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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

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.

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.

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.

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.

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

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

Formulas

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

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!

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

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?

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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