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Annuities

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

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  1. Annuities Section 5.3

  2. 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. • 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.

  3. Calculating short-term annuities • 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.

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

  5. Ordinary Annuity and Annuity Due • 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.

  6. Calculating Long Term Annuities • 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

  7. Continuing the calculation of a long term annuity • 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

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

  9. Example • 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!

  10. Sinking Funds • 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

  11. Sinking Fund Example • 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?

  12. Real – Life Example • 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

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