Calculated value dilemma

Hi all

I am designing a database for my charity (we are a local special needs
charity) to replace our existing one, and my main goals are to make
the database accurate, easy to use, and able to expand (the current
database is not!).

The database will hold contact and membership details for about 300
families, staff/volunteers and other contacts, along with event
booking (we hold about 5 a week) and payment details.

So, for each family we need to see how much they owe, plus a full list
of all their transactions. I am considering having a transaction
table, which details every payment anyone has ever made.

IDEA 1
To follow normalisation rules I should avoid having calculated values
in the table, and I should therefore simply calculate the balance
whenever I need it by querying the table. I guess as the database
grows I would have to periodically 'consolidate' the balance to
maintain efficiency.

However, I see each updated balance as a snapshot in time - when I
look back to figure out 'who paid what when', I want to be 100%
confident that the database will return the same balance it would have
returned at the time. If a rogue entry were to somehow find its way
into the table, it would mess all the balances up

IDEA 2
As keeping historically accurate info is important, 'hard-coding' the
balance in the table seems to me like it might be the most appropriate
solution, even though it breaks the normalisation rules.

Any suggestions would be greatly appreciated!

Mark

PS I've never designed a database before, so please let me know if
I'm completely barking up the wrong tree!