问一道题：NO.PZ2018113001000010

问题如下：

The portfolio has a market value of $100 million. The initial stock allocation is 60% and bond allocation is 40%. The initial beta is 1.10, and the modified duration is 5.5.

A portfolio manager has two objectives:

1. Altering stock and bond allocation to 50% and 50% for two months.
2. Altering beta to 0.95 and duration to 5.0 for two months.

A stock index futures contract that expires in two months is priced at $150,000 and has a beta of 1.05. A bond futures contract that expires in two months is priced at $125,000 and has a duration of 5.75.

In order to achieve two objectives, the manager shoud

选项：

sell 48 stock futures contracts and sell 35 bond futures contracts.

sell 70 stock futures contracts and buy 77 bond futures contracts.

sell 118 stock futures contracts and buy 42 bond futures contracts.

解释：

C is correct.

考点：调整头寸及beta & duration

解析：

现在股票头寸=0.6(100,000,000)= $60,000,000，债券头寸=0.4(100,000,000)=$40,000,000

目标股票头寸=0.5(100,000,000)= $50,000,000,债券头寸=0.5(100,000,000)= $50,000,000

为了达到目标，需要减少10,000,000股票，增加10,000,000债券。我们首先将股票和债券的头寸调整至目标头寸，再调整它们的beta和duration。

1.调整头寸

需要的股票期货合约(先转成cash)：

$N_{\text{s}f}={(\frac{0.0-1.10}{1.05})}{(\frac{10,000,000}{150,000})}=-69.84$

因此需要卖出70份股票期货合约。

需要的债券期货合约（cash转成债券）：

$N_{\text{b}f}={(\frac{5.5-0}{5.75})}{(\frac{10,000,000}{125,000})}=76.52$

因此需要买入77份债券期货合约。

2.调整beta和duration

需要的股票期货合约= ${(\frac{0.95-1.10}{1.05})}{(\frac{50,000,000}{150,000})}=-47.62$，即卖出48份股票期货合约。

需要的债券期货合约= ${(\frac{5.0-5.5}{5.75})}{(\frac{50,000,000}{125,000})}=-34.78$ ，即卖出35份债券期货合约。

综上，为了达到两个目标，需要卖出70+48=118份股票期货合约，买入77-35=42份债券期货合约。

如果大家是先调整beta和duration，再调整头寸也是一样的，只不过可能因为四舍五入的关系，最后导致合约数量有一点偏差，但实际上都是可行的。