QUESTION 11 [Total = 5 marks]

Having studied Fixed Income Securities, you are now working as an analyst for a well known bond fund.Your manager asks you to replicate the JP Morgan T-Bond Index using a tracking error minimization approach.You are to replicate this index as closely as possible using a medium duration Treasury bond (M-BOND) and a long duration Treasury bond (L-BOND).These expire in approximately 5.6 years' and 29.1 years' time respectively.The following variance-covariance matrix, based on daily returns over the preceding six months, is given to you to use in your replication:

M-BOND

L-BOND

Aggregate-bond index

M-BOND

0.0037%

0.0052%

0.0032%

L-BOND

0.0052%

0.0114%

0.0057%

Aggregate-bond index

0.0032%

0.0057%

0.0036%

Note: As usual, variances are given on the diagonal, e.g. the variance of M-BOND is 0.0037%.As usual, covariances appear in the non-diagonal elements, e.g. the covariance of M-BOND and L-BOND is 0.0052%.

(a) (2 marks) It can be shown that the optimal weights for the replicating portfolio are 78.7% for M-BOND and 21.3% for L-BOND.Carefully explain what it means for this portfolio to be the optimal replicating portfolio.Given these optimal weights, provide an example of a replicating portfolio where the weights are not optimal.

(b) (1 mark) Explain what is meant by a rebalancing of a passively managed portfolio and explain when a rebalancing is required.

(c) (2 marks) You implement the tracking error minimization approach with a weight of 78.7% for M-BOND and 21.3% for L-BOND, for the replicating portfolio.You then observe that in the first month your replicating portfolio outperforms the index by 30 basis points.In the second month, your replicating portfolio underperforms the index by 25 basis points.In the third month, your replicating portfolio outperforms the index by 16 basis points.

Based on these observations, what is the out-of-sample tracking error for your portfolio?