BPC| 植物大战僵尸中的投资Investment in Plants vs Zombies

《植物大战僵尸》这款游戏中有一个花园，玩家可以在花园里种花，等花种大之后就可以将花卖掉赚钱。笔者觉得，这样的模式和现实生活中投资模型有相似之处，便准备通过游戏，来计算出种花的利润率。（其实只是想玩游戏）

In the game "Plants vs. Zombies", there is a garden in which players can plant flowers and sell them after maturity. In the author's opinion, the method of growing flowers is similar to the realistic model of investment. This idea induces the author to investigate the profit rate of growing flowers. 

最开始我们需要知道怎么样来计算利润。我们知道利润(profit)是由收入(revenue)减去成本(cost)而得的，而利润率则是利润除以成本的百分比。那么，投资(investment)的成本包括了哪些呢？

第一，投资需要资金的投入，而这些资金的使用并非没有成本。使用资金的成本我们称之为机会成本(opportunity cost)，因为如果这些资金没有被用于投资，它们也可以被存入银行吃利息。因此，使用资金进行投资就相当于放弃了使用这笔钱存储赚取利息的机会。所以，这些资金存储到银行可以获取的利息也应该被算作是成本的一部分。同时，投资的资产由于使用的磨损(wear and tear)，会产生贬值(depreciation)。
First of all, we need to know how to calculate profit. Profit is calculated by revenue less cost, and then, we can get the profit rate by dividing profit with the cost. Then, we should know more about the cost and revenue of investment.

花园里的投资

而投资的收入主要也是有两种来源。首先，投资的资产最后会被卖出，卖出与买进价格的差价也会被计入到收入中。假如卖出的价格比买进的价格要高，这就增加了利润。最后也是最重要的一部分，是投资的资产产生的资本边际产出(marginal product of capital, or, MPK)，即多投资一单位的资本，生产规模扩大的量。比如假设一个工厂现有10台机器，每天生产100双鞋。增加一台机器，鞋履生产量增加为109双鞋，那么这一台机器投资的资本边际产出就是9双鞋/天。而在金融投资中，与之起相似作用的是分红(dividends)。分红与资本边际产出相似，在物品售卖时未产生价格变化的情况下，它们就成为了收入的主要来源。
There are two primary sources of costs. The most obvious cost is the opportunity cost of capital used in investment projects. The opportunity cost is the cost of the next best alternative forgone, that is, instead of investing the money, investors can save it into savings accounts to earn interest given by the bank. Therefore, once the investor chooses to invest the money into such a project, they automatically give up the chance of earning interest by saving. Thus, the opportunity cost is counted as a significant part of the cost of investment. Be    sides, because of the wear and tear during usage, there will be depreciation, which reduces the value of the capital. 
Similarly, there are mainly two sources of revenue in investment, the change in price and the marginal product of capital (MPK). It is reasonable for us to understand why we take into account the change in price as a source of revenue. If you buy a house for $400,000 and sell a house for $500,000, ceteris paribus, you make a $100,000 profit by the increase of market price. However, the essential part of the investment is the marginal product of capital induced by investment. MPK is the marginal increase in the real unit of production when capital increases by 1 unit. For instance, assume that a factory now occupies ten machines and produces 100 pairs of shoes per day. If now investment occurs and the factory has a new machine. Now it has 11 machines and produces 109 pairs of shoes, so by definition, the MPK of this investment is nine pairs of shoes per day. Dividend carries out a similar function in financial investment or rent revenue for real estate investment. What is common among them is that this is the main source of income when the investing object does not have a change in price.

我们知道，投资是一种将现在的财富转移到未来的行为，投资时买进和卖出的时间是不同的，分红也是在不同时间段分别进行的，因此我们就需要先将所有的成本和收益转换到同一时间点，才能进行盈亏计算。这里我们就需要引进一个概念来帮助计算，叫做贴现现值(present discounted value, or pdv)。投资的贴现现值计算就是将基准时间点设置为投资初始阶段，而将未来的所有收入来源进行贴现计算。之前我们讲过投资具有机会成本，也就是说假如甲投资了一万元，时间为一年，那么这一年的机会成本就是这些钱放在银行里存储所收获的利息。假设年名义利率为5%，那么这笔钱的机会成本就是10000*5%=500元。同样的，如果我们要把未来的收益贴现到现在，就需要做逆运算的处理，即将这笔收益除以(1+年名义利率5%)。如，假设一年之后的收益是500元，那么它的贴现现值就为500/(1+5%) = 476.2元。假如收益是两年之后的呢？因为银行的利率是复合利率，因此存储两年的机会成本就是10000*[(1+5%)²-1]=1025元。同样的，假如两年之后的收益是500元，那么它的贴现现值就为500/(1+5%)²=453.5元。
We know that investment is the behaviour of transferring wealth from today to the future, so that the timing for buying and selling is different, as well as the time periods of dividend or MPK. Thus, we should first convert all the costs and revenues to the same time point, then can we calculate the profit using cost and benefit. To simplify, we introduce the simplest method for intertemporal discounting, the present discounted value, or pdv. By dividing the future revenue with (1+r)^(t-1), in which r is the interest rate, and t is the time period of the occurring revenue, we can convert the future value of assets into today's value. So, the present discounted value is the future value divided by (1+interest rate)^((period-1). For example, an investment occurs in two years (period 3) with $500, so the pdv = $500 / (1+5%)² = $453.5.

2 实验计算

现在我们终于可以来研究种花的艺术了(东风快递）。花的种植分三个阶段，在第一阶段买入，第三阶段卖出。我们以以第一阶段为计算基准，二三阶段的资金变化需要进行贴现计算。
We divide the lifetime of a flower into three periods. In the first period, we buy in flower, and in the third period, the flower matures, and we sell it.

第一阶段的成本为买入价格2500元，肥料150元。收益有浇花得到的30元，和肥料的50元。因此第一阶段利润为-2570元。

第二阶段的成本为肥料150元。收益有浇花得到的30元和肥料的50元。因此第二阶段利润为-20元。

第三阶段的成本为肥料150元，收益有浇花30元，肥料1000元，和最终售出价3000元。因此，第三阶段利润为3880元。

最后，我们需要进行贴现计算。公式如下：

总利润 = (-2570)/1 + (-20)/(1+5%) + (3880)/(1+5%)² = 930.22676元

总成本 = (-2570)/1 + (-150)/(1+5%) + (-150)/(1+5%)² = -2928.912元

因此，利润率即为利润除以总成本乘以100%为31.76%。

Therefore, the total profit equals:

(-2570) / 1 + (-20) / (1+5%) + (3880) / (1+5%)² = $930.22676.

The total cost equals:

(-2570) / 1 + (-150) / (1+5%) + (-150) / (1+5%)² = - $2928.912.

The profit rate equals total profit divided by total cost.

So, the profit rate equals 31.76%.

本文中的贴现计算并未加入β常数来表示消费者行为上对即刻消费的偏好(pure impatience)。更加复杂的贴现现值计算需要加入这一常数，其计算公式为：pdv = Mt / β(1+r)^(1+t)。Mt指的是在t时间的收益，β表示的是对未来消费的不耐心，r表示的是利率，t表示的是收益的时间段。当然，这样一个公式也是过度简化的，更加复杂的公式需要使用偏微分方程进行推导，由于篇幅有限，不作介绍。
The discount value in this paper does not include a parameter β to show the time preference of consumer behaviour. The more complex version which includes this parameter should look like this, pdv = Mt /β (1+r)^(t-1). Mt represents the monetary value of revenue in t period, r represents the interest rate, and t represents the time period of revenue. Further discussion requires mathematical tools that are not in the range of this paper.

下期介绍

敬请期待！
In the next article, we will introduce the pricing mechanism of investing projects: how to price a bond? We will continue to use Plants vs. Zombies as an example. There are plants that we pick up during playing games as a random reward, which do not have a price. Thus, if they can be sold in the market, how can we calculate their prices?

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