Business Models: Applications to Capital Budgeting, Equity Value, and Return Attribution

Abstract

This chapter describes a business model in a contingent claim modeling framework. The model defines a “primitive firm” as the underlying risky asset of a firm. The firm’s revenue is generated from a fixed capital asset and the firm incurs both fixed operating costs and variable costs. In this context, the shareholders hold a retention option (paying the fixed operating costs) on the core capital asset with a series of growth options on capital investments. In this framework of two interacting options, we derive the firm value.

The chapter then provides three applications of the business model. Firstly, the chapter determines the optimal capital budgeting decision in the presence of fixed operating costs and shows how the fixed operating cost should be accounted by in an NPV calculation. Secondly, the chapter determines the values of equity value, the growth option, and the retention option as the building blocks of primitive firm value. Using a sample of firms, the chapter illustrates a method in comparing the equity values of firms in the same business sector. Thirdly, the chapter relates the change in revenue to the change in equity value, showing how the combined operating leverage and financial leverage may affect the firm valuation and risks.

Appendices

Appendix 1: Derivation of the Risk-Neutral Probability

The risk-neutral probabilities p(n,i) can be calculated from the binomial tree of V _p . Let V _p (n,i) be the firm value at node (n,i). In the upstate, the firm value is

$$ {V}_p\left(n,i\right)=\frac{ GRI\left(n,i\right)\times CA}{\rho } $$

By the definition of the binomial process of the gross return on investment,

$$ {V}_p\left(n+1,2i+1\right)={V}_p\left(n,i\right){e}^{\sigma } $$

Further, the firm pays a cash dividend of C _u = V _p (n,i) × ρ × e ^σ. Therefore, the total value of the firm V ^u _p , an instant before the dividend payment in the upstate, is

$$ {V}_p^u={V}_p\times \left(1+\rho \right)\times {e}^{\sigma } $$

(75.28)

Similarly, the total value of the firm V ^d _p , an instant before the dividend payment in the downstate, is

$$ {V}_p^d={V}_p\times \left(1+\rho \right)\times {e}^{-\sigma } $$

(75.29)

Then the risk-neutral probability p is defined as the probability that ensures the expected total return is the risk-free return.

$$ p\times {V}_p^u+\left(1-p\right)\times {V}_p^d=\left(1+{R}_F\right)\times {V}_p $$

(75.30)

Substituting V _p , V ^u _p , V ^d _p into equation above and solve for p, we have

$$ p=\frac{A-{e}^{-\sigma }}{e^{\sigma }-{e}^{-\sigma }} $$

(75.31)

Where $A=\frac{1+{R}_F}{1+\rho }$.

Appendix 2: The Model for the Fixed Operating Cost at Time T

When the firm may default the fixed operating cost, the fixed operating cost can be viewed as a perpetual debt of a risk bond. The valuation formula of the perpetual debt is given by Merton (1973).

$$ \Phi \left(V,\infty \right)=\frac{ FC}{r_f}\left\{1-\frac{{\left(\frac{2 FC}{\sigma^2V}\right)}^{\frac{2{r}_f}{\sigma^2}}}{\varGamma \left(2+\frac{2{r}_f}{\sigma^2}\right)}M\left(\frac{2{r}_f}{\sigma^2},2+\frac{2{r}_f}{\sigma^2},\frac{-2 FC}{\sigma^2V}\right)\right\}, $$

Where

• V = the primitive firm value

• FC = fixed cost per year

• r _f = risk free rate

• Γ = the gamma function (defined in the footnote)

• σ = the standard deviation of $\tilde{ GRI}$

• M (•) = the confluent hypergeometric function (defined in the footnote)

$$ \begin{array}{l}M\left(a,2+a,-\frac{2 FC}{\sigma^2V}\right)\\ {}=\frac{1}{b{r}_f}{e}^{\frac{b}{V}}\left[\begin{array}{l}-\left(a+a\right) bFC{\left(\frac{b}{V}\right)}^a+\\ {}{e}^{\frac{b}{V}} FC\left( aV\Gamma \left(2+a\right.\right)+\left(1+a\right)\left(b- aV\right)\Gamma \left(1+a,\frac{b}{V}\right)\end{array}\right]\end{array} $$

where

$$ \begin{array}{l}a=\frac{2{r}_f}{\sigma^2},b=\frac{2 FC}{\sigma^2},\Gamma (x)={\displaystyle {\int}_0^{\infty }{t}^{\kern0.1em x-1}{e}^{-t} dt,}\\ {} and\ \Gamma \left(a,x\right)={\displaystyle {\int}_0^{\infty }{t}^{\kern0.1em x-1}{e}^{-t} dt.}\end{array} $$

Appendix 3: The Valuation Model Using the Recombining Lattice

In this model specification, we assume that the GRI stochastic process follows a recombining binomial lattice:

$$ \begin{array}{l} GRI\left(n,j\right)=q\times GRI\left(n+1,j+1\right)+\left(1-q\right)\times GRI\left(n+1,j\right),\\ {}q=\frac{1-{e}^{-\sigma }}{e^{\sigma }-{e}^{-\sigma }},\mathrm{where}\kern0.5em \sigma\ \mathrm{is}\ \mathrm{the}\ \mathrm{volatility}\ \mathrm{of}\ \mathrm{GRI},\end{array} $$

where n = 0, 1,… T and j = 0, …, n.

At time T, the horizon date, consider the node (T, j); j is the state on a recombining lattice. Suppose that the firm has made k investments in the period T, where 0 ≤ k ≤ T − 1. The firm value is given by Eq. 75.32:

$$ \begin{array}{l}V\left(T,j;k\right)= Max\left[\begin{array}{l}\frac{ GRI\left(T,j\right)\cdot \left( CA+\left(k+1\right)I\right)}{\rho }- FC(T)+\\ {}\left( GRI\left(T,j\right)\times \left( CA+\left(k+1\right)I\right)- FC-I\right),\\ {}\frac{ GRI\left(T,j\right)\cdot \left( CA+ kI\right)}{\rho }- FC(T)+\\ {}\left( GRI\left(T,j\right)\times \left( CA+ kI\right)- FC\right),0\end{array}\right],\\ {}\mathrm{for}\ \mathrm{k}=0,\dots, \mathrm{T}\ \mathrm{and}\ \mathrm{j}=0,\dots, \mathrm{T}\end{array} $$

(75.32)

and CA is the initial capital asset.

Now we roll back one period. We then compare the firm value with or without making an investment I. Given that the firm at the end of the period T − 1 has already invested k times and would not invest at time T − 1, the firm value is

$$ \begin{array}{l}\frac{p\cdot V\left(T,j+1;k\right)+\left(1-p\right)\cdot V\left(T,j;k\right)}{1+{R}_F}\\ {}+ GRI\left(T-1,j\right)\cdot \left( CA+ kI\right)- FC\end{array} $$

(75.33)

If the firm at that time invests in the capital asset, then the firm value is

$$ \begin{array}{l}\frac{p\cdot V\left(T,j+1;k\right)+\left(1-p\right)\cdot V\left(T,j;k\right)}{1+{R}_F}\\ {}+ GRI\left(T-1,j\right)\cdot \left( CA+\left(k+1\right)I\right)- FC-I\end{array} $$

(75.34)

Optimal decision is to maximize the values of the firm under three possible scenarios: taking the investment, not taking the investment, or defaulting. Therefore, the value of the firm at the node (T − 1, j) with k investments is

$$ \begin{array}{l}v=V\left(T-1,j;k\right)\\ {}= Max\left[\begin{array}{l}\frac{p\cdot V\left(T,j+1;k+1\right)+\left(1-p\right)\cdot V\left(T,j;k+1\right)}{1+{R}_F}+\\ {} GRI\left(T-1,j\right)\cdot \left( CA+\left(k+1\right)I\right)- FC-I,\\ {}\frac{p\cdot V\left(T,j+1;k\right)+\left(1-p\right)\cdot V\left(T,j;k\right)}{1+{R}_F}+\\ {} GRI\left(T-1,j\right)\cdot \left( CA+ kI\right)- FC,0\end{array}\right],\\ {} for=0,1,\dots, T-1, and\ j=0,1,\dots, T-1.\end{array} $$

(75.35)

Now we can determine the firm value recursively for each n, n = T − 1, T − 2,…1.

At the initial period,

$$ \begin{array}{l}V\left(T,j;k\right)=\\ {} Max\left[\begin{array}{l}\frac{p\cdot V\left(T,j+1;k\right)+\left(1-p\right)\cdot V\left(T,j;k\right)}{1+{R}_F}+\\ {} GRI\left(T,j\right)\cdot \left( CA+ kI\right)- FC,0\end{array}\right],\\ {} for\ T=j=k=0.\end{array} $$

(75.36)

The firm value at the initial time can then be derived by recursively rolling back the firm value to the initial point, where n = 0. We follow the method of the fiber bundle modeling approach in Ho and Lee (2004a).

To illustrate, we use a simple numerical example. Following the previous numerical example, we assume that the GRI is 0.1, the capital asset CA is 30, the risk-free rate and the cost of capital are both 10 %, the risk-neutral probability is 0.425557, the volatility 30 %, the fixed cost FC is 3, and finally the investment is 1.

Given the above assumption, the binomial process is presented below.

The binomial lattice of GRI

Time	0	1	2	3
j	GRI
3				0.245960311
2			0.18221188	0.134985881
1		0.134985881	0.1	0.074081822
0	0.1	0.074081822	0.054881164	0.040656966

Given the GRI binomial lattice, we can now derive the firm value lattices. The values are derived by backward substitution. The firm value depends on the capital asset level CA, the state j, and the time n.

		Firm value	V(n,j,CA)
State j					CA
3				55.2836	32
2				14.9999	32
1				0.0000	32
0				0.0000	32
j
3				52.5780	31
2			31.0516	13.5150	31
1			5.3286	0.0000	31
0			0.0000	0.0000	31
j
3				49.8725	30
2			29.0473	12.0302	30
1		14.9802	4.6541	0.0000	30
0	6.329610	1.0230	0.0000	0.0000	30
Time n	0	1	2	3

At time 3, the firm values are derived by Eq. 75.32 for each level of outstanding capital asset level at time 3, an instant before the investment decision. Then the firm values for time 2 are derived by Eq. 75.34. Once again, the firm value depends on the outstanding CA level. The firm value at time 0 does not involve any investment decision, and therefore, it is derived by rolling back from the firm values where the CA level is 30.

Appendix 4: Input Data of the Model

The input data of the model are derived from the balance sheets and income statements of the firms.

IS	Target	Lowe’s	Wal-Mart	Darden
Revenue	39,888	22,111.1	217,799	4,021.2
Costs of sales	27,246	15,744.2	168,272	3,127.7
Gross profit	12,642	6,366.9	49,527	893.5
Gross profit margin (m)a	0.3169	0.2880	0.2274	0.2222
Fixed cost	8,883	4,053.2	36,173	407.7
Depreciation	1,079	534.1	3,290	153.9
Interest cost	464	180	1,326	31.5
Other incomes	0	24.7	2,013	0.9
Pretax incomes	2,216	1,624.3	10,751	301.3
Tax	842	601	3,897	104.2
Effective tax ratio (τ)b	0.3800	0.3700	0.3625	0.3458

1. ^aGross profit/Revenue
2. ^bTax/Pretax incomes

Balance sheet	Target	Lowe’s	Wal-Mart	Darden
Capital assets	13,533	8,653.4	45,750	1,779.5
Gross return on invest (GRI)a	2.9475	2.5552	4.7606	2.2597
LTDb	8,088	3,734	18,732	517.9
Book equity	7,860	6,674.4	35,102	1,035.2

1. ^aInitial GRI without the investment = Revenue/Capital assets
2. ^bWe assume that the firms have only one bond. This assumption can be relaxed using the information of the debt structure of a firm

Market information	Target	Lowe’s	Wal-Mart	Darden
Sharesa	902.8	775.7	4,500	176
Stock pricea	44.41	46.07	59.98	18.6
Market capitalization (equity)a	40,093	35,736	269,910	3,274
Risk free rate (Rf)b	0.06	0.06	0.06	0.06
Coupon rateb	0.06	0.06	0.06	0.06
Max investc	2,115	2,060.5	7,000	201

1. ^amarket data
2. ^bWe assume that the risk free rate and coupon rate are 6 %
3. ^cWe use the capital expenditure in cash flow statements as the Max invest