## Savings Annuities

### Learning Outcomes

• Calculate the balance on an annuity after a specific amount of time
• Calculate interest earned and amount deposited in an annuity problem

## Savings Annuity

For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the future by depositing a smaller amount of money from each paycheck into the bank. This idea is called a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings annuities.

An annuity can be described recursively in a fairly simple way. Recall that basic compound interest follows from the relationship

${{P}_{m}}=\left(1+\frac{r}{k}\right){{P}_{m-1}}$

For a savings annuity, we simply need to add a deposit, d, to the account with each compounding period:

${{P}_{m}}=\left(1+\frac{r}{k}\right){{P}_{m-1}}+d$

Taking this equation from recursive form to explicit form is a bit trickier than with compound interest. It will be easiest to see by working with an example rather than working in general.

### reading examples: the paper-and-pencil approach

In mathematics, we say the best way to read a math text is with a paper and pencil.The example below is challenging. Resist the temptation to gloss over it and cut straight to the formula given at the end. Instead, work through the example, line by line, with a pencil and paper. Do the math in each line to see how each subsequent, equivalent equation is formed. Ask questions if you can’t see how one line was rewritten algebraically into the next. Reading math with pencil in hand helps make future math less challenging, increases your ability to apply logic in the real world by forming new thought patterns, and it really pays off at test time!

### Recall Algebraic Skills

For this example, you’ll need to recall these skills in particular.

The distributive property: $a\left(b+c\right)=ab+ac$

Factoring out a greatest common factor: $m\left(a+b\right) + n\left(a+b\right)=\left(a+b\right)\left(m+n\right)$

How to multiply like bases with exponents: $a^{m-1}\cdot a=a^{m-1+1}=a^{m}$

Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the account is compounded with the same frequency as we make deposits unless stated otherwise. Write an explicit formula that represents this scenario. Generalizing this result, we get the savings annuity formula. ### Annuity Formula $P_{N}=\frac{d\left(\left(1+\frac{r}{k}\right)^{Nk}-1\right)}{\left(\frac{r}{k}\right)}$ • PN is the balance in the account after N years. • d is the regular deposit (the amount you deposit each year, each month, etc.) • r is the annual interest rate in decimal form. • k is the number of compounding periods in one year. If the compounding frequency is not explicitly stated, assume there are the same number of compounds in a year as there are deposits made in a year. For example, if the compounding frequency isn’t stated: • If you make your deposits every month, use monthly compounding, k = 12. • If you make your deposits every year, use yearly compounding, k = 1. • If you make your deposits every quarter, use quarterly compounding, k = 4. • Etc. ### When do you use this? Annuities assume that you put money in the account on a regular schedule (every month, year, quarter, etc.) and let it sit there earning interest. Compound interest assumes that you put money in the account once and let it sit there earning interest. • Compound interest: One deposit • Annuity: Many deposits. ### Recall order of operations Using the order of operations correctly is essential when using complicated formulas like the annuity formula. PEMDAS: First simplify like terms inside parentheses then handle exponents before multiplying or dividing. Do addition and subtraction outside of parentheses last. ### Examples A traditional individual retirement account (IRA) is a special type of retirement account in which the money you invest is exempt from income taxes until you withdraw it. If you deposit$100 each month into an IRA earning 6% interest, how much will you have in the account after 20 years?

This example is explained in detail here.

A conservative investment account pays 3% interest. If you deposit $5 a day into this account, how much will you have after 10 years? How much is from interest? ### Try It ## Solving For The Deposit Amount Financial planners typically recommend that you have a certain amount of savings upon retirement. If you know the future value of the account, you can solve for the monthly contribution amount that will give you the desired result. In the next example, we will show you how this works. ### Example You want to have$200,000 in your account when you retire in 30 years. Your retirement account earns 8% interest. How much do you need to deposit each month to meet your retirement goal?

View the solving of this problem in the following video.

## Solving For Time

We can solve the annuities formula for time, like we did the compounding interest formula, by using logarithms. In the next example we will work through how this is done.

### recall using a logarithm to solve for an exponent

In the following example, you’ll need to recall that you can solve for a variable contained in an exponent by taking the log of both sides of the equation.

Ex. Solve for x in the following equation

$a = b^{mx}$                                              we are solving for x, in the exponent

$log(a) = log\left(b^{mx}\right)$                            take the log of both sides

$log(a)=mx\ast log\left(b\right)$                       use the exponent property

$\frac{log(a)}{mb}=x$                                           divide away all non-x terms to isolate x

### Example

If you invest $100 each month into an account earning 3% compounded monthly, how long will it take the account to grow to$10,000?

This example is demonstrated here: