The optimization process and recommendations:

1. Avoid Subselect When Selecting MAX/MIN Per Group (query line: 9): Constant subquery results are usually not cached by the database, especially in non-recent database versions. Therefore, a constant subquery in a WHERE clause will be fully evaluated for every row the WHERE clause will examine, which can significantly impact query performance. Use the method mentioned in the example instead.
2. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.

Optimal indexes for this query:

ALTER TABLE `SerialBalance` ADD INDEX `serialbalance_idx_effec_subhe_seria_branc_zone_count` (`EffectiveDate`,`Subhead`,`SerialNumber`,`Branch`,`Zone`,`Country`);

The optimized query:

SELECT
        count(*),
        sum(SerialBalance1.balance) 
    FROM
        SerialBalance AS SerialBalance1 
    LEFT JOIN
        SerialBalance AS SerialBalance2 
            ON (
                SerialBalance2.Subhead = SerialBalance1.Subhead 
                AND SerialBalance2.SerialNumber = SerialBalance1.SerialNumber 
                AND SerialBalance2.Branch = SerialBalance1.Branch 
                AND SerialBalance2.Zone = SerialBalance1.Zone 
                AND SerialBalance2.Country = SerialBalance1.Country 
                AND SerialBalance2.EffectiveDate <= '2013-05-31'
            ) 
            AND (
                SerialBalance1.EffectiveDate < SerialBalance2.EffectiveDate
            ) 
    WHERE
        (
            SerialBalance1.balance > 0 
            AND 1 = 1
        ) 
        AND (
            SerialBalance2.EffectiveDate IS NULL
        )