What is the exact formula for this? Could someone also explain this formula to me in detail?
Suppose a nominal rate of 8% that capitalizes monthly.
So the effective rate must be higher than 8%. Find it using the formula:
Effective rate = (1 + 0.08/12) ^ (12) - 1 = 8.30%
Now imagine that nominal rate compounds daily, not monthly anymore. So,
Effective rate = (1 + 0.08/365) ^ (365) - 1 = 8.33%
Now imagine that nominal rate compounds every hour. There are 8760 hours in a year. So,
Effective rate = (1 + 0.08/8760) ^ (8760) - 1 = 8.33%
Another way to afford this is using the constant " e " :
Effective rate = e ^ (0.08) = 8.33%. Using this, we are assuming continious compounding (every milisecond)
*Note that all rates calculated are annual effective rates.
Hope this helps.
Sorry, forgot the -1
Effective rate = e^(0.08) -1
I believe that you’re conflating two ideas here.
The effective annual rate (EAR) has annual compounding, not continuous compounding.
If you have an EAR of 6% and you deposit $100, after one year you’ll have $106 (= $100 × 1.06); after 2 years you’ll have $112.36 (= $100 × 1.06²).
If you have a continuously compounded rate of 6% and you deposit $100, after 1 year you’ll have $106.18 (= $100 × e^0.06); after 2 years you’ll have $112.75 (= $100 × e^(2 × 0.06)).
The relationship between the continuous rate and the EAR is:
EAR = e^_r_cont − 1
_r_cont = ln(1 + EAR)
Ok, so is effective annual yield the same thing as the effective annual rate?
Yes.