An insurance company has liabilities of $$200$ million with duration of $15$ years. Assume the yields on all risk$-$free bonds are $5\%.$

a$.$ What position in $10-$year and $20-$year zero coupon bonds will fully immunize the liabilities of this insurance company?

b$.$ If yields on all bonds fall immediately to $4\%,$ what will be the actual change in the total value of the liabilities and the total value of the assets $($the portfolio of $10-$year and $20-$year zero coupon bonds$)$ determined in part a above