Mr. and Mrs. Collins started saving for their retirement 25 years ago. They deposited an initial amount of $30,000. In addition, they deposited $500 at the end of each month into a balanced mutual fund. At the end of that 25-year period, their mutual fund was worth-$689,620.22. What was the effective annual return, compounded continuosly, that the Collins earned on their investment?. Any help will be appreciated, and thank you.
Since there is no direct solution for the interest rate, we therefore have to use interpolation and iteration to hone in on the rate:
1-If we guess an interest rate of 7%, then the FV of all their deposits amounts to:$576,798.39. This is obviously lower than what the fund is worth.
2-If we guess an interest rate of 8%, then the FV of all their deposits amounts to $695,718.48. This is a bit higher than what their fund is worth.
3-By using interpolation between these 4 values and the actual value of their fund, we find that the actual interest rate is:7.95356% compouded monthly, which comes to just about 8.25% effective annual rate. But since they want the rate as "compounded continuously", then we take the natural log of:1.0825. This comes to: ~7.927% comp. continuously.
