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I am working on a project which needs to find portfolio return for the next m months.

To begin, let say investor hold a portfolio of $N$ stocks with weight $w_i$ invested in stock $i$, what is the investor's m-month buy and hold portfolio return given the monthly return for each stock in the portfolio? $$r_{P,1\rightarrow m}=\sum^N_{i=1}w_i(1+r_{i1})(1+r_{i2})...(1+r_{im})-1$$

where $r_{it}$ ($t=1,..., m$) is stock $i$'s return in month $t$.

In Liu and Strong (2006) paper Biases in Decomposing Holding Period Portfolio Returns, they suggest the following calculation is true:

$$\Pi_{t=1}^m (1+r_{Pt})-1 = \sum^N_{i=1}w_i(1+r_{i1})(1+r_{i2})...(1+r_{im})-1$$

$r_{Pt}$ is the overall portfolio return.

This equation on the LHS - I believe- is saying if we want to calculate the portfolio in next m-months, we can either do weighted average all stocks in the portfolio at each time period t to form portfolio return and multiply monthly portfolio return to get the return for the next m-months. This is equal to (RHS) by multiplying individual stock $i$'s next m-period return, then weighted average the stocks in the portfolio.

When I put some arbitrary numbers in it I get totally different answers.

Let say I want to learn what is the next month portfolio return is, so $t=2$. We have 2 stocks in each portfolio. The weight used with be equal weighted, this means each stock $i$ have 50% shares in this portfolio.

The respective return at $t=1$: stock $i=1$ is 0.3 and stock $i=2$ is 0.5.

The respective return at $t=2$: stock $i=1$ is 0.6 and stock $i=2$ is 0.4.

According to the LHS of the equation, we calculate the portfolio return first for $t=1$, which is $(0.5*0.3)+(0.5*0.5)= 0.4$. At $t=2$, we have $(0.5*0.6)+(0.5*0.4)= 0.5$.

LHS is then $$\Pi_{t=1}^m (1+r_{Pt})-1 = (1+0.4)*(1+0.5)-1 = 1.1$$.

If we calculate using the RHS, we will get:

$$\sum^N_{i=1}w_i(1+r_{i1})(1+r_{i2})...(1+r_{im})-1= [0.5(1+0.3)*(1+0.6)] + [0.5(1+0.5)*(1+0.4)]-1 = 1.09$$.

You can see the answer is totally different. 1.1 and 1.09, is there something wrong in my calculation? Please help. Thanks

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    $\begingroup$ I think the LHS is the definition of something called $r_{Pt}$ and the RHS is how you actually compute it. You begin by letting $m=1$ and the RHS gives you $r_{P1}$. Now you let $m=2$ and you can find $r_{P2}$, and so on. I don't understand how you used the LHS to come up with an independent value (which you then compared to the RHS). $\endgroup$ – Alex C Jun 18 '18 at 19:20
  • $\begingroup$ Hi Alex thanks for the comment. So basically $r_{Pt} $ is basically the return of the portfolio at time t. A portfolio return can be calculated by weighted average of individual stock's return,$ r_{Pt} = w_1*r_{1t} + w_2*r_{2t} $. This is standard portfolio calculation where $w_i $ are the weights for stock i's simple return. By multiplying portfolio return forward, i.e. 2 months, m=1 and m=2 you get the 2 periods returns. I.e $(1+r_{P1} ) *(1+r_{P2} ) -1$? In this calculation it has a difference of 0.01 which is not far off. Why would that be? $\endgroup$ – user34689 Jun 18 '18 at 20:15
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The error in your numerical example is in the LHS: without rebalancing, after period 1, the weights of the assets are not 50% each, but 46% and 54%. So the portfolio return in period 2 is not 50%, but 49.3%.

Addition in response to the comment

Without rebalancing: to get the portfolio return up to some point in time, multiply the total return of each asset up to this point by its initial weight and sum all terms.

With rebalancing: multiply each asset's return in a specific period with its initial weight for that period; then summing across assets will give you the portfolio return for that period. The weights for that period can, in principle, be arbitrary: the assumption is that you rebalance the portfolio at the close of the previous period to achieve these weights. (Which may be unrealistic/impractical: just think of daily data; which is why buy-and-hold returns are often preferred to evaluate strategies). Finally, chain together these period returns to arrive at the total return for the portfolio.

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    $\begingroup$ So is this equation used for buy-and-hold? I know buy and hold basically let the weight inside the portfolio float every period. Whilst rebalance means you adjust the weight back to the weight when the portfolio is formed. The RHS looks to me is rebalance because we put $w_i$ as the initial weight (when the portfolio is formed)? $\endgroup$ – user34689 Jun 19 '18 at 13:12
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    $\begingroup$ Yes, buy and hold. No rebalancing in the RHS since you only use the initial weight. $\endgroup$ – Enrico Schumann Jun 20 '18 at 7:15

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