Annuities are a series of regular deposits or payments at regular intervals for a specific time period. Annuities can be used to either determine the present value or future value of a given sum of money. Annuities are calculated using compound interest formulas.
The Future Value of an annuity is the sum of each payment's future value. They typically center around investments when large sums of money are required in the future. The Future Value contains interest.
They can be found algebraically using the following formula:
\(\textcolor{teal}{FV} = \cfrac{\textcolor{red}{R}[(1 + \textcolor{blueviolet}{i})^{\textcolor{brown}{n}}-1]}{\textcolor{blueviolet}{i}}\)
The Present Value of an annuity is the sum of each payment's current value. They typically center around loans when large sums of money are required in the present. The Present Value doesn't contain interest.
They can be found algebraically using the following formula:
\(\textcolor{tomato}{PV} = \cfrac{\textcolor{red}{R}[1-(1 + \textcolor{blueviolet}{i})^{-\textcolor{brown}{n}}]}{\textcolor{blueviolet}{i}}\)
The respective formulas for finding the periodic interest rate and # of compunding periods can be algebraically described as:
These formulas can help identify variables that can be used to determine the annuity values.
Mario deposits \($600\) at the end of every \(6\) months into a savings account paying \(6\)% compounded semi-annually. He does this for \(8\) years. What is the amount of annuity and what is the total interest earned?
We can determine that this question deals with a future value since Mario is making investments. As a result, we can use the Future Value formula.
First, we can determine the periodic interest rate:
\(\textcolor{blueviolet}{i} = \cfrac{\textcolor{green}{r}}{\textcolor{magenta}{c}}\)
\(\textcolor{blueviolet}{i} = \cfrac{\textcolor{green}{0.06}}{\textcolor{magenta}{2}}\)
\(\textcolor{blueviolet}{i} = \textcolor{blueviolet}{0.03}\)
Next, we can determine the total # of compounding periods:
\(\textcolor{brown}{n} = \textcolor{magenta}{c}\textcolor{blue}{t}\)
\(\textcolor{brown}{n} = (\textcolor{magenta}{2})(\textcolor{blue}{8})\)
\(\textcolor{brown}{n} = \textcolor{brown}{16}\)
Finally, we can plug all pertinent values into the Future Value formula:
\(\textcolor{teal}{FV} = \cfrac{\textcolor{red}{R}[(1 + \textcolor{blueviolet}{i})^{\textcolor{brown}{n}}-1]}{\textcolor{blueviolet}{i}}\)
\(\textcolor{teal}{FV} = \cfrac{\textcolor{red}{$600}[(1 + \textcolor{blueviolet}{0.03})^{\textcolor{brown}{16}}-1]}{\textcolor{blueviolet}{0.03}}\)
\(\textcolor{teal}{FV} = $20000[(1.03)^{\textcolor{brown}{16}}-1]\)
\(\textcolor{teal}{FV} = $20000[(1.604706439-1]\)
\(\textcolor{teal}{FV} = $20000[(0.604706439]\)
\(\textcolor{teal}{FV} = \textcolor{teal}{$12094.13}\)
In order to determine the total interest earned, we can multiply the regular deposit amount by the # of compounding periods. We can then subtract this amount from the Future Value:
\(\textcolor{olive}{I} = \textcolor{teal}{FV} - \textcolor{red}{R}\textcolor{brown}{n}\)
\(\textcolor{olive}{I} = \textcolor{teal}{$12094.13} - [(\textcolor{red}{$600})(\textcolor{brown}{16})]\)
\(\textcolor{olive}{I} = \textcolor{teal}{$12094.13} - $9600\)
\(\textcolor{olive}{I} = \textcolor{olive}{$2494.13}\)
Therefore, we can determine that the Future Value is \(\boldsymbol{$12094.13}\) and the total interest earned is \(\boldsymbol{$2494.13}\).
Jasmine wants to save money for retirement in an annuity. She plans to make equal monthly deposits, at the end of each month for \(25\) years in a trust account that has a guranteed interest rate of \(9\)% compounded monthly. She wants to have \($500,000\) in the account at the end of the \(25\) years.
Since Jasmine wants to save her money for a long-tem investment, we can determine that this amount will use a future value.
First, we can determine the periodic interest rate:
\(\textcolor{blueviolet}{i} = \cfrac{\textcolor{green}{r}}{\textcolor{magenta}{c}}\)
\(\textcolor{blueviolet}{i} = \cfrac{\textcolor{green}{0.09}}{\textcolor{magenta}{12}}\)
\(\textcolor{blueviolet}{i} = \textcolor{blueviolet}{0.0075} %\)
Next, we can determine the total # of compounding periods:
\(\textcolor{brown}{n} = \textcolor{magenta}{c}\textcolor{blue}{t}\)
\(\textcolor{brown}{n} = (\textcolor{magenta}{12})(\textcolor{blue}{25})\)
\(\textcolor{brown}{n} = \textcolor{brown}{300} \; [\text{periods}]\)
Finally, we can plug all pertinent values into the Future Value formula. We can then simplify the expression to determine the amount of Jasmine's monthly deposits:
\(\textcolor{teal}{FV} = \cfrac{\textcolor{red}{R}[(1 + \textcolor{blueviolet}{i})^\textcolor{brown}{n}-1]}{\textcolor{blueviolet}{i}}\)
\(\textcolor{teal}{500000} = \cfrac{\textcolor{red}{R}[(1 + \textcolor{blueviolet}{0.0075})^\textcolor{brown}{300}-1]}{\textcolor{blueviolet}{0.0075}}\)
\((\textcolor{teal}{500000})(0.0075) = \left(\cfrac{\textcolor{red}{R}[(1.0075)^{\textcolor{brown}{300}}-1]}{\cancel{\textcolor{blueviolet}{0.0075}}}\right)(\cancel{0.0075})\)
\(3750 = \textcolor{red}{R}(9.40841453-1)\)
\(\textcolor{red}{R}(8.40841453) = 3750\)
\(\cfrac{\textcolor{red}{R}(\cancel{8.40841453})}{\cancel{8.40841453}} = \cfrac{$3750}{8.40841453}\)
\(R \approx $445.98\)
Therefore, we can determine that Jasmine makes monthly deposits of \(\boldsymbol{\approx $445.98}\).
ii. In order to determine the total interest earned, we can multiply the regular deposit amount by the # of compounding periods. We can then subtract this amount from the Future Value:
\(\textcolor{olive}{I} = \textcolor{teal}{FV} - \textcolor{red}{R}\textcolor{brown}{n}\)
\(\textcolor{olive}{I} = \textcolor{teal}{500000} - [(\textcolor{red}{445.98})(\textcolor{brown}{300})]\)
\(\textcolor{olive}{I} = \textcolor{teal}{500000} - 133794\)
\(I = $366206\)
Therefore, we can determine that Jasmine's total interest earned using the trust fund is \(\boldsymbol{$366206}\).
Steve is buying a new Harley motorcycle. His monthly payments are \($650\) with interest within the payments which was charged at \(3\)% compounded monthly. What is the cash price of the motorcycle if he makes payments for \(4\) years?
Since Steve wants to make a current purchase, we can determine that this amount will use a present value.
First, we can determine the periodic interest rate:
\(\textcolor{blueviolet}{i} = \cfrac{\textcolor{green}{r}}{\textcolor{magenta}{c}}\)
\(\textcolor{blueviolet}{i} = \cfrac{\textcolor{green}{0.03}}{\textcolor{magenta}{12}}\)
\(\textcolor{blueviolet}{i} = \textcolor{blueviolet}{0.0025}\)
Next, we can determine the total # of compounding periods:
\(\textcolor{brown}{n} = \textcolor{magenta}{c}\textcolor{blue}{t}\)
\(\textcolor{brown}{n} = (\textcolor{magenta}{12})(\textcolor{blue}{4})\)
\(\textcolor{brown}{n} = \textcolor{brown}{48}\)
Finally, we can plug all pertinent values into the Present Value formula in order to determine the cash price:
\(\textcolor{tomato}{PV} = \cfrac{\textcolor{red}{R}[1-(1 + \textcolor{blueviolet}{i})^{-\textcolor{brown}{n}}]}{\textcolor{blueviolet}{i}}\)
\(\textcolor{tomato}{PV} = \cfrac{\textcolor{red}{650}[1-(1 + \textcolor{blueviolet}{0.0025})^{-\textcolor{brown}{48}}]}{\textcolor{blueviolet}{0.0025}}\)
\(\textcolor{tomato}{PV} = 260000[1-(1.0025)^{-\textcolor{brown}{48}}]\)
\(\textcolor{tomato}{PV} = 260000(1-0.887053263)\)
\(\textcolor{tomato}{PV} = 260000(0.112946737)\)
\(PV = $29366.15\)
Therefore, we can determine that the cash price is \(\boldsymbol{$29366.15}\).
Sonia purchases a new vehicle for \($27000\) at \(8.2 \%\) compounded quarterly and makes payments at the end of every \(3\) months for \(5\) years.
i. Since Sonia wants to make a current purchase, we can determine that this amount will use a present value.
First, we can determine the periodic interest rate:
\(\textcolor{blueviolet}{i} = \cfrac{\textcolor{green}{r}}{\textcolor{magenta}{c}}\)
\(\textcolor{blueviolet}{i} = \cfrac{\textcolor{green}{0.082}}{\textcolor{magenta}{4}}\)
\(\textcolor{blueviolet}{i} = \textcolor{blueviolet}{0.0205}\)
Next, we can determine the total # of compounding periods:
\(\textcolor{brown}{n} = \textcolor{magenta}{c}\textcolor{blue}{t}\)
\(\textcolor{brown}{n} = (\textcolor{magenta}{4})(\textcolor{blue}{5})\)
\(\textcolor{brown}{n} = \textcolor{brown}{20}\)
Finally, we can plug all pertinent values into the Present Value formula in order to determine the monthly payment:
\(\textcolor{tomato}{PV} = \cfrac{\textcolor{red}{R}[1-(1 + \textcolor{blueviolet}{i})^{-\textcolor{brown}{n}}]}{\textcolor{blueviolet}{i}}\)
\(\textcolor{tomato}{$27000} = \cfrac{\textcolor{red}{R}[1-(1 + \textcolor{blueviolet}{0.0205})^{-\textcolor{brown}{20}}]}{\textcolor{blueviolet}{0.0205}}\)
\((\textcolor{tomato}{$27000})(\textcolor{blueviolet}{0.0205}) = (\cfrac{\textcolor{red}{R}[1-(1.0205)^{-\textcolor{brown}{20}}]}{\cancel{\textcolor{blueviolet}{0.0205}}})(\cancel{0.0205})\)
\($553.5 = \textcolor{red}{R}(1-0.666407412)\)
\(\textcolor{red}{R}(0.333592588) = $553.5\)
\(\cfrac{\textcolor{red}{R}(\cancel{0.333592587})}{\cancel{0.333592587}} = \cfrac{$553.5}{0.333592587}\)
\(\textcolor{red}{R} = \textcolor{red}{$1659.21}\)
Therefore, we can determine that Sonia made monthly payments of \(\boldsymbol{$1659.21}\).
ii. In order to determine the total interest paid, we can multiply the monthly payment amount by the # of compounding periods. We can then subtract the Present Value from this amount:
\(\textcolor{olive}{I} = \textcolor{red}{R}\textcolor{brown}{n} - \textcolor{tomato}{PV}\)
\(\textcolor{olive}{I} = (\textcolor{red}{$1659.21})(\textcolor{brown}{20}) - \textcolor{tomato}{$27000}\)
\(\textcolor{olive}{I} = $33184.2 - \textcolor{tomato}{$27000}\)
\(\textcolor{olive}{I} = \textcolor{olive}{$6184.2}\)
Therefore, we can determine the total interest Jasmine payed was \(\boldsymbol{$6184.2}\).