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Let's say I spend $1000 = 100 units at $10/unit on a no-commission, no-load index or mutual fund with an MER of 1%. Let's say any distributions are deposited as cash. In one year's time, the fund has not gone up or down. The value of the investment is still at $10/unit.

How exactly is the 1% MER charged? Does my 100 units become 99 units? Or is the price reduced to $9.90 for everyone? Or something else? If so, how often, once a year, more often? And where is this information documented?

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The MER is deducted invisibly to you and is reflected in the share price.
The number of shares you own does not change; their price does. In your example, the share price will decline from $10 per share to $9.90 per share over the year. The reduction is included in the price shown as the daily Net Asset Value (NAV) of the shares as shown on the fund's website. You can't tell when the reduction is occurring (on a daily basis at 1%/365 per day, or monthly basis of 1%/12 per month etc).

If you are considering investing in a no-load index fund that is charging a MER of 1% in the US, don't.

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