We have information from the yield curve of government bonds as follows:

Interest rate term

$18\%$

$29\%$

$39\mathrm{.}5\%$

$510\%$

$1012\%$

XYZ Bank invests $200$ million USD in a government bond portfolio with the following available bond types:

Bond $1$: coupon bond with remaining term of $1$ year, coupon rate $8\%,$ annual interest payment, face value F $=100,000$ USD

Bond $2$: discounted bond with remaining term of $2$ years, par value F$=100,000$ USD

Bond $3$: Coupon bond with remaining term of $5$ years, coupon interest rate is $9\%,$ annual interest payment, face value F$=100,000$ USD

a$)$ Please calculate the Maculay Duration of the above bonds

b$)$ Suppose bond $3$ has a convexity of $45.$ If market interest rates increase by $200$ basis points, how much will the price of this bond increase or decrease?

c$)$ Assuming the bank's expected investment term is $2\mathrm{.}5$ years to neutralize risk, how should the bank allocate to each type of bond? Know that you must invest in bonds an amount of $40$ million USD.