Amount of an Annuity

So far, all of our calculations have been based on the idea that once you deposit a certain amount of money, the account remains untouched throughout the time period. (Lump Sum Savings) Is this always realistic?

The consistent pattern of making the same payment (P or R) over the same cycle length (n), allows us to model the situation mathematically. “Random payments over random cycles” is very difficult to model…..and very unpractical anyway…

An Annuity is a series of equal payments made at regular time intervals. (car payment, mortgage, credit cards)

Consider making regular deposits of $ into an account once a year for 10 years at 6%. Deposits will occur at the end of the year (take the year to save) Draw a diagram to illustrate the calculations

Time - Years 1000(1.06) (1.06) (1.06) 1 How much? JFMARJJASOND

The amount after 10 years is: DepositGrowth (1.06) (1.06) (1.06) (1.06) (1.06) (1.06) (1.06) (1.06) (1.06) (1.06) 0 $ $ $ $ $ $ $ $ $ $ = $

Calculate the interest earned 10 payments of $ , adds up to $ Since there is $ in the account. $ $ = $ in interest

Of course, there needs to be a quicker way……and there is

A = R[(1 + i) n – 1] i Amount of an ordinary simple annuity variables on the next page

A = Amount in dollars R = Regular payment in dollars i = interest per compounding period, as a decimal n = number of cycles

Consider making regular deposits of $ into an account once a year for 10 years at 6%. A = ? R = 1000 i = 0.06 n = 10 A = R[(1 + i) n – 1] i A = 1000[( ) 10 – 1] 0.06 A = $

Suppose you deposit $ every 6 months into an account at 4.5% compounded semi-annually. How much will you have after 3 years?

A = ?, R = A = R[(1 + i) n – 1] i 250, i =0.045 = n = 2 times a year for 3 years = 6 A = 250[( ) 6 – 1]

= 250 [0.1428] A = 250[( ) 6 – 1] = $

Consider your own savings… $50.00 / month at 4% (C:S-A), 40 years? (adjust the payments) A = 300[(1.02) 80 – 1] 0.02 = $

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