Tran's company is required to pay a liability of $79860$ three years from now. Using the strategy of immunization, Tran invests $($right now at time $0)15000$ in a bond which has an annual modified$-$ duration of $1\mathrm{.}80$ along with another investment $($right now at time $0)$ of $45000$ in a bond which has an

annual modified$-$duration of K $($where K $>0).$ Assuming an annual effective yield rate of $10\%,$ find K so that the portfolio is immunized. Give your

answer as a decimal rounded to two places $($i$.$e$.$ XXX$).$

A$\text{'}($i$)$

Note: Recall the definition of modified$-$duration or volatility: V $=$

Al$)$ Where Al$)$ is the present value. Remember that the negative sign is dropped when the value is written, so you will want to account

for this when doing your computations.