## 经济代写|宏观经济学代写Macroeconomics代考|ECON1120

2022年10月7日

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## 经济代写|宏观经济学代写Macroeconomics代考|Net present value and the WACC

The best place to start our understanding of investment is to go where all corporate finance books start: investment is decided on the basis of the net present value (NPV) of a project. If a project is started in period 0 and generates a (positive or negative) cash flow of $W_t$ in any period $t$ up until time $T$, the NPV will be given by:

$$N P V=\sum_{t=0}^T \frac{1}{(1+r)^t} W_t,$$
where $r$ is the cost of capital. Typically, one would expect to have a negative cash flow initially, as investment is undertaken, before it eventually turns positive.

The key question is whether this NPV is positive or not. If NPV $>0$, then you should invest; if $\mathrm{NPV}<0$, then you shouldn’t. This sounds simple enough, but this immediately begs the question of what interest rate should be used, particularly considering that firms use a complex mix of financing alternatives, including both equity and debt to finance their capital expenditure (CAPEX) programs. In short, what is the cost of capital for a firm? A very popular measure of the cost of capital is the so-called weighted average cost of capital (WACC), which is defined as
$$W A C C=\alpha_{e q} r_{e q}+\left(1-\alpha_{e q}\right) r_{d e b t},$$
$$\alpha_{\text {eq }}=\left(\frac{\text { Net worth }}{\text { Total Assets }}\right) .$$
Here, $r_{e q}$ is the return on equity (think dividends) and $r_{\text {debt }}$ is the return on debt issued (think interest). This is a very popular model in part because it is easy to compute. Basically, it allocates the cost of equity using as weight the fraction of your assets that the firm finances with equity. The return on equity can, in turn, be easily derived, say from a CAPM regression, as explained in the previous chapter. With a weight equal to thé sharee of ásseets thát you fináncee with debt, the formulá useés thé côst of debt. Fơr this, you may just take the interest cost of the company’s issued debt.

In practice, typically firms go through a planning cycle in which the CFO sets a WACC for the following planning cycle and units decide on those projects that have a return higher than that WACC. Only those that have a return higher than the cost of capital get the green light to go ahead. There are several issues with this procedure. Units tend to exaggerate the benefits of their projects to obtain additional resources, projects take a lot of time, and there are many tax-induced distortions (such as reporting investments as expenses to get a tax credit). A lot of corporate finance is devoted to exploring these and other issues.

## 经济代写|宏观经济学代写Macroeconomics代考|Pindyck’s option value critique

Investment is a decision in which the presence of uncertainty makes a critical difference. This is because investment is mostly irreversible. It follows that there are option-like features to the investment decision that are extremely relevant. Consider, for example, a project with an NPV of zero. Would you pay more than zero for it? Most probably yes, if the return of the project is stochastic and you have the possibility of activating the project in good scenarios. In other words, a zero NPV project has positive value if it gives you the option to call it when, and only when, it makes you money. In that sense, it is just like a call option – i.e. one in which you purchase the right to huy an asset at some later date at a given price. People are willing to pay for a call option. This line of reasoning is critical, for example, in the analysis of mining rights and oil fields. Today an oil field may not be profitable, but it sure is worth more than zero if it can be tapped when energy prices go up. ${ }^1$

One of the most studied implications of this option-like feature is when there is an option value to waiting. This is best described by an example such as the following. Suppose you can make an initial investment of $\$ 2200$, and that gives you the following stochastic payoff:$t=0 \quad t=1 \quad t=2\begin{array}{rr} & q \quad P_1=300 \quad \ldots \ P_0=200 & \ 1-q P_1=100 & \ldots\end{array}$That is, in the first period you make$\$200$ for sure, but from then onward the payoff will be affected by a shock. If the good realisation of the shock occurs (and let’s assume this happens with probability $q$ ), you get $\$ 300$forever. If you are unlucky, you get$\$100$ forever instead. Suppose $q=0.5$ and $r=0.10$. Given this information, the expected NPV of the project, if it is considered at $t=0$, is
$$N P V=-2200+\sum_{t=0}^{\infty} \frac{200}{(1.1)^t}=-2200+2200=\ 0 .$$
In other words, this is a really marginal investment opportunity. But now consider the option of waiting one period, so that the uncertainty gets resolved, and the project only happens if the good state gets realised (which happens with a $50 \%$ probability). Then we have
$$N P V=0.5\left[-\frac{2200}{1.1}+\sum_{t=1}^{\infty} \frac{300}{(1.1)^t}\right]=0.5\left[-\frac{2200}{1.1}+\frac{300}{(1.1)}\left(\frac{1}{1-\frac{1}{1.1}}\right)\right]=500 !$$

# 宏观经济学代考

## 经济代写|宏观经济学代写宏观经济学代考|净现值和WACC

$$N P V=\sum_{t=0}^T \frac{1}{(1+r)^t} W_t,$$

$$W A C C=\alpha_{e q} r_{e q}+\left(1-\alpha_{e q}\right) r_{d e b t},$$
$$\alpha_{\text {eq }}=\left(\frac{\text { Net worth }}{\text { Total Assets }}\right) .$$

## 经济代写|宏观经济学代写宏观经济学代考|Pindyck的期权价值批判

$$N P V=-2200+\sum_{t=0}^{\infty} \frac{200}{(1.1)^t}=-2200+2200=\ 0 .$$
。换句话说，这是一个真正的边际投资机会。但是现在考虑等待一段时间的选择，这样不确定性就会得到解决，项目只会在良好状态实现的情况下进行(发生的概率为$50 \%$)。然后我们有
$$N P V=0.5\left[-\frac{2200}{1.1}+\sum_{t=1}^{\infty} \frac{300}{(1.1)^t}\right]=0.5\left[-\frac{2200}{1.1}+\frac{300}{(1.1)}\left(\frac{1}{1-\frac{1}{1.1}}\right)\right]=500 !$$

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## MATLAB代写

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