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# FM Compounding Period

## 5.3 Modelling Annuity Investments using Technology

Note: if you cannot remember how to model reducing balance relations with regular repayments, revise notes for 5.1 Modelling Annuity Investments.

### Guide to Analysing Annuity Investments using Technology (Casio Graphics Calculator)

Note: if you cannot remember how to use the Casio Financial Calculator, revise notes for 3.3 Modelling Reducing Balance Systems with Regular Repayments using Technology.

• The annual interest rate should be entered/calculated as a positive value.
• The initial value (PV) should be entered/calculated as a negative value (remember we justify this by saying we must lose this amount to create the investment).
• The payment (PMT) (i.e. the amount withdrawn) should be entered/calculated as negative (we can justify this by saying we lose this amount per compounding period to the investment)
Read More »5.3 Modelling Annuity Investments using Technology

## 5.2 Modelling Annuity Investments using Tables

Note: if you cannot remember how to model reducing balance systems with regular repayments, revise notes for 5.1 Modelling Annuity Investments.

### Guidelines to use a Table for Annuity Investments

• Tables provide a convenient method for analysing annuity investments, especially when there are regular systemic changes (e.g. the interest rate or repayments change regularly).
• The columns of the table should list, the compounding period, payment, interest earned, principal addition (i.e. the amount the system has increased during each compounding period) and the balance at the end of the compounding period.
• The principal addition can be calculated by as follows:
Read More »5.2 Modelling Annuity Investments using Tables

## 5.1 Modelling Annuity Investments

### Modelling using Recursion Relations

• An annuity investment describes a system which has a positive interest rate, and positive deposits made at the end of each compounding period.
• We can use the simple recursion formula to model this system:
Read More »5.1 Modelling Annuity Investments

## 4.4 Annuities and Perpetuities

### Annuities

Note: if you cannot remember what a compound interest system is, revise notes for 2.1 Simple and Compound Interest.

• Compound interest systems with regular payments or withdrawals are often known as annuities.

### Perpetuities

• Perpetuities are a special case of annuity, which does not change value.
Read More »4.4 Annuities and Perpetuities

## 4.3 Modelling Compound Interest Systems with Regular Withdrawals using Technology

Note: if you cannot remember how to model reducing balance relations with regular repayments, revise notes for 4.1 Modelling Compound Interest Systems with Regular Withdrawals.

### Guide to Analysing Compound Interest Equations with Regular Withdrawals using Technology (Casio Graphics Calculator)

Note: if you cannot remember how to use the Casio Financial Calculator, revise notes for 3.3 Modelling Reducing Balance Systems with Regular Repayments using Technology.

• The annual interest rate should be entered/calculated as a positive value.
• The initial value (PV) should be entered/calculated as a negative value (remember we justify this by saying we must lose this amount to create the account).
Read More »4.3 Modelling Compound Interest Systems with Regular Withdrawals using Technology

## 4.2 Modelling Compound Interest Systems with Regular Withdrawals using Tables

Note: if you cannot remember how to model reducing balance systems with regular repayments, revise notes for 4.1 Modelling Compound Interest Systems with Regular Withdrawals.

### Guidelines to using a Table for Compound Interest Systems with Regular Withdrawals

• Tables provide a convenient method for analysing compound interest systems with regular withdrawals, especially when there are regular systemic changes (e.g. the interest rate or withdrawals change regularly).
• The columns of the table should list, the compounding period, withdrawal, interest earned, principal addition (i.e. the amount the system has increased/decreased during each compounding period) and the balance at the end of the compounding period.
• The principal addition can be calculated as follows:
Read More »4.2 Modelling Compound Interest Systems with Regular Withdrawals using Tables

## 3.3 Modelling Reducing Balance Systems with Regular Repayments using Technology

Note: if you cannot remember how to model reducing balance relations with regular repayments, revise notes for 3.1 Modelling Reducing Balance Systems with Regular Repayments.

### Benefits of using Technology to Model Reducing Balance Systems

• Solving reducing balance systems using recursion can become a long and tedious process when the number of compounding periods becomes large. Using technology allows us to find values much quicker in these situations.
• When solving by hand, we generally round off at each compounding period. While this is accurate for some systems, in others this causes an error which is compounded through the following compounding periods, resulting in inaccurate results.

### Guide to Analysing Reducing Balance Equations using Technology (Casio Graphics Calculator)

Read More »3.3 Modelling Reducing Balance Systems with Regular Repayments using Technology

## 3.2 Modelling Reducing Balance Systems with Regular Repayments using Tables

Note: if you cannot remember how to model reducing balance systems with regular repayments, revise notes for 3.1 Modelling Reducing Balance Systems with Regular Repayments.

### Guidelines to using a Table for Reducing Balance Systems

• Tables provide a convenient method for analysing reducing balance systems, especially when there are regular systemic changes (e.g. the interest rate or repayments change regularly).
• The columns of the table should list, the compounding period, payment, interest charged, principal addition (i.e. the amount the system has increased/decreased during each compounding period) and the balance at the end of the compounding period.
• The principal addition can be calculated by as follows:
Read More »3.2 Modelling Reducing Balance Systems with Regular Repayments using Tables

## 2.4 Predicting Future Values for Compound Interest Loans and Investments

### Predicting Future Values involving Compound Interest

• Predicting the value of a compound interest system after a large number of compounding periods is a long and tedious process using recursion. Fortunately, for systems which do not involve regular additions or withdrawals, we can use the convenient formula:

A_{n}=\left(1+\frac{r}{100}\right)^{n} A_{0}

Where A_{n} is the amount after n compounding periods, r is the interest as a percentage and A_{0} is the initial value.

• When dealing with systems which do have regular additions or withdrawals, the only method we have (in the scope of this course) is recursion.
Read More »2.4 Predicting Future Values for Compound Interest Loans and Investments