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					  Time Value of Money

Future Value and Compounding
Present Value and Discounting
Annuity and Perpetuity
Amortization Loans
Different Compounding Periods
                                1
   Principle of Time Value
Time is money.
A dollar today is more valuable than a
dollar tomorrow
Why?
Money can earn a positive rate of return
at no risk.



                                           2
               Time Line


   0           1           2           3
        i%

  CF0          CF1        CF2         CF3

Time lines show the timing of cash flows.
Tick marks are at the end of periods. Time 0 is
today; Time 1 is placed at the end of Period 1
or at the beginning of Period 2.

                                                  3
Time line for a $100 cash flow
received at the end of Year 2



 0           1           2 Year
      i%

                        100




                                  4
Time line for equal payments of
        $100 for 3 years



0         1         2             3
    i%

         -100     -100        -100




                                      5
Time line for an uneven cash flow
              stream



 0           1       2          3
       i%

-150        100      75         50



                                     6
   Time Value of Money


One of the most important
fundamental concepts in finance.
A dollar in hand today is worth
more than a dollar to be received
in the future.

      FV = PV  (1 + i )n
                                    7
What’s the value of an initial $100
deposit after 3 years if i = 10%?


 0            1          2           3
       10%

100                              FV = ?

     The process of finding FVs is
     called compounding.

                                          8
After 1 year:
   FV1 = PV + INT1 = PV + PV  i
       = PV  (1 + i )
       = $100  (1.10)
       = $110.00




                                   9
After 2 years:
   FV2 = FV1  (1 + i )
       = 110  (1.10)
       = $100  (1.10)2
       = PV  (1 + i )2
       = $121.00




                          10
After 3 years:
    FV3 =121(1 +0.10 )
        =100(1 +0.10 )2 (1+0.10)
        =100(1.10)3
        =PV(1 + i )3
        =$133.10
In general:

      FVn = PV  (1 + i )n
                                     11
    Future value equation


       FVn = PV  (1 + i )n
There are 4 variables in the
equation: FVn , PV, i and n. If any 3
variables are known, the calculator
will solve for the 4th.

                                        12
The setup to find FV:

   INPUTS   3     10    100    0
            N    I/YR   PV    PMT     FV

   OUTPUT                           -133.10




                                              13
 How much should you save now to
  have $100 in 3 years if i = 10%?


   0           1        2           3
         10%


PV = ?                             100


  Finding PVs is discounting, and it’s
  the reverse of compounding.
                                         14
Present Value Equation



   PV = FVn  (1 + i )n

                         3
                1   
   PV = $100  
                1.10 
      = $100  0.7513 = $75.13

                                 15
Financial Calculator Solution


 INPUTS   3    10              0    100
          N   I/YR    PV      PMT   FV

 OUTPUT              -75.13




                                          16
What interest rate would cause $100 to
     grow to $125.97 in 3 years?

Solve for i :

        100 (1 + i )3 = $125.97

  INPUTS    3           -100      0   125.97
            N    I/YR   PV     PMT     FV

  OUTPUT         8.0
                                               17
If sales grow at 20% per year, how
     long before sales double?

Solve for n:
          FVn = PV  (1 + i )n
            2 = 1  (1.20)n



   INPUTS          20     -1      0    2
             N    I/YR   PV      PMT   FV

   OUTPUT 3.8

                                            18
     Definition of Annuity

Annuity: a series of cash flows of an
equal amount at fixed intervals for a
specified number of periods.
Ordinary Annuity: an annuity whose
payments occur at the end of each
period.
Annuity Due: an annuity whose
payments occur at the beginning of
each period.
                                        19
What’s the difference between an
ordinary annuity and an annuity due?
Ordinary Annuity
    0              1    2       3
         i%

               PMT     PMT     PMT

Annuity Due
    0              1    2       3
         i%

   PMT         PMT     PMT
                                    20
What’s the FV of a 3-year ordinary
    annuity of $100 at 10%?


0          1       2          3
    10%

          100     100        100
                             110
                             121
                        FV = 331

                                     21
Financial Calculator Solution



INPUTS   3    10    0    -100
         N   I/YR   PV   PMT      FV

OUTPUT                          331.00




                                         22
  What’s the PV of this ordinary
            annuity?

  0            1     2             3
      10%

              100   100        100
 90.91
 82.64
 75.13
248.68 = PV

                                       23
Financial Calculator Solution



 INPUTS   3    10              100   0
          N   I/YR     PV      PMT   FV

 OUTPUT              -248.69




                                          24
           Loan Analysis
Loan amount=PV of all payments
discounted at the contractual interest rate




                                              25
   Amortization Loans


Amortized Loan: a loan that is
repaid in equal payments over its
life.
Each periodic payment includes
not only interest but also a portion
of principal.



                                       26
If you borrow a 3-year, $1,000, 10%
amortized loan, how much do you need
to pay every year?

  0                1               2                  3
        10%

1,000             PMT         PMT                PMT

  INPUTS      3         10    1000                0
              N        I/YR   PV        PMT      FV

  OUTPUT                               -402.11
                                                          27
Loan= $1,000
PV of all payments discounted at 10%=Loan of $,1000




   0               1               2                  3
        10%

1,000             PMT         PMT                PMT

  INPUTS      3         10    1000                0
              N        I/YR   PV        PMT      FV

  OUTPUT                               -402.11
                                                          28
   Amortization Schedule


 Amortization  Schedule is a table
  that shows precisely how a loan
  will be repaid.
 Construct a amortization
  schedule for the 3-year, $1,000,
  10% amortized loan.


                                      29
  Find interest charge for Year 1



Interest charge
= Initial balance  interest rate
= $1,000  (0.10) = $100




                                    30
Find principal repayment in Year 1


  Principal repayment
  = Total payment – interest charge
  = $402.11 - $100 = $302.11




                                      31
   Find end balance for Year 1


End balance
= Initial balance - principal repayment
= $1,000 - $302.11 = $697.89

Repeat these steps for Years 2 and 3
to complete the amortization table.

                                          32
     Amortization Schedule

     BEG                     PRIN    END
YR   BAL      PMT     INT    PMT     BAL

1    $1,000   $402    $100   $302    $698
2       698     402     70     332    366
3       366     402     36     366      0
              1,206    206   1,000



                                            33
     $
402.11
             Interest charge

302.11

               Principal repayments



         0        1            2      3
Level payments. Interest declines because
outstanding balance declines. Lender earns
10% on loan outstanding, which is falling.
                                             34
Different Compounding Periods

Will the FV of a lump sum be larger or
smaller if we compound more often,
holding the stated i% constant?

LARGER! If compounding is more
frequent than once a year--for
example, semiannually, quarterly, or
daily--interest is earned on interest
more often.
                                         35
Annual vs. Semiannual
Compounding
 0             1       2            3
       10%

100                                133.10

 Annually: FV3 = 100(1.10)3 = 133.10
0              1           2       3 Year
0          1   2   3       4   5   6 Period
      5%

100                                ?
                                            36
Annual vs. Semiannual
Compounding


 INPUTS   6    5     100    0
          N   I/YR   PV    PMT     FV

 OUTPUT                          -134.01




                                           37
Effective Annual Rate

 EAR is the interest rate which
 causes PV to grow to the same FV
 as under multi-period compounding.
 EAR is the actual rate of return
 investors earn, or the actual rate of
 interest borrowers pay.


                                         38
Effective Annual Rate

 The return on an investment with
 monthly payments is different from
 one with the same nominal value
 but quarterly payments.
 We must convert both into EAR
 basis to compare the rates of
 return.

                                      39
Calculating Effective Annual Rate

                        m
                 iNom 
     EAR   = 1 +       -1
                  m 
                 0 . 10  2
           = 1 +         - 1 .0
                    2 
           = ( . 05 )2 - 1 . 0
             1
           = 0 . 1025 = 10 . 25 %
                                    40
Calculating Effective Annual Rate

 Nominal Rate                        = 10%

 EARAnnual                           = 10%

 EARQ        = (1 + 0.10/4)4 - 1     = 10.38%

 EARM        = (1 + 0.10/12)12 - 1   = 10.47%

 EARD(365) = (1 + 0.10/365)365 - 1 = 10.52%

                                                41
Calculating annual nominal rate (inom) from
 effective compounding period rate (keff)

inom=keffxm
Also,
keff=inom/m




                                              42
            Perpetuities

A perpetuity is an annuity that
continues forever.
The present value of a perpetuity is:


                   PMT
           PV =
                    i
                                        43
      Growing Perpetuities

The cash flows of a growing perpetuity
grow at a constant rate forever.
The present value of a growing
perpetuity is:


                  PMT1
           PV =
                  i-g
                                         44