Suppose the indices are up 1% one day, and down 1% the next. Is the cumulative effect 0%? What if they were down 1% and then up 1%? It turns out that in both cases, the cumulative effect is not 0%, but rather negative. To understand this in the most simple manner, realize that 1% of 1.01 (after 1% gain) is more than 1% of 0.99 (after 1% loss). Therefore, the 1% loss is greater than 1% gain. And due to the communicative property of multiplication, it doesn't matter whether up or down occurred first.
The cumulative effect is 1.01*0.99 = 0.9999, not 1. To truly offset an 1% loss, the gain required would only be 1/0.99-1 = 1.010101%, which is above 1%. This residual decreases as the magnitude of the gain and loss becomes smaller. Let x be the same magnitude for the gain and loss, so x=0.01 in the case above: (1+x)*(1-x) = 1-x^2. Compared to the unchanged initiate value of 1, the difference is simply x^2. So when the gain and loss is 10%, the end result is 1% lower. When the gain and loss are 1%, or 0.01, as was the case above, the end result is 0.0001 deviated, and the cumulative effect of 1.01*0.99 = 0.9999 confirms that.
While 0.9999 may not seem much different from 1, the effects can easily add up. Supposed that the portfolio repeatedly bounced up and down 1% each day for the entire calendar year. Given 250 trading days, so 125 of such cycles, an $100 portfolio will only be worth $100*0.9999^125 = $98.76 after the year, corresponding to 1.24% loss. And because the magnitude-residual relationship is quadratic as observed above, the effects multiply quickly: given 2% back-and-forth for a year, the same $100 portfolio would only be worth $100*0.9996^125 = $95.12, or 4.88% loss.
