Converting Between Time Frequencies

Video coming soon4 worked examples

Theory

Often in finance, you will be given an interest rate for one time period (e.g., an annual rate) but you will need to apply it to a completely different time period (e.g., monthly payments). You must convert the interest rate to match the payment frequency.

1. The Division Trap (A Major Exam Warning)

You cannot simply divide an annual interest rate by 12 to find the monthly rate.

Because of the effects of compound interest, dividing by 12 will give you an incorrect, slightly larger number. The SQA examiners reported this as a highly common mistake.

2. Scaling Up (Smaller to Larger Time Periods)

If you have a monthly rate and need an annual rate, you use the standard compounding formula: $(\text{Multiplier})^n$ .

Example: To convert a monthly rate to an annual rate, you raise the monthly multiplier to the power of 12.

3. Scaling Down (Larger to Smaller Time Periods)

If you have an annual rate and need a monthly rate, you must do the mathematical reverse, using a fractional power: $(\text{Multiplier})^{\frac{1}{n}}$ .

Example: To convert an annual rate to a monthly rate, you raise the annual multiplier to the power of $\frac{1}{12}$ .

4. The Calculator Syntax Trap

When scaling down using a fractional power on your calculator, you must put brackets around the fraction (e.g., ^(1/12)).

If you type ^1/12 without brackets, your calculator will raise the number to the power of 1, and then divide the whole answer by 12, resulting in zero marks.

Worked examples

Example 1

Example 1: Scaling Up (Monthly to Annual)

A credit card company charges an effective rate of interest of 1.4% per month. Calculate the equivalent effective annual rate of interest.

Give your answer as a percentage to two decimal places.

Find the monthly multiplier: $100\% + 1.4\% = 101.4\% \rightarrow 1.014$ .
Scale up to a year: There are 12 months in a year, so we raise it to the power of 12.
$1.014^{12} = 1.18155...$
Convert back to a percentage:
$1.18155... - 1 = 0.18155...$
$0.18155... \times 100 = 18.16\%$ .

Example 2

Example 2: Scaling Down (Annual to Monthly)

A Help-to-Buy ISA offers an effective rate of interest of 5.2% per year. Calculate the equivalent effective monthly rate of interest.

Give your answer as a percentage to two decimal places.

Find the annual multiplier: $100\% + 5.2\% = 105.2\% \rightarrow 1.052$ .
Scale down to a month: We want 1 month out of the 12 in a year, so we use the power of $\frac{1}{12}$ .
$1.052^{\left(\frac{1}{12}\right)} = 1.00423...$
Convert back to a percentage:
$1.00423... - 1 = 0.00423...$
$0.00423... \times 100 = 0.42\%$ .

Example 3

Example 3: Extreme Scaling (Payday Loans)

Fiona is considering using a short-term payday loan company to borrow £400. The company advertises a seemingly small interest rate of just 0.9% per day.

(a) Calculate the equivalent effective annual rate of interest Fiona faces.
(b) If Fiona makes no repayments, calculate the total amount she will owe after exactly one year (365 days).

(a)

Daily Multiplier: $100\% + 0.9\% = 100.9\% \rightarrow 1.009$ .
Scale up to a year (365 days): $1.009^{365} = 26.402...$
Convert to percentage: $(26.402... - 1) \times 100 = 2540.24\%$
(Payday loans scale up to thousands of percent!)

(b)

$\text{Final Balance} = \text{Capital} \times \text{Annual Multiplier}$

$\text{Final Balance} = \text{\pounds}400 \times 26.402... = \text{\pounds}10,560.91$ .

Example 4

Example 4: Intermediate Conversions (Half-Year to Month)

A premium savings bond offers an effective rate of interest of 2.75% per half-year.

Calculate the equivalent effective monthly rate of interest.

Find the half-year multiplier: $100\% + 2.75\% = 102.75\% \rightarrow 1.0275$ .
Scale down to a month: There are 6 months in a half-year. To find the rate for just 1 month, we raise the multiplier to the power of $\frac{1}{6}$ .
$1.0275^{\left(\frac{1}{6}\right)} = 1.00453...$
Convert back to a percentage:
$1.00453... - 1 = 0.00453...$
$0.00453... \times 100 = 0.45\%$ .

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4 Converting Time Frequencies questions with answers and video solutions