# ACMGM097

## 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:
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## 1.3 Predicting Future Values for Depreciation Systems

Note: if you cannot remember how to model each of these types of depreciation using recurrence relation formulas and how they appear graphically, revise notes for 1.2 Forms of Depreciation.

### Predicting Values for Flat Rate Depreciation

• You may notice from the flat rate depreciation formula that this represents a system that increases by a value of d per unit in time (month, year, etc.), starting from the initial value; A_{0}. Thus, we can also express this relationship as the linear formula:

A=A_{0}+d t

Where t is the number of periods which have passed.

• This is equivalent to a linear relation with a slope of d, and a y-intercept of A_{0}.
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## 1.2 Forms of Depreciation

### Flat Rate Depreciation

• Flat rate depreciation occurs when the value of an asset decreases by a fixed amount every period (i.e. no interest occurs).
• Graphs of flat rate depreciation are linear.
• The recurrence relation formula describing flat rate depreciation is:

A_{n+1}=d+A_{n}

Note: as this is depreciation: d<0. Consequently, any graphs showing flat rate depreciation will have a negative slope.

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## 1.1 Recurrence Relations and Sequences

### Sequence

A list of numbers written down in succession.

Terms of a Sequence

The terms of a sequence can be identified by their term numbers.

The initial term or 0th term is A_{0}; The first term is A_{1};The second term is A_{2}.

A_{0}, A_{1}, A_{2}

The 𝑛th term is 𝑢; The term before the 𝑛th term is 𝑢; The term after the 𝑛th term is 𝑢.

A_{n-1}, A_{n}, A_{n+1}

Any letter, not just A, can represent the terms of a sequence.

### Recurrence Relations

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