the going rate of interest
in the market was 10%, how much would your lottery winnings really
be worth?
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Using the annuity formula:
PVannuity
PVannuity
PVannuity
PVannuity
= PJMT (PVIFAi,n)
= 50,000 (PVIFA10%,20)
= 50,000 (8.514)
= 50,000 ($425,700.00)
The calculator solution is $425,678.
So winning the "one million dollar" lottery is worth only $425,700
in today's dollars, or less than half what tottery officials claimed it
was worth.
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Perpetuities
Annuities whose payments go on for infinity are called perpetuities.
On the surface, it would seem impossible to value such an asset,
since the payments we need to value never end. Fortunately,
valuing perpetuities is actually very simple.
The formula for the value of perpetuity is:
PV0 = PMT / i
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Examples:
Suppose you were able to buy on investment that makes $500
annual payments that never end. If you require a 12% rate of return
on your investment, how much would you be willing to pay for the
perpetuity?
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PV0 = PMT / i
PV0 = $500 / 0.12
PV0 = $4,166.67
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Another way to look at this problem is to ask yourself the following
question:
"If I could place $4,166.67 into a bank account earning 12% interest,
and if l wanted to spend only the interest each year, how large would
my annual payments be?"
By simply multiplying $4,166.67 by the interest rate (0.12), you come
up with the annual interest earned on your investment, or $500. If
you leave the money in the account forever, you have just created a
perpetuity that pays $500 each year. The cost to create such
perpetuity is $4,166.67, which therefore also must be its present
value.
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Uneven Streams
When an annuity is mixed with other irregular payments we have
what is called an imbedded annuity.
To solve for the present value of mixed-cash flow streams we find
the present value of the annuity, and then add the present value of
any other cash flows.
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Assume you are evaluating cm investment that pays
$50 at the end of year 1,
$75 at the end of year 2,
$90 at the end of year 3, and
$100 at the end of years 4 through 6.
If you require a 10% rate of return on your investments, what is the
most you should be willing to pay for this investment?
In order to solve this problem, first draw a timeline like the one
below. The irregular payments in years 1 through 3 require us to find
their associated present values individually and then add them up.
The embedded annuity in years 4 through 6 allows us to use the
annuity formula.
However, the present value of an annuity formula is designed to find
the PV of an annuity where the payments begin at the end of the
first year. In this case, the annuity payments begin at the end of the
fourth year. Therefore, the formula has to be modified somewhat.
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The solution is:
PV = $50(PVIF10%,1 )+ $75(PVIF10%,2)+ $90(PVIF10%,3)+
$100(PVIFA10%,3)(PVIF10%,3)
PV=$50(0.909)+$75(0.826)+$90(.751)+$100(2.487)(0.751)
PV=$45.45+$61.95+$67.59+$186.77
PV=$361.76
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Note that the annuity formula finds the value of the annuity one
period before the first payment is made. This is because the formula
is designed to find the present value of an ordinary annuity (where
the first payment is received at the end of year 1).
Hence, multiplying the $100 payment by FVIFA 10%,3 yields a value
of $248.70, but that is really the value of the annuity at the end of
year 3, one period before the 1st $100 payment is made.
In order to obtain the proper result, we must discount $248.70 back
to time zero. This was accomplished using PVIF 10%,3 in the solution
above.
The timeline shows how each of the original cash flows is
discounted back to time zero and then added to the others.
The final result is that the investment would be worth $361.76.