Here is an other version (a bit shorter and maybe easier to read)

    public static LocalDate getBeginningPreviousAcademicYear(LocalDate currentDate) {
        int currentCivilYear = currentDate.getYear();
        int previousAcademicYear;

        // If we've started a new civil year and haven't changed the academic year yet, go back one more year.
        if (currentDate.isBefore(FIRST_DAY_OF_ACADEMIC_YEAR.toLocalDate(currentCivilYear)))
            previousAcademicYear = currentCivilYear - 2;
        else
            previousAcademicYear = currentCivilYear - 1;

        return FIRST_DAY_OF_ACADEMIC_YEAR.toLocalDate(previousAcademicYear);
    }