Acf 504 - Definition of the Dataset

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) Q_H=S_2 Q_A+(F_1-F_2 )hQ_A

where Q_A is the size of position being hedged, and Q_H is the size of one forward contract. Since we want to find the hedge ratio,h=Q_H/Q_A , Q_H could be replaced as h*Q_A.

Because S_2 and F_1are the same, the variance of W_2 equals

Var[W_2 ]=Q_A^2 σ_S^2+h^2 Q_A^2 σ_F^2-2hQ_A^2 ρσ_S σ_F

where ρ is the correlation coefficient between S_2 and F_2 and σ_F^2 is the variance of F_2.

To achieve the objective of minimizing the variance of terminal wealth, Var[W_2 ], we need to find the hedge ratio. Therefore, we differentiate Var[W_2 ]:

(∂Var[W_1])/∂h=2hQ_A^2 σ_F^2-2Q_A^2 ρσ_S σ_F=0

The hedge ratio that minimizes Var[W_2 ] is

h^*=ρ×σ_S/σ_F

Because we assume that there are the forward contracts on the stock of TESCO, which means that this is a perfect hedge to fully eliminate all risk, the hedge ratio (h^*) would be equal to 1. In this case, when we invest £100,000 on the stock of TESCO, we should short the same amount of forward contracts position for one-year on the stock to minimize the risk. We assume that one contract is for 10 shares, so the optimal number of contracts is 5,804, which means that 5,804 forward contracts should be shorted to hedge the underlying position.

N^(**)=(h^*×V_A)/V_F =(1×10,000,000 pence)/(172.29 ×10)=5,804

where V_A (=S_1×Q_A) denotes the value of position being hedged and V_F (=F_1×Q_F) denotes the value of one forward contract.

Hedging as a portfolio choice

Because Minimum-variance hedge places all the weight on risk and ignores the expected return, in this part, we are going to hedge as a portfolio choice in order to optimize the trade-off between the expected return and risk. The optimal hedge ratio maximizes therefore

u=E[W_2 ]-αVar[W_2 ]=Q_A E[S_2 ]-hQ_A (F_1-E[F_2 ])-α[Q_A^2 σ_S^2+h^2 Q_A^2 σ_F^2-2Q_A^2 hσ_S σ_F ρ]

The optimal hedge ratio obtained by differentiating formula above equals:

h ̃=(2Q_A αρσ_S σ_F+F1-E[F2])/(2Q_A ασ_F^2 )=ρ σ_S/σ_F -(E[F2]-F1)/(2Q_A ασ_F^2 )=h^*-(E[F2]-F1)/(2Q_A ασ_F^2 )

As we calculated before, h^* is equals to 1 because the TESCO’s spot price on 30 November 2015 is same as discounted forward price for the 30 November 2016. As for future price at delivery date, E[F_2 ] is close to the stock price E[S_2 ].

E[F_2 ]=E[S_2 ]=S_1×(1+E(R_s ))=167.2×(1+9.43%)=182.96

Therefore, we get the optimal hedge ratio in this case:

h ̃=1-(182.96-172.29)/(2×59,809×0.000012×(25.26)^2 )=0.988

where α is calculated as 0.00001×(10+2) and Q_A is derived from 10,000,000/167.2.

Subsequently, the optimal number of the forward contracts is

N^(**)=(h ̃×V_A)/V_F =(0.988×10,000,000)/1722.9=5,735

In conclusion, we should hedge TESCO stocks by shorting 5,735 forward contracts to minimize risk and gain the highest return.