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Wize High School Grade 11 Math Textbook > Financial Applications

Investments and Loans with Regular Payments (Ordinary Simple Annuities)

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Investments and Loans with Regular Payments

An ordinary simple annuity is a series of payments made at the end of each compounding period.
payment frequency = compounding frequency\text{payment frequency = compounding frequency}

Future Value

If RR represents the amount of each regular payment, and the interest rate ii is compounded at the same frequency as the payments, we can create a timeline:


Here, a payment of $R\$R is made at the end of each period.
  • Payment 1 earns interest for 3 periods, so its future value at the end of the 4th period is R(1+i)3R(1+i)^3
  • Payment 2 earns interest for 2 periods, so its future value at the end of the 4th period is R(1+i)2R(1+i)^2
  • Payment 3 earns interest for 1 period, so its future value at the end of the 4th period is R(1+i)R(1+i)
  • Payment 4 is made right at the end of the 4th period, so it does not earn any interest and its value is still RR.
Wize Tip
These future values are all terms of a geometric sequence with first term a=Ra=R and common ratio r=1+ir=1+i.
To find the total ending balance, we add up each of these terms -- this is a geometric series!

Future Value of Annuities Formula

We can use the geometric series formula to write a formula for the ending balance after many regular payments:

FV=a(rn−1)r−1=R((1+i)n−1)(1+i)−1=R((1+i)n−1i)\begin{aligned} FV &= \dfrac{a(r^n-1)}{r-1}\\[1.5em] &= \dfrac{R((1+i)^{n}-1)}{(1+i)-1}\\[1.5em] &= \boxed{R\left(\dfrac{(1+i)^{n}-1}{i} \right)}\\[1em] \end{aligned}

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Present Value

If instead we want to know how much future regular payments would be worth in total now, we need a present value calculation.

If RR represents the amount of each regular payment, and the interest rate ii is compounded at the same frequency as the payments, we can create a timeline:



Here, a payment of $R\$R is made at the end of each period.
  • Payment 1 is "pulled back" 1 period to its present value, given by R(1+i)−1R(1+i)^{-1}
  • Payment 2 is "pulled back" 2 periods to its present value, given by R(1+i)−2R(1+i)^{-2}, etc.
Wize Tip
This time, the first term of the geometric series is a=R(1+i)−1a=R(1+i)^{-1}, and the common ratio is r=(1+i)−1r=(1+i)^{-1} .

Present Value of Annuities Formula

Using the geometric series formula, we can simplify to get the formula:

PV=R(1−(1+i)−ni)\boxed{PV = R\left(\dfrac{1-(1+i)^{-n}}{i} \right)}
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Example: Regular Payments (Future Value)

Adam contributes $100 at the end of every month into a savings account that earns 6%/a compounded monthly.
What is the balance of the account after 24 months? Create a spreadsheet to find the answer.

Since interest is compounded monthly, we calculate the monthly interest to be: i=0.06/12=0.005i=0.06/12=0.005

The first month's payment earns no interest since it occurs at the end of the month.
At the end of every month, we multiply the previous balance by (1+i)=1.005(1+i) = 1.005.

MonthEnding Balance1$ 100.002100.00(1.005)+100=$ 200.503200.50(1.005)+100=$ 301.504301.50(1.005)+100=$ 403.01……24$ 2 543.20→first payment is at the end of month 1→multiply previous balance by 1.005 and add 100→multiply previous balance by 1.005 and add 100etc.\begin{array}{|c|c|} \hline \text{Month} & \text{Ending Balance}\\ \hline 1 & \hspace{11em} \$\ 100.00 \quad\\ \hline 2 & 100.00(1.005) +100 =\$\ 200.50\\ \hline 3 & 200.50(1.005) +100 =\$\ 301.50\\ \hline 4 & 301.50(1.005) +100 =\$\ 403.01\\ \hline \dots & \dots\\ \hline 24 & \hspace{10.5em} \$\ 2\,543.20\\ \hline \end{array} \begin{array}{l} \\ \rightarrow \text{first payment is at the end of month 1} \\ \rightarrow\text{multiply previous balance by 1.005 and add 100}\\ \rightarrow\text{multiply previous balance by 1.005 and add 100}\\ etc.\\ \\ \\ \end{array}

Technology: we can use a spreadsheet to complete the Ending Balance column.


How can we express the future value (ending balance) after the 24th month as a geometric series?
Confirm the ending balance using two different methods.

FV=100+100(1.005)+100(1.005)2+⋯+100(1.005)23\begin{aligned} FV &= 100 + 100(1.005) + 100(1.005)^2+\dots+100(1.005)^{23}\\ \end{aligned}
To evaluate, we can use the geometric series formula with first term a=100a=100, common ratio r=1.005r=1.005, and n=24n=24.
If we use the future value of an annuity formula instead, use the interest rate i=0.005i=0.005.

Geometric Series FormulaFV=a(rn−1)r−1=100(1.00524−1)1.005−1=100(1.12716−1)0.005=$2 543.20✓Annuity FormulaFV=R((1+i)n−1i)=100((1+0.005)24−10.005)=100(1.12716−10.005)=$2 543.20✓\begin{array}{cc} \begin{aligned} &\text{Geometric Series Formula}\\ FV &= \dfrac{a(r^n-1)}{r-1}\\[1em] &= \dfrac{100(1.005^{24}-1)}{1.005-1}\\[1em] &= \dfrac{100(1.12716-1)}{0.005}\\[1em] &= \boxed{\$2\,543.20} \quad \colorThree{\checkmark} \end{aligned} \hspace{10em} & \begin{aligned} &\text{Annuity Formula}\\ FV &= R\left(\dfrac{(1+i)^n-1}{i}\right)\\[1em] &= 100\left(\dfrac{(1+0.005)^{24}-1}{0.005}\right)\\[1em] &= 100\left(\dfrac{1.12716-1}{0.005}\right)\\[1em] &= \boxed{\$2\,543.20} \quad \colorThree{\checkmark} \\ \end{aligned} \end{array}

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Example: Regular Payments (Present Value)

Sharon is ready to retire and her bank offers to pay her an annuity in exchange for all of her savings.
The annuity would consist of a regular payment of $9200 every 3 months for 25 years.
If her money can earn 6.4% compounded quarterly, how much money must she have now for this to be a fair offer?

Since interest is compounded quarterly, it occurs 4x per year for 25 years: n=4×25=100n=4\times25=100
Moreover, we have to divide te annual interest rate by 4 to get: i=0.064/4=0.016i=0.064/4=0.016
Throughout our calculations, we use: (1+i)=1.016(1+i)=1.016

Using the Present Value Formula

PV=R(1−(1+i)−ni)=9200(1−1.016−1000.016)=$ 457 429.86\begin{aligned} PV &= R\left(\dfrac{1-(1+i)^{-n}}{i} \right)\\[1.5em] &= 9200\left(\dfrac{1-1.016^{-100}}{0.016} \right)\\[1.5em] &= \colorbox{yellow}{\$\,457\,429.86} \end{aligned}
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Using Technology


Method 1: Find the Present Value of Each Payment, then Sum

QuarterPV of Payment n19200(1.016)−1=$ 9 055.1229200(1.016)−2=$ 8 912.5239200(1.016)−3=$ 8 772.1649200(1.016)−4=$ 8 634.02……1009200(1.016)−100=$ 1 881.12TOTAL$ 457 429.86\begin{array}{|c|c|} \hline \text{Quarter} & \text{PV of Payment }n\\ \hline 1 & 9200(1.016)^{\bm{-1}} = \$\,9\,055.12\\ \hline 2 & 9200(1.016)^{\bm{-2}} = \$\,8\,912.52\\ \hline 3 & 9200(1.016)^{\bm{-3}} = \$\,8\,772.16\\ \hline 4 & 9200(1.016)^{\bm{-4}} = \$\,8\,634.02\\ \hline \dots & \dots\\ \hline 100 & 9200(1.016)^{\bm{-100}} = \$\,1\,881.12\\ \hline \hline \text{TOTAL} & \hspace{7.3em} \colorbox{yellow}{\$\,457\,429.86}\\ \hline \end{array}

Method 2: Accumulate the PV of Each Payment

QuarterCumulative PV (sum of first n payments)19200(1.016)−1=$ 9 055.122  9055.12+9200(1.016)−2=$ 17 967.64317967.64+9200(1.016)−3=$ 26 739.80……100=$ 457 429.86\begin{array}{|c|c|} \hline \text{Quarter} & \text{Cumulative PV (sum of first }n \text{ payments})\\ \hline 1 & \hspace{4.5em} 9200(1.016)^{\bm{-1}} = \$\,9\,055.12\\ \hline 2 & \ \ 9055.12 + 9200(1.016)^{\bm{-2}} = \$\,17\,967.64\\ \hline 3 & 17967.64 + 9200(1.016)^{\bm{-3}} = \$\,26\,739.80\\ \hline \dots & \dots\\ \hline 100 & \hspace{12em}= \colorbox{yellow}{\$\,457\,429.86}\\ \hline \end{array}

Practice: Regular Payments

Annual payments of $6000 are made for 5 years. Money earns interest of 10.3%/a compounded annually.
Use the formulas to answer the following questions.

a) What is the total future value of these payments?

b) What is the total present value of these payments?

Practice: Regular Payments

Annual payments of $6000 are made for 5 years. Money earns interest of 10.3%/a compounded annually.
Use a spreadsheet or graphing calculator to answer the following questions.

a) What is the total future value of these payments?

b) What is the total present value of these payments?

Practice: Regular Payments

Match each situation with the correct equation.
A.
PV=500(1−1.00167−150.00167)PV=500\left(\dfrac{1-1.00167^{-15}}{0.00167} \right)
B.
FV=100(1.007524−10.0075)FV=100\left(\dfrac{1.0075^{24}-1}{0.0075} \right)
C.
FV=500(1.0215−10.02)FV=500\left(\dfrac{1.02^{15}-1}{0.02} \right)
D.
100=R(1−1.0075−120.0075)100=R\left(\dfrac{1-1.0075^{-12}}{0.0075} \right)
$500 is invested every year at 2% compounded annually. What is the balance in 15 years?
How much is owed in total if you borrow money at 9% compounded monthly and agree to repay with monthly payments of $100 over 2 years?
How much money now is equivalent to monthly payments of $500 for 15 months if money can earn 2% compounded monthly?
You buy a $100 item on a credit card charging 9% compounded monthly. To pay it off in 1 year, how much will each monthly payment be?

Practice: Regular Payments

Rue is buying a car and the salesperson proposes two options, both with annual interest of 7.2% compounded monthly:

a) The car can be paid off in 8 years, and the future value of all monthly payments will amount to $30,000

b) Buy the car at its advertised price of $21,000 and pay it off with monthly payments for 10 years

Which scenario has the cheaper monthly payment?