Find the effective rate of an account that earns 5 compound

Find the effective rate of an account that earns 5 % compounded:
Round your answer to the nearest hundredth of a percent.

semi-annually:

quarterly:

monthly:

daily:

continuously:

Solution

Let us suppose the principle ammount is X. Now if 5% compounded semi annually then interest rate per 6 months

i = 5% / 6

= 0.83%, Number of interest period per year is 2

So for a year at 5% compounded semiannually the ammount of interest you earn is:-

i_a = effective annual interest rate per year.

r = Nominal interest rate per year

M = Number of interest period per year

So we have i_a = (1 + r/M)^M - 1

Put the values i_a = (1 + 5/2*100)² - 1 = (1+0.025)² - 1 = (1.025)² - 1 = 1.050625 - 1 = 0.050625 = 5.06% compounded halfyearly

Now if compounde quarterly the M = 4 then the formulae becomes

i_a = (1 + 5/2*100)⁴ - 1 = (1+0.025)⁴ - 1 = (1.025)⁴ - 1 = 1.103812890625 - 1 = 0.103812890625 = 10.38% compounded quarterly

Now if compounde monthly the M = 12 then the formulae becomes

i_a = (1 + 5/2*100)¹² - 1 = (1+0.025)¹² - 1 = (1.025)¹² - 1 = 1.3448888242462984371781349182129 - 1

= 0.34488882424629843717813491821289 = 34.48% compounded monthly

Now if compounde daily the M = 365 then the formulae becomes

i_a = (1 + 5/2*100)³⁶⁵ - 1 = (1+0.025)³⁶⁵ - 1 = (1.025)³⁶⁵ - 1 = 8206.499557696606902207435986129 = 820649.95% compounded daily