array formula: sum to date - excel

Question

I have a number of invoices:
invoice #
start
end
paid on
amount
paid to date (hardcoded)
1
01/01/2020
30/01/2020
01/02/2020
£10.00
£10.10
2
01/02/2020
20/02/2020
01/03/2020
£7.50
£17.60
3
21/02/2020
30/02/2020
01/03/2020
£2.50
£20.10
4
01/01/2000
30/01/2000
01/03/2000
£0.10
£0.10
Where the invoices
are not necessarily sorted by start and end date
are not necessarily sorted by paid date.
might not have a paid-on value.
I want to add a field called paid to date that would show me the amount I have been paid so far where it would add the amount for:
the current invoice
the invoices that were paid prior to this invoice's paid-on date.
the invoices with the same paid-on date as this invoice but with a start date <= this invoice's start date.
Effectively mirroring the hard-coded column.
This is how I do it with a query (which might not be the simplest/most elegant way of doing it)
=Index(
query(
A1:E10,
"select SUM(E)
where D is not null
and (
D < date '"& TEXT( D2,"yyyy-mm-dd")&"'
OR (
D = date '"& TEXT( D2,"yyyy-mm-dd")&"'
and
B <= date '"& TEXT( B2,"yyyy-mm-dd")&"'
)
)"
),
2, 1
)
which is all well and good. but I want to be able to do it with array-formula, so I can have it auto-generated for me.
I tried using it inside array-formula but the value is only ever generated for the first row. I guess it's misinterpreting the range I am passing as the range of the query function, ie A1:E10. is there an easy way of fixing it?
Do I need to use VLookup?
Here is a sample spreadsheet.

Answer 1

try:
=INDEX(IFNA(VLOOKUP(A2:A, {INDEX(SORT({A2:B, D2:E}, 2, 1, 3, 1),,1),
MMULT(TRANSPOSE((ROW(E2:E)<=TRANSPOSE(ROW(E2:E)))*
INDEX(SORT({A2:B, D2:E}, 2, 1, 3, 1),,4)), SIGN(E2:E))}, 2, 0)))

Answer 2

If you have Excel 365, you can Filter directly.
Note that I am using a table with structured references. Then you don't need to adjust the range references as you add/remove rows from the table.
=SUM(FILTER([amount],([paid on]<[#[paid on]])+(([paid on]=[#[paid on]])*([start]<[#start])),0),[#amount])
This part does the filtering:
...([paid on]<[#[paid on]])+(([paid on]=[#[paid on]])*([start]<[#start]))...
Note: You can use the same algorithm in Sheets:
Note that I am using regular addressing, and also using a semicolon instead of a comma for the argument separators
=sum(iferror(FILTER($E$2:$E$5;($D$2:$D$5<$D2)+(($D$2:$D$5=$D2)*($B$2:$B$5<$B2)));0);E2)