Find the effective rate corresponding to the given nominal rate. (Round your answer to the nearest hundredth of a percentage point.)
(a) 4%/year, compounded quarterly
? %
(b) 3%/year, compounded monthly
? %
(1+r)^12 = 1.03
1+r = (1.03)^1/12
1+r = 1.00246627
r = 0.00246627
r % = .247 % monthly
To find the effective rate corresponding to a given nominal rate compounded periodically, 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 4% per year, compounded quarterly:
Number of Compounding Periods = 4 (quarterly)
Nominal Rate = 4%
Effective Rate = (1 + (4% / 4)) ^ 4 - 1
Now we can calculate the value:
Effective Rate = (1 + 0.01) ^ 4 - 1
Effective Rate = (1.01) ^ 4 - 1
Effective Rate = 1.0406 - 1
Effective Rate = 0.0406
Rounded to the nearest hundredth of a percentage point, the effective rate is approximately 4.06%.
(b) For a nominal rate of 3% per year, compounded monthly:
Number of Compounding Periods = 12 (monthly)
Nominal Rate = 3%
Effective Rate = (1 + (3% / 12)) ^ 12 - 1
Now we can calculate the value:
Effective Rate = (1 + 0.0025) ^ 12 - 1
Effective Rate = (1.0025) ^ 12 - 1
Effective Rate = 1.0304 - 1
Effective Rate = 0.0304
Rounded to the nearest hundredth of a percentage point, the effective rate is approximately 3.04%.
To find the effective rate corresponding to a given nominal rate compounded quarterly or monthly, 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 4% per year, compounded quarterly, we can substitute the values into the formula:
Effective Rate = (1 + (0.04 / 4))^(4) - 1
Calculating this, we get:
Effective Rate = (1 + 0.01)^(4) - 1
= (1.01)^(4) - 1
= 1.04060432 - 1
= 0.04060432
Rounding this to the nearest hundredth of a percentage point, the effective rate is approximately 0.04%.
(b) For a nominal rate of 3% per year, compounded monthly, we can use the same formula:
Effective Rate = (1 + (0.03 / 12))^(12) - 1
Calculating this, we get:
Effective Rate = (1 + 0.0025)^(12) - 1
= (1.0025)^(12) - 1
= 1.030396 - 1
= 0.030396
Rounding this to the nearest hundredth of a percentage point, the effective rate is approximately 0.03%.
let the effective rate in each case be i
a)
so (1+i)^1 = 1.-1^4
1+i = 1.040604
i = .040604 = 4.06%
f)
3% compounded monthly --- monthly rate of .0025
1+i = 1.0025^12 = 1.0304059..
rate = .0304 or 304%