Future Value of an Ordinary Annuity

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Appendix D

       Compound
        Interest
                                             2


              Objectives
1. Understand simple interest and compound
   interest.
2. Compute and use the future value of a
   single sum.
3. Compute and use the present value of a
   single sum.
4. Compute and use the future value of an
   ordinary annuity.
5. Compute and use the future value of an
   annuity due.    Continued
                                                  3


               Objectives
6. Compute and use the present value of an
   ordinary annuity.
7. Compute and use the present value of an
   annuity due.
8. Compute and use the present value of a
   deferred ordinary annuity.
9. Explain the conceptual issues regarding the
   use of present value in financial reporting.
                                       4


Simple Interest
       Simple interest is interest
       on the original principal
       regardless of the number
       of time periods that have
                passed.



  Interest = Principal x Rate x Time
                                        5


Compound Interest
   Compound interest is the interest
   that accrues on both the principal
      and the past unpaid accrued
                interest.
                                                            6


             Compound Interest
          Value at                               Value at
         Beginning                   Compound    End of
Period   of Quarter x Rate   x Time = Interest   Quarter
1st qtr. $10,000.00 x 0.12 x 1/4 $ 300.00 $10,300.00
2nd qtr. 10,300.00 x 0.12 x 1/4    309.00 10,609.00
3rd qtr. 10,609.00 x 0.12 x 1/4    318.27 10,927.27
4th qtr. 10,927.27 x 0.12 x 1/4    327.82 11,255.09
5th qtr. 11,255.09 x 0.12 x 1/4    337.65 11,592.74
Compound interest on $10,000 at
  12% compounded quarterly for
  5 quarters………………………... $1,592.74
                                                              7

   Future Value of a Single Sum at
         Compound Interest
  One thousand dollars is invested in a savings account
   on December 31, 2004. What will be the amount in
  the savings account on December 31, 2008 if interest
       at 14% is compounded annually each year?
                             How much will be in the
   $1,000 is invested
                            savings account (the future
      on this date
                                value) on this date?


Dec. 31,    Dec. 31,     Dec. 31,     Dec. 31,     Dec. 31,
 2004        2005         2006         2007         2008
                                                            8

 Future Value of a Single Sum at
       Compound Interest
 (1)       (2)              (3)               (4)
                         Annual          Future Value
        Value at        Compound             at End
       Beginning of       Interest          of Year
Year      Year        (Col. 2 x 0.14)   (Col. 2 + Col. 3)
2005   $1,000.00         $140.00        $1,140.00
2006    1,140.00          159.60         1,299.60
2007    1,299.60          181.94         1,481.54
2008    1,481.54          207.42         1,688.96
                                                      9

 Future Value of a Single Sum at
       Compound Interest
                     Formula Approach
                     n
      ƒ = p(1 + i)
where ƒ = future value of a single sum at compound
          interest i and n periods
      p = principal sum (present value)
      i = interest rate for each of the stated time
          periods
      n = number of time periods
                                        10

Future Value of a Single Sum at
      Compound Interest
             Formula Approach

     f = p(1 + i) n
                       4
     fn=4, i=14 = (1.14)
     f = $1,000(1.688960) = $1,688.96
                                     11

Future Value of a Single Sum at
      Compound Interest
                Table Approach

  This time we will use a table to
 determine how much $1,000 will
accumulate to in four years at 14%
      compounded annually.
                                   12

Future Value of a Single Sum at
      Compound Interest
                 Table Approach

Using Table 1 (the future value
of 1) at the end of Appendix D,
determine the table value for an
   annual interest rate of 14
   percent and four periods.
                                                                    13

Future Value of a Single Sum at
      Compound Interest
                        Table Approach
n      8.0%     9.0%      10.0%      12.0%      14.0%      16.0%
1   1.080000 1.090000   1.100000   1.120000   1.140000   1.160000
2   1.166400 1.188100   1.210000   1.254400   1.299600   1.345600
3   1.259712 1.295029   1.331000   1.404928   1.481544   1.560896
4   1.360489 1.411582   1.464100   1.573519   1.688960
                                              1.688960 1.810639
5   1.469328 1.538624   1.610510   1.762342   1.925415   2.100342
6   1.586874 1.677100   1.771561   1.973823   2.194973   2.436396
                                  14

Future Value of a Single Sum at
      Compound Interest
               Table Approach

 One thousand dollars times
 1.688960 equals the future
    value, or $1,688.96.
                                                            15


    Present Value of a Single Sum
       If $1,000 is worth $1,688.96 when it earns 14%
      compounded annually for 4 years, then it follows
    that $1,688.96 to be received in 4 years from now is
            worth $1,000 now at time period zero.

     $1,000 (the
    present value)                   For $1,688.96 to be
   must be invested                 received on this date
     on this date

Dec. 31,    Dec. 31,     Dec. 31,      Dec. 31,      Dec. 31,
 2004        2005         2006          2007          2008
                                                        16


  Present Value of a Single Sum
                 Formula Approach
                1
       p = f (1 + i) n
Where p = present value of any given future value due
          in the future
      ƒ = future value
      i = interest rate for each of the stated time
          periods
      n = number of time periods
                                           17


Present Value of a Single Sum
           Formula Approach

                       1
    p n=4, i=14 =   (1 .14) 4
                              = 0.592080

    p = $1,688.96(0.592080) = $1,000.00
                                  18


Present Value of a Single Sum
          Table Approach

       Use 14% 3, the present
      Find Tableand four
       value of obtain end
      periods to1, at thethe of
             Appendix
           table value. D.
                                                                      19


    Present Value of a Single Sum
                         Table Approach

n       8.0%     9.0%       10.0%      12.0%      14.0%      16.0%
1    0.925926 0.917431    0.909091   0.892857   0.877193   0.862069
2    0.857339 0.841680    0.826446   0.797194   0.769468   0.743163
3    0.793832 0.772183    0.751315   0.711780   0.674972   0.640658
4    0.735030 0.708425    0.683013   0.635518   0.592080
                                                0.592080 0.552291
5    0.680583 0.649931    0.620921   0.567427   0.519369   0.476113
6    0.630170 0.596267    0.564474   0.506631   0.455587   0.410442
                                20


Present Value of a Single Sum
         Table Approach

           $1,688.96 times
           0.592080 equals
               $1,000.
                                                            21

              Future Value of an
              Ordinary Annuity
                                            The future value
    Debbi Whitten wants to calculate the future value of
                                             of an ordinary
       four cash flows of $1,000, each with interest
                                                annuity
    compounded annually at 14%, where the first cash is
           flow is made on December 31, 2004.  determined
                                              immediately
                                           after the last cash
                                                  flow
$1,000      $1,000        $1,000       $1,000


Dec. 31,    Dec. 31,      Dec. 31,     Dec. 31,
 2004        2005          2006         2007
                                                       22

         Future Value of an
         Ordinary Annuity
                 Formula Approach
                         n
                  (1 + i) - 1
        Fo = C
                      i
Where Fo= future value of an ordinary annuity of a
           series of cash flows of any amount
      C = amount of each cash flow
      n = number of cash flows
       i = interest rate for each of the stated time
           periods
                                               23

      Future Value of an
      Ordinary Annuity
            Formula Approach

                   (1 .14)4 – 1
Fo = n=4, i=14 =                  = 4.921144
                      0.14

Fo = $1,000(4.921144) = $4,921.14
                                      24

         Future Value of an
         Ordinary Annuity
                Table Approach
   Using the same data—four
 Go to Table 2, the future value of
   equal annual cash flows of
  an ordinary annuity of 1. Read
$1,000 beginning on December
31, 2004 and an interest rate and
the table value for n equals 4 of i
            equals 14%.
           14 percent.
                                                                     25

            Future Value of an
            Ordinary Annuity
                        Table Approach

n      8.0%     9.0%       10.0%      12.0%      14.0%      16.0%
1   1.000000 1.000000    1.000000   1.000000   1.000000   1.000000
2   2.080000 2.090000    2.100000   2.120000   2.140000   2.160000
3   3.246400 3.278100    3.310000   3.374400   3.439600   3.505600
4   4.506112 4.573129    4.641000   4.779328   4.921144
                                               4.921144 5.066496
5   5.866601 5.984711    6.105100   6.352847   6.610104   6.877135
6   7.335929 7.523335    7.715610   8.115189   8.535519   8.977477
                                         26

Future Value of an
Ordinary Annuity
   So, cash flow of $1,000 each at
    14% at the end of 2004, 2005,
  2006, and 2007 will accumulate to
     a future value of $4,921.14.



         $1,000 x 4.921144 = $4,921.14
                                                             27

            Future Value of an
              Annuity Due
                    Solutions Approach


                        How much will be in the fund on
                        this date, which is 1 period after
                         the last cash flow in the series?
$1,000     $1,000        $1,000        $1,000


Dec. 31,   Dec. 31,      Dec. 31,     Dec. 31,
 2004       2005          2006         2007
                                           28

          Future Value of an
            Annuity Due
             Solutions Approach
Step 1:
In the ordinary annuity table (Table 2),
look up the value of n + 1 cash flows at
14% or the value of 5 cash flows at 14%.
                                                                    29

            Future Value of an
              Annuity Due
                   Solutions Approach
n      8.0%     9.0%      10.0%      12.0%      14.0%      16.0%
1   1.000000 1.000000   1.000000   1.000000   1.000000   1.000000
2   2.080000 2.090000   2.100000   2.120000   2.140000   2.160000
3   3.246400 3.278100   3.310000   3.374400   3.439600   3.505600
4   4.506112 4.573129   4.641000   4.779328   4.921144   5.066496
5   5.866601 5.984711   6.105100   6.352847   6.610104
                                              6.610104 6.877135
6   7.335929 7.523335   7.715610   8.115189   8.535519   8.977477
                                                        30

           Future Value of an
             Annuity Due
                Solutions Approach
Step 1:
In the ordinary annuity table (Table 2),
look up the value of n + 1 cash flows at
14% or the value of 5 cash flows at 14%.   6.610104
Step 2:
Subtract 1 without interest.               (1.000000)
Table value                                5.610104
                                           31

          Future Value of an
            Annuity Due
                Solutions Approach
Step 3:
Multiply the amount of each cash flow
($1,000) by the table value from Step 2.

    Fd = $1,000(5.610104) = $5,610.10
                               32

Future Value of an
  Annuity Due

  So, if $1,000 is deposited
   …a cumulative total of
   annually for four years
  $5,610 can be withdrawn
  beginning on December
   on December 31, 2008.
          31, 2004…
                                                          33

           Present Value of an
            Ordinary Annuity
                     Table Approach
  Kyle Vasby wants to calculate the present value on
January 1, 2004 (one period before the first cash flow)
of four future withdrawals (cash flows) of $1,000 each,
with the first withdrawal being made on December 31,
     2004. Assume again an interest rate of 14%.

            $1,000       $1,000       $1,000       $1,000


            Dec. 31,     Dec. 31,     Dec. 31,     Dec. 31,
             2004         2005         2006         2007
                                         34

 Present Value of an
  Ordinary Annuity

Go to Table 4, the present value of an
ordinary annuity of 1. Read the table
value for n equals 4 and i equals 14%.
                                                                     35

            Present Value of an
             Ordinary Annuity
                        Table Approach

n      8.0%     9.0%       10.0%      12.0%      14.0%      16.0%
1   0.925926 0.917431    0.909091   0.892857   0.877193   0.862069
2   1.783265 1.759111    1.735537   1.690051   1.646661   1.605232
3   2.577097 2.531295    2.486852   2.401831   2.321632   2.245890
4   3.312127 3.239720    3.169865   3.037349   2.913712
                                               2.913712 2.798181
5   3.992710 3.889651    3.790787   3.604776   3.433081   3.274294
6   4.622880 4.485919    4.355261   4.111407   3.888668   3.684736
                                     36

   Present Value of an
    Ordinary Annuity
          Table Approach

   One thousand dollars times
 2.913713 equals $2,913.71. So,
the present value of this ordinary
      annuity is $2,913.71.
                                                       37


 Present Value of an Annuity Due

                 Table Approach
 Barbara Livingston wants to calculate the present
 value of an annuity on December 31, 2004, which
  will permit four annual future receipts of $1,004
each, the first to be received on December 31, 2004.


     $1,000       $1,000       $1,000       $1,000


      Dec. 31,    Dec. 31,     Dec. 31,     Dec. 31,
       2004        2005         2006         2007
                                           38


 Present Value of an Annuity Due

                Table Approach

Step 1:
In the ordinary annuity table (Table 4),
look up the value of n – 1 cash flows at
14% or the value of 3 cash flows at 14%.
                                                                    39


    Present Value of an Annuity Due

                   Table Approach

n      8.0%     9.0%      10.0%      12.0%      14.0%      16.0%
1   0.925926 0.917431   0.909091   0.892857   0.877193   0.862069
2   1.783265 1.759111   1,735537   1.690051   1.546661   1.605232
3   2.577097 2.531295   2.485852   2.402831   2.321632
                                              2.321632 2.245890
4   3.312127 3.329720   3.159865   3.037349   2.913712   2.798181
5   3.992710 3.889651   3.790787   3.604776   3.443081   3.274294
6   4.622880 4.485919   4.355261   4.111407   3.888668   3.684736
                                                      40


 Present Value of an Annuity Due

                Table Approach

Step 1:
In the ordinary annuity table (Table 4),
look up the value of n – 1 cash flows at
14% or the value of 3 cash flows at 14%.   2.321632
Step 2:
Add 1 without interest.                    1.000000
                                           3.321632
                                  41


Present Value of an Annuity Due

            Table Approach

 One thousand dollars times
 3.321632 equals $3,321.63.
 So, this is the present value
 of an ordinary annuity due.
                                                 42

Present Value of a Deferred
     Ordinary Annuity
               Table Approach


  Helen Swain buys an annuity on January 1,
 2004 that yields her four annual payments of
$1,000 each, with the first payment on January
1, 2008. The interest rate is 14% compounded
   annually. What is the cost of the annuity?
                                                                      43

         Present Value of a Deferred
              Ordinary Annuity
  The present              Table Approach
  value of the
    deferred                     $1,000    $1,000    $1,000     $1,000
   annuity is
  determined                     Jan. 1,   Jan. 1,               Jan. 1,
                                                      Jan. 1,
  on this date                    2008      2009                  2011
                                                       2010


Jan.1, Jan. 1, Jan. 1, Jan. 1,
2004 2005 2006 2007
                                   $1,000 x 2.913712(n=4,
                                      i=14) = $2,913.71
                                                           44

         Present Value of a Deferred
              Ordinary Annuity
  The present              Table Approach
  value of the
    deferred
   annuity is                      $2,913.71
  determined
  on this date



Jan.1, Jan. 1, Jan. 1, Jan. 1,
2004 2005 2006 2007
                                  $2,913.71 x 0.674972 =
                                  $1,966.67
                                          45

Present Value of a Deferred
     Ordinary Annuity
         If Helen buys an annuity for
      $1,966.67 on January 1, 2004, she
         can make four equal annual
       $1,000 withdrawals (cash flows)
        beginning on January 1, 2008.
                       46


Appendix D


             The End
47

				
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