Business Analytics Micro-Case #3: Better Decisions with Monte Carlo Simulation

Background

最近的一次 路透 article discussed how US Retailers have purchased, and are subsequently storing, goods that would be subject to an anticipated 25% tariff on Chinese imports. This in turn has driven up transportation and storage costs which is “adding pressure to quarterly results for retailers”.

在我们的3rd Analytics Micro-Case, 我们站在零售商的立场上,问自己是否能期望获得足够的增量利润,以证明我们加速(关税前)采购是合理的, given four principal unknowns:

  1. 关税金额: Although expected to be 25%, it is possible a deal may be announced and no tariff is implemented, or it could end up a different amount altogether.
  2. Expected Retail Price Increases: We know that the tariff of 25% would be applied to our purchase price (i.e. our Cost-Of-Goods-Sold), but we also know that i) different retailers have different COGS to which the tariff will be applied and ii) some of these retailers may be willing to absorb some of the tariff rather than pass it on to customers and iii) some manufacturers may be willing to absorb some of the tariff in order to maintain market share against local (US) manufacturers thereby dampening some of the price increase.
  3. Increased 存储成本如果我们怀疑,我们将不是唯一一个增加库存水平的零售商, we should also expect storage costs to increase by some indeterminate amount.
  4. Duration of Storage: The 25% tariff was originally expected to effective Jan 1, 2019, but was subsequently pushed back to March 2, 2019. 因此, 目前尚不清楚货物需要储存多长时间才能实现免税优势.

For simplicity, we ignore shipping cost increases, 资本成本和其他项目,可能包括在一个更全面的模型.

Even with just 4 unknowns, this is a fairly difficult problem or which to derive a quantitative answer. 在这个微观案例中,我们对比了传统方法(其中我们的决策模型基于平均估计), 使用蒙特卡罗(MC)模拟方法(根据各自的概率模拟所有可能的结果). 因此,我们展示了MC模拟如何在决策过程中防止不必要的乐观.

是基线

这个模型, assume our pre-tariff price is $100, with a corresponding 40% COGS (which would be subject to the tariff). 还假设存储成本在基线模型中每单位/月增加了1美元.

因此, our pre-tariff margin can be seen to be $59.

This basic margin formula is deployed in all subsequent calculations, however the relevant Price, COGS and Storage is adjusted according to our model specifications.

Now, assume that we have duly conducted our research and concluded that we expect:

  1. The tariff is most likely to be implemented at 25%
  2. Retail prices will increase by 10% on average.
  3. Storage costs to increase by 20%.
  4. Stockpiling averages 4 (total) months of inventory carry.

因此,在没有任何库存累积的情况下,我们预计税后利润率为59美元:

In our simplified example, 预期的价格上涨正好抵消了关税的影响,利润没有变化. 我们是否建立库存的决定取决于我们是否相信我们的利润率会高于59美元的基准利润率.

Traditional (Single-Point Estimate) Approach

To calculate our expected return using the Traditional Approach, we use the average expected increases in Price and in Storage costs, plus the expected four months storage costs. (The tariff can be ignored as we are importing goods before the tariff takes effect.) Which can be calculated as:

It seems that with our average calculations approach, we expect to make a $65.20 margin on each unit compared to our baseline, which is $6.20 more than our expected margin without the stockpiling effort. This sounds impressive, it is a healthy 6.2% higher margin than could be expected without the stockpiling effort.

然而, 这种方法没有认识到每个变量固有的风险或可变性, and so produces a relatively uninformed result.

Monte Carlo Simulation Approach

Monte Carlo simulations 是否有一类广泛的计算算法依赖于重复随机抽样来获得数值结果 (维基百科). 具体地说, 我们的MC模型从4个未知变量中随机选择一个值,该值与该变量的分配概率一致, and calculates the corresponding improved margin for that combination of inputs. 我们这样做了10万次,以生成改进后的利润率的概率分布.

Input Distributions

We augment the research conclusions for each of our 4 unknowns as above, with a probability distribution that maintains the average as stated. 定义概率分布需要将特定的专业知识应用到分布中, which we assume has been applied below:

Tariff Distribution

关税金额当前位置我们认为,25%的关税有80%的可能性被实施,而达成20%的协议将保持现有的0%关税价值. To represent this numerically, we draw samples (with replacement) from the set {0%, 25%} with a probability of drawing each as {20%, 80%}.

 

Price Increases

Expected Price Increases: A 25% tariff only applies to our COGS, which is 40% of our Price. 因此, if we wanted to fully pass the tariff on to our customers, our Price would need to increase by 10% (.4 * .25). 我们假设正态分布的平均增长率为10%,标准差为4%. 因此, 我们有95%的信心,我们实现的价格涨幅将大约在2%-18%之间(10% +- 1).96 * 4%).

存储成本

Increased 存储成本:这里我们假设一个正态分布,平均值为20%,标准差为5%(因此在存储中95% CI为10%-30%增加).

Duration of Storage

Duration of Storage: The duration (in months) that the tariff-free purchases needs to be held. 这里我们使用泊松分布,因为我们想要持续时间为零和一个小概率的大持续时间. We use a mean of 4 months.

Calculating Likely Margin

With the assumptions and probability distributions laid out, 我们从4个输入变量中随机选择一个,然后计算相关的改善裕度, 100,000倍. We now count the instances of each level of margin improvement.

Based on the approach, we can expect a per unit, mean incremental margin of $3.42 with a 95% confidence interval ranging from $-7.5 to $9.097. 在我们预期的结果范围内,这是一个相当大的似是而非的边际范围,应该被视为一个相当有风险的命题.

The median, per unit incremental margin is $5.058.

This mean incremental margin of $3.42 is considerably less than the $6.使用单点估计方法,并结合可能的保证金变动范围,预计增加20美元.60 vs $0) it a considerably riskier undertaking than previously understood. 这在很大程度上是由于我们考虑了完全不征收关税(20%)的可能性, 但一个没有考虑到这种结果可能性的模型肯定是不够的.

Safeguarding Against Unwarranted Optimism

使用平均值来确定预期的输出值并不考虑可能值的范围,因此这样的决策是在不考虑所涉及的风险的情况下做出的. By using MC simulation, we were not only forced to consider the range of possible input values, but our output variable was considerably more information-rich, which leads a more informed decision made.

使用平均值来确定结果往往过于乐观,因为模型很少考虑到负面事件所产生的不成比例的高影响. 在我们的案例中, 不征收关税将对利润率的改善产生负面影响,而与基线(我们模拟的20%)相比 , 低于平均水平的价格和高于平均水平的存储时间的组合——占我们模拟的18%左右. Both of these subtleties are lost with single-point estimates but fortunately, with MC simulation we can guard against such unwarranted and inadvertent optimism.

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