Because the cost of the old machine has already been incurred and the machine has no resale value, it is irrelevant to the value of initial investments in the upgrade. Meanwhile, the expert service fee for assessing the upgrade has incurred and is irrelevant. So the value of the investment in the machine upgrade =  $700,000

Cash inflows relevant to this upgrade include the cost reduction amount of the increased unit sales during the upgrade’s life. For the worst state of economy, it should be  40,000 * $3 = $120,000 per year for 8 years. For the average state of economy, it should be 70,000 * $4 = $280,000 per year for 6 years. For the best state of economy, it should be 100,000 * $7 = $700,000 per year for 4 years.

Thus, we can calculate the NPV of the upgrade for each state of the economy. The company’s cost of capital is 8 percent. 

We can get the PVIFA (Present Value Interest Factor of Annuity) for all three states: PV of $1 annuity =1/r-(1/r*(1+r)^n), where r = 8%, n = 8,6,4 respectively

For the worst state of economy, PV of $1 annuity = 5.7466, NPV = -700,000 + 120,000 * 5.7466 = -10,403

For the average state of economy, PV of $1 annuity = 4.6229, NPV = -700,000 + 280,000 * 4.6229 = 594,406

For the best state of economy, PV of $1 annuity = 3.3121, NPV = -700,000 +700,000 * 3.3121 = 1618,489

1. Using the average state as the base case, we can conduct the sensitivity analysis as follows:

1) Initial Outlay.

NPV = 0 = (IO)+ PV(CFs)

IO = PV(CFs) = 280,000 * 4.6229 = $1294,406

Answer: (1294,406-700,000)/700,000 = 84.91%

2) Unit sales

NPV = 0 = (700,000)+ 4.6229 * (Increase in unit sales * 4)

Increase in unit sales = (700,000/4.6229)/4 =37855

Answer: (37855-70000)/70000 = -45.92%

3) Cost of capital

NPV = 0 = (700,000)+ Annuity factor * 280,000

Annuity factor = 2.5 = 1/r-(1/r*(1+r)^6) 

r = 16%

Answer: (16%-8%)/8% =100%

4) Duration

Payback period = 2+(700,000-560,000)/280,000 = 2.5

Answer: (2.5-6)/6 = -58.33%

3.

The update should proceed for the following 2 reasons:

1) According to the information, the average state is the most likely outcome, then it is very likely that the update would bring a positive NPV of $594,406, and in the very rare case of the worst state of economy, the NPV is negatively, but a relatively small amount.

2) Because the sensitivity analysis in question 2 shows that all the variables are not very sensitive, it is likely to maintain a positive NPV even if there are fluctuations.

So far, we only discussed the NPV of upgrading, but there could be other measures to evaluate this investment, such as IRR and payback period. For the further investigation to improve the quality of this recommendation, I suggest they could compute these measures to see whether it accords with our recommendation.

评估费已经支付，属于Sunk costs，不会对未来的现金流造成影响。所以不算。

没事，有用就好。

Q1:

2.5 =[(1/r)－(1/(r×(1+r)^6) ]  

r=16%

请问这个保留两位小数的话是什么结果呢？我算不出来呢

这个我也不是计算的 我是直接在annuity table中找的最近值，可以Google，有不同精确度的。这里看Time period =6时，annuity = 2.5左右，得到的就是r= 16%附近。

不用会算，会看table像我一样找出来就行。

Q2：

2) Unit sales

NPV = 0 = (700,000)+ 4.6229 * (Increase in unit sales * 4)

Increase in unit sales = (700,000/4.6229)/4 =37855

Answer: (37855-70000)/70000 = -45.92%

这个是不是应该这样才是正确的：Answer: (37855-700,000)/700,000 = -94.59%

不对，700,000是investment value,单位是$，这里的37855是 unit sales，单位是（个），这是两个东西。你看题目，70,000是average state本身的预估increase in unit sales。

Increase in unit sales * 4=PMT, 我这样理解对吗？

可以这样理解，就是每年的net cash flows

Q3:请问下次怎样才能找到你呢？你解题的非常清楚呢，容易看懂

平台上还有很多其他优秀的老师的。有缘再见啦。

Q1: Part b : sensivity analysis

Saving in unit production cost=$4

有必要把这一点作为变量吗？再求出它的变化情况？

这个没有太大的实际意义，但加上也是可以的。

Q2:

Part c

计算各种经济情况的概率，以获得真正的预期净现值。

请问这点需要怎么算呀？

这个删掉吧。不过这个的意思就是expected value的定义。expected outcome = outcome 1 * probability of outcome 1 + outcome 2 * probability of outcome 2 +……文中不是有三种state of economy，也就是三种outcome嘛。但这个建议可以不写，不是什么考点。就写建议IRR，payback period验证。

不用计算也计算不了（信息不足），就说你建议这样子further investigate。一般来说，还是以average state去看，因为它是可能性最大的情况

你看题目： suggest further investigations to improve the quality of the recommendation. 你只用说方法。再说了已知信息不足，也计算不了。你可以Google一下计算这些IRR和pb period所需要的信息有哪些。