The
Time Value of Money
A Shilling on hand today is worth more than a Shilling to be received in
the future because the Shilling on hand today can be invested to earn interest
to yield more than a Shillingin the future. The Time Value of Money mathematics
quantify the value of a Shilling through time. This, of course, depends upon
the rate of return or interest rate which can be earned on the investment.
The Time Value of Money has applications in many areas of Corporate
Finance including Capital Budgeting, Bond Valuation, and Stock Valuation. For
example, a bond typically pays interest periodically until maturity at which
time the face value of the bond is also repaid. The value of the bond today,
thus, depends upon what these future cash flows are worth in today's dollars.
The Time Value of Money concepts will be grouped into two areas: Future
Value and Present Value. Future Value describes the process of finding what an
investment today will grow to in the future. Present Value describes the
process of determining what a cash flow to be received in the future is worth
in today's dollars.
Future
Value
The Future
Value of a cash flow represents the amount, at some time in the future, that an
investment made today will grow to if it is invested to earn a specific
interest rate. For example, if you were to deposit Shs100 today in a bank
account to earn an interest rate of 10% compounded annually, this investment
will grow to Shs110 in one year. This can be shown as follows:
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Year 1
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Shs100(1 + 0.10) = Shs110
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At the end of two years, the initial investment will have grown to Shs121.
Notice that the investment earned Shs11 in interest during the second year,
whereas, it only earned Shs10 in interest during the first year. Thus, in the
second year, interest was earned not only on the initial investment of Shs100
but also on the Shs10 in interest that was paid at the end of the first year.
This occurs because the interest rate in the example is a compound interest
rate.
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Compound
Interest
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Under
compound interest, interest is earned not only on the initial principal but
also on the accumulated interest. Interest begins to be earned on the accumulated
interest as soon as it is paid, which occurs at the end of each compounding
period. This is in contrast to simple interest, under which interest
is only earned on the initial principal.
Valuations should generally be based on compound
interest because, after the interest has been paid, the full amount, i.e.,
the initial principal plus interest, could be withdrawn and reinvested
elsewhere. Thus, interest on the new investment would be earned on the full
amount.
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The interest rate in the example is 10% compounded annually. This
implies that interest is paid annually. Thus the balance in the account was Shs110
at the end of the first year. Thus, in the second year the account pays 10% on
the initial principal of Shs100 and the Shs10 of interest earned in the first
year. Thus, the Shs121 balance in the account after two years can be computed
as follows:
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Year 2
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Shs110(1+0.10) = Shs121 or
Shs100(1+0.10)(1+0.10) = Shs121 or
Shs100(1+0.10)2 = Shs121
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If the money was left in the account for one more year, interest would
be earned on Shs121, i.e., the initial principal of Shs100, the Shs10 in
interest paid at the end of year 1, and the Shs11 in interest paid at the end
of year 2. Thus the balance in the account at the end of year three is Shs133.10.
This can be computed as follows:
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Year 3
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Shs121(1+0.10) = Shs133.10 or
Shs100(1+0.10) (1+0.10) (1+0.10) =
Shs133.10 or
Shs100 (1+0.10)3 = Shs133.10
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A pattern should be becoming apparent. The Future Value of an initial
investment at a given interest rate compounded annually at any point in the
future can be found using the following equation:
where
- FVt = the Future Value at the end of year t,
- CF0 = the initial investment,
- r = the annually compounded interest rate, and
- t = the number of years.
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Future Value Example
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Find the Future Value at the end of 4 years of Shs100
invested today at an interest rate 10%.
Solution:
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Present
Value
Present Value describes the process of determining what a cash flow to
be received in the future is worth in today's dollars. Therefore, the Present
Value of a future cash flow represents the amount of money today which, if
invested at a particular interest rate, will grow to the amount of the future
cash flow at that time in the future. The process of finding present values is
called Discounting and the interest rate used to calculate present
values is called the discount rate. For example, the Present Value of Shs100
to be received one year from now is Shs90.91 if the discount rate is 10%
compounded annually. This can be demonstrated as follows: (Refer to the Future Value page if you are unfamiliar with the
calculations.)
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One Year
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Shs90.91(1 + 0.10) = Shs100 or
Shs90.91 = Shs100/(1 + 0.10)
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Notice that the Future Value Equation is used to describe the relationship
between the present value and the future value. Thus, the Present Value of Shs100
to be received in two years can be shown to be Shs82.64 if the discount rate is
10%.
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Two Years
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Shs82.64(1 + 0.10)2 = Shs100
or
Shs82.64 = Shs100/(1 + 0.10)2
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A pattern should be becoming apparent. The following equation can be
used to calculate the Present Value of a future cash flow given the discount
rate and number of years in the future that the cash flow occurs. (This
equation can be obtained algebraically from the Future Value Equation.)
where
- PV = Present Value
- CFt = Future Cash Flow which occurs t years from now
- r = the interest or discount rate
- t = the number of years
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Present Value Example
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Find the Present Value of Shs100 to be received 3
years from today if the interest rate is 10%.
Solution:
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Annuities
An Annuity is a cash flow stream which adheres to a specific pattern.
Namely, an Annuity is a cash flow stream in which the cash flows are level (i.e.,
all of the cash flows are equal) and the cash flows occur at a regular
interval. The annuity cash flows are called annuity payments or simply payments.
Thus, the following cash flow stream is an annuity.
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Figure 1
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While, the following cash flow stream is not an annuity because the
payments do not occur at a regular interval.
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Figure 2
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When a cash flow stream is of the form given in Figure 1, i.e.,
an annuity, the process of finding the Present Value or Future Value of the cash flow stream is greatly
simplified.
Present Value of an Annuity
The Present Value of an Annuity is equal to the sum of the present values
of the annuity payments. This can be found in one step through the use of the
following equation:
where
- PVA = The Present Value of the Annuity
- PMT = The Annuity Payment
- r = The Interest or Discount Rate
- t = The Number of Years (also the Number of Annuity Payments)
Consider the annuity of Shs100 per year for five years given in Figure
1. If the discount rate is equal to 10%, then the Present Value of the
Annuity can be found as follows:
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Present Value of the Annuity
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Other Compounding Periods
In the real world, interest rates are often compounded more often than
once per year. By convention, interest rates are quoted on an annual basis. An
interest rate, quoted on an annual basis, which is compounded more often than
once per year is called a nominal rate, stated rate, quoted rate, or annual
percentage rate (APR). For example, mortgages typically require monthly
payments and, therefore, the interest rates quoted on mortgages are compounded
monthly. Thus, the nominal interest rate on a mortgage might be 12% compounded
monthly. However, the relevant rate for valuations is the periodic rate. The
periodic rate is computed by dividing the nominal rate by the number of
compounding periods per year.
where
- r = the rate per period,
- rnom = the nominal rate, and
- m = the number of compounding periods per year.
Thus a 12% nominal rate compounded monthly is equivalent to a periodic
rate of 1% per month.
The following sections of this page demonstrate how to convert a nominal
rate into an equivalent rate that is compounded annually and provide versions
of the Present Value and Future Value formulas for use with interest rates
compounded more often than once per year. The page concludes with a discussion
of continuous compounding.
EAR - Effective or
Equivalent Annual Rate
The Effective or Equivalent Annual Rate (EAR) is the interest rate
compounded annually that is equivalent to a nominal rate compounded more than
once per year. In other words, present and future values computed using the EAR
will be the same as those computed using the nominal rate. The EAR is computed
as follows:
- EAR = the Equivalent or Effective Annual Rate,
- rnom = the nominal interest rate,
- m = the number of compounding periods per year, and
Moreover, it is not proper to directly compare interest rates which have
a particular compounding frequency with those that have a different compounding
frequency, e.g.,, comparing 10.1% compounded semiannually with 10%
compounded quarterly. This problem can be overcome by finding the EAR for each
of the rates and then comparing the EARs.
First, let's find the EAR for 10.1% compounded semiannually. Here, m
equals 2.
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EAR for 10.1% compounded
semiannually
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Now, let's find the EAR for 10% compounded quarterly. Here m = 4.
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EAR for 10% compounded quarterly
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Thus, we see that 10% compounded quarterly is actually a higher interest
rate than 10.1% compounded semiannually. Given a choice, we would prefer to invest
at 10% compounded quarterly.
Present Value
The Present Value of a future cash flow when the interest rate is
compounded m times per year can be calculated as follows:
where
- PV = the Present Value,
- CFt = the cash flow which occurs at the end of year t,
- rnom = the nominal interest rate,
- m = the number of compounding periods per year, and
- t = the number of years.
- Thus, mt = the number of compounding periods in t years.
In the earlier discussion of Present Value
the interest rate was compounded annually and there was one compounding period
per year. In that case m = 1. Thus, our earlier Present Value formula is
actually just a special case of this formula since under annual compounding the
rate per period is the same as the nominal rate.
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Present Value Example
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Find the Present Value of Shs100 to be received 3
years from today if the interest rate is 12% compounded quarterly.
Solution:
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Future Value
The Future Value of a future cash flow when the interest rate is
compounded m times per year can be calculated as follows:
where
- FV = the Future Value,
- CF0 = the cash flow which occurs at time 0,
- rnom = the nominal interest rate,
- m = the number of compounding periods per year, and
- t = the number of years.
- Thus, mt = the number of compounding periods in t years.
Thus, the earlier Future Value formula is actually just a special case
of this formula since under annual compounding (i.e., when m = 1) the
rate per period is the same as the nominal rate.
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Future Value Example
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Find the Future Value of 3 years from now of Shs100
invested today at an interest rate of 10% compounded semiannually.
Solution:
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Present Value of an
Annuity
The Present Value of an Annuity when the payments occur m times per year
and the interest rate is compounded m times per year can be calculated as
follows:
where
- PVA = the Present Value,
- PMT = the Annuity Payment which occurs m times per year,
- rnom = the nominal interest rate,
- m = the number of compounding periods per year, and
- t = the number of years.
- Thus, mt = the number of payments and compounding periods in t years.
This formula can only be applied when the frequency of the annuity
payments is the same as the compounding period for the interest rate. For
example, if the annuity has quarterly payments the interest rate must be
compounded quarterly (m = 4).
Thus, the earlier Present Value on an Annuity formula is actually just a
special case of this formula since under annual compounding (i.e., when
m = 1) the rate per period is the same as the nominal rate.
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Present Value of an Annuity
Example
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Find the Present Value of an annuity of Shs100 per
month for 2 years if the interest rate is 12% compounded monthly.
Solution:
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Future Value of an Annuity
The Future Value of an Annuity when the payments occur m times per year
and the interest rate is compounded m times per year can be calculated as
follows:
where
- FVAt = the Future Value of the annuity at the end of year t,
- PMT = the Annuity Payment which occurs m times per year,
- rnom = the nominal interest rate,
- m = the number of compounding periods per year, and
- t = the number of years.
- Thus, mt = the number of payments and compounding periods in t years.
This formula can only be applied when the frequency of the annuity
payments is the same as the compounding period for the interest rate. For
example, if the annuity has quarterly payments the interest rate must be
compounded quarterly (m = 4). As with the earlier formula, the Future Value is
computed at the end of the period in which the last annuity payment occurs.
Thus, the earlier Future Value on an Annuity formula is actually just a
special case of this formula since under annual compounding (i.e., when
m = 1) the rate per period is the same as the nominal rate.
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Future Value of an Annuity Example
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Find the Future Value at the end of 3 years of an
annuity of Shs100 per quarter for 3 years if the interest rate is 8%
compounded quarterly.
Solution:
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