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# FM Compound interest investments and loans

## 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.
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## 2.3 Nominal and Effective Interests

### Nominal Interest

• Interest rates are generally quoted annual rates (i.e. per year). This figure is known as the nominal interest.
• If interest is compounded annually, this is equal to the interest per compounding period.
• If interest is compounded more frequently however, the interest per compounding period is calculated by dividing the nominal interest by the number of compounding periods per year:

r_{\text {compounding period }}=\frac{r}{n}

Where r is the nominal interest rate and n is the number of compounding periods in a year.

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## 2.2 Analysis of Compound Interest

### Compound Interest with Regular Additions or Withdrawals

Note: if you cannot remember the basic recursive formula for a compound interest system, revise notes for 2.1 Simple and Compound Interest.

• A compound interest system with regular additions or withdrawals has both a percentage increase/decrease and a fixed amount increase/decrease at each compounding period.
• This type of system can be modelled using the following recursive formula:
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## 2.1 Simple and Compound Interest

Note: the formulas used in this topic describe the simplest form of these systems. Future notes will analyse situations involving regular withdrawals or additions.

### Simple Interest

• Simple interest describes a system that increases or decreases by a fixed amount after each period of time. That amount is calculated as a percentage of the initial value of the system.
• Graphs for this type of system are linear.
• This type of system can be modelled using the following recursive formula:

A_{n+1}=I A_{0}+A_{n}

Where I is the interest rate as a decimal, A_{0} is the initial value and A_{n} is the value after n periods of time.

• A non-recursive formula can also be used:
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