Idiosyncratic risk and the cost of capital: the case of electricity networks

Abstract

We analyze the treatment and impact of idiosyncratic or firm-specific risk in regulation. Regulatory authorities regularly ignore firm-specific characteristics, such as size or asset ages, implying different risk exposure in incentive regulation. In contrast, it is common to apply only a single benchmark, the weighted average cost of capital, uniformly to all firms. This will lead to implicit discrimination. We combine models of firm-specific risk, liquidity management and regulatory rate setting to investigate impacts on capital costs. We focus on the example of the impact of component failures for electricity network operators. In a simulation model for Germany, we find that capital costs increase by $\sim$0.2 to 3.0 % points depending on the size of the firm (in the range of 3–40 % of total cost of capital). Regulation of monopolistic bottlenecks should take these risks into account to avoid implicit discrimination.

Appendices

Appendix 1

The optimal simultaneous choice of up-front capital provision $B$ and choice of credit lines $L$ is implicitly defined by solving the firm’s objective function contained in Eq. (9). Usually, $s_{1}<< s_{2}$, which does not restrict the following results:

$$\begin{aligned}&\max _{B,L} \left[ \int _{-\infty }^{-L} {\left[ {\left( {x+L} \right) \left( {r+u} \right) } \right] } f\left( x \right) dx+\left( {B-L} \right) s_{1}\right. \nonumber \\&\qquad \quad \left. +\int _{-L}^{-B} {\left[ {\left( {x+B} \right) \left( {r+s_{2}} \right) } \right] } f\left( x \right) dx-Br \right] \end{aligned}$$

Optimization with respect to $L$ and $B$ gives the following results:

$$\begin{aligned} \frac{\partial E}{\partial L}&= \frac{\partial \left( {\int _{-\infty }^{-L} {\left[ {\left( {x+L} \right) \left( {r+u} \right) } \right] } f\left( x \right) dx+\left( {B-L} \right) s_{1} +\int _{-L}^{-B} {\left[ {\left( {x+B} \right) \left( {r+s_{2} } \right) } \right] } f\left( x \right) dx-Br} \right) }{\partial L} \nonumber \\&= -s_{1} +\left( {r+u} \right) \left( {F\left( {-L} \right) -Lf\left( {-L} \right) } \right) +\left( {r+s_{2}} \right) Bf\left( {-L} \right) +\left( {u-s_{2}} \right) \frac{\partial \int _{-\infty }^{-L} {xf\left( x \right) dx} }{\partial L} \end{aligned}$$

Assuming normal distribution characteristics for stochastic cash flows $x$, an implicit solution of the optimization is possible:

$$\begin{aligned} &= -s_1 +\left( {r+u} \right) \left( {F\left( {-L} \right) -Lf\left( {-L} \right) } \right) +\left( {r+s_{2} } \right) Bf\left( {-L} \right) \\& +\left( {u-s_{2} } \right) \frac{\partial \left( {\mu F\left( {-L} \right) -\sigma ^{2}f\left( {-L} \right) } \right) }{\partial L} \\ &= -s_1 +\left( {r+u} \right) \left( {F\left( {-L} \right) -Lf\left( {-L} \right) } \right) +\left( {r+s_2 } \right) Bf\left( {-L} \right) \\&+\left( {u-s_2 } \right) \left( {\mu F\left( {-L} \right) -\sigma ^{2}f^{{\prime }}\left( {-L} \right) } \right) \\&\mathop {=}\limits ^{\begin{array}{c} f^{{\prime }}\left( {-L} \right) =\left( {-\frac{-L-\mu }{\sigma ^{2}}} \right) f\left( {-L} \right) \\ \Leftrightarrow \sigma ^{2}f^{{\prime }}\left( {-L} \right) =\left( {L+\mu } \right) f\left( {-L} \right) \\ \Leftrightarrow -Lf\left( {-L} \right) =\mu f\left( {-L} \right) -\sigma ^{2}f^{{\prime }}\left( {-L} \right) \\ \end{array}}&-s_{1} +\left( {r+u} \right) F\left( {-L} \right) -\left( {r+u} \right) Lf\left( {-L} \right) \\&+\left( {r+s_{2} } \right) Bf\left( {-L} \right) +\left( {u-s_{2} } \right) Lf\left( {-L} \right) \\ &= -s_{1} +\left( {r+u} \right) F\left( {-L} \right) -\left( {r+u} \right) Bf\left( {-L} \right) -\left( {r+s_{2}} \right) Lf\left( {-L} \right) \\ \frac{\partial E}{\partial L}&= 0 \end{aligned}$$

Algebraic transformation leads to $L=B+\frac{\left( {r+u} \right) F\left( {-L} \right) -s_{1} }{\left( {r+s_{2} } \right) f\left( {-L} \right) }$.

The solution for $B$ is of course analogous:

$$\begin{aligned} \frac{\partial E}{\partial B}&= \frac{\partial \left( {\int _{-\infty }^{-L} {\left[ {\left( {x+L} \right) \left( {r+u} \right) } \right] } f\left( x \right) dx+\left( {B-L} \right) s_{1} +\int _{-L}^{-B} {\left[ {\left( {x+B} \right) \left( {r+s_{2}} \right) } \right] } f\left( x \right) dx-Br} \right) }{\partial B} \\&= s_{1} -r+\left( {r+s_{2} } \right) \frac{\partial \int _{-\infty }^{-B} {xf\left( x \right) dx} }{\partial B}+\left( {r+s_{2}} \right) \left( {F\left( {-B} \right) -Bf\left( {-B} \right) -F\left( {-L} \right) } \right) \end{aligned}$$

Again, assuming normally distributed cash flows will make the following solution possible:

$$\begin{aligned} =\left( {r+s_{2}} \right) \left( {F\left( {-B} \right) -F\left( {-L} \right) } \right) +s_{1} -r. \end{aligned}$$

$\frac{\partial E}{\partial B}=0:$ Algebraic transformation leads to:

$$\begin{aligned} F\left( {-B} \right) =F\left( {-L} \right) +\frac{r-s_{1} }{r+s_{2} } \end{aligned}$$

Appendix 2

$$\begin{aligned} K\left( {1-F\left( {t_1 } \right) } \right)&= sf\left( {t_1 } \right) G\left( {t_1 } \right) -F\left( {t_1 } \right) \left( {1-F\left( {t_1 } \right) } \right) \\ \Leftrightarrow \;K_s&= \frac{f\left( {t_1 } \right) G\left( {t_1 } \right) -F\left( {t_1 } \right) \left( {1-F\left( {t_1 } \right) } \right) }{\left( {1-F\left( {t_1 } \right) } \right) } \\ \Leftrightarrow \;K_s&= \frac{m\left[ {t_1 -\frac{1}{2}m\left( {t_1 -t_a } \right) ^{2}} \right] -m\left( {t_1 -t_a } \right) \left( {1-m\left( {t_1 -t_a } \right) } \right) }{\left( {1-m\left( {t_1 -t_a } \right) } \right) }\qquad \qquad \qquad \end{aligned}$$

$$\begin{aligned} \Leftrightarrow \;\left( {\frac{1}{2}m} \right) t_1^2 +\left( {K_s -mt_a } \right) t_1 +\left( {\frac{1}{2}mt_a^2 +t_a -\frac{K_s }{m}-K_s t_a } \right) =0 \end{aligned}$$

Solving this quadratic equation gives:

$$\begin{aligned} t_{1_{1} /1_{2}}^*\!=\!t_{a} \!-\!\frac{K_{s} }{m}\pm \frac{\sqrt{K_{s} ^{2}+2K_{s} -2mt_{a} }}{m}\;\Rightarrow t_{1_{1} /1_{2} }^*\!=\!t_{a} +\frac{\sqrt{K_{s} ^{2}+2K_{s} -2mt_{a}}\!-\!K_{s}}{m}\;. \end{aligned}$$

Appendix 3

It is sufficient to show that total cost is convex in replacement age:

$$\begin{aligned} \mu&= n\left( {\left( {K+s} \right) \frac{m(t_{j} -t_{a} )}{t_{j} -\frac{m}{2}\left( {t_{j} -t_{a} } \right) ^{2}}+K\frac{1-m(t_{j} -t_{a} )}{t_{j} -\frac{m}{2}\left( {t_{j} -t_{a} } \right) ^{2}}} \right) \\&= n\left( {s\frac{m(t_{j} -t_{a} )}{t_{j} -\frac{m}{2}\left( {t_{j} -t_{a} } \right) ^{2}}+K\frac{1}{t_{j} -\frac{m}{2}\left( {t_{j} -t_{a} } \right) ^{2}}} \right) \end{aligned}$$

This is done by showing that the second derivative of this composition of polynomials is positive:

$$\begin{aligned} \frac{\partial \mu }{\partial t_{j} }&= n\left( {s\frac{m\left( {t_{j} -\frac{m}{2}\left( {t_{j} -t_{a} } \right) ^{2}} \right) -m(t_{j} -t_{a} )\left( {1-m\left( {t_{j} -t_{a} } \right) } \right) }{\left( {t_{j} -\frac{m}{2}\left( {t_{j} -t_{a} } \right) ^{2}} \right) ^{2}}+K\frac{-\left( {1-m\left( {t_{j} -t_{a} } \right) } \right) }{\left( {t_{j} -\frac{m}{2}\left( {t_{j} -t_{a} } \right) ^{2}} \right) ^{2}}} \right) \\ \frac{\partial ^{2}\mu }{\partial t_{j} ^{2}}&= n\left( {sm\frac{t_{a} +\frac{m}{2}\left( {t_{j} -t_{a} } \right) ^{2}}{\left( {t_{j} -\frac{m}{2}\left( {t_{j} -t_{a} } \right) ^{2}} \right) ^{2}}+K\frac{m\left( {t_{j} -t_{a} } \right) -1}{\left( {t_{j} -\frac{m}{2}\left( {t_{j} -t_{a} } \right) ^{2}} \right) ^{2}}} \right) ^{\prime } \\&= n\left( \underbrace{sm\frac{m\left( {t_{j} -t_{a} } \right) \left( {t_{j} -\frac{m}{2}\left( {t_{j} -t_{a}} \right) ^{2}} \right) -2\left( {t_{a} +\frac{m}{2}\left( {t_{j} -t_{a}} \right) ^{2}} \right) \underbrace{\left( {1-m\left( {t_{j} -t_{a}} \right) } \right) }_{<0}}{\left( {t_{j} -\frac{m}{2}\left( {t_{j} -t_{a}} \right) ^{2}} \right) ^{3}}}_{>0} \right. \\&\left. +\underbrace{K\left( {\frac{m\left( {t_{j} -\frac{m}{2}\left( {t_{j} -t_{a} } \right) ^{2}} \right) +2\left( {1+m\left( {t_{j} -t_{a} } \right) } \right) \left( {1-m\left( {t_{j} -t_{a} } \right) } \right) }{\left( {t_{j} -\frac{m}{2}\left( {t_{j} -t_{a} } \right) ^{2}} \right) ^{3}}} \right) }_{>0} \right) . \end{aligned}$$

By inspecting the following transformation, this can easily be seen. $\mu$ is convex in $t_{j}$.

Appendix 4

This network structure is based on Obergünner (2005), who presents the components of a standard distribution network operator in German electricity distribution. The network components (or asset clusters) are as follows: transformers, circuit breakers, overhead lines (medium and low voltage), cables (medium and low voltage), relay stations. This structure consists of: six transformer stations (with 12 transformers and 90 circuit breakers), 600 relay stations, and 2,000 km of cables and overhead lines.

Transformer stations (with the transformers and circuit breakers), which are connected to the high voltage grid, transform electricity to medium voltage and feed into medium voltage rings. Industry customers are directly connected to transformer stations. Based on the analytical network modeling approach in Consentec, IAEW, RZVN and Frontier Economics (2008), which has also been used by the German regulator for various regulatory purposes, we determine that five medium voltage rings, each 20 km in length, are connected to each transformer station. A medium voltage ring consists of cables and overhead lines, which connect several relay stations per ring. At each relay station, electricity is transformed to low voltage and fed into low voltage cables and overhead lines which provide low voltage customers via a string structure. Each ring includes 20 relay stations, from which four low voltage lines per relay station (each 600 m in length) connect households, commercial customers and others (i.e., traffic, agriculture, public institutions). This corresponds to a typical distribution network operator’s structure.

Appendix 5

The 80 % survival age indicates the age that 80 % of assets will attain without replacement or major repair (the value of the cumulative distribution function CDF attaining 0.2), the 20 % survival age indicates the age at which 80 % of assets in one cluster will have failed (CDF = 0.8). The estimates for these survival ages reflect practitioners’ current expectations of asset lifetimes.⁴³Based on these assumptions, Weibull failure distribution laws for the conditional failure rate have been estimated (see Fig. 5 for cumulated distribution functions).

Fig. 5

Cumulative failure distribution based on Weibull law

Appendix 6

The customer segment definitions were taken from BNetzA (2008, p. 45), drawing on the overall yearly consumptions from BDEW (2008). The number of customers was derived from the data gathered with reference to the share of total line length (2,000 km compared to $\sim$1,600,000 km overall in Germany).

Survival ages were deduced from project experience and the data gathered. AVG failure duration was taken from VDN (2005), and Weibull parameters were estimated based on averages. Asset replacement cost can be found in Consentec et al. (citeyearcon08, p. 113), and additional unplanned replacement costs were estimated at 20 % of planned costs based on project experience. Penalties were calculated based on €10/kWh, customer-specific data in Table 2, average failure durations and assumptions concerning network structure and redundancy (see Table 3).

Table 2 Customer specific data

Table 3 Component specific data