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## Learn the math behind your money

The world of finance is literally FULL of mathematical models, formulas, and systems. There's a reason that many word problems in math class involve making change, calculating interest rates, or auditing lemonade stands. There's no avoiding math when it comes to money. Fortunately, most of what the average person needs to know is straightforward. However, it is absolutely necessary to understand certain key concepts in order to be successful financially, whether that means saving money for the future, or to avoid being a victim of a quick-talking salesman.

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## Finite Mathematics : Mathematics of Finance

Study concepts, example questions & explanations for finite mathematics, all finite mathematics resources, example questions, example question #1 : simple interest.

What is the total interest made after 4 years on a simple interest loan that accumulates 13% each year and has original amount was $1350.

To solve this problem, first recall the formula to calculate simple interest.

From the question it's known that,

Thus the equation becomes,

Therefore $702 in interested accumulated after four years.

## Example Question #1 : Mathematics Of Finance

Money is deposited in a bank account at 7% interest per year, compounded bimonthly. If no money is deposited into or withdrawn from the account, then how long will it take for the money to double?

9 years 8 months

9 years 6 months

9 years 10 months

10 years 0 months

10 years 2 months

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## Mathematical Finance

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Introduction

Mathematical finance is a branch of applied mathematics. Mathematical finance is also known as quantitative finance. It has a prominent use in many different parts of economics, especially in banking. Financial Mathematics includes a whole range of problems related to calculating the final value of capital. Capital is the value (in the form of money, property or human resources), which is invested in the production or other economic activity with the main purpose to increase value or to make a profit. The capital can be invested in the production or non-production sector. Its main purpose, however, is always the same, and that is the increasing basic value or the production of other goods.

With mathematical finance it is possible to answer the following questions:

Is it better to buy a good today or tomorrow?

Should I invest money in the bank?

Do my earnings in dollars convert into foreign currency?

Would it be financially wiser to acquire a new car for cash, credit or a lease?

What are the best securities in which to invest money?

When is it advisable to take a bank loan?

Importance of Mathematical Finance

In terms of limited financial resources, it is necessary to make the wisest decision about where to invest. With the help of mathematical finance, it is possible to increase the profitability of a company. It involves making the right decisions at the right time. Once the wrong decision has been made, it is very difficult to put the company back on the right path. This can make a tremendous difference in the following areas:

- Businesses and individuals
- Banks, insurers, and other financial institutions
- Legislation
- Education of staff and citizens

Simple and Compound Interest Calculation

Simple and compound interest are the most common methods of calculation interest.

The General definition for interest is the fee for using borrowed money. It can be explained as an expense for the person who borrows money and income for the person who lends money. Interest can be charged on principal amount at a certain rate for a certain period. For instance, 12% per year, 3% per quarter or 1% per month etc.

By definition, the principal amount means the amount of money that is originally borrowed from an individual or a financial institution and it does not include interest. In real life, the interest is charged using one of two methods- simple or compound calculation.

Simple Interest (Is) = P × i × t

P is the principle amount;

i represents the interest rate per period;

t is the time for which the money is borrowed or lent.

Compound Interest (Ic) = P × (1 + i) n – P

i represents the compound interest rate per period;

n refers to the number of periods.

References:

- An Introduction to Quantitative Finance, Stephen Blyth
- Quantitative Methods for Business, David R. Anderson, Dennis J. Sweeney, Thomas A. Williams, Jeffrey D. Camm, James J. Cochran
- Quantitative Methods for Finance, Terry Watsham (Author), Keith Parramore

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## 24HourAnswers Online Mathematical Finance Tutors

Our tutors are just as dedicated to your success in class as you are, so they are available around the clock to assist you with questions, homework, exam preparation and any Mathematical Finance related assignments you need extra help completing.

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## Part 7: Financial Mathematics | Free Worksheet

Has Financial Maths not accrued your interest? Well, after you read this article, it will have! In this article, we dsicuss everything you need to know to master financial maths, including types of interest, modelling investments and loans and harder questions.

- Share this article

Mastering financial maths is an extremely important skill, not only in High School mathematics, but also in later life.

By being able to adeptly solve financial mathematics questions, students can efficiently put their Series and Sequences, AP and GP knowledge to the test.

Further, grasping the fundamentals of financial mathematics allows students also gain deeper insight into how interest, super, loans, etc., are calculated, preparing them for later life.

## In this article we discuss:

- The NESA Syllabus Outcomes

## Assumed Knowledge

Types of interest.

- Modelling Investments and Loans

## Harder Financial Mathematics

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## NESA Syllabus Outcomes: Financial Maths

NESA requires students to be proficient in the following syllabus outcomes:

- Solve compound interest problems involving financial decisions, including a home loan, a savings account, a car loan or superannuation
- Solve problems involving compound interest loans or investments, eg determining the future value of an investment or loan, the number of compounding periods for an investment to exceed a given value and/or the interest rate needed for an investment to exceed a given value (ACMGM096)
- Recognise a reducing balance loan as a compound interest loan with periodic repayments, and solve problems including the amount owing on a reducing balance loan after each payment is made
- Solve problems involving financial decisions, including a home loan, a savings account, a car loan or superannuation

Students should already be familiar with basic series and sequences concepts. This includes being able to prove an AP or GP as well as determine the common differences, ratios, sums, etc.

Students can refresh their knowledge on basic trigonometric functions in the following Year 12 Advanced Subject Guide:

- Series and Sequences

## Simple Interest

Simple interest is calculated on the investment only and depends intrinsically on the amount of time the money is invested. The interest is usually calculated annually and is a percentage of the original investment.

The formula for simple interest is as follows:

\( I = Prn \)

Where \( I \) simple interest, \( P \) principal amount, \( r \) interest rate, \( n \) number of time periods.

It is important to note that \( r \) would be expressed as a decimal. I.e. if the interest rate is \( 3 \% \ p.a \), \( r = 0.03 \).

## Compound Interest

On the other hand, compound interest is the interest earned from both the principal amount as well as any past interest accumulated.

As compound interest also takes into account any past interest accumulated, it would be significantly more than simple interest calculated at the same interest rate.

The formula for compound interest is as follows:

\begin{align} A_n = P(1 + r)^{n} \end{align}

Where \( A_n \) amount at the end of the \( n^{th} \) period, \( P = \) principal amount, \( r = \) interest rate, \( n = \) number of time periods.

As above, it is important to note that would be expressed as a decimal. I.e. if the interest rate is \( 3 \% \ p.a., r = 0.03 \).

Further, the interest rate must correspond to the time period used.

For example, if the interest rate is given per annum, but our question requires us to compound monthly, we must then divide the interest rate by 12 (as there are 12 months) to obtain \( r \).

## Modelling Investment and Loans

An annuity is a regular series of equal investments which are subject to compound interest. These investments are generally made at the end of a time period meaning that no interest is earned until the next compounding period.

With annuities, we are often required to find the final value of an annuity or the present value of an annuity.

The general formula for these two annuities is as follows:

\begin{align} F = a \Big{(} \frac{(1 + r)^{n} – 1}{r} \Big{)} \end{align}

Where \( F = \) the final value of the annuity at the end of the \( n^{th} \) period, \( a = \) equal investments made, \( r = \) interest rate, \( n = \) number of time periods.

\begin{align} P = \frac{F}{(1 + r)^{k}} = \frac{a}{r} \Big{(} 1 – \frac{1}{(1 + r)^{k}} \Big{)} \end{align}

Where \( P = \) the present/current value of the annuity at the end of a \( k^{th} \) period, \( a = \) equal investments made, \( r = \) interest rate, \( n = \) total number of time periods.

However, generally annuities are generally represented through standard tables, one for the present value of an annuity (PVA) and one for a final value of an annuity (FVA). These standard tables will use $1 as the base value.

The FVA table will represent how the final value of an annuity differs depending on the interest rate and the number of time periods.

Conversely, the PVA table will represent the current value of the annuity and how it differs depending on the interest rate and the number of time periods. Hence, the formulae described above will seldom need to be used.

Example partial tables are shown below:

## Recurrence Relation

A recurrence relation is when the equation is expressed as a function of the preceding terms.

In most cases, the amount left at the end of a time period is expressed as a function of the previous period’s balance.

When constructing a recurrence relation, you must make sure to define the initial terms of the correctly. In particular, you have to be careful if you are analysing the amount left at the beginning or the end of the time period. In most cases, it is best to choose the end of the time period.

An example is as follows:

If I repay \( M \) on my loan each year compounded annually at \( 100r \ \% \ p.a. \):

Let \( A_{n} = \) the amount left in my loan at the end of each year and \( L = \) the original amount of the loan.

Recurrence relations are an integral part of a many financial mathematics questions but are most commonly used to determine repayments needed to be made on a loan, etc. Other types of questions which will use recurrence relations include:

## Superannuation

Superannuation questions involve regular investments made into a fund for time periods. The fund pays interest which is compounded every period.

These sorts of questions often want us to determine the amount left in the account at the end of \( n \) time periods, i.e. we need to find what \( A_{n} \) is.

To do this we use the recurrence relation described above.

Sally opens a new super account and deposits of \( $885 \). At the beginning of each month, her employer matches this deposit. The interest for her account is compounded monthly at a rate of \( 2.65 \ \% \ p.a. \) After 35 years, how much will Sally have in her account?

\begin{align} A_{1} &= 885 \\ A_{2} &= 885 \Bigg{(} 1 + \Big{(} 1 + \frac{2.65}{100 \times 12} \Big{)} \Bigg{)} \end{align}

Following this recurrence relation:

## Home Loans/Time Repayments

Questions which involve repayments on loans also use the recurrence relation but are a little bit more difficult when compared to superannuation problems. These repayments are made on a timely, regular basis (most often monthly) and are equal.

When solving these questions, we must take into account the interest left on the previous balance of our loan. I.e. the loan keeps growing in value the longer we take to repay — it does not stagnant as the original amount.

Thus, in general, after repayments of \( \$ M \) on a loan worth \( \$ L \) :

Where \( A_{n} \) the balance left in my loan at the end of each time period.

Roger borrows $300,000 at an interest rate of \( 5.35 \% \ p.a \). He borrowed the money at the beginning of January, 2012. At the end of each month he pays back an instalment of \( \$ M \). What is the value of \( M \) if the loan is repaid after 20 years?

When \( n = 240, \ A_{n} = 0 \). Therefore, we rearrange and solve for \( M \):

\begin{align} M &= \frac{ \big{(} 1 + \frac{5.35}{100 \times 12} \big{)}^{240} (300 \ 000)}{ \Bigg{[} \frac{\big{(} 1 + \frac{5.35}{100 \times 12} \big{)}^{240} – 1}{ \big{(} 1 + \frac{5.35}{100 \times 12} \big{)} – 1} \Bigg{]}} \\ &= \$ 2038.33 \end{align}

1. The total value of an investment after earning simple interest for 7 years is $30,598. If original investment was $25,000, what was the interest rate?

2. Veronica opens a savings account and deposits $798 fortnightly. The bank gives Veronica an interest rate of \( 1.15 \% \ p.a. \) Calculate the amount, to the nearest dollar, left in her account after 3 years if the account collects compound interest calculated monthly.

3. Tim’s employer deposits $700 into a super account at the beginning of every month. The interest earned on the account is \( 4.65 \% \ p.a. \) compounding quarterly. If Tim works for this employer for 24 years, how much will he collect when he retires (to the nearest dollar)?

1. \( 3.20 \% \)

2. \( \$ 63 \ 299 \)

3. \( \$122 \ 419 \)

## Boost your Financial Maths skills before your HSC Exams now!

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