CM1 Foundation · Theory of Interest Rates
Relate nominal and effective rates under p-thly compounding.
Recommended before this lesson — you can still read and practice below:
Real contracts rarely state interest as a single annual effective rate. Banks quote nominal rates convertible p-thly — for example, “8% per annum convertible quarterly”. That phrase does not mean the effective annual rate is 8%; it means 8% is split into equal compound periods per year.
This lesson connects the nominal rate to the effective annual rate you already use in .
If interest is credited times per year at nominal rate per annum, each sub-period rate is . Compounding over one full year gives
The effective annual rate is whatever single factor reproduces one year of p-thly compounding. The nominal rate is a quoting convention; is the rate that actually governs year-long accumulation.
| Convention | | Sub-period | |------------|-----|------------| | Annual | 1 | 1 year | | Semi-annual | 2 | 6 months | | Quarterly | 4 | 3 months | | Monthly | 12 | 1 month |
CM1 questions usually state explicitly. “Convertible half-yearly” means ; “convertible monthly” means .
Problem. A deposit earns per annum convertible quarterly. Find the effective annual rate (four decimal places as a rate).
Solution.
So p.a. effective — higher than the quoted 8% because interest compounds four times per year.
Given , the equivalent nominal rate convertible p-thly is
This is the rate that, when split into equal parts and compounded, reproduces over one year.
Problem. Effective annual rate is . Find (semi-annual nominal, percentage to two decimals).
Solution.
The semi-annual nominal is below 10% because each half-year sub-rate compounds twice to reach 10% effective.
You should be able to (1) translate “ convertible p-thly” into a sub-period rate, (2) convert between and in either direction, and (3) explain why more frequent compounding increases for a fixed nominal rate. Practice and mastery checks below test that.
If per annum convertible quarterly, the effective annual rate is:
per annum convertible half-yearly. Find to four decimal places.
convertible quarterly. Find as a percentage to two decimal places.
convertible monthly implies is:
Nominal rate divided by gives the rate per:
convertible quarterly. Find to four decimal places.
half-yearly. Sub-period rate as percentage (one decimal).
monthly nominal. Effective as percentage (two decimals).
Effective . Find as percentage (two decimals).
quarterly. Effective as percentage (two decimals).
Effective . Find as percentage (two decimals).
Pass each skill by answering correctly on 2 of 3 checks (fewer if fewer are available). Practice questions above are rehearsal — only mastery checks count toward progress. Retry wrong answers until you succeed.
0/2 verified
The formula means that is:
If , then is quoted:
Same nominal with larger gives effective that is:
Locked — complete prior skills first
Effective annual rate . Find (monthly nominal) as a percentage to two decimal places.
. Quarterly sub-period rate as percentage.
effective. as percentage (two decimals).
Locked — complete prior skills first
Holding fixed, increasing (more frequent compounding) makes the effective annual rate :
. Effective as percentage (two decimals).
Converting effective to nominal uses:
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