The effect of synergy enhancement on information technology portfolio selection

Abstract

This paper investigates how firms can use synergy to optimize their information technology portfolios. We begin by developing a framework for the portfolio selection by identifying three types of information technology synergy. Next, we use this framework to examine the impact of different types of synergy on the portfolio selection. Analytical models are developed to illustrate the roles of different types of the synergy, and analytical and computational methods are used to investigate the impact of the synergy. The analysis in this paper provides conditions in which synergy enhancement results in a more efficient or a less efficient portfolio. Our study establishes that firms with higher risk thresholds are more likely to obtain more efficient information technology portfolios by enhancing synergy, whereas firms with lower risk thresholds are less likely to benefit from enhancing synergy.

Appendix

Proof for Proposition 1A

The expected return from a portfolio (1 − x, x) is given by

$$ RT(1 - x,x) = r_{1} (1 - x) + r_{2} x + \beta r_{1} x(1 - x), $$

where $\beta = \beta_{12} .$

One way of obtaining the efficient frontier is to require $RT(1 - x,x) = R_{0} ,$ where R ₀ is a particular expected return desired and then obtain the variance. For a two-fund problem, $RT(1 - x,x) = R_{0}$ is sufficient to determine the portfolio (1 − $x^{*}$,$x^{*}$).

From $RT(1 - x,x) = R_{0} ,$ we have

$$x = \frac{1}{{2\beta r_{1} }}\left[ {r_{2} - r_{1} + \beta r_{1} \pm \sqrt {(r_{2} - r_{1} + \beta r_{1} )^{2} - 4(R_{0} - r_{1} )\beta r_{1} } } \right]. $$

We assume that β is a small number; in this case, x has only one solution within the interval [0,1]:

$$ x^{*} = \frac{1}{{2\beta r_{1} }}\left[ {r_{2} - r_{1} + \beta r_{1} - \sqrt {(r_{2} - r_{1} + \beta r_{1} )^{2} - 4(R_{0} - r_{1} )\beta r_{1} } } \right]. $$

Consider $R_0 = r_1+\varepsilon$ for an arbitrarily small positive number $\varepsilon > 0.$ Then

$$ x^{*} = 1 - \frac{\varepsilon }{{r_{2} - r_{1} (1 + \beta_{12} )}} + O(\varepsilon^{2} ). $$

Then the variance v is

$$ \begin{aligned} v & = x^{{*2}} \sigma _{2}^{2} + 2x^{*} (1 - x^{*} )(1 + \beta x^{*} )\sigma _{1} \sigma _{2} \rho + (1 - x^{*} )^{2} (1 + \beta x^{*} )^{2} \sigma _{1}^{2} \\ & = \sigma _{2}^{2} + 2\varepsilon \frac{{\sigma _{1} \sigma _{2} \rho (1 + \beta ) - \sigma _{2}^{2} }}{{r_{2} - r_{1} (1 + \beta )}} + O(\varepsilon ^{2} ). \\ \end{aligned} $$

Now we obtain competitive statistics for the variance with respect to β:

$$ \frac{\partial v}{\partial \beta } = 2\varepsilon \frac{{\sigma_{1} \sigma_{2} \rho r_{2} - \sigma_{2}^{2} r_{1} }}{{[r_{2} - r_{1} (\beta + 1)]^{2} }} + O(\varepsilon^{2} ). $$

Thus, when $\frac{{r_{1} /\sigma_{1} }}{{r_{2} /\sigma_{2} }} > \rho ,\frac{\partial v}{\partial \beta } < 0.$ □

Proof for Proposition 1B

The expected return from a portfolio (1 − x, x) is given by

$$ RT(1 - x,x) = r_{1} (1 - x) + r_{2} x + \beta r_{1} x(1 - x), $$

where $\beta = \beta_{12} .$

One way of obtaining the efficient frontier is to require $RT(1 - x,x) = R_{0} ,$ where R ₀ is a particular expected return desired and then obtain the variance. For a two-fund problem, $RT(1 - x,x) = R_{0}$ is sufficient to determine the portfolio (1 − $x^{*}$, $x^{*}$)

From $RT(1 - x,x) = R_{0}$, we have

$$ x = \frac{1}{{2\beta r_{1} }}\left[ {r_{2} - r_{1} + \beta r_{1} \pm \sqrt {(r_{2} - r_{1} + \beta r_{1} )^{2} \, - \, 4(R_{0} - r_{1} )\beta r_{1} } } \right]. $$

We assume that β is a small number; in this case, x has only one solution within the interval [0,1]:

$$ x^{*} = \frac{1}{{2\beta r_{1} }}\left[ {r_{2} - r_{1} + \beta r_{1} - \sqrt {(r_{2} - r_{1} + \beta r_{1} )^{2} \, - \, 4(R_{0} - r_{1} )\beta r_{1} } } \right]. $$

Consider $R_{0} = r_{1} + \varepsilon$ for an arbitrarily small positive number $\varepsilon > 0.$ Then $x^{*} = \frac{\varepsilon }{{r_{2} - r_{1} (1 - \beta_{12} )}} + O(\varepsilon^{2} ).$

Then the variance v is

$$ \begin{gathered} v = x^{*2} \sigma_{2}^{2} + 2x^{*}(1 - x^{*})(1 + \beta_{{}} x^{*})\sigma_{1}^{{}} \sigma_{2}^{{}} \rho + (1 - x)^{*2} (1 + \beta x)^{*2} \sigma_{1}^{2} \\ = \sigma_{1}^{2} + 2\varepsilon \frac{{\sigma_{1}^{{}} \sigma_{2}^{{}} \rho + (\beta_{{}} - 1)\sigma_{1}^{2} }}{{r_{2} - r_{1} (1 - \beta )}} + O(\varepsilon^{2} ). \\ \end{gathered} $$

Now we obtain competitive statistics for the variance with respect to β:

$$ \frac{\partial v}{\partial \beta } = 2\varepsilon \frac{{\sigma_{1} \sigma_{2} \rho r_{2} - \sigma_{1}^{2} r_{2} }}{{[r_{2} - r_{1} (1 - \beta )]^{2} }} + O(\varepsilon^{2} ). $$

Thus, when $\frac{{r_{1} /\sigma_{1} }}{{r_{2} /\sigma_{2} }} < \frac{1}{\rho },\frac{\partial v}{\partial \delta } > 0.$ □

Proof for Proposition 2A

We assume symmetric super-additive synergy, where $\beta^{\prime } = \beta_{12}^{\prime } = \beta_{21}^{\prime } .$ The expected return from a portfolio (1 − x, x) is given by

$$ RT(1 - x,x) = r_{1} (1 - x) + r_{2} x + \beta^{\prime } r_{1} (1 - x) + \beta^{\prime } r_{2} x(1 - x). $$

One way of obtaining the efficient frontier is to require $RT(1 - x,x) = R_{0} ,$ where R ₀ is a particular expected return desired and then obtain the variance. For a two-fund problem, $RT(1 - x,x) = R_{0}$ is sufficient to determine the portfolio (1 − $x^{*}, x^{*}$).

From $RT(1 - x,x) = R_{0} ,$ we have

$$ x = \frac{1}{{2\beta^{\prime } (r_{1} + r_{2} )}}\left[ {r_{2} - r_{1} + \beta^{\prime } (r_{1} + r_{2} ) \pm \sqrt {(r_{2} - r_{1} + \beta^{\prime } (r_{1} + r_{2} ))^{2} - 4\beta^{\prime } (R_{0} - r_{1} )(r_{1} + r_{2} )} } \right]. $$

We assume that $\beta^{\prime }$ is a small number $\left( {\beta^{\prime } < \frac{{r_{2} - r_{1} }}{{r_{2} + r_{1} }}} \right);$ in this case, x has only one solution within the interval [0,1]:

$$ x^{*} = \frac{1}{{2\beta^{\prime } (r_{1} + r_{2} )}}\left\{ {r_{2} - r_{1} + \beta^{\prime } (r_{1} + r_{2} ) - \sqrt {\left[ {r_{2} - r_{1} + \beta^{\prime } (r_{1} + r_{2} )} \right]^{2} - 4\beta^{\prime } (R_{0} - r_{1} )(r_{1} + r_{2} )} } \right\}. $$

Consider $R_{0} = r_{2} - \varepsilon ,$ where r is an arbitrarily small positive number $\varepsilon > 0.$ Then

$$ x^{*} = 1 - \frac{\varepsilon }{{r_{2} - r_{1} - \beta^{\prime}(r_{1} + r_{2} )}} + O(\varepsilon^{2} ). $$

Then the variance v is

$$ \begin{aligned} v &= (1 - x)^{*2} (1 + \beta^{\prime }x)^{*2} \sigma_{1}^{2} + x^{*2} (1 + \beta^{\prime } - \beta^{\prime} x)^{*2} \sigma_{2}^{2} + 2x^{*}(1 - x^{*})(1 + \beta^{\prime }x^{*})(1 + \beta^{\prime } - \beta^{\prime } x^{*})\sigma_{1}\sigma_{2} \rho \\ &= \sigma_{2}^{2} + 2\varepsilon\frac{{\sigma_{1}^{{}} \sigma_{2}^{{}} \rho (1 + \beta^{\prime } ) -\sigma_{2}^{2} }}{{r_{2} - r_{1} - \beta^{\prime } (r_{1} + r_{2})}} + O(\varepsilon^{2} ). \\ \end{aligned} $$

Now we obtain the competitive statistics for the variance with respect to $\beta^{\prime}:$

$$ \frac{\partial v}{{\partial \beta^{\prime } }} = 4\varepsilon \frac{{\sigma_{1} \sigma_{2} \rho r_{2} - \sigma_{2}^{2} r_{1} }}{{\left[ {r_{2} - r_{1} - \beta^{\prime } (r_{1} + r_{2} )} \right]^{2} }} + O(\varepsilon^{2} ). $$

Thus, when $\frac{{r_{1} /\sigma_{1} }}{{r_{2} /\sigma_{2} }} > \rho ,\frac{\partial v}{{\partial \beta^{\prime } }} < 0.$ □

Proof for Proposition 2B

We assume symmetric super-additive synergy, where $\beta^{\prime } = \beta_{12}^{\prime } = \beta_{21}^{\prime } .$ The expected return from a portfolio (1 − x, x) is given by

$$ RT(1 - x,x) = r_{1} (1 - x) + r_{2} x + \beta^{\prime } r_{1} (1 - x) + \beta^{\prime } r_{2} x(1 - x). $$

One way of obtaining the efficient frontier is to require $RT(1 - x,x) = R_{0} ,$ where R ₀ is a particular expected return desired and then obtain the variance. For a two-fund problem, $RT(1 - x,x) = R_{0}$ is sufficient to determine the portfolio (1 − $x^{*}$, $x^{*}$).

From $RT(1 - x,x) = R_{0}$, we have

$$ x = \frac{1}{{2\beta^{\prime } (r_{1} + r_{2} )}}\left\{ {r_{2} - r_{1} + \beta^{\prime } (r_{1} + r_{2} ) \pm \sqrt {\left[ {r_{2} - r_{1} + \beta^{\prime } (r_{1} + r_{2} )} \right]^{2} - 4\beta^{\prime } (R_{0} - r_{1} )(r_{1} + r_{2} )} } \right\}. $$

We assume that β is a small number ($\delta \le \frac{{r_{2} - r_{1} }}{{r_{1} + r_{2} }}$); in this case, x has only one solution within the interval [0,1]:

$$ x^{*} = \frac{1}{{2\beta^{\prime } (r_{1} + r_{2} )}}\left\{ {r_{2} - r_{1} + \beta^{\prime } (r_{1} + r_{2} ) - \sqrt {\left[ {r_{2} - r_{1} + \beta^{\prime } (r_{1} + r_{2} )} \right]^{2} - 4\beta^{\prime } (R_{0} - r_{1} )(r_{1} + r_{2} )} } \right\}. $$

Consider $R_{0} = r_{1} + \varepsilon ,$ where r is an arbitrarily small positive number $\varepsilon > 0.$ Then

$$ x^{*} = \frac{\varepsilon }{{r_{2} - r_{1} - \beta^{\prime } (r_{1} + r_{2} )}} + O(\varepsilon^{2} ). $$

Then the variance v is

$$ v = \sigma_{1}^{2} + 2\varepsilon \frac{{\sigma_{1} \sigma_{2} \rho (1 + \beta^{\prime } ) + \sigma_{1}^{2} (\beta^{\prime } - 1)}}{{r_{2} - r_{1} + \beta^{\prime } (r_{1} + r_{2} )}} + O(\varepsilon^{2} ). $$

Now we obtain competitive statistics for the variance with respect to β:

$$ \frac{\partial v}{{\partial \beta^{\prime } }} = 4\varepsilon \frac{{\sigma_{1} \sigma_{2} \rho r_{1} - \sigma_{1}^{2} r_{2} }}{{\left[ {r_{2} - r_{1} + \beta^{\prime } (r_{1} + r_{2} )} \right]^{2} }} + O(\varepsilon^{2} ). $$

Thus, when $\frac{{r_{1} /\sigma_{1} }}{{r_{2} /\sigma_{2} }} < \frac{1}{\rho },\frac{\partial v}{{\partial \beta^{\prime } }} > 0.$ □