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Math help from the Learning Centre

This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

Introduction to Annuities

What is an annuity? It is a series of equal payments which occurs at equal time intervals - usually monthly, quarterly, semi-annually or annually. There are two important terms of annuities - ordinary and simple.

Ordinary Annuity : Has payments at the end of each time interval. There is no payment at the start of the term of the annuity. 

Simple Annuity : Frequency of the payment is the same as the frequency of the compound interest.

Below are the definitions of the terms we use for ordinary simple annuities:

\( S_n \ or \ FV = \) future value at \( n \) payment

\( \ \ \ \ \ \ \ \ \ \ \ \ \ R = \) regular payment amount

\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = \frac{j}{m} = \) periodic rate of interest

\( \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ j = \) nominal annual rate of interest

\( \ \ \ \ \ \ \ \ \ \ \ \ \ m = \) compounded frequency daily: \( m = 365 \); monthly: \( m = 12 \); quarterly \(m = 4\)

\(\ \ \ \ \ \ \ \ \ \ \  \ \ \  n = (length \ of \ period) \times m = \) time 

\(A_n \ or \ PV = \) present value at \( n \) payment  

Future Value and Present Value of an Annuity

The future value (\( FV \) or \( S_{n} \)) of an annuity is the value of a current asset at a future date based on an assumed rate of growth. The future value is important to investors and financial planners as they use it to estimate how much an investment made today will be worth in the future. 

\[ S_{n} = \frac{R[(1+i)^n - 1]}{i} \]

Suppose you invested $1000 per quarter over a 15 year period. If money earns an annual rate of 6.5% compounded quarterly, what would be the value at the end of the time period?

Solution:

\( \Longrightarrow S_{n} = ? \) | \( R = $1000 \) | \( i = \frac{6.5\%}{4} = 0.01625 \) | \( n = 15*4 = 60 \)

\( \Longrightarrow S_{60} = \frac{1000[(1+0.01625)^{60} - 1]}{0.01625} = $10, 033.67 \) 

The present value of an annuity is the amount of cash today equivalent in value to a payment, or to a stream of payments, to be received in the future. It is calculated using the following formula:

\[ A_{n} = \frac{R[1- (1+i)^{-n}]}{i} \]

Suppose you with to be able to withdraw $3000 at the end of each month for two years. How much money must you deposit now at 2.75% interest compounded monthly?

Solution:

\( \Longrightarrow A_{n} = ? \) | \( R = $3000 \) | \( i = \frac{2.75\%}{12} = 0.002291666 \) | \( n = 2*12 = 24 \)

\( \Longrightarrow A_{24} = \frac{3000[1-(1+0.002291666)^{-24}]}{0.002291666} = $69, 977.66 \)

Calculating the Payment of an Annuity

Since there are different formulas for the present and future values of annuities. The first step is to determine whether the given value of the question is the future or present value. If we rearranged the formulas to solve for the payment, we have the following.

\[R=\frac{i\cdot S_n}{\left(1+i\right)^n-1}\]

\[R=\frac{i\cdot A_n}{1-\left(1+i\right)^{-n}}\]


Let's try the following example.

How much would you have to pay into an account at the end of every 6 months to accumulate $10,000 in eight years if interest is 3% compounded semi-annually?

Solution:

To solve any annuity, we need to pick out the important pieces of information from the question. 

  • Payment is at the end of the period which implies this is an ordinary annuity.
  • Payment is every 6 months and compounding is semi-annual. Since it is the same this is a simple annuity.
  • The nominal interest rate, j, is 3% compounded semi-annually.
  • Thus, the periodic interest rate is \(i=\frac{0.03}{2}=0.015\).
  • There are payments every 6 months for eight years, so \(n=16\).
  • We are trying to accumulate to $10,000. Since this is a value after all the payments, this is the future value.

\begin{align}R&=\frac{i\cdot S_n}{\left(1+i\right)^n-1}\\&=\frac{0.015\cdot 10,000}{\left(1+0.015\right)^{16}-1}\\&=557.65\end{align}

Therefore, a regular payment of $557.65 every 6 months for eight years will be required to accumulate to $10,000.

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