Present Value, Future Value, and Annuity

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					                                Present Value and Future Value



                                            Formulas


Future Value (single period)     Value n = Value 0 *(1 + r ) n             or Value n = Value 0 * FVFn


                                                         (1 + r ) n - 1
Future Value (annuity)           Value n = CFannuity   *                or Value n = CFannuity * FVFannuity
                                                               r


                                                           1
Present Value (single period)    Value 0 = Value n *                       or Value 0 = Value n * PVFn
                                                       (1 + r ) n

                                                                  1
                                                         1-
                                                              (1 + r ) n
Present Value (annuity)          Value 0 = CFannuity *                      or Value 0 = CFannuity * PVFannuity
                                                                r




Assumption of Cash Flows
Deferred basis: The cash flows are assumed to take place at the end of the year. Almost all
present value and future value tables that you will ever see are on a deferred basis.
Due basis: The cash flows are assumed to take place at the beginning of the year.


Remember: The annuity present value table simply sums up the values from the single period
          table.


When working a time value of money problem, ask yourself two questions:
1. What is the unknown - the value today (Value0) or the value in the future (Value n)?
       If the unknown is the value today, then the problem is a present value type of problem.
       If the unknown is the value in the future, the problem is a future value type of problem.
2. Is there an annuity involved?
         An annuity is defined as a fixed amount of money in consecutive periods. If no annuity
         is involved, you will use the single period tables (or formulas).
Time Value of Money                                                               Page 2


Problems: (Some may be worked with the tables; others must be worked with a calculator
          that has a yx function.)
1. If you have 100 rabbits now that are growing at a rate of 10% a year, how many rabbits will
   you have three years from now?

   The unknown amount lies in the future, therefore the problem is a future value problem.
   There is no annuity involved, so it is a single period type of problem. Therefore, the
   appropriate formula is:

        Value n = Value 0 *(1 + r) n
                 = 100 * (1 + 010) 3
                               .
                          .
                 = 100 * 1331
                      .
                 = 1331 rabbits
                            (Assuming that you can have 1/10 of a rabbit!)

2. If you have $100 in the bank now that is growing at a rate of 4% a year, how many dollars
   will you have in the bank 10 years from now?

        Value n = Value 0 * (1 + r ) n
                 = $ 100 * (1+.04) 10
                            .
                 = $ 100 * 1480
                 = $ 148.00

3. A company currently pays a dividend of $10.00 per share. Determine how much the
   company’s dividend will be in 10 years, assuming a 5% annual rate of growth.

        Value n = Value 0 * (1 + r) n
                = $ 10 * 10510
                          .
                          .
                = $ 10 * 1629
                = $ 16.29

4. How much will you have in your savings account at the end of five years if you deposit
   $1,000 now and it compounds at a rate of 1 1/2% quarterly?

        Value n = Value 0 * (1 + r) n
                = $ 1,000 * 101520
                             .
                             .
                = $ 1,000 * 1347
                = $ 1,347.00
Time Value of Money                                                                 Page 3


5. How much must you have in the bank now in order to have $100 in the bank five years from
   now? (Assume that the money you have in the bank will grow at a rate of 4% per year.)

                                  1
        Value 0 = Value n *
                              (1 + r ) n
                               1
                = $ 100 *
                          (104) 5
                            .
                = $ 100 * 0.822
                = $ 82.20



6. You would like to put some money aside for your newborn baby’s college education. How
   much must you deposit in a savings account now if the account grows at a rate of 6% a year
   and you desire a balance of $20,000 eighteen years from now?

                                1
        Value 0 = Value n *
                            (1 + r ) n
                                  1
                = $ 20,000 *
                             (1+.06)18
                = $ 20,000 * 0.350
                = $ 7,000



7. If you put $100 in the bank in each of the next five years (you put the money in at the end of
   each year), how much money will you have in the bank at the end of the fifth year, assuming
   a 4% rate of interest.

                              (1 + r ) n - 1
        Value n = CFannuity *
                                    r
                                 5
                            .
                          (104 ) - 1
                = $ 100 *
                              .04
                = $ 100 * 5.416
                       .
                = $ 54160
Time Value of Money                                                                Page 4


8. You have just won the state of Kentucky’s lottery for $1 million. However, when you read
   the fine print, you are actually to receive $50,000 per year for the next 20 years. How much
   does the state of Kentucky have to put in the bank today to allow you to withdraw $50,000
   per year, assuming a 6% interest rate?

                                     1
                             1-
                                 (1 + r ) n
        Value 0 = CFannuity *
                                   r
                                         1
                              1-
                                   (1+.06) 20
                = $ 50,000 *
                                     .06
                               .
                = $ 50,000 * 11470
                = $ 573,500



9. What is the present value of $10,000 to be received (as per gift agreement) ten years from
   now assuming a 6% required rate of return?

        Value 0 = CFn * PFVn
                                   1
        Value 0 = $ 10,000 *         10
                                 .
                               (106)
                = $ 10,000 * 0.558
                = $ 5,580.00



10. How much must you have now in a savings account that grows at a rate of 6% per year to be
    able to withdraw $30,000 a year at the end of each of the next thirty years?

                                      1
                               1-
                                 (1 + r) n
        Value 0 = CFannuity *
                                    r
                                        1
                              1-
                                  (1+.06) 30
                = $ 30,000 *
                                     .06
                = $ 30,000 * 13.765
                = $ 412,950.00
Time Value of Money                                                                 Page 5


11. What must be paid for an investment which provides $1,000 annually for ten years plus
    $5,000 after ten years and yields 10%?

        Value 0 = CF * PFVannuity + Value n * PFVn
                             .
                = $ 1,000 * 6145 + $5,000 * 0.386
                = $6,145.00 + $1,930.00
                = $8,075.00

12. The monthly payment on a 24-month installment auto loan (payments due at the end of each
    month) is $100. The interest rate on this loan is 1% per month. What is the amount
    borrowed?

        Value 0 = CFannuity * PVFannuity
                = $100 * 21.243
                = $ 2,124.30

13. What will be the price of a ten-year bond be if it has to yield 8% to the buyer? The bond has
    a face value of $1,000 and pays $10 interest on a quarterly basis.

        Value 0 = CFannuity * PVFannuity + Vn * PVFn
                = $ 10 * 27.355 + $ 1,000 * 0.453
                       .         .
                = $ 27355 + $ 45300
                       .
                = $ 72655


14. What will the price of a bond in #13 be if it is to yield 6% to the buyer?

        Value 0 = CFannuity * PVFannuity + Vn * PVFn
                = $ 10 * 29.916 + $1,000 * 0.551
                                  .
                = $ 299.16 + $ 55100
                       .
                = $ 85016


15. You are ready to buy a new car. Given an annual percentage rate of 8%, how much can you
    borrow if you can afford to repay $300 each month for 60 months (paid at the end of each
    month)?

        Value 0 = CFannuity * PVFannuity
                = $300* 49.318
                = $14,795.40

				
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