· Client B’s portfolio holds $40 million of U.S. large-cap value stocks with a portfolio beta of 1.06. This client wants to shift $22 million from value to growth stocks with a target beta of 1.21. Allison will implement this shift using S&P/Barra Growth and S&P/Barra Value futures contracts.

Price of December S&P/Barra Growth futures contract

$117,475

Price of December S&P/Barra Value futures contract

$120,875

Beta of S&P/Barra Growth futures contract

1.15

Beta of S&P/Barra Value futures contract

1.03

When implementing the shift from value to growth stocks for Client B, the number of S&P/Barra Value future contracts Allison shorts will be closest to:

I know how to get the answer for the question, however are we expected to know how to complete the client’s objective by converting to growth and getting the overall portfolio beta to 1.21 while still having $18mm allocated to value and $22mm allocated to growth?

If so, I’m not able to figure out how to do this mathematically.

It seems that the target beta of 1.21 is only for growth stocks, not the overall position. You may after compute an overall beta approximated by a proper composite (22/40% weight in growth stocks and 18/40% weight in value stock) to find the beta of the overall position. More precisely, to estimate the overall beta:

Choose a period for the beta estimation (1 year for instance)

Choose a frequency of rebalacing (let us assume daily to simplify)

Compute 1year returns for value and growth stocks

Compute an overall return (daily) = 22/40 * returns(growth daily)+18/40*returns(value daily) => then you have your composite, or “benchmark” with your target weights (strategic weight)

Use the global portfolio and regress it against the benchmark (assume one factor model). If needed, simulate your portfolio global returns