You have one payment due in three years in the amount of PLN $20000.$ The interest rate on the market is $8\%$ per annum $($regardless of the length of the investment$).$ Also available on the market are coupon bonds with a time to maturity of exactly two years with a nominal value of PLN $1000$ and an annual coupon of $4\%,$ and zero coupon $5-$year bonds with a nominal value of PLN $10000.$ The yield to maturity on both of these bonds is also $8\%.$

a$)$ Calculate the present value of this liability.

b$)$ What is the durations of this liability $($with one payment$)?$

c$)$ Calculate the current prices of both bonds $($for the coupon bond, assume we are just after the coupon payment$).$

d$)$ Calculate the durations of these bonds.

e$)$ Assume you are constructing a portfolio from both of these bonds.

Remembering that the duration of this portfolio is equal to the weighted average of the value of these bonds in the portfolio, calculate what the share of each bond in portfolio should be if the duration of this portfolio must equal to the duration of the liability. Remember that the shares add up to $100\%.$ Calculate the share of both types of bonds in this portfolio.

f$)$ Knowing that the number of bonds in this portfolio will equal the present value of the liability multiplied by the share of the bond in the portfolio and divided by the current bond price, calculate the number of each bond type in your portfolio.

If you have done tasks a$)-$f$)$ then you have applied some of the bond investment strategy discussed in the lecture. Provide the name this strategy and briefly write what roles of the conditions in $(e)$ and $(f)$ are.