Postgres 13

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CREATE TABLE tbl (
  date    date
, PR      int
, amount  numeric
, total   numeric
, balance numeric
, expected_value text
, PRIMARY KEY(pr, date)
);

INSERT INTO tbl
VALUES
    ('2020-1-1', 1,  1.0,  100.0, 1.0, '100'),
    ('2020-1-2', 1,  2.0,  220.0, 3.0, '320'),
    ('2020-1-2', 2,  3.0,  270.0, 4.5, '430'),  -- different pr!
    ('2020-1-3', 1, -1.5, -172.5, 1.5, '160'),
    ('2020-1-4', 1,  3.0,  270.0, 4.5, '430'),
    ('2020-1-4', 2,  3.0,  270.0, 4.5, '430'),  -- different pr!
    ('2020-1-5', 1,  1.0,   85.0, 5.5, '515'),
    ('2020-1-6', 1,  2.0,  202.0, 7.5, '717'),
    ('2020-1-7', 1, -4.0, -463.0, 3.5, '334.6'),
    ('2020-1-8', 1, -0.5, - 55.0, 3.0, null),
    ('2020-1-9', 1,  2.0,  214.0, 5.0, null);
11 rows affected

CREATE OR REPLACE FUNCTION f_special_running_sum (_state numeric, _total numeric, _amount numeric, _prev_balance numeric)
  RETURNS numeric
  LANGUAGE sql IMMUTABLE AS
'SELECT CASE WHEN _amount > 0 THEN _state + _total
             ELSE _state * (1 + _amount / _prev_balance) END';

CREATE OR REPLACE AGGREGATE special_running_sum (_total numeric, _amount numeric, _prev_balance numeric) (
  sfunc    = f_special_running_sum 
, stype    = numeric
, initcond = '0'
);
-- query
SELECT *
    , special_running_sum(total, amount, prev_balance) OVER (PARTITION BY pr ORDER BY date)
FROM   (
   SELECT * -- pr, date, amount, total
        , lag(balance, 1, '1') OVER (PARTITION BY pr ORDER BY date) AS prev_balance
   FROM   tbl
   ) t


date
pr
amount
total
balance
expected_value
prev_balance
special_running_sum


2020-01-01
1
1.0
100.0
1.0
100
1
100.0

2020-01-02
1
2.0
220.0
3.0
320
1.0
320.0

2020-01-03
1
-1.5
-172.5
1.5
160
3.0
160.000000000000000000000

2020-01-04
1
3.0
270.0
4.5
430
1.5
430.000000000000000000000

2020-01-05
1
1.0
85.0
5.5
515
4.5
515.000000000000000000000

2020-01-06
1
2.0
202.0
7.5
717
5.5
717.000000000000000000000

2020-01-07
1
-4.0
-463.0
3.5
334.6
7.5
334.60000000000000000239000000000000000000000

2020-01-08
1
-0.5
-55.0
3.0
null
3.5
286.8000000000000000030045714285714285714354000000000000000000000

2020-01-09
1
2.0
214.0
5.0
null
3.0
500.8000000000000000030045714285714285714354000000000000000000000

2020-01-02
2
3.0
270.0
4.5
430
1
270.0

2020-01-04
2
3.0
270.0
4.5
430
4.5
540.0

date	pr	amount	total	balance	expected_value	prev_balance	special_running_sum
2020-01-01	1	1.0	100.0	1.0	100	1	100.0
2020-01-02	1	2.0	220.0	3.0	320	1.0	320.0
2020-01-03	1	-1.5	-172.5	1.5	160	3.0	160.000000000000000000000
2020-01-04	1	3.0	270.0	4.5	430	1.5	430.000000000000000000000
2020-01-05	1	1.0	85.0	5.5	515	4.5	515.000000000000000000000
2020-01-06	1	2.0	202.0	7.5	717	5.5	717.000000000000000000000
2020-01-07	1	-4.0	-463.0	3.5	334.6	7.5	334.60000000000000000239000000000000000000000
2020-01-08	1	-0.5	-55.0	3.0	null	3.5	286.8000000000000000030045714285714285714354000000000000000000000
2020-01-09	1	2.0	214.0	5.0	null	3.0	500.8000000000000000030045714285714285714354000000000000000000000
2020-01-02	2	3.0	270.0	4.5	430	1	270.0
2020-01-04	2	3.0	270.0	4.5	430	4.5	540.0