Matthew has two options to save money for his retirement. His two best options are 3.95% compounded monthly or 4% compounded annually.
A principal amount P at an annual interest rate of r, after N years increases to `P_f = P*(1+r)^N` ...(1).
As the period of compounding changes, the final amount changes. If the interest is compounded after n months, the interest rate to be used in formula (1) changes to r/n, and the number of terms N increases to n*N.
For an amount P at an interest rate of 3.95% compounded monthly, the rate of interest to be used in (1) is 3.95/12% and the number of terms is equal to 12*N. This gives the amount P after N years as
= `P*(1.003291)^(12*N) `
The same amount P at an interest rate of 4% compounded annually after N years increases to P*1.04^N.
It is seen that P*(1.040223)^N > P*1.04^N. This shows that even though the interest rate of 3.95% seems to be lower than 4%, as the compounding in the former is every month, a larger interest is earned in the same time period when money is invested at 3.95% compounded monthly.
Matthew should invest his money in the account offering 3.95% interest compounded every month.