# CH 05 IM 7th BFM by liwenting

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```									                                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
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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