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There's two studies that test the same thing in different markets (i.e. they apply the identical methodology). They state:

1) "$R_{mt}$ is the equally weighted average stock return in the dual-listed portfolio."

2) "$R_{mt}$ is the average return of the equal-weighted market portfolio."

To find the equally weighted portfolio do I average the prices of each constituent and THEN take the log returns of this averaged price? Or, do I just average the log returns of each stock at each $t$ to get this $R_{mt}$? Pretty sure I average the returns based on what they've said but want to be 100\% sure before making all my results!

They both give very similar results.

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    $\begingroup$ Can you give references to these papers? $\endgroup$ – user1157 Feb 7 '14 at 13:42
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It looks like 1 and 2 are different portfolios of companies.

1 is a portfolio of dual-listed companies, and 
2 is a portfolio of everything in the "market". 

Once you have constructed these these portfolios, let's say you put the returns for every time step into a vector, call it r, then the average return would be mean(r).

You need some clarification as to what "equally-weighted portfolio" means in this case in order to construct your portfolios. For example, if you simply assume to buy the same # of shares of every stock, you may have a situation like this:

Assume the whole market consists of stocks A, B, and C.

stock            price
A                 10
B                 25
C                 50

if you buy 1 share of each stock, then your total portfolio will worth 85 dollars, with $50 (59%) being from stock C, $25 (29%) from stock B, and $10 (12%) from stock A. So you can see even if you bought the same % of shares, you do not have an equally weighted portfolio. Your portfolio is much more sensitive to fluctuations in stock C than it is to fluctuations in stock A. If stock A goes to zero, you only lose 12% of your portfolio, but if stock C goes to zero, you lose 59%.

As far as I can tell they do ask you to use log-returns. I don't think it is necessary to use log returns calculate average portfolio returns. If you do use log returns, remember there is a difference between log returns and arithmetic returns: http://en.wikipedia.org/wiki/Rate_of_return#Arithmetic_and_logarithmic_return.

Easiest way to find the returns of the equally weighted portfolio would be to adjust your prices so that start price of each asset is equal to 1. Then you pretend that you buy one of each asset and look at the returns for you time period. This would be the same as assuming that you are investing the same $ amount in each asset regardless of the share price.

If these are your prices for asset A and the first 4 time points:

50.50 @ t = 1
50.75 @ t = 2
50.80 @ t = 3
50.95 @ t = 4

after adjusting the prices you would have

1       @ t = 1
1.00495 @ t = 2
1.00594 @ t = 3
1.00891 @ t = 4

So no you can see that your return for this asset over the first 4 time periods is:

1.00891 - 1 = 0.00891 or 0.89%

Do this for all the assets and you will have your equally weighted portfolio.

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  • $\begingroup$ I don't understand how this is easier? In any programming language if we have a matrix of returns we can just do something like rowMeans(matrixofreturns) and this will be our equally weighted portfolio, but what you've suggested seems like it'll take 30 minutes to program. $\endgroup$ – user2921 Oct 3 '12 at 1:55
  • $\begingroup$ i think you'll need to brush up on your mathematical programming. Assuming you start with a matrix of returns, where each column holds prices for a particular asset, and each row is a price at a time point, then this is about 3 to 4 lines of code in matlab or numpy. also log returns are not arithmetic returns. Usually when people are talking about money, they want to know exactly how much money they would make/lose. see here: en.wikipedia.org/wiki/… $\endgroup$ – jeffery_the_wind Oct 3 '12 at 3:17
  • $\begingroup$ I was asking specifically about the statement in the academic paper that I quoted and what it was they wanted me to do (my question was whether they wanted me to just take the row means of the log returns matrix). $\endgroup$ – user2921 Oct 3 '12 at 3:21
  • $\begingroup$ I see now, your question was a little hard to understand. Please see my updated answer. $\endgroup$ – jeffery_the_wind Oct 3 '12 at 12:32
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An equally-weighted portfolio is equally dollar-weighted, not share-weighted, so the above answer is based on a mistaken assumption.

The total return of an equally-weighted portfolio is the average return of all constituents at each period. For example, say you have these two stocks starting at these prices:

A  $10
B  $20

So, you might start with 10 shares of A (\$100) and 5 shares of B (\$100), giving you a $200 portfolio.

The next day, say the prices are now:

A  $12 
B  $18

giving you a total portfolio value of \$210, a 5% (=10/200) return. However, this portfolio is no longer equally-weighted since you now have \$120 (=\$12 * 10 shares) of A and \$90 (=\$18 * 5 shares) of B. This means you have to rebalance by selling 1.25 shares of A ((\$210 total portfolio value/2 number of constituents) /\$12 -> 8.75 new number of A shares) and buying 0.83333 shares of B ((210/2) /\$18 -> 5.83333), giving you $105 of each constituent. If you cannot buy fractional shares, this will introduce tracking error as you will not be able to exactly maintain equal-weighting.

Another way to figure this is: the return for A is %20 (=12/10-1) and the return for B is _10% (=20/18-1), so the portfolio return is 5% (average=(20% + _10%)/2).

To further complicate matters, returns are usually best stated as "total return" which includes things like dividends and spin-offs. For example, say we have the scenario for the first two periods as stated above but that B's price drop on the 2nd day is accompanied by a \$1/share dividend payment. Now the total value of the portfolio would be \$215 (= (12*\$10)+(5*\$18)+5*\$1), so each constituent should now be worth $107.5. This translates to 8.95833 shares of A and 5.97222 shares of B.

In this case, the total return is 7.5% (\$215/\$200-1), which also is the average of the 20% return on A and the _5% return on B (=((5*\$1)+5*\$18))/\$100-1).

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