Abstract
The information needed for capital budgeting is generally not known with certainty. Therefore, capital-budgeting procedures under conditions of uncertainty should be developed to improve the precision of assessment of the value of risky investment projects. The three alternative methods are introduced to analyze the capital budgeting decisions under uncertainty. Statistical distribution methods, decision tree methods, and simulation methods are three interrelated methods of analysis for capital-budgeting decisions under uncertainty because all of them allow the statistical distributions of the net present values to be explicitly estimated. Under these methods, an interval rather than a point estimate of the expected NPV will be presented. Hence, the stochastic methods discussed in this paper are generally more objective and general than the traditional capital budgeting methods under the assumption of certainty in cash flows.
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Notes
- 1.
Substituting Equation70.2b and letting ρτt = 1, we have
$$ \begin{aligned} {{\sigma}^2}_{{NPV}} & = {{{{[\mathop{\sum}\limits_{{t = 1}}^N \frac{{{\mathop{\sigma}\nolimits_{{Xt}}^2 }}}{{(1 + k\mathop{)}\nolimits^{{2t}} }} + 2 \mathop{\sum}\limits_{{\tau = 1}}^N \mathop{\sum}\limits_{{t = 1}}^N \frac{{{{\sigma}_t}}}{{\left( {(1 + k\mathop{)}\nolimits^t } \right)}}\,\frac{{{{\sigma}_{\tau }}}}{{\left( {(1 + k\mathop{)}\nolimits^{\tau } } \right)}}]}}^{{1/2}}}} \\ & = {{[{{\left( {\sum\limits_{{t = 1}}^N {\frac{{{{\sigma}_t}}}{{{{{\left( {1 + k} \right)}}^t}}}} } \right)}^2}]}^{{1/2}}} = \sum\limits_{{t = 1}}^N {\frac{{{{\sigma}_t}}}{{{{{\left( {1 + k} \right)}}^t}}}} \\ \end{aligned} $$ - 2.
Substituting Equation70.2b and letting ρτt = 0, we have
$$ {{\sigma}_{{NPV}}} = [\sum\limits_{{t = 1}}^N {\frac{{\sigma_t^2}}{{{{{\left( {1 + k} \right)}}^{{2t}}}}}{{]}^{{1/2}}}} $$ - 3.
Assume the net cash flow at time t, Xt, is related to the sources as follows
$$ {{\mathrm{X}}_{\mathrm{t}}} = {{\mathrm{Y}}_{\mathrm{t}}} + {{\mathrm{Z}}^{{(1)}}}_{\mathrm{t}} + {{\mathrm{Z}}^{{(2)}}}_{\mathrm{t}} + \ldots + {{\mathrm{Z}}^{{(\mathrm{m})}}}_{\mathrm{t}} $$where Yt, Z (1)t ,…, and Z (m)t are following normal distribution. The random variables for Yt are mutually independent, Cov(Yt, Yτ) = 0, while the random variables for Z (1)t ,…, and Z (m)t are perfectly positive correlated, Cov(Z (h)t , Z(h) τ) = \( {{\sigma}_{{{{Z}_{\tau }}^{{(h)}}}}}{{\sigma}_{{{{Z}_t}^{{(h)}}}}} \)for h = 1,…,m.
Then variance of NPV is defined as follows:
$$ {{\sigma}^2}_{{NPV}} = {\left[\mathop{\sum}\limits_{{t = 1}}^N \frac{{{\mathop{\sigma}\nolimits_{{Xt}}^2 }}}{{(1 + k\mathop{)}\nolimits^{{2t}} }} + 2 \mathop{\sum}\limits_{{\tau = 1}}^N \mathop{\sum}\limits_{{t = 1}}^N \frac{{{Cov (\mathop{X}\nolimits_{\tau }, \mathop{X}\nolimits_t )}}}{{(1 + k\mathop{)}\nolimits^{\tau } (1 + k\mathop{)}\nolimits^t }}\right]} $$where \( \mathop{{ \sigma }}\nolimits_{{Xt}}^2 = \mathop{{ \sigma }}\nolimits_{{Yt}}^2 + \mathop{{ \ \sigma }}\nolimits_{{{{Z}_t}^{{(1)}}}}^2 + \mathop{{ \ \sigma }}\nolimits_{{{{Z}_t}^{{(2)}}}}^2 + \ldots + \mathop{{ \ \sigma }}\nolimits_{{{{Z}_t}^{{(m)}}}}^2 \)
$$ \eqalign{ Cov (\mathop{X}\nolimits_{\tau }, \mathop{X}\nolimits_t ) = Cov ({{Y}_{\tau }} + {{Z}_{\tau }}^{{(1)}} + {{Z}_{\tau }}^{{(2)}} + \ldots + {{Z}_{\tau }}^{{(m)}},\cr {{Y}_t}+ {{Z}_t}^{{(1)}} + {{Z}_t}^{{(2)}} + \ldots + {{Z}_t}^{{(m)}}) \hfill \cr = Cov (\mathop{Y}\nolimits_{\tau }, \mathop{Y}\nolimits_t ) + Cov ({{Z}_{\tau }}^{{(1)}},{{Z}_t}^{{(1)}}) + \ldots\cr + Cov ({{Z}_{\tau }}^{{(m)}},{{Z}_{\tau }}^{{(m)}}) \hfill \\}<!endgathered> $$Substituting the covariance and variance of Yt, Z (1)t ,…, and Z (m)t , we have
$$ \begin{gathered} {{\sigma}^2}_{{NPV}} = {[\mathop{\sum}\limits_{{t = 1}}^N \frac{{{\mathop{{ ~ \sigma }}\nolimits_{{Yt}}^2 + \mathop{{ ~ \sigma }}\nolimits_{{{{Z}_t}^{{(1)}}}}^2 + \mathop{{ ~ \sigma }}\nolimits_{{{{Z}_t}^{{(2)}}}}^2 + \ldots + \mathop{{ ~ \sigma }}\nolimits_{{{{Z}_t}^{{(m)}}}}^2 }}}{{(1 + k\mathop{)}\nolimits^{{2t}} }} + 2 \mathop{\sum}\limits_{{h = 1}}^m \mathop{\sum}\limits_{{\tau = 1}}^N \mathop{\sum}\limits_{{t = \tau + 1}}^N \frac{{{{{\sigma}_{{{{Z}_{\tau }}^{{(h)}}}}}{{\sigma}_{{{{Z}_t}^{{(h)}}}}}}}}{{(1 + k\mathop{)}\nolimits^{\tau } (1 + k\mathop{)}\nolimits^t }}]} \hfill \\ = {[\mathop{\sum}\limits_{{t = 1}}^N \frac{{{\mathop{{ ~ \sigma }}\nolimits_{{Yt}}^2 }}}{{(1 + k\mathop{)}\nolimits^{{2t}} }} + \mathop{\sum}\limits_{{h = 1}}^m (\mathop{\sum}\limits_{{t = 1}}^N \frac{{{\mathop{{ ~ \sigma }}\nolimits_{{{{Z}_t}^{{(h)}}}}^2 }}}{{(1 + k\mathop{)}\nolimits^{{2t}} }}) + 2\mathop{\sum}\limits_{{h = 1}}^m \mathop{\sum}\limits_{{\tau = 1}}^N \mathop{\sum}\limits_{{t = \tau + 1}}^N \frac{{{{{\sigma}_{{{{Z}_{\tau }}^{{(h)}}}}}{{\sigma}_{{{{Z}_t}^{{(h)}}}}}}}}{{(1 + k\mathop{)}\nolimits^{\tau } (1 + k\mathop{)}\nolimits^t }}]} \hfill \\ = {[\mathop{\sum}\limits_{{t = 1}}^N \frac{{{\mathop{{ ~ \sigma }}\nolimits_{{Yt}}^2 }}}{{(1 + k\mathop{)}\nolimits^{{2t}} }} + \mathop{\sum}\limits_{{h = 1}}^m [\mathop{\sum}\limits_{{t = 1}}^N \frac{{{\mathop{{ ~ \sigma }}\nolimits_{{{{Z}_t}^{{(h)}}}}^2 }}}{{(1 + k\mathop{)}\nolimits^{{2t}} }} + 2\mathop{\sum}\limits_{{\tau = 1}}^N \mathop{\sum}\limits_{{t = \tau + 1}}^N \frac{{{{{\sigma}_{{{{Z}_{\tau }}^{{(h)}}}}}{{\sigma}_{{{{Z}_t}^{{(h)}}}}}}}}{{(1 + k\mathop{)}\nolimits^{\tau } (1 + k\mathop{)}\nolimits^t }}]}] \hfill \\ = {\mathop{\sum}\limits_{{t = 1}}^N \frac{{{\mathop{\sigma}\nolimits_{{yt}}^2 }}}{{(1 + k\mathop{)}\nolimits^{{2t}} }}} + \mathop{\sum}\limits_{{h = 1}}^m {(\mathop{\sum}\limits_{{t = 1}}^N \mathop{{\frac{{\mathop{\sigma}\nolimits_{{zt}}^{{(h)}} }}{{(1 + k\mathop{)}\nolimits^t }})}}\nolimits^2 } \hfill \\ \end{gathered} $$ - 4.
The detailed discussion about capital budgeting decisions can be found in Lee et al. (2009), Lee and Lee (2006), and Lee (2009).
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Lee, CF., Tai, T. (2013). The Statistical Distribution Method, the Decision-Tree Method and Simulation Method for Capital Budgeting Decisions. In: Lee, CF., Lee, A. (eds) Encyclopedia of Finance. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-5360-4_70
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