Annuities
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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

Annuities

Section 5.3


Introduction

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

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

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

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

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

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

Formulas

  • ORDINARY ANNUITY

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


Example

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

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

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

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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