Annuities Section 5.3   Annuity:  A sequence of equal regular payments into an account where each payment receives compound interest. Example of a Short-term annuity:  Christmas club account. Using our previous formulas for FV, we would need to compute a new value with every new deposit. Example 1:  Deposit \$150 at the beginning of each month starting July 1st  until Dec. 1st. Earning 9 ¼ % compounded monthly.  How much will you have?         FV of first deposit (July)=         FV of second deposit (Aug) =         FV of third deposit (Sept) =         FV of fourth deposit (Oct) =         FV of fifth deposit (Nov) =         Take money out Dec. 1 =   Payment period of an annuity is the time between payments (deposits). The Term of an annuity is the time from the beginning of the first payment to the end of last payment. Simple annuities have the same compounding period as payment period.   This textbook only uses simple annuities. Annuity Due = payments at the beginning of each period Ordinary Annuity = Payments at the end of each period Notice how tedious it was to compute short-term annuities. (Christmas Club Ex.) We computed the FV of each payment, then added all the FVs. Simplified Formulas:         FV(Ordinary Annuity) = pymt [(1 + i)n – 1]                                                                       i         FV(Annuity Due) = [FV(ordinary)] (1 + i)         FV(Annuity Due)  =  pymt  [(1 + i)n – 1 ] (1 + i)                                                                   i         pymt = amount of each payment         i= periodic rate =       (    interest rate     )                                         (# of periods a year)         n = number of pymts = (# of periods a year)(# of years)   Tax Deferred Annuities   (TDA) are set up in order to save for retirement.  It is automatically deducted from a person’s paycheck until retirement.  Federal and State tax deductions are computed after the annuity payment has been taken out   Exercise 2:  Long term simple annuity (ordinary) If I start at 42 years old to have \$150 deducted from my paycheck every month and deposit it into an account that pays 12% whose term ends on my 62nd birthday.  How much will I have?  Find FV. i = .12/12 = .01          n = (12)(20) = 240     pymt = \$150         FV = 150 [(1 + .01)240 – 1]  =  \$148,388.30                                       .01 How much of that is interest? Own contributions           (\$150)(240)  =  \$36,000                                             FV – contributions  =  interest earned         \$148,388.30 – \$36,000  =  \$112,388.30 interest! Sinking Funds: An annuity in which the FV is a specific amount.  Usually set up for college education, down payments on house.   Example 3:   How much must be deducted from Mrs. Olson’s paycheck if she wants to set up a sinking fund to pay for her baby’s college education, \$30, 000 at the end of 18 years.  Paychecks are cut bi-weekly in an account paying 8 ¾ %?         Biweekly means every two weeks = 26 weeks/year           i = .0875/26 = .003365385              n =(26)(18) = 468                  FV = \$30,000           \$30,000 = pymt [(1 + .003365385)468 – 1 ] = pymt(1134.4868)                                              .003365385             \$  30,000    = pymt = \$26.4436 = \$26.44           1134.4868   Actually earned:           FV = \$26.44(1134.4868) = \$29,995.83  because we rounded down \$26.4436   Present Value of an annuity refers to the lump sum that must be deposited at the beginning of an annuity’s term @ the same interest rate and compounding period that would yield the same amount as an annuity.   Example 4:   What is the present Value of the previous example; ie what lump sum must be deposited to earn \$29,995.83 @ 8 ¾ % compounded bi-weekly for 18 years?  Find P.           FV = \$29,995.83       I = .0875/26 = .003365385 n = (26)(18) = 468           \$29,996.83 = P(1 + .003365385)468 = P(4.817984483)           \$ 29,995.83     = P = \$6225.80                  4.817984483   Exercise 5:   What is the present value of an annuity that has \$150 monthly payments for 25 years @ 11% interest.   Instead of finding FV of the annuity and then finding the PV as in example 4, we can set both FV equal to each other to find P.   You would have to deposit \$24,347.93 right now to earn as much as having \$150 deducted from you paycheck every month earning 11% interest for 25 years. Back to Finance, Geometry and Logic Main Page Back to the Survey of Math Ideas Home Page