In section 3.5 of Hull 8th edition, Hull describes in equation (3.5) that in general N*= β V_A/V_F Which, compared to equation (3.3), N* = h* V_A/V_F suggests that h* is equal to β. He then goes on to explain that this is not surprising as h* is the slope of the best-fit line when changes in the portfolio are regressed against changes in the futures price of the index and β is the slope of the best-fit line when the return from the portfolio is regressed against the return for the index. However, by definition of h* given in section 3.4: h* = ρ σ_S/σ_F where σ_S is defined to be the standard deviation of the ABSOLUTE change in the value of the stock price and similarly σ_F is the standard deviation of the absolute change in the value of the forward price. Considering specifically By definition of CAPM, β is defined as: β = ρ σ'_S/σ'_F where σ'_S is defined to be the standard deviation of the RETURNS of the stock price and similarly σ'_F is the standard deviation of the returns of the forward price (assuming that the standard deviation of returns on the forward price is the same as that for the underlying index for which we are hedging). Thus it follows that: σ'_S = Var(∆S/S)^(0.5) = 1/S * Var(∆S)^0.5 = 1/S * σ_S using σ_S = the standard deviation of ∆S (absolute change) as discussed above. Doing a similar breakdown for σ'_F it implies: β = ρ σ'_S/σ'_F = ρ σ_S/σ_F * F/S = h* F/S So that β and h* are not equal unless the forward price is equal to the current spot price. As far as I can see this also implies that, ignoring tailing the hedge, N* = h* Q_A/Q_F = β Q_A / Q_F * S / F = β V_A / V_F which makes sense to me. However if we then consider tailing the hedge for futures rather than for forwards: N* = h* V_A / V_F = β * V_A / V_F * S / F which is not equal to the β * V_A / V_F suggested in equation 3.5 by Hull, why is this the case? Can you explain why Hull claims h* = β? Thanks.
I haven't done the algebra, but maybe the apparent mismatch between absolute and relative units is accounted for through the values of rho, which will take different numerical values depending on whether you're looking at the correlation between the actual values (for h) or the percentage returns (for beta). You'd probably have to write things in terms of covariances to check this.
