Find the nominal rate compunded semi anually that is equivalent to 5.2 compunded anually
(1+r/2)^2 = 1+.052
r = 0.05134 = 5.134%
(1 + r/2)^2 = 1 + .052
1 + r/2 = √1.052
To find the nominal rate compounded semi-annually that is equivalent to 5.2% compounded annually, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (initial investment)
r = the annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case, we want to find the nominal rate compounded semi-annually, so n = 2 (twice a year).
Let's assume the principal amount is $1, so P = 1.
We know that A = P(1 + r/n)^(nt), so we can rewrite the formula as:
A = 1(1 + r/2)^(2t)
Substituting the given values into the formula, we get:
1 + 0.052/2)^(2t) = 1.052
Next, we can solve for (1 + r/2)^(2t):
(1 + r/2)^(2t) = 1.052
Taking the square root of both sides:
1 + r/2 = √(1.052)
Subtracting 1 from both sides:
r/2 = √(1.052) - 1
Now, we can solve for r by multiplying both sides by 2:
r = 2(√(1.052) - 1)
Calculating the right-hand side of the equation, we get:
r ≈ 0.0408
Therefore, the nominal rate compounded semi-annually that is equivalent to 5.2% compounded annually is approximately 4.08%.