% This is lecture no. 8 of BU8201/: Business Finance.

\chapter{The Basics of Capital Budgeting}%
\label{cha:the_basics_of_capital_budgeting}

\section{Introduction}%
\label{sec:introduction}

Capital budgeting is used to decide which projects should or should not be ventured into.
This decision will be decided based on how much money the project will theoretically bring in, as well as how risky the project is.

There are multiple methods to decide how to estimate the yield of a potential project.

\subsection{Normal \& Non-normal Cash Flow Streams}%
\label{sub:normal_&_non_normal_cash_flow_streams}

To estimate the yield of any project, we need to first determine all the cash flows of the project.
There are two main classifications of the patterns of the cash flows in a project.

In a normal cash flow stream, there is first an upfront cost (a negative cash flow stream) followed by a series of positive cash inflows.
In this case, there is only one change of sign.

In a non-normal cash flow stream, most commonly, there is also an upfront cost first, followed by a series of positive cash flows, followed by a cost to close the project.
This results in minimally two changes of sign.

\subsection{Independent \& Mutually Exclusive Projects}%
\label{sub:independent_&_mutually_exclusive_projects}

Another basic way to compare projects is to assess whether they are independent or mutually exclusive.
They are independent if the cash flows of one are unaffected by the acceptance of the other; both projects can be accepted.
They are mutually exclusive if one is accepted, and as a result, the other would have to be rejected; only one of the projects can be accepted.

\section{Payback Period}%
\label{sec:payback_period}

\begin{definition}
	Payback Period.
	The number of years required to recover a project's cost.
	It is calculated by adding the project's cash inflows to its cost until the cumulative cash flow for the project turns positive.
\end{definition}

\subsection{Discounted Payback Period}%
\label{sub:discounted_payback_period}

The discounted payback period is congruent to the payback period, except for the fact that is uses discounted cash flows rather than the raw cash flows.

\section{Net Present Value (NPV)}%
\label{sec:net_present_value_npv_}

\begin{definition}
	Net Present Value (NPV).
	The sum of the present values of all the cash inflows and outflows of a project.
	It is calculated using the following formula:
	\begin{align*}
		\textit{NPV} & = \sum_{t=0}^{n} \frac{\textit{CF}_t}{(1+r)^t}\\
			     & = \textit{CF}_0 + \frac{\textit{CF}_1}{(1+r)^1} + \frac{\textit{CF}_2}{(1+r)^2} + \dotsb + \frac{\textit{CF}_n}{(1+r)^n}
	\end{align*}
	Where $\textit{CF}_0$ represents the initial investment, and $r$ represents the cost of capital, or the WACC as calculated from lecture number \ref{cha:the_cost_of_capital}.
\end{definition}

The rationale behind the NPV method is that it calculates the present value of all the inflows and deducts from it the present value of all the costs, resulting in the net gain in wealth.
If projects are independent, accept all projects which have a positive NPV.
If projects are mutually exclusive, accept the project with the higher NPV.

A NPV profile is the graph where on the $x$-axis, the WACC is varied.
The resulting NPV is represented on the $y$-axis.

\section{Internal Rate of Return (IRR)}%
\label{sec:internal_rate_of_return_irr_}

\begin{definition}
	Internal Rate of Return (IRR).
	It is the discount rate that forces the NPV to be 0.
	It is calculated using the following formula:
	\[
		0 = \sum_{t=0}^{n} \frac{\textit{CF}_t}{(1+\textit{IRR})^t}\\
	.\]
	IRR is the expected annual rate of return that the firm will earn if it invests in the project and receives the given cash flows.
\end{definition}

If the IRR is larger than the WACC, then the project's return exceeds its costs and there is some return left over to boost stockholder's returns.
If the projects are independent, accept all projects which have an IRR that exceeds the WACC.
If the projects are mutually exclusive, accept the project with a higher IRR.

\subsection{Comparing NPV and IRR}%
\label{sub:comparing_npv_and_irr}

Although it may seem as though the process of determining which project to choose for both the IRR and NPV are congruent and only differ in the steps taken, there are differences.

If projects are independent, both methods will yield the same accept/reject decisions.
If the projects are mutually exclusive, and the discount rate is larger than the crossover rate, both methods lead to the same decision.
However, if the projects are mutually exclusive, and the discount rate is lower than the crossover rate, they will lead to different decisions.

In the end, you should base decisions on NPV as NPV is superior.
NPV assumes that reinvestment of intermediate cash flows will be at the WACC rate which is more realistic, whereas IRR assumes that reinvestment will be at the IRR rate.
NPV also gives a better indication of how much shareholders' wealth is affected when evaluating projects with different scale.

\subsection{Modified IRR (MIRR)}%
\label{sub:modified_irr_mirr_}

\begin{definition}
	Modified Internal Rate of Return (MIRR).
	It is the discount rate that causes the present values of a project's terminal value to equal the present value of its costs.
	The terminal value is found by compounding inflows at cost of capital.
	The MIRR is calculated will the following formula:
	\[
		\sum_{t=0}^n \frac{\textit{COF}}{(1+r)^t} = \frac{\sum_{t=0}^n \textit{CIF}(1+r)^{n-t}}{(1+\textit{MIRR})^n}
	.\]
\end{definition}

MIRR assumes reinvestment at WACC rates, but does not solve the problem of scale.
It should be thought of as a IRR-\textit{plus} or a NPV-\textit{minus}.
