凭栏袖手
第一部分 问题的提出
最近为物业公司做了个利润达标奖金方案，其计算有点复杂，大体是这样：
例一，利润达标奖金= （年度实际经营利润-基准利润）×奖金计提率× 考核计提倍率
其中，考核计提倍率是关键，这样算：
|
年度经营利润 增长率P |
P＞90%部分 |
90%≥P＞60%部分 |
60%≥P＞30%部分 |
30%≥P＞0部分 |
|
考核计提倍率 |
1.5 |
1.3 |
1.2 |
1 |
定义：P=（年度实际经营利润-基准利润）/年度目标经营利润
实际上要分段计算，尽管公式不是很严密，但意思是清楚的。
领导要看看是否合理，就拟了一组假设数据试算：
设：年度目标经营利润=500万基准利润=350万 奖金计提率=20%
年度实际经营利润=550万、570万、590万、、810万
年度实际经营利润=550万时，P=(550-350)/500=200/500=40%
因此 30%≥P＞0部分为500*30%=150 对应考核计提倍率 1
60%≥P＞30%部分为500*（40%-30%）=50 对应考核计提倍率 1.2
利润达标奖金= 150*20%*1+50*20%*1.2=30+12=42(万元）
同样可计算其他情况，利用Excel 易得结果。
我觉得这样算太麻烦了，有简单的计算方法吗？最好是一个公式。
比方说，类似计算个人所得税的那个神奇的公式：
例二，个人所得税=ROUND(5*MAX(年度应纳税所得额*{0.6;2;4;5;6;7;9}%
-{0;504;3384;6384;10584;17184;36384}),2) （公式一）
经过一番努力，我还真找到了一个这样的公式：
先定义: 实际超基准利润=年度实际经营利润-基准利润
利润达标奖金=
ROUND(MAX(实际超基准利润*{1;1.2;1.3;1.5}*20%-{0;6;12;30}),2) （公式二）
利用Excel 易得
神奇吗？为什么会这样？作为一个唯物论者，我需要一个证明。
百度过后发现，大家只满足于运用，但字里行间都有些疑惑，用得不大放心。找不到证明，只好我自己来。
第二部分 把问题抽象化并给出证明
我要证明一个一般化的命题：
命题一和命题二等价，下面先证推论一：
下面我们分三步来证明命题一：
1、当m=2 命题一成立
2、当k=2命题一成立
3、假设 1）、当m=h+1 k=l命题一成立
2）、当m=h k=l+1命题一成立
证明3）、当 m=h+1 k=l+1命题一成立
我们用的是双重数学归纳法。
1、当 m=2 命题一成立
2、当 k=2命题一成立
3、假设 1）、当 m=h+1 k=l命题一成立
2）、当 m=h k=l+1 命题一成立
证明 3）、 当 m=h+1 k=l+1 命题一成立
第三部分命题二的应用
我们先利用命题二的结论推导例二的公式一：
个人所得税=ROUND(5*MAX(年度应纳税所得额*{0.6;2;4;5;6;7;9}%
-{0;504;3384;6384;10584;17184;36384}),2)
其实质是Excel函数的运用，四舍五入函数加一个MAX函数的数组运算，其核心是后者。
新个人所得税法于2019年1月1日起施行，居民个人工资、薪金所得预扣预缴税款按照下表计算
对照命题二知，此时，n=6。
由命题二有(设年度应纳税所得额为a)：
这就是公式一。
下面我们看公式二：
由命题二有(设实际超基准利润为a=P*年度目标经营利润=P*500)：
总之，命题二适用于所有阶梯分段累进计算，大家放心用吧。前不久，我司的人力总对他初三刚毕业的小孩用类似例一的绩效考核方式计算期末成绩的奖金，说是有一万多，兑现了，该小孩十分激动，但拿着钱在自己房间呆了不到半小时，就出来把钱交还给了人力总，说是要求委托理财。等小孩回到房间去后，人力总夫妇俩在客厅里差点笑岔了气。
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