Consider a pension fund at $01/01/2023$ that requires a payout of $$100,000,000$ at the end on $31/12/2025.$ The fund would like to invest into a portfolio that delivers this payout for sure. The yield curve is flat at y $=10\%$ across maturities, so that the net present value of the liability is given by $$68,301,345\mathrm{.}54.$

The fund considers two bonds. Bond A is a one year pure discount bond, while Bond B is a $5$ year bond with coupons paid annually at the rate of $5\%.$ All payouts and investment decisions take place at the start of each year.

$(1)$ Construct a portfolio using the two bonds that immunizes the investment to market movements when funds need to be reinvested on $01/01/2024.$

$(2)$ Show that if any cash flow received at the end of $2023$ and $2024$ can be invested for two years at a yield of $8\%,$ the pension fund will receive $$100,000,000$ at the

on $31/12/2025.$