Why Buy-and-Hold and Rebalanced Portfolio Returns Differ
Summary
The document compares two ways of combining the annualized returns of a two-asset portfolio. Weighting each asset’s annualized return assumes the portfolio weights are maintained over time through rebalancing. Weighting the assets’ cumulative returns instead describes an initial allocation that is held without rebalancing, with dividends reinvested in the originating asset. As the assets earn different returns, their portfolio weights drift, so the two approaches need not produce the same annualized result.
The answers illustrate this drift with a portfolio that ends with a larger share in the better-performing asset. They also caution that a rebalanced portfolio’s return depends on when and how rebalancing occurs; periodic rebalancing generally requires a time-series simulation. The weighted annualized-return calculation is only an approximation when weights are kept near their targets. The document does not provide a full return calculation or discuss fees, taxes, or other portfolio constraints.
Key ideas
- Weighting asset annualized returns assumes portfolio weights are maintained through rebalancing.
- A buy-and-hold allocation allows weights to drift as asset values change.
- Reinvested dividends remain invested in the asset that paid them in the example.
- Rebalanced returns depend on the timing and rules used to rebalance.
- A detailed simulation can model a specified rebalancing schedule.
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# Help Calculating a Portfolio Returns # Help Calculating a Portfolio Returns 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. 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? ## Answer by nbbo2 (score 4, accepted) https://quant.stackexchange.com/a/81069 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). ## Answer by Enrico Schumann (score 5) https://quant.stackexchange.com/a/81067 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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