% This is lecture no. 9 of BU8201/: Business Finance.

\chapter[Free Cash Flows]{Cash Flow Estimation, Risk Analysis, \& The Optimal Capital Budget}%
\label{cha:cash_flow_estimation_risk_analysis_&_the_optimal_capital_budget}

\section{Cash Flow Estimation}%
\label{sec:cash_flow_estimation}

Accounting figures such as net income and operating income are different cash flows.
What matters most to investors are cash flows and not accounting numbers.

In the capital budgeting context, there should only be consideration for all relevant cash flows; i.e. incremental cash flows that will be incurred if and only if a project is taken up.

\subsection{The Free Cash Flow Equation (FCF)}%
\label{sub:the_free_cash_flow_equation}

\begin{definition}
	Free Cash Flow (FCF).
	They are the cash flows available to shareholders and debtholders less the need for re-investment to operate and produce future cash flows.
	It can be calculated using the following formula:
	\[
	\textit{FCF} = [\textit{EBIT}(1-T) + \text{Depreciation}] - \left[\substack{\text{Capital} \\ \text{Expenditure}} + \Delta \textit{NOWC}\right]
	.\]
	It determines the real cash flows within a project.
\end{definition}

\noindent
Recall from lecture \ref{cha:financial_markets_&_institutions_financial_statements_cash_flow_taxes_&_interest_rates} that the NOWC is calculated as follows:
\begin{align*}
	\substack{\text{Net Operating} \\ \text{Working Capital (NOWC)}}
	& =
	\substack{\text{Operating Current} \\ \text{Assets}}
	-
	\substack{\text{Operating} \\ \text{Current Liabilities}}
	\\
	& =
	\substack{\text{Operating Current} \\ \text{Assets}}
	-
	\left(\substack{\text{Accrued} \\ \text{Wages \& Taxes}} + \substack{\text{Accounts} \\ \text{Payable}}\right)
	\\
	& =
	\substack{\text{Operating Current} \\ \text{Assets}}
	-
	\left(\substack{\text{Current} \\ \text{Liabilities}} + \substack{\text{Notes} \\ \text{Payable}}\right)
\end{align*}

\noindent
Also recall from lecture \ref{cha:financial_markets_&_institutions_financial_statements_cash_flow_taxes_&_interest_rates} that the EBIT is calculated as follows:
\[
\text{Operating income (or EBIT)}
=
\text{Sales Revenues} - \text{Operating Costs}
.\]
And that depreciation and amortisation are important components of operating costs.
Therefore the first square bracket of the FCF equation can be rewritten as follows:
\begin{align*}
	 [\textit{EBIT}(1-T) + \text{Depr.}] & = \left[\left(\substack{\text{Sales} \\ \text{Revenues}} - \substack{\text{Operating} \\ \text{Costs}}\right)(1-T) + \text{Depr.}\right] \\
					     & = \left[\left(\substack{\text{Sales} \\ \text{Revenues}} - \substack{\text{Operating} \\ \text{Costs} \\ \text{Less Depr.}} - \text{Depr.}\right)(1-T) + \text{Depr.}\right] \\
					     & = \left[\substack{\text{Sales} \\ \text{Revenues}}(1-T) - \substack{\text{Operating} \\ \text{Costs} \\ \text{Less Depr.}}(1-T) - \text{Depr.}(T)\right]
\end{align*}

The point of the above is that depreciation is part of the EBIT, and the point of adding it to the end of the first square bracket is that FCF does not consider anything that is not an actual cash transaction.
Depreciation is only on the books and is only realised when disposed or sold.

\subsection{Depreciation of Assets}%
\label{sub:depreciation_of_assets}

When a new project requires a new asset, the purchase of this asset will be brought under capital expenditure in the FCF equation.
After which the asset will most likely depreciate, unless the firm is some kind of really famous firm and there is history associated with it.

Straight line depreciation is where the price of the asset depreciates at a constant rate.
While the asset is owned by the firm, the book value of the asset will drop to reflect the effect of depreciation on the asset.
However, even when the book value of an asset is 0, it may still have a salvage value.

The salvage value will be exercised when the project finishes as one of the terminal cash flows.
There are two terminal cash flows.
Recovery of NOWC and after tax salvage value of assets obtained via capital expenditure.
After tax salvage value is calculated as follows:
\[
	\text{After Tax SV} = \text{SV} - T(\text{SV} - \text{Remaining Book Value})
.\]

\section{Leasing \& Buying}%
\label{sec:leasing_&_buying}

To obtain use of assets, a company can buy the asset or lease the asset.
Leasing is similar to renting.

Say a company has already determined that acquiring the asset is a postitive NPV project.
Now, the firm has to decide how to acquire the asset, it has the following options:
\begin{enumerate}
	\item Lease the asset
	\item Borrow funds to buy the asset
	\item Use available cash on hand to buy the asset
	\item Raise equity to buy asset
\end{enumerate}

\begin{exmp}
	Lease vs. Borrow-and-buy.
	If the firm borrows money to buy the asset:
	\begin{itemize}
		\item New machine costs \$1,200,000.
		\item Can be financed with a 10\% loan to be amortised over four years with end of year repayments.
		\item 3-year MACRS class life; 4-year economic life.
		\item Maintenance of \$25,000 per year, payable at beginning of each year.
		\item Salveage value in year 4 = \$125,000.
	\end{itemize}

	\noindent
	If the firm leases the asset:
	\begin{itemize}
		\item 4-year lease includes maintenance.
		\item Lease payment is \$340,000 per year, payable at beginning of each year.
		\item Tax rate = 40\%.
	\end{itemize}
	Cost of debt should be used to discount the cash flows:
	\[
		r = r_d(1-T)
	.\]
	Cost of owning analysis: Depreciation Schedule.
	\bigbreak
	\begin{center}
		\begin{tabular}{c c r r}
			Year & MACRS Rate & Depreciation Expense & End of Year Book Value\\
			\hline
			1 & 0.33 & \$396,000 & \$804,000\\
			2 & 0.45 & 540,000 & 264,000\\
			3 & 0.15 & 180,000 & 84,000\\
			4 & 0.07 & 84,000 & 0\\
			\hline
			  & 1.00 & \$1,200,000 &\\
		\end{tabular}
	\end{center}

	\noindent
	Cash flows(the below is in thousands of dollars):

	\begin{center}
		\begin{tabular}{l c c c c c}
			Year & 0 & 1 & 2 & 3 & 4\\
			\hline
			Cost of asset & -1,200 & & & &\\
			Depr. Tax savings & & 158.4 & 216.0 & 72.0 & 33.6\\
			Maintenance (AT) & -15.0 & -15.0 & -15.0 & -15.0 &\\
			Residual Value (AT) & & & & & 75.0\\
			\hline
			Free Cash Flow & -1,215.0 & 143.4 & 201.0 & 57.0 & 108.6\\
		\end{tabular}
	\end{center}
	Therefore, the present value of th cost of owning at 6\% discount rate is -\$766.948.
	Note that depreciation is tax deductible, as well as the maintenance payment.

	\bigbreak
	\noindent
	Cost of Leasing Analysis:
	\begin{center}
		\begin{tikzpicture}
			\draw [|->](0,0)--(4.5,0);
			% The above describes the line; from cord 1 to 2, such that |---> is the design
			\draw (0,-.1)--(0,.1);
			\draw (1,-.2)--(1,.2);
			\draw (2,-.2)--(2,.2);
			\draw (3,-.2)--(3,.2);
			\draw (4,-.2)--(4,.2);
			% The above generates the tickmarks
			\node[align=left,above] at (0,0.2) {0};
			\node[align=left,above] at (1,0.2) {1};
			\node[align=left,above] at (2,0.2) {2};
			\node[align=left,above] at (3,0.2) {3};
			\node[align=left,above] at (4,0.2) {4};
			% The above are the numbers representing time above the line.
			\node[align=left,below] at (0,-0.2) {-\$204};
			\node[align=left,below] at (1,-0.2) {-\$204};
			\node[align=left,below] at (2,-0.2) {-\$204};
			\node[align=left,below] at (3,-0.2) {-\$204};
			% The above describes the payments below the time line.
		\end{tikzpicture}
	\end{center}
	Each lease payment of \$340 is deductible, so the after-tax cost of the lease is:
	\[
		(1-T)(\$340) = \$204
	.\]
	Therefore, the present value cost of leasing at a discount rate of 6\% is -\$749.294.
	The net advantage of leasing is therefore \$17.654.
	The firm should lease the asset.
\end{exmp}
