NO.PZ2022123002000055
问题如下：
Ari Patheo, a portfolio manager for Astute Investments, manages a
280 million U.S. dollar (USD) investment portfolio. Astute’s investment
committee has recently become more risk averse in anticipation of a major
announcement regarding monetary policy. To reflect this view, Patheo wishes to
temporarily make the following changes in the portfolio:
decrease the portfolio’s equity
allocation and decrease its equity beta;
increase the portfolio’s bond allocation and decrease its modified
duration.
The portfolio’s
current and target characteristics are shown in Exhibit 1.
Exhibit 1 Investment Portfolio
Characteristics
Patheo does not want to incur
high trading costs for a temporary reallocation and decides to use the
following futures contracts to achieve the portfolio targets.
equity futures - currently
priced at USD 129,000 per contract (after accounting for the multiplier), with
an equity beta of 0.97;
bond futures - currently priced at USD 103,000 per contract, with a
modified duration of 7.70 and a yield beta of 1.00.
Determine the
action (buy or sell) and the number of futures contracts required to achieve
the:
i. equity targets.
ii. bond targets.
Show
your
calculations.
选项：
解释：
Correct Answer:
i. Equity targets
Patheo effectively
needs to sell $28 million of stock by converting it to cash using stock index
futures and buy $28 million of bonds by using bond futures. This would
effectively convert the stock into cash and then convert that cash into bonds.
Of course, this entire series of transactions will be synthetic; the actual
stock and bonds in the portfolio will stay in place.
In order to
achieve the equity targets, Patheo must determine the number of equity futures
necessary to:
1. Reduce the equity allocation by $28 million and
2. Decrease the equity beta to 0.90
In both cases
Patheo will rely on the following relationship: Nfe = [(βT – βP)
/ (βf)] × (E / fe)
To Reduce the
Equity Allocation by $28 million:
Patheo wants to
reduce equities by USD 28,000,000, so the target beta is the beta of cash,
which is assumed to be zero. The portfolio’s current beta is 1.08 and the
futures’ beta is 0.97.
Therefore, Nfe = [(0 – 1.08) / (0.97)] × (28,000,000 /
129,000) = –241.67.
Patheo should sell 242 equity
futures contracts.
To Decrease the
Equity Beta to 0.90:
Next, Patheo needs
to decrease the equity beta from 1.08 to 0.90 on what is now a USD 154,000,000
equity portfolio.
Therefore Nfe = [(0.90 – 1.08) / (0.97)] × (154,000,000 /
129,000) = –221.53
Patheo should sell 222 equity
futures contracts.
To achieve the
equity targets, Patheo should sell 242 + 222 = 464 equity futures
contracts.
ii. Bond targets
In order to
achieve the bond targets, Patheo must determine the number of bond futures
necessary to:
1. Increase the bond allocation by $28 million and
2. Decrease the modified duration to 6.0
In both cases
Patheo will rely on the following relationship:
Nfb
= [(MDURT – MDURP) / MDURf] × (B/fb)
To Increase the
Bond Allocation by $28 Million:
Patheo wants to
increase bond exposure by USD 28,000,000. The starting position for this is the
synthetic cash which has been raised by the sale of equity futures, so the
modified duration of this component is zero.
Therefore Nfb = [(7.20 – 0.00) / 7.70] × (28,000,000 /
103,000) = 254.19
Patheo should buy 254 bond futures
contracts.
To Decrease the
Modified Duration to 6.0:
Next, Patheo needs
to change the modified duration from 7.20 to 6.00 on what is now a USD
126,000,000 bond portfolio.
Nfb = [(6.00 – 7.20) / 7.70] × (126,000,000 / 103,000) =
–190.64
Patheo should sell 191 bond futures
contracts.
To achieve the
bond targets, Patheo should buy 254 – 191 = 63 bond futures contracts.
如果不经过把equity换成cash，可以根据目标beta一步到位吗？计算出需要多少份Nf
{(0.9-1.08)/0.97}/(28/0.129)
