The Time Value of Money
 The Time Value of Money – Basics
Firms, as well as individuals, are confronted with opportunities to earn
positive rates of return on their money, either through investments in
projects (physical assets) or in interest-bearing or appreciating securities and
deposits (financial assets). Therefore, the timing of cash inflows and
outflows has important economic consequences. Time value of money is
based upon the premise that a dollar today is worth more than a dollar to be
received at some future date due to the fact that the sooner a dollar is
received the quicker it can be invested to earn a positive return.
Evaluating financial transactions requires valuing uncertain future cash
flows. Translating a value (cash flow) to the present is referred to as
discounting (finding the present value). Translating a value (cash flow) to
the future is referred to as compounding (finding the future value). Time
value of money concepts are applied to single cash flows (lump sums),
series of equal cash flows (annuities), series of equal cash flows that last
forever (perpetuities), and series of unequal cash flows (uneven cash flow
series). Time lines are frequently used to illustrate graphically when the
cash flows occur.
The principal is the amount borrowed or invested. Interest is the
compensation for the opportunity cost of funds (the time value of money)
and the uncertainty of repayment of the amount borrowed; that is, it
represents both the price of time and the price of risk. The price of time is
compensation for the opportunity cost of funds and the price of risk is
compensation for bearing risk; together they constitute the required rate of
return for the investment.
Interest is compound interest if interest is paid on both the principal and
any accumulated interest. Most financial transactions involve compound
interest or compound rate of return (yield).
Compounding (discounting) periods per year refers to the number of
times per year a particular cash flow or cash flows are compounded
(discounted).
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 Future Value of a Lump Sum
The process of calculating a future value is called "compounding." In its
most simple form, the relationship between the present value [PV], future
value [FV], interest rate [i] and time [n] variables can be stated as follows:
An amount to be received in the future [FV] is equal to the present value
[PV] and the interest earned during the investment period [n]. That is:
FV = PV + Interest, where Interest = PV * i
FV = PV + [PV * i] = PV * [1 + i]
Example: Assume that you currently have $100 and can invest it for one
year at six percent. How much money will you have at the end of the year?
FV [End of Year 1] = PV * [1 + i]
FV [End of Year 1] = $100 * [1 + .06] = $100 * 1.06 = $106
Therefore, at the end of the year you will have accumulated $106 (which
consists of the original $100 of principal and $6 of interest income).
Now assume that you leave your $106 on deposit to earn six percent for
another year. How much money will you have accumulated by the end of
the second year? Stated differently, what is the future value of $100 that is
invested at six percent for two years?
FV [End of Year 2] = PV [End of Year 1] * [1 + i]
FV [End of Year 2] = $106 * [1 + .06] = $106 * 1.06 = $112.36
or
FV [End of Year 2] = PV * [1 + i] * [1 + i] = PV * [1 + i]2
FV [End of Year 2] = $100 * [1 + .06]2 = $100 * 1.1236 = $112.36
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The second expression can be rewritten and summarized as follows:
FV = PV * [1 + i]n ,
where n is the total number of years over which the current amount (present
value) earns interest.
In the event that interest is earned more frequently than annually, then the
time variable [n] and the interest rate variable [i] must be modified to
represent the number of interest-earning periods and the periodic interest
rate involved. That is:
FV = PV * [1 + (i/m)]n*m ,
where [m] represents the number of compounding (interest-earning) periods
per year [n]. For example, if interest is paid semiannually (every six
months), then the interest rate [i] must be divided by two (2) in order to
calculate a semiannual interest rate, and the time variable [n] must be
doubled because there are two interest-earning periods per year.
Example: Assume that you currently have $100 and would like to invest
your funds for one year at 6% (compounded semiannually). How much
money would you have at the end of the year?
FV [End of Year 1] = $100 * [1 + (.06/2)]1*2 = $100 * [1.03]2
FV [End of Year 1] = $100 * 1.0609 = $106.09
The amount of interest earned (and therefore the future value accumulated)
when interest is compounded more frequently than annual is always greater
than the amount of interest earned (and the future value accumulated) when
interest is compounded annually, all other things remaining constant, since
you are earning interest on interest more frequently.
In the future value of a lump sum formula, the interest-rate term [1 + i]n is
called the "Future Value Interest Factor for i and n" which can be written
as [FVIFi,n]. Therefore, the future value of a lump sum formula (assuming
annual compounding) can be written as:
FVn = PV * [FVIFi,n]
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Some individuals use interest tables that have FVIF’s calculated for various
combinations of interest rate and time period. A sample section of a FVIF
table is shown below:
Future Value of $1 at the End of n Periods:
Number of
Periods
5%
8%
10%
20%
1
5
8
10
50
1.0500
1.2763
1.4775
1.6289
11.467
1.0800
1.4693
1.8509
2.1589
46.902
1.1000
1.6105
2.1436
2.5937
117.39
1.2000
2.4883
4.2998
6.1917
9100.4
Note: It is recommended that you obtain the FVIF’s using the “formula”
method since the “tables” will not help you if you need to find the FVIF
when, for example, n = 11.5 years and i = 6.75%. The FVIF in this case is
found as follows:
FVIF6.75%,11.5 = [1 + .0675]11.5 = [1.0675]11.5 = 2.1194865
For compounding periods other than annual, the future value of a lump sum
formula can be written as:
FVn = PV * [FVIF i/m,n*m]
The calculations can be made easier with the help of either a financial
calculator or a spreadsheet program. Using a Hewlett-Packard 10B
calculator, for example, we calculate the future value of $1,000 invested 10
years at 6% with the following keystrokes:
1000
+/-
PV
10
N
6
I/YR
FV
4
PV is the present value, N is the number of periods, I/YR is the interest rate
per period, and FV is the future value. You will notice that we changed the
sign on the PV. This is due to the way the financial function is programmed,
assuming that the present value is the cash outflow. The changing of the
sign for cash outflows is required in most (but not all) financial calculators.
In Microsoft Excel, the calculation uses the worksheet function FV:
=FV(rate,nper,pmt,pv,type)
Since there are no other cash flows in this problem, PMT (which represents
periodic cash flows, such as a mortgage payment) is entered as a zero. To
calculate the FV, the function requires the following inputs:
=FV(.06,10,0,-1000,0)
If interest is compounded continuously (that is, instantaneously), the
compound factor uses the exponential function, e, the inverse of
the natural logarithm (ln). To find the future value of a lump sum that is
compounded continuously, the following formula is used:
FV = PV * (e)i*n , where e  2.7183, a constant.
For example, suppose you want to calculate the future value of $2,000
invested for twenty years at 12%, with interest compounded continuously.
The future value is:
FV20 = $2,000 * (e).12*20 = $2,000 * (2.7183)2.4
FV20 = $2,000 (11.02317638) = $22,046.35
 Determining the Unknown Interest Rate (solving for i)
Assume that you have $2,000 to invest today and in four years you would
like the investment to be worth $3,500. What compound annual rate of
interest would you need to earn? We know the present value (PV = $2,000),
the future value (FV = $3,500) and the number of years (n= 4). Using the
basic valuation equation:
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FV = PV * [1 + i]n ,
and substituting the known values of FV, PV, and n,
$3,500 = $2,000 (1 + i)4 .
Rearranging the terms, we see that the ratio of the future value to the present
value is equal to the compound factor for four periods at some unknown
interest rate:
$3,500/$2,000 = (1 + i)4 or 1.7500 = (1 + i)4 ,
where 1.7500 is the future value interest factor [FVIF i, 4].
Therefore, we have one equation with one unknown, i. We can determine
the unknown interest rate either mathematically or by using the table of
interest factors. Using the table of interest factors, we see that for four
periods, the interest rate that produces an interest factor closest to 1.7500 is
approximately 15% per year.
We can determine the interest rate more precisely, however, by solving for i
using the following equation:
i = [(FV/PV)1/n] - 1.0
Therefore, the exact interest rate that will cause $2,000 to grow to $3,500 at
the end of four years is:
i = [($3,500/$2,000)1/4] – 1.0
i = [(1.75).25] – 1.0 = 1.1501633 – 1.0 = .1501633 = 15.01633%
Therefore, if you invested $2,000 in an investment that pays 15.01633%
compounded interest per year for four years, you would have $3,500 at the
end of the fourth year.
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A financial calculator can also be used to solve for i. Using a HP 10B, solve
for i as follows:
2,000
3,500
4
+/-
PV
FV
N
I/Y
Application: Determining growth rate in dividends for a stock.
There are many applications in which we need to determine the rate of
change in values over a period of time. If values are increasing over time,
we refer to the rate of change as the growth rate. We can use the
information about the starting value (the PV), the ending value (the FV), and
the number of periods to determine the rate of growth of values over this
time period. If we wish to determine the rate of growth in the values, we
solve for the unknown interest rate. Consider the growth rate of dividends
for ACME Corporation. ACME paid dividends of $2.84 per share in 1995
and $4.00 in 2000. We have dividends for two different points in time: 1995
and 2000. With 1995 dividends as the present value, 2000 dividends as the
future value, n = 5 and we can solve for the compound annual growth rate as
follows:
i = g = ($4.00/$2.84)1/5 - 1 = 7.089855%
Therefore, ACME's dividends grew at a rate of over 7% per year over this
five year period.
 Determining the Number of Compounding Periods (solving for n)
Let's say that you place $500 in a savings account that pays 8% compounded
interest per year. How long would it take for that savings account balance to
reach $800? In this case, we know the present value (PV = $500), the future
value (FV = $800), and the interest rate (i = 8% per year). What we need to
determine is the number of periods (n) required for $500 to grow to $800 at
an interest rate of 8%. Let's start with the basic valuation equation and insert
the known values of PV, FV, and i:
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FV = PV (1 + i)n = PV * [FVIF8%,n]
$800 = $500 [FVIF8%,n]
Rearranging terms,
[FVIF8%,n] = $800/$500 = 1.6000
Therefore, the FVIF8%,n is 1.6000.
Similar to the determination of the unknown interest rate, we can estimate
the number of periods by using the table of interest factors. If using the
table of interest factors, look down the column for the 8% interest rate to
find the factor closest to 1.6000. Then look across the row containing this
factor to find n. From the table of FVIF’s, we see that the n that corresponds
to a factor of 1.6000 for an 8% interest rate is between 6 and 7 periods.
Therefore, we know that it will take somewhere between 6 and 7 years for a
single deposit of $500 to grow to $800 at an 8% rate of interest.
Using a HP-10B calculator, we can solve for the exact n as follows:
500
800
8
+/-
PV
FV
I/YR
N
We can also use the following formula to solve for n:
n = [ln(FV/PV)]/[ln(1+i)] ,
where ln stands for the natural logarithm.
In the example above,
n = [ln(800/500)]/[ln(1.08)]
n = [ln(1.6000)]/[ln(1.08)]
n = (.4700036292/.0769610411) = 6.107 years (6 years, 39 days)
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