What annually compounded rate per year will give the same value at the end of 10 years as 7% p.a. simple interest?

To solve this problem, we need to equate the future value of an investment under simple interest and compound interest and solve for the compound interest rate.

Step 1: Calculate the future value with simple interest The formula for simple interest is FV = P(1 + rt), where FV is the future value, P is the principal amount, r is the rate of interest, and t is the time in years.

Given that r = 7% or 0.07 and t = 10 years, and assuming P = 1 for simplicity, we can calculate the future value as follows:

FV = 1(1 + 0.07*10) = 1.7

Step 2: Equate the future value with compound interest and solve for the rate The formula for compound interest is FV = P(1 + r/n)^(nt), where n is the number of times interest is compounded per year.

Given that FV = 1.7, P = 1, n = 1 (since the interest is compounded annually), and t = 10, we can set up the following equation and solve for r:

1.7 = 1(1 + r/1)^(1*10)

Taking the 10th root of both sides gives:

1.7^(1/10) = 1 + r

Subtracting 1 from both sides gives:

r = 1.7^(1/10) - 1

Calculating the right side gives the annual compound interest rate that will give the same future value as 7% per year simple interest after 10 years.