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					                                                    1




Mathematics
of Finance            With a Financial Calculator



              Presentation
                                                   2
Compound Value
Parameters:
Interest rate (i)
Amount that is invested, present value (PV)
Time money remains invested (n)
Future value of the investment in n years (FVn)
Periodic equal payment (or deposit) (PMT)
                                                      3
Compound Value
Future Value of a Lump Sum (one time payment):
Value at some time in the future of an investment
Interest compounds: earn interest on interest in later
 years.
Future value in one year is present value plus the
 interest that is earned over the year.
                                                 4
Compound Value
Future Value of a Lump Sum (one time payment):
In General

               FVn = PV(1+ i)n
                                                           5
Compound Value
Present Value of a Lump Sum (one time payment):
Value today of an amount to be received or paid in
 the future.
                           FVn
                    PV =
                         (1 + i)n

  Example: Expect to receive $100 in eight years. If can
           invest at 10%, what is it worth today?
                                                               6
Compound Value
Present Value of a Lump Sum (one time payment):
Value today of an amount to be received or paid in
 the future.
                         FV8
                  PV =
                       (1 + i)8

  Example: Expect to receive $100 in EIGHT years. If can
           invest at 10%, what is it worth today?
                   0    1    2   3    4    5    6    7     8

                   ?                                     $100
                                                               7
Compound Value
Present Value of a Lump Sum (one time payment):
Value today of an amount to be received or paid in
 the future.
                         FV8
                  PV =
                       (1 + i)8

  Example: Expect to receive $100 in EIGHT years. If can
           invest at 10%, what is it worth today?
                   0    1    2   3    4    5    6    7     8

                   ?                                     $100
                                                                          8
Financial Calculator
Setting Display
Should show at least 2
 decimal places on dollar                 0.00
 amounts and 4 decimal
 places on percentages                                  P/YR
                                      N   I/YR PV PMT FV


                                     CLEAR ALL
                                      INPUT

                                             7      8           9     
                                             4      5           6     x

                                             1      2           3     –
                                          BEG/END              DISP
                                             0      .           =     +
                  HP10B Calculator
                                                                          9
Financial Calculator
Setting Display
Should show at least 2
 decimal places on dollar                 0.00
 amounts and 4 decimal
 places on percentages                                  P/YR
                                      N   I/YR PV PMT FV


                                     CLEAR ALL
                                      INPUT

                                             7      8           9     
                                             4      5           6     x
                             1
                                             1      2           3     –
                                          BEG/END              DISP
                                             0      .           =     +
                  HP10B Calculator
                                                                          10
Financial Calculator
Setting Display
Should show at least 2
 decimal places on dollar                 0.00
 amounts and 4 decimal
 places on percentages                                  P/YR
                                      N   I/YR PV PMT FV


                                     CLEAR ALL
                                      INPUT

                                             7      8           9     
                                             4      5           6     x
                             1
                                             1      2           3     –
                                 2        BEG/END              DISP
                                             0      .           =     +
                  HP10B Calculator
                                                                          11
Financial Calculator
Setting Display
Should show at least 2
 decimal places on dollar                 0.0000
 amounts and 4 decimal
 places on percentages                                  P/YR
                                      N   I/YR PV PMT FV


                                     CLEAR ALL
                                      INPUT

                                 3           7      8           9     
                                             4      5           6     x
                             1
                                             1      2           3     –
                                 2        BEG/END              DISP
                                             0      .           =     +
                  HP10B Calculator
                                                                         12
Financial Calculator
Clearing Memory
Financial calculators contain
 a number of memory                      0.0000
 registers. These registers
 should be cleared to                                  P/YR
 prevent carry-over errors.          N   I/YR PV PMT FV


                                    CLEAR ALL
                                     INPUT

                                            7      8           9     
                                            4      5           6     x

                                            1      2           3     –
                                         BEG/END              DISP
                                            0      .           =     +
                 HP10B Calculator
                                                                         13
Financial Calculator
Clearing Memory
Financial calculators contain
 a number of memory                      0.0000
 registers. These registers
 should be cleared to                                  P/YR
 prevent carry-over errors.          N   I/YR PV PMT FV


                                    CLEAR ALL
                                     INPUT

                                            7      8           9     

                             1              4      5           6     x

                                            1      2           3     –
                                         BEG/END              DISP
                                            0      .           =     +
                 HP10B Calculator
                                                                         14
Financial Calculator
Clearing Memory
Financial calculators contain
 a number of memory                      0.0000
 registers. These registers
 should be cleared to                                  P/YR
 prevent carry-over errors.          N   I/YR PV PMT FV

                             2
                                    CLEAR ALL
                                     INPUT

                                            7      8           9     

                             1              4      5           6     x

                                            1      2           3     –
                                         BEG/END              DISP
                                            0      .           =     +
                 HP10B Calculator
                                                                        15
Financial Calculator
Setting Compounding Frequency
Compounding should be set
 to annual, i.e. P/YR=1, not            0.0000
 the factory setting of 12.
                                                      P/YR
                                    N   I/YR PV PMT FV


                                   CLEAR ALL
                                    INPUT

                                           7      8           9     
                                           4      5           6     x

                                           1      2           3     –
                                        BEG/END              DISP
                                           0      .           =     +
                HP10B Calculator
                                                                        16
Financial Calculator
Setting Compounding Frequency
Compounding should be set
 to annual, i.e. P/YR=1, not            1.0000
 the factory setting of 12.
                                                      P/YR
                                    N   I/YR PV PMT FV


                                   CLEAR ALL
                                    INPUT

                                           7      8           9     
                           1
                                           4      5           6     x

                                           1      2           3     –
                                        BEG/END              DISP
                                           0      .           =     +
                HP10B Calculator
                                                                        17
Financial Calculator
Setting Compounding Frequency
Compounding should be set
 to annual, i.e. P/YR=1, not            1.0000
 the factory setting of 12.
                                                      P/YR
                                    N   I/YR PV PMT FV


                                   CLEAR ALL
                                    INPUT

                                           7      8           9     
                           1
                                           4      5           6     x
                           2
                                           1      2           3     –
                                        BEG/END              DISP
                                           0      .           =     +
                HP10B Calculator
                                                                        18
Financial Calculator
Setting Compounding Frequency
Compounding should be set
 to annual, i.e. P/YR=1, not            1.0000
 the factory setting of 12.
                                                      P/YR
                                    N   I/YR PV PMT FV


                           3       CLEAR ALL
                                    INPUT

                                           7      8           9     
                           1
                                           4      5           6     x
                           2
                                           1      2           3     –
                                        BEG/END              DISP
                                           0      .           =     +
                HP10B Calculator
                                                                         19
Financial Calculator
Setting Compounding Frequency
Compounding should be set
 to annual, i.e. P/YR=1, not             1 P/Yr
 the factory setting of 12.
To check setting CLEAR the                            P/YR

 calculator (holding down the        N   I/YR PV PMT FV
 CLEAR ALL key)
                                    CLEAR ALL
                                     INPUT
                            2
                                            7      8           9     
                            1               4      5           6     x

                                            1      2           3     –
                                         BEG/END              DISP
                                            0      .           =     +
                 HP10B Calculator
                                                           20
Financial Calculator Solution
Present Value of a Lump Sum (one time payment):
Previous Example:
  Example: Expect to receive $100 in EIGHT years. If can
           invest at 10%, what is it worth today?
   0    1    2   3       4   5   6   7    8     9   10

   ?                                     $100

    Using Formula:
         100
  PV = (1+.1)8 = 46.65
                                                           21
Financial Calculator Solution
Present Value of a Lump Sum (one time payment):
Previous Example:
  Example: Expect to receive $100 in EIGHT years. If can
           invest at 10%, what is it worth today?
   0    1    2   3    4    5    6   7     8      9   10

   ?                                     $100


                                        8.0000


                                    N   I/YR PV PMT FV

                                    8
                                                                22
Financial Calculator Solution
Present Value of a Lump Sum (one time payment):
Previous Example:
  Example: Expect to receive $100 in EIGHT years. If can
           invest at 10%, what is it worth today?
   0    1    2      3    4    5       6   7      8     9   10

   ?                                           $100


                                              10.000


                                          N   I/YR PV PMT FV
                 Enter the Interest
                 Rate as a WHOLE #        8 10
                                                           23
Financial Calculator Solution
Present Value of a Lump Sum (one time payment):
Previous Example:
  Example: Expect to receive $100 in EIGHT years. If can
           invest at 10%, what is it worth today?
   0    1    2   3    4    5    6   7      8    9   10

   ?                                     $100


                                        100.0000


                                    N   I/YR PV PMT FV

                                    8 10            100
                                                             24
Financial Calculator Solution
Present Value of a Lump Sum (one time payment):
Previous Example:
  Example: Expect to receive $100 in EIGHT years. If can
           invest at 10%, what is it worth today?
   0    1    2   3    4    5    6   7      8       9   10

   ?                                     $100


                                        - 46.65


                                    N   I/YR PV PMT FV

                                    8 10       ?       100
                                                                 25
Compound Value
Present Value of a Lump Sum (one time payment):
Previous Example:
  Example: Expect to receive $100 in EIGHT years. If can
           invest at 10%, what is it worth today?
    0    1     2    3     4    5     6   7      8      9   10

    ?                                         $100

   Additional Calculator Notes:              - 46.65
  Can change any or all parameters
  without reentering others

                                         N   I/YR PV PMT FV

                                         8    10           100
                                                                 26
Compound Value
Present Value of a Lump Sum (one time payment):
Previous Example:
  Example: Expect to receive $100 in EIGHT years. If can
           invest at 10%, what is it worth today?
    0    1     2    3     4    5     6   7     8       9   10

    ?                                         $100

   Additional Calculator Notes:              5.0000
  Can change any or all parameters
  without reentering others

        Change Interest rate to          N   I/YR PV PMT FV
        5%                               8    10           100
                                                   5
                                                                     27
Compound Value
Present Value of a Lump Sum (one time payment):
Previous Example:
  Example: Expect to receive $100 in EIGHT years. If can
           invest at 10%, what is it worth today?
    0    1     2    3     4    5     6   7      8          9   10

    ?                                         $100

   Additional Calculator Notes:              - 67.68
  Can change any or all parameters
  without reentering others

        Change Interest rate to          N   I/YR PV PMT FV
        5%                               8    10       ?       100
                                                   5
                                                              28
Compound Value
Present Value of a Lump Sum (one time payment):
Previous Example:
  Example: Expect to receive $100 in EIGHT years. If can
           invest at 10%, what is it worth today?
    0    1     2    3    4     5    6   7     8      9   10

    ?                                        $100

   Additional Calculator Notes:             0.0000
  Can check the number entered in
  each memory location using the
  recall (RCL) key.
                                        N   I/YR PV PMT FV

                                            RCL
                                                              29
Compound Value
Present Value of a Lump Sum (one time payment):
Previous Example:
  Example: Expect to receive $100 in EIGHT years. If can
           invest at 10%, what is it worth today?
    0    1     2    3    4     5    6   7     8      9   10

    ?                                        $100

   Additional Calculator Notes:             8.0000
  Can check the number entered in
  each memory location using the
  recall (RCL) key.
                                        N   I/YR PV PMT FV

        Check setting for years:            RCL
                                                      30
Compound Value
Solve for other parameters (I/YR)
Given any three of the following: PV, FV, i and n, the
 fourth can be computed.
                                                          31
Compound Value
Solve for other parameters (I/YR)
Given any three of the following: PV, FV, i and n, the
 fourth can be computed.
 Example: A $200 investment has grown to $230 over two years.
          What is the ANNUAL return on this investment?
        0                  1                  2

      $200                                  $230
                                                          32
Compound Value
Solve for other parameters (I/YR)
Given any three of the following: PV, FV, i and n, the
 fourth can be computed.
 Example: A $200 investment has grown to $230 over two years.
          What is the ANNUAL return on this investment?
        0                  1                  2

      $200                                  $230
                                                          33
Compound Value
Solve for other parameters (I/YR)
Given any three of the following: PV, FV, i and n, the
 fourth can be computed.
 Example: A $200 investment has grown to $230 over two years.
          What is the ANNUAL return on this investment?
        0                    1                2

      $200                                  $230

                      0.00
                or FVn = PV(1+ i)n
                  N   I/YR PV PMT FV
                                                          34
Compound Value
Solve for other parameters (I/YR)
Given any three of the following: PV, FV, i and n, the
 fourth can be computed.
 Example: A $200 investment has grown to $230 over two years.
          What is the ANNUAL return on this investment?
        0                    1                2

      $200                                  $230

                      2.00
                or FVn = PV(1+ i)n
                  N   I/YR PV PMT FV

                  2
                                                             35
Compound Value
Solve for other parameters (I/YR)
Given any three of the following: PV, FV, i and n, the
 fourth can be computed.
 Example: A $200 investment has grown to $230 over two years.
          What is the ANNUAL return on this investment?
        0                   1                   2

      $200                                   $230

                     2.00            When Entering inflows
                                   n and outflows of cash,
                or FVn = PV(1+ i) enter as follows:
                                     (-) = cash outflow
                  N I/YR PV PMT FV
                                     (+) = cash inflow
                 2
                                                             36
Compound Value
Solve for other parameters (I/YR)
Given any three of the following: PV, FV, i and n, the
 fourth can be computed.
 Example: A $200 investment has grown to $230 over two years.
          What is the ANNUAL return on this investment?
        0                  1                    2

      $200                                   $230

                     – 200.00        When Entering inflows
                                   n and outflows of cash,
                or FVn = PV(1+ i) enter as follows:
                                     (-) = cash outflow
                  N I/YR PV PMT FV
                                     (+) = cash inflow
                 2       -200
                                                          37
Compound Value
Solve for other parameters (I/YR)
Given any three of the following: PV, FV, i and n, the
 fourth can be computed.
 Example: A $200 investment has grown to $230 over two years.
          What is the ANNUAL return on this investment?
        0                      1              2

      $200                                  $230

                      230.00
                or FVn = PV(1+ i)n
                  N   I/YR PV PMT FV

                  2       -200     230
                                                          38
Compound Value
Solve for other parameters (I/YR)
Given any three of the following: PV, FV, i and n, the
 fourth can be computed.
 Example: A $200 investment has grown to $230 over two years.
          What is the ANNUAL return on this investment?
        0                     1               2

      $200                                  $230

                      7.24
                or FVn = PV(1+ i)n
                  N   I/YR PV PMT FV

                  2    ?     -200   230
                                                            39
Compound Value
Solve for other parameters (N)
Given any three of the following: PV, FV, i and n, the
 forth can be computed.
  Example: How long will it take for a $300 investment to
           grow to $500 if 6% annual interest is earned?
                                                            40
Compound Value
Solve for other parameters (N)
Given any three of the following: PV, FV, i and n, the
 forth can be computed.
  Example: How long will it take for a $300 investment to
           grow to $500 if 6% annual interest is earned?
        0                    1                   N

      $300                                     $500
                                                            41
Compound Value
Solve for other parameters (N)
Given any three of the following: PV, FV, i and n, the
 forth can be computed.
  Example: How long will it take for a $300 investment to
           grow to $500 if 6% annual interest is earned?
        0                    1                   N

      $300                                     $500
                                                            42
Compound Value
Solve for other parameters (N)
Given any three of the following: PV, FV, i and n, the
 forth can be computed.
  Example: How long will it take for a $300 investment to
           grow to $500 if 6% annual interest is earned?
        0                    1                   N

      $300                                     $500

                       – 300.00


                   N   I/YR PV PMT FV

                           -300
                                                            43
Compound Value
Solve for other parameters (N)
Given any three of the following: PV, FV, i and n, the
 forth can be computed.
  Example: How long will it take for a $300 investment to
           grow to $500 if 6% annual interest is earned?
        0                       1                N

      $300                                     $500

                       500.00


                   N   I/YR PV PMT FV

                           -300     500
                                                            44
Compound Value
Solve for other parameters (N)
Given any three of the following: PV, FV, i and n, the
 forth can be computed.
  Example: How long will it take for a $300 investment to
           grow to $500 if 6% annual interest is earned?
        0                      1                 N

      $300                                     $500

                       6.00


                   N   I/YR PV PMT FV

                        6     -300   500
                                                            45
Compound Value
Solve for other parameters (N)
Given any three of the following: PV, FV, i and n, the
 forth can be computed.
  Example: How long will it take for a $300 investment to
           grow to $500 if 6% annual interest is earned?
        0                      1                 N

      $300                                     $500

                       8.77


                   N   I/YR PV PMT FV

                   ?    6     -300   500
                                                        46
Non-Annual Compounding
All equations and calculator solutions thus far have
 assumed compounding occurs ONCE a year.
                                                               47
Non-Annual Compounding
All equations and calculator solutions thus far have
 assumed compounding occurs ONCE a year.
Example: Deposit $1,000 at 10% nominal annual interest rate.
         How much will you have at end of 1 year?
ANNUAL COMPOUNDING
      0                                      1

    $1,000
                  $1,000(1.1)
                                         $1,100
SEMI-ANNUAL COMPOUNDING
      0                6 months              1

    $1,000
                                                               48
Non-Annual Compounding
All equations and calculator solutions thus far have
 assumed compounding occurs ONCE a year.
Example: Deposit $1,000 at 10% nominal annual interest rate.
         How much will you have at end of 1 year?
ANNUAL COMPOUNDING
      0                                      1

    $1,000
                     $1,000(1.1)
                                         $1,100
                                                 Earn 10%/2=5%
SEMI-ANNUAL COMPOUNDING                          each compounding
                                                 period
      0                     6 months         1

    $1,000
             $1,000(1.05)
                             $1,050
                                                                          49
Non-Annual Compounding
All equations and calculator solutions thus far have
 assumed compounding occurs ONCE a year.
Example: Deposit $1,000 at 10% nominal annual interest rate.
         How much will you have at end of 1 year?
ANNUAL COMPOUNDING
      0                                                  1

    $1,000
                     $1,000(1.1)
                                                      $1,100
                                                             Earn 10%/2=5%
SEMI-ANNUAL COMPOUNDING                                      each compounding
                                                             period
      0                     6 months                     1

    $1,000
             $1,000(1.05)
                             $1,050
                                       $1,050(1.05)
                                                        $1,102.50
                                                          50
Non-Annual Compounding
All equations and calculator solutions thus far have
 assumed compounding occurs ONCE a year.
When compounding more than once a year, must
 adjust formula
                         i mn     m = # of compounding
            FVn =   PV(1+m)           periods in a year
                                                              51
Non-Annual Compounding
All equations and calculator solutions thus far have
 assumed compounding occurs ONCE a year.
When compounding more than once a year, must
 adjust formula
                          i mn        m = # of compounding
             FVn =   PV(1+m)              periods in a year


Example: Deposit $1,800 at 8% nominal annual interest rate,
         compounded quarterly. How much will you have at
         end of 3 years?
                                                                       52
Financial Calculator Solutions
Setting Compounding Frequency
Calculator makes
 adjustments for differing             4.0000
 compounding periods based
 on the setting of P/YR           xP/YR              P/YR

For Quarterly compounding         N      I/YR PV PMT FV
 set P/YR = 4
                          3       CLEAR ALL
                                   INPUT

                                           7     8           9     
                          1
                                           4     5           6     x
                          2
                                           1     2           3     –
                                       BEG/END              DISP
                                           0     .           =     +
               HP10B Calculator
                                                                       53
Financial Calculator Solutions
Setting Compounding Frequency
Calculator makes
 adjustments for differing             8.0000
 compounding periods based
 on the setting of P/YR           xP/YR              P/YR

For Quarterly compounding         N      I/YR PV PMT FV
 set P/YR = 4
I/YR/YR is automatically         CLEAR ALL

 adjusted by the P/YR              INPUT
 setting.                                  7     8           9     
                                           4     5           6     x

                                           1     2           3     –
                                       BEG/END              DISP
                                           0     .           =     +
               HP10B Calculator
                                                                        54
Financial Calculator Solutions
Setting Compounding Frequency
Calculator makes
 adjustments for differing              12.0000
 compounding periods based
 on the setting of P/YR            xP/YR              P/YR

For Quarterly compounding          N      I/YR PV PMT FV
 set P/YR = 4
I/YR/YR is automatically          CLEAR ALL

 adjusted by the P/YR      3        INPUT
 setting.                                   7     8           9     
To adjust N by P/YR enter                  4     5           6     x
 the number of years on the
 xP/YR key.                                 1     2           3     –
                           2            BEG/END              DISP
                                                                         1
                                            0     .           =     +
                HP10B Calculator
                                                                  55
Non-Annual Compounding
All equations and calculator solutions thus far have
 assumed compounding occurs ONCE a year.
When compounding more than once a year, must
 adjust formula
                               i mn      m = # of compounding
               FVn =      PV(1+m)            periods in a year


Example: Deposit $1,800 at 8% nominal annual interest rate,
         compounded quarterly. How much will you have at
         end of 3 years?                               P/Yr = 4
                                                 12.00

       Enter Years using Shift [xP/YR]
                                         xP/YR           P/YR
       combination
                                          N      I/YR PV PMT FV
                                          3
                                                                56
Non-Annual Compounding
All equations and calculator solutions thus far have
 assumed compounding occurs ONCE a year.
When compounding more than once a year, must
 adjust formula
                           i mn        m = # of compounding
              FVn =   PV(1+m)              periods in a year


Example: Deposit $1,800 at 8% nominal annual interest rate,
         compounded quarterly. How much will you have at
         end of 3 years?                               P/Yr = 4
                                               8.00

                                       xP/YR          P/YR
                                        N      I/YR PV PMT FV
                                        3       8
                                                                   57
Non-Annual Compounding
All equations and calculator solutions thus far have
 assumed compounding occurs ONCE a year.
When compounding more than once a year, must
 adjust formula
                           i mn        m = # of compounding
              FVn =   PV(1+m)              periods in a year


Example: Deposit $1,800 at 8% nominal annual interest rate,
         compounded quarterly. How much will you have at
         end of 3 years?                               P/Yr = 4
                                               – 1800.00

                                       xP/YR                P/YR
                                        N      I/YR PV PMT FV
                                        3       8   -1800
                                                                       58
Non-Annual Compounding
All equations and calculator solutions thus far have
 assumed compounding occurs ONCE a year.
When compounding more than once a year, must
 adjust formula
                           i mn        m = # of compounding
              FVn =   PV(1+m)              periods in a year


Example: Deposit $1,800 at 8% nominal annual interest rate,
         compounded quarterly. How much will you have at
         end of 3 years?                               P/Yr = 4
                                               2,282.84

                                       xP/YR                P/YR
                                        N      I/YR PV PMT FV
                                        3       8   -1800          ?
                                                                  59
Financial Calculator Solutions
                           Automatic
Alternative Settings
                                                       P/Yr = 4
Calculator make                   2,282.84
 compounding adjustments
 automatically based on    xP/YR                P/YR
 P/YR setting.              N      I/YR PV PMT FV
                            3       8   -1800           ?
                                                               60
Financial Calculator Solutions
                                 Automatic
Alternative Settings
                                                    P/Yr = 4
Calculator make                    2,282.84
 compounding adjustments
 automatically based on       xP/YR          P/YR
 P/YR setting.                  N I/YR PV PMT       FV
You can keep P/YR=1 and        3 8 -1800            ?
 make the adjustments to N
 and I/YR manually.          Manual
Advantage: should never                            P/Yr = 1
 need to change P/YR,               2,282.84
 therefore fewer errors on
 later problems.                             P/YR
If change P/YR, always         N I/YR PV PMT       FV
 change back to 1 P/YR after    12 2 -1800           ?
 doing problem.
                                                             61
Future Value of an Annuity
Annuity- string of deposits with constant value and
 fixed interval.
      0             1              2              3

     $0            $100          $100          $100


 Compute FV3
   How much would this account have in it at the end of 3
   years if interest were earned at a rate of 8% annually?
                                                             62
Future Value of an Annuity
Annuity- string of deposits with constant value and
 fixed interval.
      0             1                  2          3

     $0            $100           $100         $100
 Compute FV3
   How much would this account have in it at the end of 3
   years if interest were earned at a rate of 8% annually?

                                3.00


                            N   I/YR PV PMT FV
                            3
                                                             63
Future Value of an Annuity
Annuity- string of deposits with constant value and
 fixed interval.
      0             1                  2          3

     $0            $100           $100         $100
 Compute FV3
   How much would this account have in it at the end of 3
   years if interest were earned at a rate of 8% annually?

                                8.00


                            N   I/YR PV PMT FV
                            3    8
                                                             64
Future Value of an Annuity
Annuity- string of deposits with constant value and
 fixed interval.
      0             1                2            3

     $0            $100           $100           $100
 Compute FV3
   How much would this account have in it at the end of 3
   years if interest were earned at a rate of 8% annually?

                                –100.00


                            N   I/YR PV PMT FV
                            3    8        -100
                                                             65
Future Value of an Annuity
Annuity- string of deposits with constant value and
 fixed interval.
       0               1              2               3

     $0              $100          $100              $100
 Compute FV3
   How much would this account have in it at the end of 3
   years if interest were earned at a rate of 8% annually?

                                 324.64
 NOTE:
 PV = 0 since the cashflow
 in time period 0 = $0       N   I/YR PV PMT FV
                             3    8       -100   ?
                                                    66
Future Value of an Annuity
Example
Susan is able to save $980/yr for retirement. She
 makes these deposits at the end of each year. If she
 invests her savings at 12% compounded annually,
 how much will she have upon retirement in 45
 years?
                                                    67
Future Value of an Annuity
Example
Susan is able to save $980/yr for retirement. She
 makes these deposits at the end of each year. If she
 invests her savings at 12% compounded annually,
 how much will she have upon retirement in 45
 years?
        0     1      2     3      44     45

             $980   $980   $980   $980   $980
                                                            68
Future Value of an Annuity
Example
Susan is able to save $980/yr for retirement. She
 makes these deposits at the end of each year. If she
 invests her savings at 12% compounded annually,
 how much will she have upon retirement in 45
 years?
        0     1      2     3      44        45

             $980   $980   $980   $980     $980


                                      1,331,065.43


                                  N    I/YR PV PMT FV

                                  45 12          -980   ?
                                                                69
Future Value of an Annuity
Example #1a
Susan will make equal quarterly payments totaling
 $980/yr for retirement. She makes these deposits at
 the end of each quarter. If she invests her savings at
 12% compounded quarterly, how much will she have
 upon retirement in 45 years?
    0            1             2              45



                                                          P/Yr = 1

                        $245           1,661,944.10


                                   N     I/YR PV PMT FV

                                   180    3        -245    ?
                                                            70
Present Value of an Annuity
  How much would the following cash flows be worth to you
  today if you could earn 8% on your deposits?
      0            1             2             3

    $0            $100          $100         $100
                                                             71
Present Value of an Annuity
  How much would the following cash flows be worth to you
  today if you could earn 8% on your deposits?
      0             1              2                 3

    $0             $100           $100           $100
   $92.60 $100/(1.08)
                      $100 / (1.08)2
   $85.73
                                     $100 / (1.08)3
   $79.38
  $257.71

                                            257.71


                                        N   I/YR PV PMT FV

                                        3    8   ? -100
                                                     72
Present Value of an Annuity
Loan Amortization
Borrow $1,000 today, how much would the annual
 payments be if you are required to repay in two
 years and the interest rate is 10%?




                                    –576.19


                                N   I/YR PV PMT FV

                                2    10 1,000 ?
                                                    73
Present Value of an Annuity
Example #1a
Bob borrows $5,000 from his children to purchase a
 used car. He agrees to make payments at the end of
 each month for the next 5 years. If the interest rate
 on this loan is 6%, what is the amount of the
 payments?
                                                    74
Present Value of an Annuity
Example #1a
Bob borrows $5,000 from his children to purchase a
 used car. He agrees to make payments at the end of
 each month for the next 5 years. If the interest rate
 on this loan is 6%, what is the amount of the
 payments?
      0             1                   5

   $5,000
                                                     75
Present Value of an Annuity
Example #1a
Bob borrows $5,000 from his children to purchase a
 used car. He agrees to make payments at the end of
 each month for the next 5 years. If the interest rate
 on this loan is 6%, what is the amount of the
 payments?
      0             1                   5

   $5,000

                                    –96.66


                                  N I/YR PV PMT FV

                                  60 0.5 5,000 ?
                                                     76
Present Value of an Annuity
Example #1a
Bob borrows $5,000 from his children to purchase a
 used car. He agrees to make payments at the end of
 each month for the next 5 years. If the interest rate
 on this loan is 6%, what is the amount of the
 payments?
      0             1                   5

   $5,000

                                    –96.66


                                  N I/YR PV PMT FV

                                  60 0.5 5,000 ?
                                                          77
Annuity Due
Two Types of Annuities
Ordinary Annuity - Payments (or deposits) occur at
 the end of the period
    0                  1                   2      FV = $205
   $0                $100               $100


Annuity Due - Payments (or deposits) occur at the
 beginning of the period

    0                  1                   2

  $100                 $100             FV = ?
Each payment (or deposit) for an annuity due earns one
  additional period interest.
                                                     78
Annuity Due
Solving Annuity Due
Annuity Due - Payments (or deposits) occur at the
 beginning of the period

    0                 1                  2

  $100             $100              FV = ?


           FV AD = FV (ordinary) (1+i)


           PV AD = PV (ordinary) (1+i)
                                                                      79




             0.00                                – 215.25
             BEGIN                                BEGIN

                           P/YR
         N   I/YR PV PMT FV                  N   I/YR PV PMT FV

                                             2    5         100   ?
        CLEAR ALL
         INPUT

                7      8           9     
                4      5           6     x
1
                1      2           3     –
    2        BEG/END              DISP
                0      .           =     +
                                                    80
Additional Problems
Problem #1
  Compute the monthly payments on a 30 year
  mortgage for a $120,000 loan at 8% annual interest,
  compounded monthly.
                                                    81
Additional Problems
Problem #2
  You have determined that your budget will only allow
  you to make a $700 monthly mortgage payment. If
  interest rates are currently 6% and mortgage terms
  are typically 30 years, what price range home should
  you be searching for if your downpayment is
  $15,000?

				
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