A banker forecasts that interest rates will rise over a $5-$year period with a force of interest function given by ${}_t=0\mathrm{.}04+\frac{0\mathrm{.}2t}{t+1}$ for $0t5.$

$($a$)$ According to this scheme, what is the average annual compound effective rate for the $5-$year period?

$($b$)$ What are the equivalent annual effective rates for each of years $1,2,3,4,5?$

$($c$)$ What is the present value at $t=2$ of $1000$ due at $t=4?$