对于题目中的“show your calculation"应该怎么简单写过程呢？

问题如下：

Russell Hood is a portfolio manager at Bullseye Asset Management. He oversees two funds: a balanced fund and a tactical equity fund. The balanced fund has a current market value of USD 700 million and a current allocation of 65% equity (equity beta of 1.12) and 35% fixed income (modified duration of 6.55).

Hood believes that reported earnings in the upcoming quarter will be better than anticipated by the market. Based on this short-term market view, he decides to use futures contracts to adjust the allocation of the balanced fund to 80% equity and 20% fixed income for the next three months. Hood wants to maintain the balanced fund’s current equity beta and modified duration.

Hood plans to use the Aries equity index futures contract and the Taurus bond futures contract to execute his transaction. He gathers the data shown in Exhibit 1 for the two contracts. Both contracts expire three months from today. The risk-free rate is 1.85%.

Exhibit 1 Selected Futures Contract Data

A. Determine, to achieve Hood’s adjusted allocation, the number of:

i. Taurus contracts he should sell.

ii. Aries contracts he should buy.

Show your calculations.

选项：

解释：

Correct Answer:

Hood needs to:

sell 971 Taurus contracts and ii. buy 1,283 Aries contracts.

Hood wants to shift 15 percentage points of his USD 700 million portfolio, or USD 105 million, from fixed income to equity. Therefore, he effectively needs to sell USD 105 million of bonds by converting them to cash using bond futures and buy USD 105 million of stocks using equity index futures. This would effectively convert the bonds into cash and then convert that cash into equity.

i. To reduce the fixed-income allocation to 20% from 35%, the number of Taurus futures Hood needs to sell is:

N_bf = [(MDUR_T – MDUR_B)/MDUR_f] x (B/f_b)

N_bf = number of bond futures contracts

MDUR_T = target modified duration

MDUR_B = modified duration of existing position

MDUR_f = implied modified duration of futures

B = market value of portfolio to be reallocated

f_b = bond futures price

MDUR_T is zero in this case, as Hood is effectively converting USD 105 million of the fixed income portfolio into synthetic cash rather than actual cash. Also, the Taurus contract has a yield beta of 1.00, which indicates that its sensitivity to interest rate changes is identical to that of the bonds. Therefore:

N_bf = [(0.00 – 6.55)/7.15] x (105,000,000/99,100) = –970.62 contracts, or sell 971 contracts

ii. To increase the equity allocation to 80% from 65%, Hood needs to use that synthetic cash to buy Aries futures as follows:

N_sf = [(B_T – B_S)/B_f] x (S/f_S)

N_sf = number of equity futures contracts

B_T = target beta

B_S = beta of existing position

B_f = futures beta

S = market value of portfolio to be reallocated

f_S = equity index futures price

In this case, BT is 1.12 and BS is zero, as Hood needs to take the existing synthetic cash position generated above (beta equal to zero) and effectively convert it to an equity position that will match the beta of the current equity portfolio. Therefore:

N_sf = [(1.12 – 0)/0.97] x (105,000,000/94,505) = +1,282.86 contracts, or buy 1,283 contracts