Interest Rates Over Different Time Periods and Effective Interest Rates Revision Notes for VCE SSCE General Mathematics

Interest Rates Over Different Time Periods and Effective Interest Rates

Understanding nominal and compounding period interest rates

When you take out a loan or open a savings account, the interest rate is typically quoted as an annual figure, known as the nominal interest rate (or interest rate per annum). However, interest is often calculated and added to your account more frequently than once a year - perhaps monthly, quarterly, or even daily.

The compounding period refers to how often interest is calculated and added to your balance. For example, if interest is calculated monthly, the compounding period is one month. Understanding these concepts is crucial because the frequency of compounding affects how much interest you'll earn or pay over time.

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The frequency of compounding can significantly impact the total amount of interest you earn or pay. Even with the same nominal rate, more frequent compounding periods will result in higher actual returns for investments or higher costs for loans.

Converting nominal rates to compounding period rates

To work with different compounding periods, we need to convert the annual nominal rate into the rate for each compounding period. This conversion is straightforward: we divide the annual rate by the number of compounding periods in one year.

Standard compounding periods

When converting interest rates, we use these standard values for the number of periods per year (represented by $p$ ):

Monthly compounding: $p = 12$ (12 months in a year)
Quarterly compounding: $p = 4$ (4 quarters in a year, where each quarter equals 3 months)
Fortnightly compounding: $p = 26$ (26 fortnights in a year)
Weekly compounding: $p = 52$ (52 weeks in a year)
Daily compounding: $p = 365$ (365 days in a year)

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Some of these are approximations - for instance, there are actually slightly more than 52 weeks in a year - but these standard values are used for consistency in financial calculations.

Conversion formula

To convert a nominal interest rate to a compounding period interest rate, simply divide by $p$ :

$\text{Compounding period rate} = \frac{\text{Nominal rate}}{p}$

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Worked Example: Converting a Nominal Rate

Let's say an investment account pays $4.68\%$ per annum. We can convert this to different compounding period rates:

a) Monthly rate

Divide by $p = 12$

$\text{Monthly interest rate} = \frac{4.68}{12} = 0.39\%$

b) Fortnightly rate

Divide by $p = 26$

$\text{Fortnightly interest rate} = \frac{4.68}{26} = 0.18\%$

c) Quarterly rate

Divide by $p = 4$

$\text{Quarterly interest rate} = \frac{4.68}{4} = 1.17\%$

Recurrence relations with different compounding periods

Once we understand how to convert interest rates, we can model loans and investments that compound at different frequencies using recurrence relations. This requires updating our formula for the growth multiplier, $R$ .

The growth multiplier for different compounding periods

Previously, for annual compounding, we used $R = 1 + \frac{r}{100}$ , where $r$ is the interest rate percentage. When interest compounds more frequently than annually, we modify this formula:

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Growth Multiplier for Different Compounding Periods

$R = 1 + \frac{r}{100 \times p}$

where:

$r$ is the annual nominal interest rate
$p$ is the number of compounding periods per year

For annual compounding (when interest is calculated once per year), we simply use $p = 1$ .

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Worked Example: Writing Recurrence Relations for Different Compounding Periods

Brian borrows $5000 from a bank with an interest rate of $4.5\%$ per annum. Let's write recurrence relations to model the value of his loan under different compounding scenarios. Let $V_n$ represent the value of the loan after $n$ compounding periods.

a) Yearly compounding

Define the variable: Let $V_n$ be the value of Brian's loan after $n$ years
Calculate $R$ with $r = 4.5$ and $p = 1$ :

$R = 1 + \frac{4.5}{100 \times 1} = 1.045$

Write the recurrence relation:

$V_0 = 5000, \quad V_{n+1} = 1.045V_n$

b) Quarterly compounding

Define the variable: Let $V_n$ be the value of Brian's loan after $n$ quarters
Calculate $R$ with $r = 4.5$ and $p = 4$ :

$R = 1 + \frac{4.5}{100 \times 4} = 1.01125$

Write the recurrence relation:

$V_0 = 5000, \quad V_{n+1} = 1.01125V_n$

c) Monthly compounding

Define the variable: Let $V_n$ be the value of Brian's loan after $n$ months
Calculate $R$ with $r = 4.5$ and $p = 12$ :

$R = 1 + \frac{4.5}{100 \times 12} = 1.00375$

Write the recurrence relation:

$V_0 = 5000, \quad V_{n+1} = 1.00375V_n$

Notice that the growth multiplier $R$ gets smaller as the compounding period gets shorter, but because interest is applied more frequently, the overall effect can still result in more interest being paid or earned over time.

Modelling investments with monthly compounding

Let's work through a complete example that shows how to model an investment, write its recurrence relation, develop a rule, and use it to find future values.

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Worked Example: Investment with Monthly Compounding

A principal value of $10,000 is invested in an account earning compound interest monthly at the rate of $9\%$ per annum. Let $V_n$ be the value of the investment after $n$ months.

a) Calculate the growth multiplier, $R$

Since interest compounds monthly, $p = 12$ :

$R = 1 + \frac{9}{100 \times 12} = 1.0075$

b) Write down a recurrence relation

Substitute the initial value and growth multiplier:

$V_0 = 10000, \quad V_{n+1} = 1.0075V_n$

c) Write down a rule for the value after $n$ months

Using the general form for geometric sequences:

$V_n = 1.0075^n \times 10000$

d) Find the value after 4 years

First, convert 4 years to months: $4 \text{ years} = 48 \text{ months}$

Substitute $n = 48$ into the rule:

$V_{48} = 1.0075^{48} \times 10000$

$V_{48} = \$14,314.05$

This shows that after 4 years, the investment will have grown to approximately $14,314.

Understanding effective interest rates

When comparing different investment or loan options with varying compounding periods, it's difficult to make a fair comparison using just the nominal rates. This is where effective interest rates become useful.

The effective interest rate represents the actual percentage increase in value over one year, taking into account the compounding frequency. It allows us to compare different financial products on an equal basis, regardless of how often they compound.

Why compounding frequency matters

Consider two investments both offering $4.8\%$ per annum:

Investment A compounds quarterly
Investment B compounds monthly

Even though they have the same nominal rate, Investment B will earn slightly more interest because the interest is calculated and added more frequently, allowing you to earn "interest on interest" more often.

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Example Comparison

If you invest $5000 at $4.8\%$ p.a.:

With quarterly compounding, after one year you'll have $5244.35 (interest of $244.35)
With monthly compounding, after one year you'll have $5245.35 (interest of $245.35)

The monthly compounding earns $1 more due to the more frequent calculation of interest.

Calculating effective interest rates

The effective interest rate can be calculated using this formula:

$\text{Effective rate} = \frac{\text{Total interest in one year}}{\text{Principal}} \times 100\%$

Alternatively, we can use a more direct formula:

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Effective Interest Rate Formula

$r_{\text{eff}} = \left[\left(1 + \frac{r}{100 \times n}\right)^n - 1\right] \times 100\%$

where:

$r$ is the nominal interest rate per annum
$n$ is the number of times interest compounds each year
$r_{\text{eff}}$ is the effective annual interest rate

Important note: In this formula, $n$ represents the number of compounding periods (which we've been calling $p$ elsewhere). This notation aligns with the VCAA formula sheet.

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Worked Example: Comparing Loans Using Effective Interest Rates

Brooke wants to borrow $20,000 that she will repay entirely after one year. She's considering two options:

Option A: $5.95\%$ per annum, compounding weekly
Option B: $6\%$ per annum, compounding quarterly

a) Calculate the effective interest rate for each option

For Option A: $r = 5.95\%$ and $n = 52$

$r_{\text{eff}} = \left[\left(1 + \frac{5.95}{100 \times 52}\right)^{52} - 1\right] \times 100\%$

$r_{\text{eff}} = 6.13\%$

For Option B: $r = 6\%$ and $n = 4$

$r_{\text{eff}} = \left[\left(1 + \frac{6}{100 \times 4}\right)^4 - 1\right] \times 100\%$

$r_{\text{eff}} = 6.14\%$

b) Which option is best and why?

Since Brooke is borrowing money, she wants to pay the least amount of interest. Option A has the lower effective interest rate ( $6.13\%$ compared to $6.14\%$ ), so she should choose Option A.

Notice that even though Option A has a lower nominal rate ( $5.95\%$ vs $6\%$ ), this advantage is partially offset by the more frequent compounding (weekly vs quarterly). However, Option A still works out slightly better overall.

Using a CAS calculator to find effective interest rates

While you can calculate effective interest rates manually using the formula, a CAS calculator can perform this calculation much more quickly and reduce the chance of errors.

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Worked Example: Using a CAS Calculator

Marissa has $10,000 to invest in an account earning $4.5\%$ per annum, compounding monthly. Let's find the effective interest rate using a CAS calculator.

On a TI calculator:

Press the menu button and select 8: Finance ▶ 5: Interest Conversion ▶ 2: Effective interest rate
This pastes the eff(...) command, which takes two parameters:
- The nominal rate
- The number of times interest compounds per year
Enter: eff(4.5, 12)
Press enter to calculate

On a Casio calculator:

Select Interactive > Financial > Interest Conversion > ConvEff
Enter the number of compounding periods per year: 12
Enter the nominal rate: 4.5
Press EXE to calculate

Result: The effective interest rate for this investment is 4.594%.

This means that even though the nominal rate is $4.5\%$ , the monthly compounding results in an actual annual return of approximately $4.594\%$ .

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Exam Tip

When using your CAS calculator in an exam, make sure you're familiar with how to access the interest conversion functions. Practice using both methods (manual calculation and calculator) so you can check your work and understand what the calculator is doing.

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Key Points to Remember:

The nominal interest rate is the quoted annual rate, while the compounding period determines how often interest is actually calculated and added
To convert a nominal rate to a compounding period rate, divide by the number of periods per year: monthly ( $p=12$ ), quarterly ( $p=4$ ), fortnightly ( $p=26$ ), weekly ( $p=52$ ), or daily ( $p=365$ )
The growth multiplier for different compounding periods is $R = 1 + \frac{r}{100 \times p}$ , where $r$ is the nominal annual rate and $p$ is the number of compounding periods per year
The effective interest rate shows the actual percentage increase over one year and allows fair comparison between different investment or loan options, regardless of their compounding frequency
For borrowers, a lower effective interest rate is better; for investors, a higher effective interest rate is better
Use the formula $r_{\text{eff}} = \left[\left(1 + \frac{r}{100 \times n}\right)^n - 1\right] \times 100\%$ or your CAS calculator to find effective interest rates

Interest Rates Over Different Time Periods and Effective Interest Rates (VCE SSCE General Mathematics): Revision Notes