Find the effective rate corresponding to the given nominal rate. (Round your answer to the nearest hundredth of a percentage point.)
(a) 7%/year, compounded daily
%
(b) 6%/year, compounded monthly
%
(1+r)^365 =(1.07)
1+r = 1.07^1/365
1+r =1.00018538
r = 0.0018538
r = .185% daily
(1+r)^12 = 1.06
1+r = (1.06)^1/12
1+r = 1.004867551
r = 0.004867551
r = .486% monthly
I disagree with angela's answer
When they ask for the effective rate, they are asking for the annual rate which is equivalent to the given compounded rate . This effective rate is often called APR
7% compounded dails
i = .07/365 = .00019178
let the effective rate be r
1+r = 1.00019178)^365
1+r = 1.0725
r = .0725
the effective rate is 7.25%
for the 2nd ...
i = .06/12 = .005
1+r = 1.005^12 = 1.06168
r = .06168 or appr 6.168%
To find the effective rate, we can use the formula:
Effective rate = (1 + (nominal rate / number of compounding periods))^number of compounding periods - 1
(a) For a nominal rate of 7% compounded daily, the number of compounding periods is 365 since it is compounded daily.
Effective rate = (1 + (7%/365))^365 - 1
Calculating this gives us:
Effective rate ≈ (1 + 0.00019178)^365 - 1 ≈ 1.0725 - 1 ≈ 0.0725
Rounded to the nearest hundredth of a percentage point, the effective rate is approximately 7.25%.
(b) For a nominal rate of 6% compounded monthly, the number of compounding periods is 12 since it is compounded monthly.
Effective rate = (1 + (6%/12))^12 - 1
Calculating this gives us:
Effective rate ≈ (1 + 0.005)^12 - 1 ≈ 1.061677 - 1 ≈ 0.061677
Rounded to the nearest hundredth of a percentage point, the effective rate is approximately 6.17%.
To find the effective rate corresponding to a given nominal rate compounded daily or monthly, we can use the formula:
Effective Rate = (1 + Nominal Rate / n)^n - 1
Where,
- Nominal Rate is the given rate in decimal form,
- n is the number of compounding periods in a year (365 for daily compounding and 12 for monthly compounding).
Let's calculate the effective rates:
For (a) 7%/year, compounded daily:
Nominal Rate = 7% = 0.07
n = 365
Effective Rate = (1 + 0.07 / 365)^365 - 1
Effective Rate ≈ 0.0716 or 7.16%
Therefore, the effective rate corresponding to 7%/year compounded daily is approximately 7.16%.
For (b) 6%/year, compounded monthly:
Nominal Rate = 6% = 0.06
n = 12
Effective Rate = (1 + 0.06 / 12)^12 - 1
Effective Rate ≈ 0.0617 or 6.17%
Therefore, the effective rate corresponding to 6%/year compounded monthly is approximately 6.17%.