Using managerial revenue and cost estimates to value early stage real option investments

Abstract

Real options analysis is widely recognized as a superior method for valuing projects with managerial flexibilities. Yet, its adoption remains limited due to varied difficulties in its implementation. In this work, a real options approach that utilizes managerial cash-flow estimates to value early stage project investments is proposed. Our model is based on the assumption that managers can provide pessimistic, likely and optimistic sales and gross margin percent estimates. A market sector indicator is introduced, which is assumed to be correlated to a tradeable market index, which drives the project’s sales estimates. Another indicator, assumed partially correlated to the sales indicator drives the gross margin percent estimates. In this way a cash-flow process can be modelled that is partially correlated to a traded market index. This provides the mechanism for valuing real options of the cash-flow in a financially consistent manner. The method requires minimal subjective input of model parameters and is very easy to implement, based on simple managerial estimates.

Appendices

Appendix 1: Triangular Distributions

In this Appendix, we collect some standard results on triangular distributions. Assuming a triangular density function for a random variable, A, we have

$$\begin{aligned} f_A(y) = {\left\{ \begin{array}{ll} {\displaystyle a\frac{y-y_-}{y_0-y_-}}, &{} y_-< y \le y_0, \\ {\displaystyle a\frac{y_+-y}{y_+-y_0}}, &{} y_0 < y \le y_+, \\ 0, &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

(22)

Here, the normalizing constant $a=2/(y_+-y_-)$. The corresponding distribution function is

$$\begin{aligned} F_A(y) = {\left\{ \begin{array}{ll} 0, &{} y \le y_-, \\ {\displaystyle \frac{a}{2}\frac{(y-y_-)^2}{y_0-y_-}}, &{} y_-< y \le y_0, \\ {\displaystyle 1 - \frac{a}{2}\frac{(y_+-y)^2}{y_+-y_0}}, &{} y_0 < y \le y_+, \\ 1, &{} y > y_+. \end{array}\right. } \end{aligned}$$

(23)

It’s inverse is given by

$$\begin{aligned} F_A^{-1}\left( p \right) = {\left\{ \begin{array}{ll} {\displaystyle y_- + \sqrt{(y_+-y_-)(y_0-y_-)\, p}}\;, &{} p \le p_c, \\ {\displaystyle y_+ - \sqrt{(y_+-y_-)(y_+-y_0)\, (1-p)}}\;, &{} p > p_c, \end{array}\right. } \end{aligned}$$

(24)

where the critical point $p_c =(y_0-y_-)/(y_+-y_-)$.

Figure 7 shows a typical configuration. Note that $y_-$, $y_0$ and $y_+$ carry the interpretation of the project value in a pessimistic or poor scenario, a likely or moderate scenario and an optimistic or good scenario.

Fig. 7

A random variable with a triangular pdf

Appendix 2: Proof of results

In this Appendix we provide concise proofs of Propositions 1 through 4. Where applicable the proofs are done for sales only and, for convenience we write $\varphi ^S(.)$ as $\varphi (.)$.

1.1 Proof of Proposition 1

We seek $\varphi (.)$ such that ${\mathbb {P}}(\varphi (X_T)\le s | \mathcal F_0) = F^*(s)$. Since,

$$\begin{aligned} X_T|_{\mathcal F_0} \mathop {=}\limits ^{d} X_0 + \sqrt{T} Z \quad \text {where} \quad Z\underset{\scriptstyle {\mathbb {P}}}{\sim } \mathcal N(0,1), \end{aligned}$$

we have that

$$\begin{aligned} {\mathbb {P}}(\varphi (X_T)\le s | \mathcal F_0)&= {\mathbb {P}} \left( X_0 + \sqrt{T} Z \le {\varphi }^{-1}(s) \right) \\&= \varPhi \left( \frac{{\varphi }^{-1}(s)-X_0}{\sqrt{T}}\right) \triangleq F^*(s)\,. \end{aligned}$$

Consequently, if $F^*(.)$ is invertible then

$$\begin{aligned} \varphi (X_T) = F^{*-1}\left( \varPhi \left( \frac{X_T-X_0}{\sqrt{T}}\right) \right) \end{aligned}$$

and the proof is complete.$\square$

1.2 Proof of Proposition 2

We define the joint distribution function of sales and GM% as U(s, m). Thus,

$$\begin{aligned} U(s,m)=P(S\le s,\,M \le m) \end{aligned}$$

and recalling that $S=\varphi ^S(X_T)$ and $M=\varphi ^M(Y_T)$, and $\varphi ^S(X_T) = F^{*-1}\left( \varPhi \left( \frac{X_T-X_0}{\sqrt{T}}\right) \right)$ and $\varphi ^M(Y_T) = G^{*-1}\left( \varPhi \left( \frac{Y_T-Y_0}{\sqrt{T}}\right) \right)$, we have

$$\begin{aligned} U(s,m)&=P\left( F^{*-1}\left( \varPhi \left( \frac{X_T-X_0}{\sqrt{T}}\right) \right) \le s, \, G^{*-1}\left( \varPhi \left( \frac{Y_T-Y_0}{\sqrt{T}}\right) \right) \le m\right) \\&=P\left( \frac{X_T-X_0}{\sqrt{T}} \le \varPhi ^{-1}(F^*(s)),\, \frac{Y_T-Y_0}{\sqrt{T}} \le \varPhi ^{-1}(G^*(m)) \right) \\&=\varPhi _{\varOmega _{\rho _{SM}}} \left( \varPhi ^{-1}(F^*(s)), \, \varPhi ^{-1}(G^*(m)) \right) \end{aligned}$$

where $\varPhi _{\varOmega _{\rho }}$ represents the standard joint normal distribution for correlation $\rho$. The PDF $u(s,m) = \partial _{s,m} U(s,m)$ can now be computed to give

$$\begin{aligned} u(s,m)=\phi _{\varOmega _{\rho _{SM}}} \left( \varPhi ^{-1} \left( F^*(s) \right) , \varPhi ^{-1} \left( G^*(m) \right) \right) \frac{f^*(s)}{\phi \left( \varPhi ^{-1}\left( F^*(s)\right) \right) } \frac{g^*(m)}{\phi \left( \varPhi ^{-1}\left( G^*(m)\right) \right) } \end{aligned}$$

where $\phi _{\varOmega _{\rho }}$ represents the standard joint normal density with correlation $\rho$, and $\phi$ is the standard normal density.$\square$

1.3 Proof of Proposition 3

Here, we compute the probability distribution of the project value conditional on the market sector indicator $S_t$ at time t. To this end we have

$$\begin{aligned} F_{S_T | X_t}&= {\mathbb {P}}( S_T = \varphi (X_T) \le s | X_t ) \\&={\mathbb {P}}\left( X_t + \sqrt{(}T-t) Z \le \varphi ^{-1}(s) \right) \\&=\varPhi \left( \frac{\varphi ^{-1}(s) - X_t}{\sqrt{T-t}} \right) \\&=\varPhi \left( \frac{\varPhi ^{-1}\left( F^*(s) \right) \sqrt{T}+X_0-X_t}{\sqrt{T-t}} \right) \end{aligned}$$

where $Z\underset{\scriptstyle {\mathbb {P}}}{\sim } \mathcal N(0,1)$ and the proof is complete.$\square$

1.4 Proof of Proposition 4

Condition on the market sector index at time t, the risk-neutral probability distribution function can be computed as follows

$$\begin{aligned} \widehat{F}_{S_T | X_t }(s)&= {\mathbb {Q}}\left( \varphi (X_{T}) \le s \, | X_t \right) \\&= {\mathbb {Q}}\left( X_t + \widehat{\nu }(T-t)+ \sqrt{T-t} Z \le \varphi ^{-1}(s) \right) \\&= {\mathbb {Q}} \left( Z \le \frac{\varPhi ^{-1}\left( F^*(s) \right) \sqrt{T} + X_0 - X_t - \widehat{\nu }(T-t)}{\sqrt{T-t}} \right) \end{aligned}$$

Here, $Z\underset{\scriptstyle {\mathbb {Q}}}{\sim } \mathcal N(0,1)$. Finally,

$$\begin{aligned} \widehat{F}_{S_T|X_t}(s) = \varPhi \left( \frac{\varPhi ^{-1}\left( F^*(s) \right) \sqrt{T}+X_0-X_t-\widehat{\nu }(T-t)}{\sqrt{T-t}} \right) \end{aligned}$$

and the proof is complete.$\square$