I love autopay and paperless. I will have 23 automatic payments drafted from my checking account in the next 30 days. Of those, at least 21 are paperless. For all the paperless ones, I get an email in advance with either the amount owed and the date the payment will be drawn or a link to login and check my statement.
I think it's very safe. I've been on autopay for as much as possible for at least a decade. I've never had any problems. Zero.
I've been switching over to paperless as well for a while now. I used to feel like CRLLS and wanted to get a paper statement in the mail. I switched over when I realized that an email in my inbox serves all the same purposes, just without the paper.
I like it so much that one financial institution I have that does not do autopay very well is less appealing to me for that reason and I might stop doing business with them.
I still get year end tax stuff in the mail. I keep pretty good track of tax stuff, but I might have a tax effect that I forget about through the year, or didn't make a note of in my tax records, or whatever.
To keep things safe, I do the following:
1. Strong password on my email.
2. Paperless statements on most things.
3. Bank and credit card alerts on card not present transactions, transactions over a nominal amount, etc.
4. Regular Quicken tracking with bills scheduled, so I know what, how much, and when to expect bills.
5. Quicken downloaded transactions for most accounts, so I can confirm the actual debits and credits match.
6. Only a handful of few checking/savings accounts, so I can keep track.
7. Most everything goes into and out of a central checking account.
8. Mail any payments with checks from a postal box rather than my mailbox (bordering on paranoid, I agree).
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I use a Fidelity Visa for everything and get 2% back plus some tax benefits so I think of it as a 2.66% card.
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My Dad has had a couple of times where transactions have posted on his checking account that he didn't recall making or were somewhat scammy. Every time, I go to his bank, they immediately issue a full provisional credit, and reissue him a new debit card in about two minutes while I wait. The provisional credit has always become permanent.
So even though this is only a debit card at an FDIC-insured bank, the protection there in our experience has been excellent, and the minor hassle a few times hasn't been any big deal.
(I have full POA on the account so I can legally and properly do those things for him.)