Suppose you have a position with a current market value of $50,000 in 30-year zero coupon bonds. You can use a 7-year zero to hedge. Compute the amount of the 7-year zero to sell to minimize risk:

(i) using the duration approximation.

(ii) by minimizing the portfolio VAR (note that this can be achieved by trial and error until the marginal VAR becomes zero, or using "solver").

(iii) Explain why the two numbers are different. [This must be explained in terms of the covariance matrix. It may help to plot the volatility of yield changes across maturities and to consider correlations. Discuss conditions under which the duration approximation is exact.]