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Financial Management



                      Notes         2.3.1  Future Value of Annuity of   1

                                    If you open a savings account that compounds interest each month, and at the end of each month
                                    you deposit   100 in the savings account, your deposits are the rents of an annuity. After 1 year,
                                    you will have 12 deposits of   100 each, and a total of   1200, but the account will have more than
                                      1200 in it because each deposit earns interest. If the interest rate is 6 per cent a year, compounded
                                    monthly, your balance is   1233.56. The future value of an annuity or amount of annuity is the
                                    sum accumulated in the future from all the rents paid and the interest earned by the rents. The
                                    abbreviation FV is used for the future value of an annuity to differentiate it from the lower case
                                    fv used for the future value of  1.

                                    To obtain a table of future values of annuities, we assume payments of  1 each period made into
                                    a fund that earns 8 per cent interest compounded each period. The following diagram illustrates
                                    an annuity of four payments of  1, each paid at the end of each period, with interest of 8 per cent
                                    compounded each period.
                                                                       Figure  2.2











                                    Notice that there are four rents and four periods, each rent is paid at the end of each period. At
                                    the end of the first period,  1 is deposited and earns interest for three periods. The next rent earns
                                    interest for two periods, and so on. The amount at the end of the fourth period can be determined
                                    by calculating the future value of each individual  1 deposit as follows:
                                             Future value of   1 at 8% for 3 periods =   1.25971
                                             Future value of   1 at 8% for 2 periods =   1.16640
                                              Future value of   1 at 8% for 1 period =   1.08000
                                             The fourth rent of   1 earns no interest =   1.0000
                                                                Total for 4 rents =   4.50611
                                    The formula for the future value of an annuity of  1 can be used to produce tables for a variety
                                    of periods and interest rates

                                                                        (1  + 1) n  - 1
                                                                    Fv  =
                                                                            i
                                           Example: In the beginning of 2006, the directors of Molloy Corporation decided that
                                    plant facilities will have to be expanded in a few years. The company plans to invest:   50,000
                                    every year, starting on June 30, 2006, into a trust fund that earns 11 per cent interest compounded
                                    annually. How much money will be in the fund on June 30, 2010, after the last deposit has been
                                    made?
                                    Solution: The first deposit is made at the end of the first 1-year period, and there is a total of
                                    5 periods. The last deposit, made on June 30, 2010 earns no interest.The investment is an ordinary
                                    annuity with n =5 and i =11 per cent. From Table Future Value of Annuity  1 we find that the
                                    amount of an ordinary annuity of  1 is 6.22780.

                                                               FV = Rent × f (n =5, i =11%)
                                                                  =   50,000 × 6.22780 =   311,390




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