Используйте агрегат sum()
в качестве оконной функции, например:
with my_table (created_at, amount, register_name) as (
values
('2019-01-01', -150.01, 'Front'),
('2019-01-01', 38.10, 'Back'),
('2019-01-02', -150.01, 'Front'),
('2019-01-02', 38.10, 'Back')
)
select
created_at, amount, register_name,
sum(amount) over (partition by register_name order by created_at)
from my_table
order by created_at, register_name;
created_at | amount | register_name | sum
------------+---------+---------------+---------
2019-01-01 | 38.10 | Back | 38.10
2019-01-01 | -150.01 | Front | -150.01
2019-01-02 | 38.10 | Back | 76.20
2019-01-02 | -150.01 | Front | -300.02
(4 rows)