CH 05 IM 7th BFM by liwenting

VIEWS: 17 PAGES: 4

									                                CHAPTER 5
            The Time Value of Money
                          CHAPTER ORIENTATION
In this chapter the concept of a time value of money is introduced, that is, a dollar today is
worth more than a dollar received a year from now. Thus if we are to logically compare
projects and financial strategies, we must either move all dollar flows back to the present or
out to some common future date.


                              CHAPTER OUTLINE
I.     Compound interest results when the interest paid on the investment during the first
       period is added to the principal and during the second period the interest is earned on
       the original principal plus the interest earned during the first period.
       A.      Mathematically, the future value of an investment if compounded annually at
               a rate of i for n years will be
                      FVn      =   PV (l + i)n
                      where n    = the number of years during which the compounding
                                       occurs
                             i   = the annual interest (or discount) rate
                           PV = the present value or original amount invested at the
                                       beginning of the first period
                          FVn    = the future value of the investment at the end of n
                                       years
               1.     The future value of an investment can be increased by either
                      increasing the number of years we let it compound or by
                      compounding it at a higher rate.
               2.     If the compounded period is less than one year, the future value of an
                      investment can be determined as follows:
                                                i  mn
                      FVn         =      PV 1 + m
                                                  
                      Where m         = the number of times compounding occurs during the
                                            year




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II.    Determining the present value, that is, the value in today's dollars of a sum of money
       to be received in the future, involves nothing other than inverse compounding. The
       differences in these techniques come about merely from the investor's point of view.
       A.     Mathematically, the present value of a sum of money to be received in the
              future can be determined with the following equation:
                                               1 
                      PV           =       FVn       n
                                              (1 + i) 
                      where:   n       =    the number of years until payment will be received,
                               i       =    the opportunity rate or discount rate
                               PV      =    the present value of the future sum of money
                               FVn     =    the future value of the investment at the end of n
                                                years
              1.      The present value of a future sum of money is inversely related to
                      both the number of years until the payment will be received and the
                      opportunity rate.
III.   An annuity is a series of equal dollar payments for a specified number of years.
       Because annuities occur frequently in finance, for example, bond interest payments,
       we treat them specially.
       A.     A compound annuity involves depositing or investing an equal sum of money
              at the end of each year for a certain number of years and allowing it to grow.
              1.      This can be done by using our compounding equation and
                      compounding each one of the individual deposits to the future or by
                      using the following compound annuity equation:
                                                n 1        
                      FVn          =       PMT   (1  i) t 
                                                            
                                                t 0        
                      where:   PMT          = the annuity value deposited at the end of each
                                                  year
                               i            = the annual interest (or discount) rate
                               n            = the number of years for which the annuity will
                                                  last
                               FVn          = the future value of the annuity at the end of the
                                                  nth year
       B.     Pension funds, insurance obligation, and interest received from bonds all
              involve annuities. To compare these financial instruments we would like to
              know the present value of each of these annuities.
              1.      This can be done by using our present value equation and discounting
                      each one of the individual cash flows back to the present or by using
                      the following present value of an annuity equation:
                                                   n          1 
                      PV               =      PMT  
                                                                     
                                                   t 1    (1  i) t 
                                                                      


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                      where:       PMT    =   the annuity withdrawn at the end of each year
                                   i      =   the annual interest or discount rate
                                   PV     =   the present value of the future annuity
                                   n      =   the number of years for which the annuity will
                                                  last
      C.     This procedure of solving for PMT, the annuity value when i, n, and PV are
             known, is also the procedure used to determine what payments are associated
             with paying off a loan in equal installments. Loans paid off in this way, in
             periodic payments, are called amortized loans. Here again we know three of
             the four values in the annuity equation and are solving for a value of PMT,
             the annual annuity.
IV.   Annuities due are really just ordinary annuities where all the annuity payments have
      been shifted forward by one year, compounding them and determining their present
      value is actually quite simple. Because an annuity due merely shifts the payments
      from the end of the year to the beginning of the year, we now compound the cash
      flows for one additional year.
V.    A perpetuity is an annuity that continues forever, that is every year from now on this
      investment pays the same dollar amount.
      A.     An example of a perpetuity is preferred stock which yields a constant dollar
             dividend infinitely.
      B.     The following equation can be used to determine the present value of a
             perpetuity:
                                     pp
             PV                =      i
             where:    PV =          the present value of the perpetuity
                       pp =          the constant dollar amount provided by the perpetuity
                       i  =          the annual interest or discount rate
VI.   (Financial Tables)
V.    Spreadsheets and the Time Value of Money.
      A.     While there are several competing spreadsheets, the most popular one is
             Microsoft Excel. Just as with the keystroke calculations on a financial
             calculator, a spreadsheet can make easy work of most common financial
             calculations. Listed below are some of the most common functions used with
             Excel when moving money through time:
      Calculation:             Formula:
      Present Value        = PV(rate, number of periods, payment, future value, type)
      Future Value         = FV(rate, number of periods, payment, present value, type)
      Payment              = PMT(rate, number of periods, present value, future value,
              type)
      Number of Periods    = NPER(rate, payment, present value, future value, type)
      Interest Rate        = RATE(number of periods, payment, present value, future
                              value, type, guess)



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where: rate              = i, the interest rate or discount rate
       number of periods = n, the number of years or periods
       payment           = PMT, the annuity payment deposited or received at the
                                end of each period
       future value      = FV, the future value of the investment at the end of n
                                periods or years
       present value     = PV, the present value of the future sum of money
       type              = when the payment is made, (0 if omitted)
       0                 = at end of period
       1                 = at beginning of period
       guess             = a starting point when calculating the interest rate, if
                                omitted, the calculations begin with a value of 0.1 or
                                10%




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