Hello,I recently tried to backtest the performance of a beta hedged portfolio, consisting of a given asset [$]A[$] and a proxy for the market portfolio [$]M[$].Usual approach:Let the risk premium of asset A be defined by a linear regression on the market risk premium: [$]r(A)-r_f = \alpha + \beta*(r(M)-r_f)+\epsilon [$].Then, for a given number of shares [$]N[$] of asset [$]A[$], the Hedging Factor is defined as [$]HF_0 = N*A_0/M_0*\beta[$]. The value of the corresponding 'beta-neutral portfolio' at time [$]t[$] is [$]P_t = N*A_t-HF_0*M_t[$].I want to check whether the returns of the resulting Portfolio, [$]r(P)[$], are in fact uncorrelated with [$]r(M)[$], the market returns. [$]P[$] takes positive and negative values over time. My problem is: How do I compute the return of an 'asset' [$]P[$] which takes negative values? Defining [$]r(P)_t = ln(P_t/P_{t-1})[$] obviously does not work if [$]sgn(P_t) \neq sgn(P_{t-1})[$].My idea:I could simply calculate the correlation coefficient of [$]\Delta P = P_t-P_{t-1}[$] and [$]\Delta M = M_t-M_{t-1}[$] and compare it to the correlation coefficient of [$]\Delta A[$] and [$]\Delta M[$].I would really appreciate any other suggestions.

Are you rebalancing the long and short positions or is this a static hedge ? Why not run a Durbin Watson test on the residuals ? Don't you just want to see if the alpha is signifcant ?

Thank you for your quick reply!Measuring the significance of alpha is of course a very interesting thing to do, but not my focus at the moment.I am considering both a dynamic and static hedge. My primary goal is to assess whether beta hedging does in fact eliminate the market risk of a given asset. I want to do this for multiple estimation horizons and rebalancing frequencies.If the beta hedge works, the (beta-neutral) portfolio and market returns should be uncorrelated. To check whether this is the case, I need the portfolio returns. My question is: How do I measure them? Or: Is there another approach to evaluating the performance of market risk elimination?

you should work directly with the returns of the long and short portfolio.the portfolio return is given by [$] r_p = \sum_i w_i r_i [$]. Question is what are the weights [$]w_i[$] ? If your portfolio is initially cash neutral then they are 1 and -1 respectively. Otherwise it will be the relative values of the long and short portfolios relative to the net invested amount.

I have trouble turning your last sentence into math. Can you give me the weights for the following simple example:[$]Asset_1(t_1)[$]: 498[$]Asset_2(t_1)[$]: -502[$]P(t_1) = Asset_1(t_1)+Asset_2(t_1) = -4[$]Thanks!

QuoteAsset1(t1): 498 Asset2(t1): -502OK, but how much Cash do you have ? [or, equivalently, what is the Net Invested Amount?NetInvested Amount = Asset1 + Asset2 + Cash]

QuoteOriginally posted by: acastaldoOK, but how much Cash do you have ? It seems the least arbitrary to start with zero cash. If the investment is set up at [$]t_1[$] (and not changed over time) and interest rates are zero, I have 4$ cash all the time.

QuoteOriginally posted by: JustusQuantQuoteOriginally posted by: acastaldoOK, but how much Cash do you have ? I am assuming to start with zero cash. If the investment is set up at [$]t_1[$] (and not changed over time) and interest rates are zero, I have 4$ cash all the time.you can't have $4 of cash all the time - your capital is growing or shrinking depending on the the returns from your l/s position.the portfolio [$] \pi[$] at any time is [$]\pi = \sum_i n_i S_i + C [$] the change in the portfolio is given by[$]\delta\pi = \sum_i n_i\delta S_i + \delta C[$][$]\delta\pi = \sum_i n_i\delta S_i + r\delta t C [$][$]\frac{\delta\pi}{\pi}= \sum_i \frac{n_i \delta S_i }{\pi}+\frac{r\delta t C}{\pi}[$][$]\frac{\delta\pi}{\pi}= \sum_i w_i r_i +\frac{r\delta t C}{\pi}[$]where [$]w_i = \frac{n_i S_i}{\pi}[$] and [$]r_i = \frac{\delta S_i}{\S_i}[$]So the amount of capital you have effectively controls the volatility of the portfolio.

QuoteOriginally posted by: daveangelyou can't have $4 of cash all the time - your capital is growing or shrinking depending on the the returns from your l/s position.If the P&L of the portfolio positions is realized every day you are right. If, as I assumed, interest rates are zero and I do not change the portfolio over time, [$]\Delta C = 0[$] and the P&L of the individual positions affects [$]\pi[$], but not [$]C[$]. [$]C[$] is in this case determined by the amount of cash (not) required to initially set up the portfolio.If [$]\pi[$] can take negative values, I think it should be replaced by [$]|\pi|[$] in the definition of [$]\frac{\Delta\pi}{\pi}[$]. Otherwise the portfolio return has the wrong sign.In this context I am asking myself the following: is it intuitively plausible to interpret a $2 gain on a $5 short portfolio and a $2 gain on a $5 long portfolio equivalently in terms of returns?Given my assumptions, [$]\pi_t[$] is the cumulated profit at time [$]t[$]. A $5 short portfolio therefore means that, overall, the portfolio value decreased by $5.

QuoteOriginally posted by: JustusQuantQuoteOriginally posted by: daveangelyou can't have $4 of cash all the time - your capital is growing or shrinking depending on the the returns from your l/s position.If the P&L of the portfolio positions is realized every day you are right. If, as I assumed, interest rates are zero and I do not change the portfolio over time, [$]\Delta C = 0[$] and the P&L of the individual positions affects [$]\pi[$], but not [$]C[$]. [$]C[$] is in this case determined by the amount of cash (not) required to initially set up the portfolio.If [$]\pi[$] can take negative values, I think it should be replaced by [$]|\pi|[$] in the definition of [$]\frac{\Delta\pi}{\pi}[$]. Otherwise the portfolio return has the wrong sign.In this context I am asking myself the following: is it intuitively plausible to interpret a $2 gain on a $5 short portfolio and a $2 gain on a $5 long portfolio equivalently in terms of returns?Given my assumptions, [$]\pi_t[$] is the cumulated profit at time [$]t[$]. A $5 short portfolio therefore means that, overall, the portfolio value decreased by $5.the portfolio made a loss .... its not the value of the portfolio but the change in the value of the portfolioyou need capital to hold a long short position. The amount of capital you hold will determine the volatility of the rates of return of your capital.

Quoteyou need capital to hold a long short position.Or if you have a Prime Broker who will let me hold $498 of stock long and short $502 of stock without my putting up any capital please let me know the name and phone number.

QuoteOriginally posted by: acastaldoQuoteyou need capital to hold a long short position.Or if you have a Prime Broker who will let me hold $498 of stock long and short $502 of stock without my putting up any capital please let me know the name and phone number.You might be pleased to hear that my special broker does not charge transaction fees either.QuoteOriginally posted by: daveangelthe portfolio made a loss .... its not the value of the portfolio but the change in the value of the portfolioyou need capital to hold a long short position. The amount of capital you hold will determine the volatility of the rates of return of your capital.I understand that, for given long and short positions, volatility decreases as my capital increases. But shouldn't the correlation of the l/s portfolio returns and the market returns be independent of my required capital? If it is, it seems okay to abstain from certainly more realistic (but arbitrary) assumptions about capital requirements in my analysis.

QuoteOriginally posted by: JustusQuantQuoteOriginally posted by: acastaldoQuoteyou need capital to hold a long short position.Or if you have a Prime Broker who will let me hold $498 of stock long and short $502 of stock without my putting up any capital please let me know the name and phone number.You might be pleased to hear that my special broker does not charge transaction fees either.QuoteOriginally posted by: daveangelthe portfolio made a loss .... its not the value of the portfolio but the change in the value of the portfolioyou need capital to hold a long short position. The amount of capital you hold will determine the volatility of the rates of return of your capital.I understand that, for given long and short positions, volatility decreases as my capital increases. But shouldn't the correlation of the l/s portfolio returns and the market returns be independent of my required capital? If it is, it seems okay to abstain from certainly more realistic (but arbitrary) assumptions about capital requirements in my analysis.It sounds like you are describing a 'zero-investment' portfolio, which yes, as I understand it is a valid tool to asses raw return. So for example if I want to asses the returns of a small cap strategy I would construct the zero investment portfolio as that which is established by my going short 1$ of large cap names and taking my 1$ realised from this short sale and investing it in small cap names. The resultant value of these positions netted out is then considered the return due to this factor.Of course this assumes zero risk free return and zero transaction costs or capital requirements (in the same way as a physicist assumes a perfect vacuum, this is the 'perfect broker').You can also use this to include such costs however by modelling the whole thing in terms of ongoing cash flows. For example I imagine borrowing my initial capital from the bank (with attendant cost) and using this as collateral for my 1$ short sale and covering the first periods cost, I take the <1$ proceeds from the short sale (net of the fee) and buy the <1$ long position (again where the actualy amount of stock I buy is less than 1$ since I have payed transaction costs). At this point if I just hold the portfolio then the return is the net of my <1$(initial) short, my <1$(initial) long, and interest/fees to the bank and share-lender. If fees and interest rates are constant then they should not alter the correlation of the returns with the market (from the point of view of the zero investment portfolio), but you cant separate the effect of fees/liquidity and interest rate variations from the 'real market' data (which you are obliged to use) so this could be a source model bias.