Future Value of an Annuity Find the future value of each...

Future Value of an Annuity Find the future value of each annuity. See Example 9. Individual Retirement Account Starting on his 40 th birthday, Michael deposits $\$ 2000$ per year in an Individual Retirement Account until age 65 (last payment at age 64 ). Find the total amount in the account if he had a guaranteed interest rate of $2 \%$ compounded annually.

Future Value of an Annuity Find the future value of each annuity. See Example 9.
Individual Retirement Account Starting on his 40 th birthday, Michael deposits $\$ 2000$ per year in an Individual Retirement Account until age 65 (last payment at age 64 ). Find the total amount in the account if he had a guaranteed interest rate of
$2 \%$ compounded annually.

Key Concepts

Annuity

An annuity is a series of equal payments made at regular intervals over a period of time. In financial contexts, these payments can be deposits or withdrawals from an investment or savings account. The concept of an annuity is central to understanding how regular contributions accumulate value when combined with interest over time.

Compound Interest

Compound interest refers to the process where interest earned on an investment is reinvested, so that in subsequent periods, interest is earned on both the original principal and the accumulated interest. This mechanism is foundational in finance as it demonstrates how money can grow at an exponential rate over time, particularly in scenarios involving recurring payments or deposits.

Future Value of an Annuity

The future value of an annuity is the total amount of money that will accumulate over time with a series of regular contributions, accounting for compound interest. It is calculated using formulas that incorporate the payment amount, the interest rate, and the number of periods, allowing one to determine the end balance of investments such as retirement accounts or savings plans.

Payment Timing (Ordinary Annuity vs. Annuity Due)

The timing of payments in an annuity can impact the total accumulated value. In an ordinary annuity, payments are made at the end of each period, meaning each deposit earns interest for one period less than in an annuity due, where payments are made at the beginning of each period. This distinction is crucial when applying the correct formula to calculate the future value of the annuity.

Transcript

00:01 So in this problem, we're told that this person, michael, is going to initially deposit $2 ,000 when he's 40 years old.

00:07 And he's going to continue to earn interest over the next 25 years.

00:11 And he's getting 2 % interest.

00:13 What we want to do is figure out, well, how much money would he have in total? well, in this case, because it's 2 % interest, we know that this is going to represent a geometric series.

00:22 So what we want to do first is set up our summation.

00:25 So i'll have our summation symbol.

00:27 And at the bottom we'll let x equal to one.

00:30 That one is representing that first year when he was 40, he had $2 ,000 in it.

00:34 Well, he's going to be saving for 25 years, meaning there would be the 25 terms in the sequence or in the series.

00:41 So our upper limit will be 25.

00:43 So now we just have to come up with our formula.

00:46 Well, our initial value is 2000.

00:49 That would be our first term.

00:51 And then, remember, he's earning 2 % interest.

00:54 But that means he's going to take what he had before, all 100%, plus he's going to earn an additional 2%, which means he would have 102%.

01:03 So as a decimal, that would be 1 .02...