Simple, Compound, APR, APY

Simple vs Compound Interest

Simple interest takes into account the principal, interest rate and period, but ignores the effect of accumulated interest.
Compound interest accounts for accumulated interest: each period interest is generated on both the principal and the interest generated thus far up till the previous period.

In each interest gathering period:

simple: principal stays the same
compound: principal grows

Example

Principal = $100
annual interest rate = 5%
period = 5 years

Compounding generates greater interest versus simple, for the same parameters.

APR & APY

APR

APR represents the cost of borrowing or the interest rate on a loan without considering compounding.
Uses simple interest in its calculation, ignoring any compounding effects.
APR = Periodic Interest Rate * Number of Periods in Year
APR will always be equal to or lesser than APY on a like-for-like basis in terms of interest rates.

APR is typically used to compare different loan options or credit products. By using APR in their advertising, banks can make their loan products appear more attractive to potential borrowers because the APR does not reflect the impact of compounding over time.

APY

APY reflects the actual annualized return or cost on an investment or debt position, taking into account the compounding effect.
The value of the position grows faster because every time the interest is calculated on the new balance, which includes the previous balance and the previous interest earned.

When comparing interest rates on a like-for-like basis, if both APR and APY are provided, the APY will always be equal to or higher than the APR due to the inclusion of compounding.
However, banks typically advertise loan products using APR, as it makes the interest rate appear lower and more enticing to borrowers.

Why have both APY and APY?

convenient depending on context to use one vs the other

Converting APR to APY

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Simple vs Compound Interest

Simple interest takes into account the principal, interest rate and period, but ignores the effect of accumulated interest.

Compound interest accounts for accumulated interest: each period interest is generated on both the principal and the interest generated thus far up till the previous period.

In each interest gathering period:

simple: principal stays the same
compound: principal grows

Example

Principal = $100

annual interest rate = 5%

period = 5 years

Compounding generates greater interest versus simple, for the same parameters.

APR & APY

APR

APR represents the cost of borrowing or the interest rate on a loan without considering compounding.

Uses simple interest in its calculation, ignoring any compounding effects.

APR = Periodic Interest Rate * Number of Periods in Year

APR will always be equal to or lesser than APY on a like-for-like basis in terms of interest rates.

APY

APY = (1 + \frac{r}{n})^n - 1

, where r is the periodic interest rate, and n is the no. of periods.

APY reflects the actual annualized return or cost on an investment or debt position, taking into account the compounding effect.

The value of the position grows faster because every time the interest is calculated on the new balance, which includes the previous balance and the previous interest earned.

When comparing interest rates on a like-for-like basis, if both APR and APY are provided, the APY will always be equal to or higher than the APR due to the inclusion of compounding.
However, banks typically advertise loan products using APR, as it makes the interest rate appear lower and more enticing to borrowers.

Why have both APY and APY?

convenient depending on context to use one vs the other

Converting APR to APY