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# Annuities - PowerPoint PPT Presentation

Annuities. Section 5.3. Introduction. Let’s say you want to save money to go on a vacation, or you want to save money now for your baby’s college education. A strategy for saving a little bit of money in the present and having a big payoff in the future is called an annuity .

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### Annuities

Section 5.3

• Let’s say you want to save money to go on a vacation, or you want to save money now for your baby’s college education.

• A strategy for saving a little bit of money in the present and having a big payoff in the future is called an annuity.

• An annuity is an account in which equal regular payments are made.

• There are two basic questions with annuities:

• Determine how much money will accumulate over time given that equal payments are made.

• Determine what periodic payments will be necessary to obtain a specific amount in a given time period.

• Claire wants to take a nice vacation trip, so she begins setting aside \$250 per month. If she deposits this money on the first of each month in a savings account that pays 6% interest compounded monthly, how much will she have at the end of 10 months?

• Claire’s first payment will earn 10 months interest. So F = 250(1 + .06/12)12(10/12). Note that the time t is 10/12. Therefore F = 250(1.005)10 = \$262.79.

• Claire’s second payment will earn 9 months interest. Thus F = 250(1.005)9 = \$261.48.

Totaling up the future value column, we see that Claire has \$2569.80 to use for her vacation. She earned \$69.80 in interest.

• There are two types of annuity formulas.

• One formula is based on the payments being made at the end of the payment period. This called ordinary annuity.

• The annuity due is when payments are made at the beginning of the payment period.

• We will derive the ordinary annuity formula first.

• The previous example reflects what actually happens to an annuity.

• The problem is what if the annuity is for 30 years.

• Future Value of the 1st payment for an ordinary annuity is

• F1 = PMT(1+r/n)m-1

• The future value of the next to last payment is Fm-1 = PMT(1+r/n)

• The future value of the last payment is Fm = PMT.

• The total future value F = F1 + F2 + F3 + … + Fm-1 + Fm

• The future value is

• Eq1

• Now multiply the equation above by (1+r/n)

• Eq2

• Take Eq2 – Eq1

• Note that m = nt. Simplifying gives the ordinary annuity future value formula

• ORDINARY ANNUITY

• ANNUITY DUE – receives one more period of compounding than the ordinary annuity so the formula is

• Find the future value of an ordinary annuity with a term of 25 years, payment period is monthly with payment size of \$50. Annual interest is 6%.

• F = \$34,649.70

• Note: We only put in \$15,000. This means that interest earned was \$19,649.70!

• A sinking fund is when we know the future value of the annuity and we wish to compute the monthly payment.

• For an ordinary unity this formula is

• For an annuity due the formula is

• Suppose you decide to use a sinking fund to save \$10,000 for a car. If you plan to make 60 monthly payments (5 years) and you receive 12% annual interest, what is the required payment for an ordinary annuity?

• In 18 years you would like to have \$50,000 saved for your child’s college education. At 6% annual interest, compounded monthly, what monthly deposit must be made to accomplish this goal?

• The question does not specify when the payments will be made so we use both formulas for comparison.

• For the ordinary annuity

• For the annuity due