You are managing interest rate risk in an investment portfolio. Your liabilities consist of $250$ bonds of type A$.$ Each of these bonds has a face value of $$1,000,$ pays coupons semiannually, and has two years to maturity. The annual coupon rate is $4\%,$ and the YTM is $5\%$ APR. Your assets consist of cash and $180$ bonds of type B$.$ Each of these bonds has a face value of $$1,000,$ has $4$ years remaining until maturity, and pays coupons semiannually. The annual coupon rate is $5\%$ and the YTM is $7\%$ APR. The amount of cash in your portfolio is such that the current market value of assets is equal to the current market value of liabilities. $($a$)$ What is the Macaulay duration of bonds A and B$?$ What is the modified duration of bonds A and B$?($b$)$ What is the modified duration of your assets and the modified duration of your liabilities? $($c$)$ Using only modified duration, compute the approximate change in the value of your assets and the approximate change in the value of your liabilities $($in $\%$ and in dollars$)$ in response to a $200$ bps upward shift in interest rates $($YTMs$)$ along the entire yield curve. $($d$)$ Compute the actual change in the value of your assets and liabilities $($in dollars$)$ in response to a $200$ bps upward shift in interest rates $($YTMs$)$ along the entire yield curve. Is the decrease in the value of assets and the decrease in the value of liabilities $($in absolute terms$)$ smaller or larger than that predicted by the duration approximation in part $($c$)?$ Explain why. How much capital do you need to contribute after the shift in YTMs so that the value of assets is again equal to the value of liabilities?