a$)$ An investment of $$4000$ is made into a fund that pays interest under the i$.$i$.$d$.$ varying interest rate model with a log$-$normal distribution. The annual rates have mean $(j=8\%)$ and standard deviation $(s=12\%.)$

Calculate the expected value of the accumulation after $12$ years.

Calculate the probability that the accumulated value after $12$ years will be less than $95\%$ of the expected value from $(1).($Hint: first calculate $,)$

b$)$ A man purchased a bond with $$1000$ face value having $10\%$ coupon rate. he gets $$1100$ after $4$ years from the bond. If the interest rate is $12\%,$ then find out the present value of the bond.