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I'm having issues understanding why these two methods for calculating a portfolios annualized returns aren't matching.

Lets take for example a portfolio comprised of 2 assets.

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Now the first way I thought to calculate the portfolio's annualized return is to take the weight of each asset and multiple it by each assets annualized return. So for example (0.5 x 0.15) + (0.5 x 0.05) = 0.1. This is a 10% annualized return for the portfolio over 10 years.

The second way I thought to calculate the portfolio's annualized return was to take the weight of each asset and multiple it by each assets cumulative return. Once we have the portfolio's cumulative return we would then annualize that return to get the portfolios annualized return. So for example (0.5 x 3.0456) + (0.5 x 0.6289) = 1.8372. Then (1 + 1.8372)^(1/10) - 1 = 0.1099. This comes out to a 10.99% annualized return over 10 years, which is different than the answer from the first way.

Why don't these two annualized returns match each other? What am I missing?

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  • $\begingroup$ when you do the (1 + 1.8372)^(1/10), that's the average annualized return. I don't know how you calculated the annualized return in the table but I imagine that's it's exact rather than average ? So, you wouldn't expect them to be the same. $\endgroup$ Commented Nov 5, 2024 at 1:58

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As mentioned the first method assumes continuous rebalancing to 50/50 weights.

In the second method you assume Buy and Hold: you initially split your money 50% 50%, buy the two assets and just keep them without ever taking the initiative to sell or buy. If there are dividends you reinvest them in the same asset that they came from. On the last day you will have 406/(406+162) = approx 71% in Asset 1 and 29% in Asset 2.

Of course, in the case of rebalancing the return will depend on the exact times the rebalancing takes place. A detailed simulation is needed (often you simulate End of Quarter or End of Year rebalancing, though there are other rebalance algorithms also). Method 1 is only an approximation, assuming frequent enough rebalancing that the weights are "close enough" to 50 50 at all times. (Or if you prefer that the outperformance of 1 asset over the other takes place gradually compared to the rebalancing).

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  • $\begingroup$ That explains my confusion I thought the first method also assumed buy and hold with no rebalancing. Are you familiar with any sources or books that mention the rebalancing frequency for the first method? $\endgroup$ Commented Nov 5, 2024 at 18:30
  • $\begingroup$ I think the first method is continuous rebalancing, like Black Scholes, at every instant (in the limit) you buy a little something and sell a little something. Not a procedure you can do in real life. $\endgroup$ Commented Nov 6, 2024 at 9:16
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In your first method, you assume that the weights remain at 50% for all years, i.e. you assume you rebalance. Without rebalancing, the weights would change over time since the first assets grows faster.

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