# Effective Annual Rate with continuous componding

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.