Download Chapter (Eng. Eco) 006

Engineering
Economics
Module No. 06
Time Value of Money
By
Muhammad Shahid Iqbal
The Time Value of Money
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A fundamental idea in finance that money that one has now is
worth more than money one will receive in the future.
Because money can earn interest or be invested, it is worth more
to an economic actor if it is available immediately.
The time value of money is money's potential to grow in value
over time.
Because of this potential, money that's available in the present is
considered more valuable than the same amount in the future.
For example, 100 dollars of today's money invested for one year
and earning 5 percent interest will be worth 105 dollars after one
year.
Therefore, 100 dollars paid now or 105 dollars paid exactly one
year from now both have the same value to the recipient who
assumes 5 percent interest.
The Concepts Interest
Interest is a charge for borrowing money, usually stated as
a percentage of the amount borrowed over a specific period
of time.
 Simple Interest Rate: The total interest earned or charged
is linearly proportional to the initial amount of the loan
(principal), the interest rate and the number of interest
periods for which the principal is committed.
I=(P)(i)(N)
where
 P = principal amount lent or borrowed
 N = number of interest periods ( e.g., years )
 i = interest rate per interest period
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The Concepts Interest
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Compound Interest Rate: Whenever the interest charge
for any interest period is based on the remaining principal
amount plus any accumulated interest charges up to the
beginning of that period. Compound interest is always
assumed in TVM problems.
Period
Amount Owed
Beginning of the
period
Interest Amount
per period @ 10%
Amount Owed at
end of period
1
1000
100
1100
2
1100
110
1210
3
1210
121
1331
The Time Value of Money
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If a investor invests a sum of RS. 1000 in a fixed deposit
with interest rate 15% compounded annually. The
accumulated amount at the end of the year is
Year end
0
1
2
3
4
5
interest
15.00
17.25
19.84
22.81
26.24
Compounded amount
100.00
115.00
132.25
152.09
174.90
201.14
The Time Value of Money
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Alternatively If we want RS. 100 at the end of 5 years what is
the amount that we deposit now at a given interest rate say 15%
Year end
Present worth
Compounded amount
0
1
86.96
100
100
2
75.61
100
3
65.75
100
4
57.18
100
5
49.72
100
Present Value
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Present Value is an amount today that is equivalent to a
future payment, or series of payments, that has been
discounted by an appropriate interest rate.
The future amount can be a single sum that will be
received at the end of the last period, as a series of equallyspaced payments (an annuity), or both.
Since money has time value, the present value of a
promised future amount is worth less the longer you have
to wait to receive it.
The time value of money principle says that future dollars
are not worth as much as dollars today.
Present Value
PV = FV/(1 + r)t
FV = Future Value
PV = Present Value
r = the interest rate per period
t= the number of compounding periods
What is the present value of $8,000 to be paid at the end of three years if
the correct (risk adjusted interest rate) is 11%?
PV = 8,000/(1.11)3 = 8,000/1.36 = 5,849
I will give you $1000 in 5 years. How much money should you give me
now to make it fair to me. You think a good interest rate would be 6%
FV= PV ( 1 + i )N
1000 = PV ( 1 + .06)5
1000 = PV (1.338)
1000 / 1.338 = PV
747.38 = PV
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Future Value
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Future Value is the amount of money that an investment with a
fixed, compounded interest rate will grow to by some future
date.
The investment can be a single sum deposited at the beginning of
the first period, a series of equally-spaced payments (an annuity),
or both. Since money has time value, we naturally expect the
future value to be greater than the present value.
The difference between the two depends on the number of
compounding periods involved and the going interest rate.
Future Value
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What is the future value of $34 in 5 years if the
interest rate is 5%? (i=.05)
FV= PV ( 1 + r )t
FV= 34 ( 1+ .05 ) 5
FV= 34 (1.2762815)
FV= 43.39.
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Determine Future Value Compounded Monthly
What is the future value of $34 in 5 years if the
interest rate is 5%? (i equals .05 divided by 12,
because there are 12 months per year. So
0.05/12=.004166, so i=.004166)
FV= PV ( 1 + i )N
FV= 34 ( 1+ .004166 )60
FV= 34 (1.283307)
FV= 43.63.
Equal Payment series compounded amount
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If we want to find the future worth of n equal payments
which are made at the end of year of the nth period at an
interest rate of i compounded annually.
(1 +i)n – 1
FV= A
i
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A person who is now 35 years old is planning for his
retired life. He plans to invest an equal sum of Rs. 10,000
at the end of every next 25 years. The bank gives 20%
interest rate compounded annually. Find the maturity
value of his account when he is 60 years old.
Equal Payment series Sinking Fund
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If we want to find the equivalent amount A that should be
deposited at the end of year of the nth period to realize a
future sum F at an interest rate of i compounded annually.
i
A = FA
(1 +i)n – 1
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A company has to replace the present facility after 15
years of Rs. 500,000. it plans to deposit an equal amount
at the end of every year for 15 years at an interest rate of
18% compounded annually. Find the equivalent amount
that must be deposited at the end of every year for the
next 15 years.
Equal Payment Series Present Worth Amount
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If we want to find the present worth of an equal payment
made at the end of year of the nth period at an interest rate
of i compounded annually.
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P=A
(1 +i)n – 1
i (1 + i)n
A company wants to set up a reserve to an annual
equivalent amount of Rs. 10,00,000 for next 20 years.
The reserve is assumed to grow at the rate of 15%
annually. Find the single payment that must be paid now
as the reserve amount.
Equal Payment Series Capital Recovery Amount
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If we want to find the annual equivalent amount A which
is to be recovered at the end of year of the nth period for a
loan P which is sanctioned now at an interest rate of i
compounded annually.
i(1 + i)n
A= P
(1 +i)n – 1
A Bank give a loan to a company worth Rs. 10,00,000 at
an interest rate of 18% compounded annually. This
amount should be repaid in 15 yearly equal installments.
Find the installment amount that company has to pay to
the bank.
Effective interest Rate
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Let i be the nominal interest rate compounded annually.
But in practice compounding may occur less than a year
like monthly, quarterly or semi-annually. The formula to
compute effective interest rate is
Effective interest rate, R = (1 + i/C)C -1
A person invests Rs. 5,000 in a bank at a nominal interest
rate of 12% for 10 years. The compounding is quarterly.
Find the maturity amount of the deposit after 10 years.
Effective interest rate, R = (1 + i/C)C -1
F = P(1 + R)n