Final answer:
To find the equivalent annual interest rate with continuous compounding for a 6% dividend added every six months, one must solve the continuous compounding formula for the unknown annual rate. The calculation involves taking the natural logarithm of 1.06 and doubling the result to find the annual rate.
Step-by-step explanation:
To calculate the equivalent annual interest rate with continuous compounding, when a 6% dividend is added every six months, we can use the formula for continuous compounding interest:
A = Pert
Where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate, t is the time the money is invested for, and e is the base of the natural logarithm.
Since the dividend is 6% every six months, the semi-annual rate is 3%. To find the equivalent annual rate, we can set up the equation with 'r' being the unknown annual rate we want to find:
1.06 = er(1/2)
Solving for r gives us the equivalent annual interest rate with continuous compounding. Remember, to solve for r, you would take the natural logarithm of both sides of the equation:
ln(1.06) = (r/2) ln(e)
Since ln(e) equals 1, we simply solve for r to find the equivalent annual interest rate.
The exact calculation is:
r = 2 * ln(1.06)
Continuous compounding can make a significant difference in investments over time as compared to simple or periodic compounding.