Managerial Economics _ Business Strategy

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					   Engineering
   Economics

    Module No. 06
Time Value of Money
        By
 Muhammad Shahid Iqbal
            The Time Value of Money
   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
                The Concepts Interest
   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        Interest Amount   Amount Owed at
                   Beginning of the   per period @ 10%    end of period
                        period

         1            1000                 100              1100

         2            1100                 110              1210

         3            1210                 121              1331
               The Time Value of Money
   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           interest      Compounded amount
          0                                  100.00
          1                15.00             115.00
           2               17.25               132.25
           3               19.84               152.09
           4               22.81               174.90
           5               26.24               201.14
               The Time Value of Money
   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                                      100
       1             86.96                    100
       2             75.61                    100
       3             65.75                    100
       4             57.18                    100
       5             49.72                    100
                       Present Value
   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 equally-
    spaced 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
                         Future Value
   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
   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.
   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
   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
   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
   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
   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
    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.

                   (1 +i)n – 1
        P=A
                   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

   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
   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

				
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posted:10/2/2011
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