Sally knows she can earn a nominal rate of 6% convertible monthly in a retirement account, and she decides she can afford to save $1,500 from her paycheck every month. How can you use geometric series to simplify the calculation of finding the future value of all these payments? How much money will Sally have saved in 30 years?

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#### Guidance

An annuity is a series of equal payments that occur periodically. The word annuity comes from annual which means yearly. You will start by working with payments that occur once at the end of each year and then delve deeper to payments that occur monthly or any period.

Assume an investor saves \begin{align*}R\end{align*} dollars at the end of each year for \begin{align*}t\end{align*} years in an account that earns \begin{align*}i\end{align*} interest per period.

- The first payment \begin{align*}R\end{align*} will be in the bank account for \begin{align*}t-1\end{align*} years and grow to be: \begin{align*}R(1+i)^{t-1}\end{align*}
- The second payment \begin{align*}R\end{align*} will be in the bank account for \begin{align*}t-2\end{align*} years and grow to be: \begin{align*}R(1+i)^{t-2}\end{align*}
- This pattern continues until the last payment of \begin{align*}R\end{align*} that is deposited in the account right at \begin{align*}t\end{align*} years, so it doesn’t earn any interest at all.

The account balance at this point in the future (Future Value, \begin{align*}FV\end{align*}) is the sum of each individual \begin{align*}FV\end{align*} of all the payments:

\begin{align*}FV=R+R(1+i)^1+R(1+i)^2+ \cdots + R(1+i)^{t-2}+R(1+i)^{t-1}\end{align*}

Recall that a geometric series with initial value \begin{align*}a\end{align*} and common ratio \begin{align*}r\end{align*} with \begin{align*}n\end{align*} terms has sum:

\begin{align*}a+ar+ar^2+ \cdots+ar^{n-1}=a \cdot \frac{1-r^n}{1-r}\end{align*}

So, a geometric series with starting value \begin{align*}R\end{align*} and common ratio \begin{align*}(1+i)\end{align*} has sum:

\begin{align*}FV &= R \cdot \frac{1-(1+i)^n}{1-(1+i)}\\ &= R \cdot \frac{1-(1+i)^n}{-i}\\ &= R \cdot \frac{(1+i)^n-1}{i}\end{align*}

This formula describes the relationship between \begin{align*}FV\end{align*} (the account balance in the future), \begin{align*}R\end{align*} (the annual payment), \begin{align*}n\end{align*} (the number of years) and \begin{align*}i\end{align*} (the interest per year).

It is extraordinarily flexible and will work even when payments occur monthly instead of yearly by rethinking what, \begin{align*}R, \ i\end{align*} and \begin{align*}n\end{align*} mean. The resulting Future Value will still be correct. If \begin{align*}R\end{align*} is monthly payments, then \begin{align*}i\end{align*} is the interest rate per month and \begin{align*}n\end{align*} is the number of months.

**Example A**

What will the future value of his IRA (special type of savings account) be if Lenny saves $5,000 a year at the end of each year for 35 years at an interest rate of 4%?

**Solution: \begin{align*}R=5,000, \ i=0.04, \ n=35, \ FV=?\end{align*}**

**\begin{align*}FV &= R \cdot \frac{(1+i)^n-1}{i}\\
FV &= 5,000 \cdot \frac{(1+0.04)^{35}-1}{0.04}\\
FV &= \$368,281.12\end{align*}**

**Example B**

How long does Mariah need to save if she wants to retire with a million dollars and saves $10,000 a year at 5% interest?

**Solution: \begin{align*}FV=1,000,000, \ R=10,000, \ i=0.05, \ n=?\end{align*}**

**\begin{align*}FV &= R \cdot \frac{(1+i)^n-1}{i}\\
1,000,000 &= 10,000 \cdot \frac{(1+0.05)^n-1}{0.05}\\
100 &= \frac{(1+0.05)^n-1}{0.05}\\
5 &= (1+0.05)^n-1\\
6 &= (1+0.05)^n\\
n &= \frac{\ln \ 6}{\ln \ 1.05} \approx 36.7 \ years\end{align*}**

**Example C**

How much will Peter need to save each month if he wants to buy an $8,000 car with cash in 5 years? He can earn a nominal interest rate of 12% compounded monthly.

**Solution:** In this situation you will do all calculations in months instead of years. An adjustment in the interest rate and the time is required and the answer needs to be clearly interpreted at the end.

\begin{align*}FV = 8,000, \ R=?, \ i=\frac{0.12}{12}=0.01, \ n=5 \cdot 12=60\end{align*}

\begin{align*}FV &= R \cdot \frac{(1+i)^n-1}{i}\\ 8,000 &= R \cdot \frac{(1+0.01)^{60}-1}{0.01}\\ R &= \frac{8,000 \cdot 0.01}{(1+0.01)^{60}-1} \approx 97.96\end{align*}

Peter will need to save about $97.96 every month.

**Concept Problem Revisited**

Sally wanted to know how much she will have if she can earn 6% in a retirement account and she decides to save $1,500 from her paycheck every month.

\begin{align*}FV = ?, \ i=\frac{0.06}{12}=0.005, \ n=30 \cdot 12=360, \ R=1,500\end{align*}

\begin{align*}FV &= R \cdot \frac{(1+i)^n-1}{i}\\ FV &= 1,500 \cdot \frac{(1+0.005)^{360}-1}{0.005}\\ FV &\approx 1,506,772.56\end{align*}

#### Vocabulary

An ** annuity** is a series of equal payments that occur periodically.

#### Guided Practice

1. At the end of each quarter, Fermin makes a $200 deposit into a mutual fund. If his investment earns 8.1% interest compounded quarterly, what will his annuity be worth in 15 years?

2. What interest rate compounded semi-annually is required to grow a $25 semi-annual payment to $500 in 8 years?

3. How many years will it take to save $1,000 if Maria saves $20 every month at a 5% monthly interest rate?

**Answers:**

1. Quarterly means 4 times per year.

\begin{align*}FV = ?, \ R=200, \ i=\frac{0.081}{4}, \ n=60\end{align*}

\begin{align*}FV = 200 \cdot \frac{\left(1+\frac{0.081}{4}\right)^{60}-1}{\frac{0.081}{4}} \approx \$23,008.71\end{align*}

2. \begin{align*}FV = 500, \ R=25, \ n=16, \ i=?\end{align*}

\begin{align*}500 = 25 \cdot \frac{(1+i)^{16}-1}{i} \ or \ 0=500-25 \cdot \frac{(1+i)^{16}-1}{i}\end{align*}

At this point it is not easy to solve using regular algebra. Systematic guess and check will work. This is an opportunity to use the solver function on the TI-83/84 calculator. Go to <MATH>, scroll down to find the <Solver> option and press <ENTER>. Enter the equation with \begin{align*}x\end{align*} instead of \begin{align*}i\end{align*} enter a guess where it says “\begin{align*}X=\end{align*}”, and let the calculator approximate the solution by pressing <ALPHA> then <ENTER>. You should get the answer:

\begin{align*}i \approx 0.029009\end{align*}

Remember that the calculation is in semi-annual time periods, so the yearly nominal interest rate is \begin{align*}2 \cdot 0.029009 \approx 0.05802=5.802 \%\end{align*}

3. \begin{align*}FV=1,000, \ R=20, \ i=0.05, \ n=?\end{align*}. Note that the calculation will be done in months. At the end you will convert your answer to years.

\begin{align*}FV &= R \cdot \frac{(1+i)^n-1}{i}\\ 1000 &= 20 \cdot \frac{(1+0.05)^n-1}{0.05}\\ 2.5 &= (1+0.05)^n-1\\ 3.5 &= (1.05)^n\\ n &= \frac{\ln \ 3.5}{\ln \ 1.05} \approx 25.68 \ months\end{align*}

It will take about 2.140 years.

#### Practice

1. At the end of each month, Rose makes a $400 deposit into a mutual fund. If her investment earns 6.1% interest compounded monthly, what will her annuity be worth in 30 years?

2. What interest rate compounded quarterly is required to grow a $40 quarterly payment to $1000 in 5 years?

3. How many years will it take to save $10,000 if Sal saves $50 every month at a 2% monthly interest rate?

4. How much will Bob need to save each month if he wants to buy a $33,000 car with cash in 5 years? He can earn a nominal interest rate of 12% compounded monthly.

5. What will the future value of his IRA be if Cal saves $5,000 a year at the end of each year for 35 years at an interest rate of 8%?

6. How long does Kathy need to save if she wants to retire with four million dollars and saves $10,000 a year at 8% interest?

7. What interest rate compounded monthly is required to grow a $416 monthly payment to $80,000 in 10 years?

8. Every six months, Shanice makes a $1000 deposit into a mutual fund. If her investment earns 5% interest compounded semi-annually, what will her annuity be worth in 25 years?

9. How much will Jen need to save each month if she wants to put $60,000 down on a house in 5 years? She can earn a nominal interest rate of 8% compounded monthly.

10. How long does Adrian need to save if she wants to retire with three million dollars and saves $5,000 a year at 10% interest?

11. What will the future value of her IRA be if Vanessa saves $3,000 a year at the end of each year for 40 years at an interest rate of 6.7%?

12. At the end of each quarter, Justin makes a $1,500 deposit into a mutual fund. If his investment earns 4.5% interest compounded quarterly, what will her annuity be worth in 35 years?

13. What will the future value of his IRA be if Ted saves $3,500 a year at the end of each year for 25 years at an interest rate of 5.8%?

14. What interest rate compounded monthly is required to grow a $300 monthly payment to $1,000,000 in 35 years?

15. How much will Katie need to save each month if she wants to put $55,000 down in cash on a house in 2 years? She can earn a nominal interest rate of 6% compounded monthly.