†
‡
Abstract
We analyse the transmission investment problem by modelling the decision of a private investor holding a perpetual option to construct a transmission line. Using the real options approach, we determine both the optimal investment timing and line capacity under uncertain congestion rents.
In particular, we find that if consumers at the importing node were to limit the investor’s liability, then not only would the option to invest be worth more, but also investment would be triggered sooner in a higher capacity line. We use price data from PJM and NYISO in numerical experiments to provide intuition for the optimal decision.
Keywords: Transmission investment, real options, mean-reverting process
Given the dependence of modern society on electricity, increased reliability of the transmission grid can result in super-linear payoffs. Timely transmission grid investments can benefit both society, e.g., via increased reliability, and investors, e.g., via profits in the form of congestion rents. Unreliability was not an issue under the regulated paradigm because transmission investment was governed by regional reliability councils (RRCs). Indeed, the primary objective of RRCs was grid reliability rather than cost reduction. In deregulated electricity industries with functioning markets, transmission investment is supposed to be undertaken by profit-maximising private investors; thus, the investment situation has changed dramatically. After the deregulation of North American electricity markets in 1996, inadequate investment in high-voltage transmission lines has resulted in less reliable transmission networks in the
US. The 2003 northeastern US blackout is a recent example of the perils from lack of investment in the transmission grid. The interconnectivity of the grids serves to worsen the problem by transmitting the effects of a local failure far enough to result in a cascading failure. Thus, the reliability of the transmission grid is important for the proper functioning of the society.
∗ Research Report No. 285, Department of Statistical Science, University College London. Date: November 2007
† Department of Statistical Science, University College London, London WC1E 6BT, United Kingdom, e-mail: afzal@stats.ucl.ac.uk
‡ Systems and Information Engineering Department, University of Virginia, Charlottesville, VA 22904, USA, e-mail: himanshu@virginia.edu
1

Transmission Investment Timing and Sizing under Uncertainty 2
Reasons for the lack of timely and adequate transmission investment in deregulated markets include the complexity of transmission system expansion, the lumpiness of investment projects, and the lack of economic incentives for private investors. Since there is little that can be done to reduce the complexity or lumpiness, we focus on studying the monetary incentives available to merchant investors as analysing these incentives will help shed light on the transmission investment problem, which in turn will address grid reliability.
In this paper, we take the perspective of a TRO that has the right, but not the obligation, to invest and has discretion over the capacity of the transmission line. We assume that the TRO holds rights only in one direction (uni-directional rights) since the TRO will build a line to take advantage of a large nodal price difference, which is unlikely to change sign. Due to the flexibility in investment and uncertain congestion rents, the real options approach seems appropriate for analysing the problem (see
[8]), but the traditional real options approach assumes fixed investment cost and capacity. The work on operational flexibility touches on the issue of capacity sizing, but does not fully develop the theory
(see [15]). This gap in the real options literature is addressed by [7], which extends the theory to allow for selecting from among mutually exclusive projects. We use the approach in [4], which allows for the examination of alternative, discrete capacity choices.
1
Since the construction of the line can potentially lower the prices for consumers at the importing node of the transmission line, they should find it beneficial to guarantee a minimum payment to the
TRO. Therefore, our aim is to characterise the optimal investment timing and line capacity under both limited and unlimited liability. Under the former, the investor is not responsible for operating losses, while under the latter, the investor may incur operating losses. Although only the limited liability case is of interest, for the sake of exposition and completeness, we present both cases. The limited liability case seems plausible and is in fact realistic as many governments offer investment guarantees for publicprivate projects that lower the risk for investors (see [2]). Furthermore, [18] indicates that a revenue guarantee precipitates investment within the context of nuclear power plants.
Our approach is similar to [1], [11], and [14] in advocating the merchant investor model. Thus far, works such as [9], [10], [16], and [17] have used derivatives, but have not employed real price data.
Another application of real options obtains a partial differential equation (PDE) written on the electricity prices at the two nodes and then solves it subject to boundary conditions to find a Black-Scholes-type formula for the value of the constructed transmission line (see [6]). It does not, however, find the value of the option to invest, the optimal investment threshold, or the determination of the line capacity. We address some of these gaps through our paper by finding the optimal investment timing and sizing using real options and market data. Of course, the inelasticity of demand, seasonality in consumption, and economic growth also have significant effects on the timing and sizing decisions, but such attributes are beyond the scope of the model we develop. Our contribution lies in extending the methodology of [4] to a case where the TRO has limited liability.
The structure of this paper is as follows:
1 For continuous capacity choice, [3] would be more appropriate.

Transmission Investment Timing and Sizing under Uncertainty 3
• Section 2 justifies the model’s assumptions and formulates the TRO’s investment problem
• Section 3 introduces the data used in our case study and presents the results of some numerical examples
• Section 4 summarises the results of this paper and offers directions for future research in this area
Before formulating the TRO’s problem, we first make the following simplifying assumptions:
• Infinite Lifetime: A transmission line has an infinite lifetime once constructed due to the possibility of ongoing maintenance upgrades.
• Uni-Directional Rights: The TRO collects rents only in one direction. If the flows were to reverse, then we assume that the TRO loses money (this assumption makes the algebra and analytics easier and is justified economically earlier).
• Congestion Revenue Rent: It is not known in advance what the ex post nodal price difference will be; however, the ex ante nodal price difference can be used to approximate ex post parameters.
• Number of Choices: There are n = 3 discrete line capacities, K i
, such that K
1
< K
2
< K
3
, with investment costs I
1
< I
2
< I
3 and variable operating costs c
1
< c
2
< c
3
. Although we make this assumption, our analysis can be extended to any finite number of options at the cost of increased algebraic complexity.
• Effect of Construction on Prices: Higher-capacity lines reduce the nodal price difference more, an effect we mimic by altering the variable operating costs accordingly rather than by making three separate congestion rent models, i.e., larger lines also have higher operating costs, c i
.
In the financial literature, a convenient stochastic process used to model asset prices is the geometric
Brownian motion (GBM), which assumes that successive percentage price changes are independent and identically normally distributed. This is reasonably valid over a sufficiently long time horizon and could be used to model the ex post electricity price at each node. However, the model would then have two stochastic processes, which would be unnecessarily complicated. It is, instead, possible to model the ex post nodal price difference directly as a mean-reverting (MR) process, i.e., successive absolute price differences between the two nodes rise or fall to a long-run mean, X , with reversion rate η . In other words, if X t
= P
B t
− P
A t is the instantaneous congestion rent at time t , then dX t
= η ( X − X t
) dt + σdz t
, where A ( B ) is the exporting (importing) node, σ is the volatility, and { z t
, t ≥ 0 } is a standard Brownian motion process. It should be noted that the selection of the MR process is motivated largely by the fact that it eases the investment and sizing analysis. In reality, since congestion rents experience occasional

Transmission Investment Timing and Sizing under Uncertainty 4 spikes, a mean-reverting jump-diffusion (MRJD) process would be more appropriate. In fact, a recent paper uses this stochastic process to model jointly the natural logarithms of electricity and natural gas prices in determining the NPV and option value of a power plant (see [5]). Again, since using the MRJD process would complicate the capacity sizing evaluation 2 , we choose to utilise the simple MR process to model the congestion rents. We compensate somewhat for missing the spikes by using a volatility that is matched to the data, an approach that is shown to result in over-valuation of a 300 MW power plant of between -2% and 13% in [5].
If the ex post price difference between the exporting and importing nodes were deterministic, i.e., equal to P
B
− P
A
, and there were no flexibility over timing, then the TRO could use the discounted cash flow (DCF) method to make its investment decision. Specifically, each year the investor receives the congestion rent along the line, which is equal to the ex post price difference between the two nodes,
X = P
B
− P
A
(in US$/MWh), times the size of the line, K (in MW of power that it can transmit), times the number of hours in a year, h (equal to 8760). The TRO would calculate the net present value
(NPV) of the investment as V ( P
A
, P
B
, K ) − I =
( P
B
− P
A
µ
) Kh
− cKh r
− I , where µ is the discount rate per annum 3 and I is the investment cost, which depends on the line’s capacity. Here, V ( P
A
, P
B
, K ) is the present value (PV) of net cash flows from operating this line.
Under the assumption of a stochastic congestion rent following a MR process, the instantaneous net cash flow at time t is a simple linear function of X t
− c i
: v i
( X t
) = ( X t
− c i
) K i h (1)
Using the fact that the conditional expected value of the congestion rent at time t given the current value, X , is E [ X t
| X ] = X + ( X − X ) e − ηt , we now find the expected PV of net revenues as follows:
⇒
V
V i i
(
(
X
X
) =
) =
Z
0
∞
Z
0
∞
⇒ V i
( X ) = K i h
⇒ V i
( X ) = K i h
E [ X t
| X ] K i he − µt dt −
XK i he − µt dt +
Z
Z
∞ c i
K i he − rt dt
0
∞
( X − X ) K i he − ( η + µ ) t dt −
·
·
η
η
X
+
X
+
µ
µ
+
+
X d
µ
¸ i
−
η
0
X
+ µ
− c r i
¸
Z
0
∞ c i
K i he − rt dt
(2)
In the last line of equation 2, we let d i
≡ X
µ
− X
η + µ
− c r i . Equation 2 also states that the expected NPV is negative if X < − d i
( η + µ ), in which case the transmission line would not be constructed. In order to do the investment analysis under a “now or never” assumption, the TRO simply has to find the expected
2 In [5], even the value of an existing plant has to be found via numerical integration.
3 This may be determined by the capital asset pricing model (CAPM) to reflect the risk of the project, i.e., µ = r + λ ( r m
− r ), where r is the risk-free rate of return, λ is the sensitivity of the project’s returns to those of the market, and r m is the market’s expected rate of return.

Transmission Investment Timing and Sizing under Uncertainty 5
NPV of each line as follows:
V i
( X ) − I i
= K i h
·
X
η + µ
+ d i
¸
− I i
(3)
The TRO then selects the line with the highest NPV at the current congestion rent, X . However, since this approach ignores the option to wait before taking advantage of the right time to invest, it under-values the investment opportunity.
Since congestion rents are subject to considerable uncertainty and investment projects may be postponed indefinitely, real options are beneficial to analysing the TRO’s discrete capacity selection problem. In order to value the option to invest in transmission line i , F i
( X ), we use the approach outlined in [8].
We first construct a risk-free portfolio, Φ i
= F i
( X ) − F i
0 ( X ) X , which consists of one unit of the option to invest and is short F i
0 ( X ) units of the underlying commodity, which is the congestion rent in this case. Then, via a “no arbitrage” argument, we equate the instantaneous rate of return on the risk-free portfolio to the expected appreciation of the portfolio less the dividend payment 4 on the short position: r Φ i dt = E [ d Φ i
] − δF i
0 ( X ) Xdt
⇒ rF i
( X ) dt − rF i
0 ( X ) Xdt = E [ dF i
( X ) − F i
0 ( X ) dX ] − δF i
0 ( X ) Xdt
From Itˆo’s Lemma, we obtain:
(4) dF i
( X ) = F i
0 ( X ) dX + 1
2
F i
00 ( X )( dX ) 2
Together, equations 4 and 5 imply:
(5)
1
2
F i
00 ( X ) σ 2 + ( r − δ ) F i
0 ( X ) X − rF i
( X ) = 0 (6)
In order to simplify equation 6, we set r = δ , which in the context of CAPM implies that the growth rate of the underlying commodity’s value is equal to its expected market risk premium (see [19] for a similar approach). The general solution to the resulting ordinary differential equation is:
F i
( X ) = A 0 i 1 e β
1
X + A 0 i 2 e β
2
X (7)
Here, A 0 i 1 and A 0 i 2 are positive endogenous constants, and β
1
=
√
2 r
σ
> 0 and β
2
= −
√
2 r
σ
< 0 are exogenous constants that are the roots to the characteristic quadratic equation associated with equations
6 and 7. Since the option to invest must be worthless when congestion rents are very negative, i.e., lim
X →−∞
F i
( X ) = 0, we conclude that A 0 i 2
= 0.
According to [7], the value of the option to invest when given a finite number of discrete capacity choices, F ( X ), involves finding the option values of investment in each project i , F i
( X ) = A 0 i 1 e β
1
X
(following the approach of equation 7) and then selecting the project j with the largest coefficient, i.e., j = arg max i
A 0 i 1
. If the current congestion rent is less than the associated threshold, X 0 j
, then the TRO
4 Here, δ is the dividend rate on congestion rent.

Transmission Investment Timing and Sizing under Uncertainty 6 should wait until the threshold is reached before investing in a line of capacity K j
. However, if the current congestion rent is greater than X 0 j
, then the TRO should invest in the transmission line that maximises its expected NPV. In other words, once the investment threshold is surpassed, the TRO may end up selecting a capacity that is larger than K j
.
While the aforementioned approach takes uncertainty into account, it does not consider what would happen at the indifference points, i.e., at the congestion rents where the expected NPVs of two trans-
0
12 is the indifference point between projects 1 and 2. In
0
12 and is greater than X 0 j
, then the NPV rule would not be able to arrive at a conclusion. Therefore, the TRO would prefer to wait for more information about the congestion rent before making a decision. It is shown in [4] that the optimal policy to follow at such an indifference point is to wait until a lower price threshold is reached before investing in the smaller project or to invest in the larger project if an upper price threshold is reached with waiting being optimal otherwise. We apply the results of this insight to investigate the TRO’s discrete investment and capacity sizing problem under uncertainty with both unlimited and limited liability.
2.3.1
Unlimited Liability
If the TRO is given the choice of three mutually exclusive transmission line capacities, then its first step is to compute the option value of each project independently.
5 Specifically, it uses the appropriate value-matching and smooth-pasting conditions to determine A 0 i 1 and X i
0 for each capacity size:
⇒ A 0 i 1 e
F
β
1 i
( X
X
0 i i
0
=
) = V i
K i h h
( X
X
0 i
η + µ i
0 ) − I i i
+ d i
− I i
(8)
F i
0 ( X i
0 ) = V i
0 ( X i
0 )
⇒ β
1
A 0 i 1 e β
1
X
0 i
= K i h
η + µ
Solving equations 8 and 9 simultaneously yields:
X i
0
=
1
β
1
+
I i
( η + µ )
K i h
− d i
( η + µ )
(9)
(10)
A 0 i 1
= e − β
1
X
0 i
K i h
β
1
( η + µ )
There are four investment cases depending on the relative ordering of the A 0 i 1
, i = 1 , 2 , 3:
(11)
1. First, if A 0
31
= max i
A 0 i 1
, then F ( X ) = F
3
( X ), i.e., the TRO should consider only the largest transmission capacity and ignore the other two (see [7]). Once the threshold X 0
3 is reached, the
TRO optimally invests I
3 in the line of capacity K
3
.
5 We assume that the length of the line, ` , is fixed and accounted for in the investment cost, I i
.

Transmission Investment Timing and Sizing under Uncertainty 7
2. If A 0
21
= max i
A 0 i 1
, then the TRO should disregard the smallest line and consider investing only in the two larger ones. As a result, the value of the option to invest is dichotomous:
F ( X ) =



A
C
0
21
0 e e β
1
β
1
X
X + D 0 e β
2
X if if
X < X
X 0
6
0
2
< X < X 0
7
(12)
Here, X 0
2
< X 0
6
< 0
23 and X 0
7
> 0
23 are congestion rent thresholds that define the region
0
23
, in which it is optimal to wait for more information before investing. From this point, if the congestion rent decreases (increases) to X 0
6
( X 0
7
), then the TRO invests in line 2 (3).
6 In order to find the two positive endogenous constants,
C 0 and D 0 , along with the indifference investment thresholds, X 0
6 and X 0
7
, we write the following value-matching and smooth-pasting conditions:
⇒ C 0 e β
1
X
0
6
F ( X 0
6
) = V
2
( X
+ D 0 e β
2
X
0
6
= K
2
0
6
) − I
2 h h
X
0
6
η + µ
+ d
2 i
− I
2
(13)
F 0 ( X 0
6
) = V
2
0 ( X 0
6
)
⇒ β
1
C 0 e β
1
X
0
6
+ β
2
D 0 e β
2
X
0
6
=
K
2 h
η + µ
(14)
⇒ C 0 e β
1
X
0
7
F ( X 0
7
) = V
3
( X
+ D 0 e β
2
X
0
7
= K
3
0
7
) − I
3 h h
X
0
7
η + µ
+ d
3 i
− I
3
(15)
F 0 ( X 0
7
) = V
3
0 ( X 0
7
)
⇒ β
1
C 0 e β
1
X
0
7
+ β
2
D 0 e β
2
X
0
7
=
K
3 h
η + µ
(16)
Since equations 13 to 16 are highly non-linear, there are no analytical solutions to the four unknowns. However, the aforementioned system may be solved numerically for specific parameter values.
3. Next, if A 0
11
> A 0
31
> A 0
21
, then the TRO should disregard the second-smallest line and consider investing only in lines 1 and 3. Again, the value of the option to invest is dichotomous as the second project is skipped altogether. This case can be developed as the one before.
4. Finally, if A 0
11
> A 0
21
> A 0
31
, then all three projects should be considered for investment. Since this case is unlikely, we describe it in the appendix (see Section A).
2.3.2
Limited Liability
By “limited liability,” we mean that the TRO does not incur losses from operating the transmission line.
As a result, its instantaneous net cash flow at any time t is a convex function of X t
− c i
: v i
( X t
) = ( X t
− c i
) + K i h
6 Of course, if X 0
2
≤ X ≤ X 0
6 or X ≥ X 0
7
, then the TRO invests immediately in line 2 or 3, respectively.
(17)

Transmission Investment Timing and Sizing under Uncertainty 8
Due to the operational flexibility that does not hold the TRO liable for any adverse flow along its line, the expected PV of net revenues is no longer given by equation 2, but as follows:

V i
( X ) =


B
B i 1 i 2 e e
β
1
β
2
X
X + K i h h
X
η + µ
+ d i i if X < c i otherwise
(18)
The first case in equation 18 corresponds to a situation in which the congestion rent is less than the variable cost of operating the line. Consequently, the TRO accrues no cash flows and incurs no losses due to the revenue guarantee. However, since TRO still has the option to earn revenue in the future
(without incurring any switchover cost), the expected PV of its existing line is worth B i 1 e β
1
X , an increasing function of the congestion rent. Here, B i 1 is a positive endogenous constant, and β
1 is defined as before. By contrast, the second case in equation 18 consists of two components: the first is the option to “turn off” costlessly the cash flows should X decrease sufficiently, and the second is simply the expected PV of net revenues from an active transmission line. Again, B i 2 is a positive endogenous constant, and β
2
< 0 is defined as before. Following the method of [15], we determine these endogenous constants using value-matching and smooth-pasting conditions that guarantee that both V i
( X ) and
V i
0 ( X ) are continuous at c i
:
⇒ B i 1 e β
1 c i
V i
( c − i
) = V i
( c + i
)
= B i 2 e β
2 c i + K i h h c i
η + µ
+ d i i
(19)
V i
0 ( c
− i
) = V i
0 ( c
+ i
)
⇒ β
1
B i 1 e β
1 c i = β
2
B i 2 e β
2 c i + K i h
η + µ
(20)
Solving simultaneously, we obtain:
B i 1
=
K i hβ
2
β
2 e
−
− β
1 ci
β
1 h c i
η + µ
+ d i
−
β
2
1
( η + µ ) i
B i 2
=
K i hβ
1
β
2 e
−
− β
2 ci
β
1 h c i
η + µ
+ d i
−
β
1
1
( η + µ ) i
(21)
(22)
Next, the value of the option to invest in each project i has the form F i
( X ) = A i 1 e β
1
X as before. Now,
A i 1 and X i must be found numerically because the value-matching and smooth-pasting conditions are highly non-linear (we assume that X i
> c i when investment occurs):
⇒ A i 1 e β
1
X i
F i
= K i
( X i h
) = V i h X i
η + µ
+
( X d i i i
) − I i
− I i
+ B i 2 e β
2
X i (23)
F i
0 ( X i
) = V i
0 ( X i
)
⇒ β
1
A i 1 e β
1
X i
= K i h
η + µ
+ β
2
B i 2 e β
2
X i
(24)
After having obtained the X i and A i 1 for specific parameter values, we can still apply the algorithm of [4]. However, in determining dichotomous option value functions, we need to include the embedded optionality within an active transmission line. There are four cases as in Section 2.3.1, but since only two seem likely to occur, we shall focus on them:

Transmission Investment Timing and Sizing under Uncertainty 9
1. First, if A
31
= max i
A i 1
, then F ( X ) = F
3
( X ), i.e., the TRO should consider only the largest transmission capacity and skip the other two. Once the threshold X
3 is reached, the TRO optimally invests I
3 in the line of capacity K
3
.
2. If A
21
= max i
A i 1
, then the TRO should disgregard the smallest line and consider investing only in the two larger ones. As a result, the value of the option to invest is dichotomous:


F ( X ) =

A
21
Ce e
β
1
β
1
X
X + De β
2
X if if
X < X
X
6
2
< X < X
7
(25)
In order to find the two positive endogenous constants, C and D , along with the indifference investment thresholds, X
6 and X
7
, we write the following value-matching and smooth-pasting conditions:
⇒ Ce β
1
X
6
+ De β
2
F
X
6
( X
6
) = V h
= K
2 h
2
( X
X
6
η + µ
6
) − I i
2
+ d
2
− I
2
+ B
22 e β
2
X
6
(26)
F 0 ( X
6
) = V
2
0 ( X
2
)
⇒ β
1
Ce β
1
X
6 + β
2
De β
2
X
6 =
K
2 h
η + µ
+ β
2
B
22 e β
2
X
6 (27)
⇒ Ce β
1
X
7 + De β
2
F
X
7
( X
=
7
) = V h
K
3 h
3
( X
X
7
η + µ
7
) − I i
3
+ d
3
− I
3
+ B
32 e β
2
X
7 (28)
F 0 ( X
7
) = V
3
0 ( X
7
)
⇒ β
1
Ce β
1
X
7 + β
2
De β
2
X
7 = K
3 h
η + µ
+ β
2
B
32 e β
2
X
7 (29)
Equations 26 to 29 are highly non-linear and similar to equations 13 through 16 except for extra terms on the right-hand sides. Since there are no analytical solutions to the four unknowns, the aforementioned system may be solved numerically for specific parameter values.
In order to illustrate the intuitive aspects of the TRO’s decision-making problem, we use numerical examples based on data from the New York ISO (NYISO) and Pennsylvania-New Jersey-Maryland
(PJM) control areas.
For our numerical examples, we require data on ex post congestion rents, investment costs, and financial parameters. Since we take the perspective of a TRO that has the right to build a transmission line from
NYISO to PJM, we use ex ante , i.e., existing, nodal price differences between the two control areas to

Transmission Investment Timing and Sizing under Uncertainty 10 approximate the ex post congestion rents. In particular, if X t is the ex post nodal price difference and Y t is the ex ante one, then we use the latter to estimate the parameters for the former because the former has not been realised yet. In order to do this, we make the following assumptions:
• The ex post mean-reversion parameter, η , and volatility, σ , are the same as the ex ante ones, η
Y
, and σ
Y
, respectively.
• Due to the construction of the transmission line, the long-run mean of the ex post nodal price difference, X , is only one-half that of the ex ante one, Y . Since we have no data on observed ex post nodal price differences, this is only an approximation.
Hence, we estimate the parameters for the current, ex ante nodal price difference process, dY t
= η
Y
( Y −
Y t
) dt + σ
Y dz t
.
Given daily nodal prices for NYISO and PJM between January 1997 and June 2002, we subtract the
NYISO, P
A t
, price from the PJM, P
B t
, to obtain the daily nodal price difference, Y t
:
Y t
= P
B t
− P
A t
(30)
The calculated nodal price differences are plotted in Figure 1. We then estimate parameters for the simple MR process (such as η
Y
, Y , and σ
Y
) by running the following ordinary least-squares (OLS) regression:
(31) Y t
− Y t − 1
= α
0
+ α
1
Y t − 1
+ ² t
Then, the parameters may be estimated as follows:
ˆ
= −
α
0
α
1
(32)
η ˆ = − ln(1 + ˆ
1
) (33)
σ ˆ
Y
σ
² s
α
1
)
α
1
) 2 − 1
(34)
σ
² is the standard error of regression (SER). By running the OLS regression for the available data, we estimate ˆ = 2 .
η
Y
= 0 .
44, and ˆ = 9 .
43. These imply that ˆ = 1 .
η = 0 .
44, and
ˆ = 9 .
43. The scatter plot and fitted regression line are shown in Figure 2. Since the spikes in the data are modelled only implicitly by the volatility, we obtain a relatively low coefficient of determination
( R 2 ) of 0.18. However, the slope of the fitted regression line is highly statistically significant with a p -value of virtually zero. A simulated sample path assuming a simple MR process using the estimated parameters and Y
0
= 2 .
22 is plotted in Figure 1 to show the potential drawback of not modelling the spikes explicitly: approximating the spikes by a higher volatility parameter may lead to undervaluation of the asset and, therefore, suggest delaying the investment decision or selecting a smaller capacity.

Transmission Investment Timing and Sizing under Uncertainty
200
Nodal Price Difference: NYI to PJM
Data
Simulated sample path
150
100
50
0
11
−50
0 200 400 600
Day
800 1000 1200 1400
Figure 1: Daily Nodal Price Differences: NYISO to PJM
We now apply the methodology of Section 2.3.1 to the data presented in Section 3.1. In order to keep the problem tractable, we focus on the following three capacity sizes: 302 MW, 796 MW, and
1060 MW. Investment costs for lines of length 250 miles with these rated capacities are US$100 million, US$132.5 million, and US$210 million, respectively, which are obtained from the US EIA (see http://www.eia.doe.gov/cneaf/pubs html/feat trans capacity/table2.html
). Finally, the financial data are taken to be r = 0 .
06 and µ = 0 .
10, while variable operating costs are assumed to be c
1
= 1, c
2
= 1 .
05, and c
3
= 1 .
10.
From a simple plot of the three lines’ NPVs (see Figure 3), we ascertain that the smallest line is
0
12
= 9 .
12. Even then, since line
1 would offer a negative profit, no investment would occur. The choice is then clearly between lines 2
0
23
= 24 .
69. This indifference point between the two projects becomes important only if the volatility is reasonably small, i.e., less than 2.68 in this case. In that range, A 0
21
> A 0
31
, which means that it is optimal to invest directly in the 796 MW line for a given range of congestion rent, i.e., if
X 0
2
≤ X ≤ X 0
6
, and to invest directly in the 1096 MW line for X ≥ X 0
7
. If the congestion rent is in any other range, then it is optimal to wait for more information. Specifically, an interesting situation is if the congestion rent is near the indifference point: if 23 .
63 = X 0
6
< X < X 0
7
= 25 .
67, then no action is taken until the congestion rent either increases or decreases to the upper or lower threshold of the indifference zone, respectively (see Figure 3). In the former (latter) case, investment in the 1096 MW (796 MW) line is triggered. Since the current nodal price difference is Y
0
= 2 .
22, we assume that the ex post congestion rent will be less than this, which is also less than the lower investment threshold in the 796 MW line alone ( X 0
2
= 22 .
77); consequently, it would be acceptable to analyse this investment opportunity using

Transmission Investment Timing and Sizing under Uncertainty 12
Estimation of Parameters for Mean−Reverting Process
200
150
100
50
0
−50
−100
−150
−50 0 50 100 150
Lagged Nodal Price Difference (US$/MWh)
200
Figure 2: Autoregression of Daily Nodal Price Differences the approach of [7]. However, if the congestion rent were in the indifference zone, (23 .
63 , 25 .
67), then we would erroneously commit to either the 796 MW or the 1096 MW line without enough information. For example, if X
0
0
23 would have a NPV of
= 24 .
i
V
3
0
23
) − I
3
= K
3 h
0
23
η + µ
+ d
3
− I
3
= US$124.5 million. On the other hand, if the approach of [4] is used, then no investment would occur yet, but the investment opportunity would have an option value of F ( ˜ 0
23
) = C 0 e β
1
0
23
+ D 0 e β
2
0
23
= US$125.6 million. Hence, not taking into account the uncertainty around the indifference zone would be like throwing away US$1.11 million.
For a moderate level of volatility ( σ > 2 .
68), however, the larger project is more profitable from a real options perspective. Given the current value of σ = 9 .
43, we obtain A 0
31
> A 0
21
> A 0
11
, which according to the algorithm developed in [4] implies that the two smaller projects can be ignored. In other words, the TRO can use standard real options analysis to determine the value and trigger congestion rent for investment in line 3 only. We find that X 0
3
= 44 .
66 here, and the value of the option to invest at this threshold is worth US$468 million (see Figure 4). By performing sensitivity analysis for the level of volatility, we find regions in which it is optimal to wait or invest immediately in one of the transmission lines (see Figure 5). As discussed earlier, it is only for a low level of volatility that the investment region is dichotomous around the indifference point. Intuitively, when the volatility is low, the congestion rent is more likely to linger in the indifference zone. Due to this reason, we do not observe a dichotomous investment region for high levels of volatility.
When we introduce limited liability for the TRO, the results change slightly since there is less risk involved with any investment. Indeed, the customers at the importing node are willing to ensure that the TRO does not lose money. As a result, the NPV functions are non-linear as well. For low levels of

Transmission Investment Timing and Sizing under Uncertainty
0.6
0.4
0.2
0
2 x 10
8
Value of Option to Invest in Transmission Capacity
(unlimited liability,
σ
= 2.68)
1.8
1.6
1.4
1.2
V
1
(X) − I
1
V
2
(X) − I
2
V
3
(X) − I
3
F(X), X
≤
X
0
2
F(X), X
0
6
< X < X
0
7
X
0
2
X
0
6
X
0
7
1
0.8
16 18 20 22
X (US$/MWh)
24 26 28
13
Figure 3: NPV and Value of the Option to Invest under Discrete Capacity Choice: Unlimited Liability with Low Volatility volatility, the 796 MW line may be still be selected. From Figure 6, it can be seen that the indifference zone shifts to (22 .
91 , 25 .
41). This is because ˜
23 is now decreasing with the volatility (see Figure 8) as higher levels of volatility diminish the downside of investing in line 3 without affecting its increased profitability (see Figure 7). Furthermore, the investment threshold for line 3 is less sensitive to σ under limited liability, which is a consequence of the lower risk due to the guarantee on operating cash flows. Here, for σ = 2 .
96, if the congestion rent were in the indifference zone, then we would erroneously commit either to the 796 MW or 1096 MW line without enough information. For example, if X
0
X
23
= 24 .
i have a NPV of V
3 23
) − I
3
= K
3 h X
23
η + µ
+ d
3
+ B
32 e β
2
X
23 − I
3
= US$123.7 million. By contrast, the approach of [4] would recommend postponing investment to retain the option, which would be worth
F ( ˜
23
) = Ce β
1
X
23 + De β
2
X
23 = US$125 million. Hence, not taking into account the uncertainty around the indifference zone would be like throwing away US$1.3 million.
Certainly, by offering limited liability, the ISO provides a lucrative incentive for the TRO to invest.
In effect, the importing customers are sharing some of their welfare gains with the TRO. We can quantify the value of this limited liability given that X
0
= 0 as follows:
F = max i
A i 1
− max i
A 0 i 1
(35)
We plot F as a function of the volatility in Figure 9 and note that the value of limited liability increases with σ to almost 15% of the value of the option to invest when X
0
= 0 and σ = 9 .
43. Indeed, the benefit of limited liability is that provides protection against losses, which may be more likely in a more volatile setting. At the same time, it does not detract the opportunity to profit from high congestion rents, which are also more likely to occur when σ is high. Consequently, limited liability not only increases the

Transmission Investment Timing and Sizing under Uncertainty
−2
−4
−6
−10
2
0
8 x 10
8
Value of Option to Invest in Transmission Capacity
(unlimited liability,
σ
= 9.43)
6
4
V
1
(X) − I
1
V
2
(X) − I
2
V
3
(X) − I
3
F(X)
X
0
3
0 10 20 30
X (US$/MWh)
40 50 60
14
Figure 4: NPV and Value of the Option to Invest under Discrete Capacity Choice: Unlimited Liability with High Volatility value of the investment opportunity, but also induces investment in transmission capacity sooner.
The deregulation of the electricity industry has attempted to improve economic efficiency in a traditionally state-regulated sector by providing price signals where feasible to promote more timely investment, operation, and consumption. However, the backbone of such a decentralised framework, the transmission network, has suffered from a lack of investment, which may have severe consequences for network reliability and functioning of modern societies. Efforts to tackle the investment problem via markets imply that merchant investors must be shown adequate monetary incentives.
Here, we take the perspective of a TRO to investigate its transmission investment and capacity sizing problem via the real options approach. We find that although the TRO would be willing to pay more for the perpetual option to invest in transmission lines than suggested by the DCF, it would, nevertheless, not exercise this right until anticipated ex post congestion rents would increase by more than their current levels. Still, once these congestion rents are reached, the TRO would construct the line of the highest possible capacity from among its alternatives with investment being precipitated by the offering of limited liability by the ISO. Hence, offering limited liability might be an effective policy tool that induces private investment in order to reduce electricity prices at the importing node by hedging some of the risks for TROs.
For future work, we would like to model congestion rents more realistically with jumps. Indeed, using a higher volatility parameter is only an approximation for the persistence of jumps. Finally, we would also like to consider the case of bi-directional rights.

Transmission Investment Timing and Sizing under Uncertainty 15
35
30
25
20
45
40
15
10
5
2 3
X
0
2
X
0
12
~
X
0
6
X
Investment Thresholds
0
7
(X
0
3
for
σ
> 2.68)
X
0
23
~
Invest in 796 MW line
4 5
Invest in 1060 MW
6
σ line
Wait
7 8 9 10
Figure 5: Investment Thresholds under Discrete Capacity Choice with Unlimited Liability
[1] Barmack, M, P Griffes, E Kahn, and S Oren (2003), “Performance Incentives for Transmission,”
The Electricity Journal 16(3): 9–22.
[2] Brand˜ao, LE and EC Saraiva (2007), “Valuing Government Guarantees in Toll Road Projects,” in
Proceedings of the 11th Annual Real Options Conference, Berkeley, CA, USA (6–9 June 2007).
[3] Dangl, T (1999), “Investment and Capacity Choice under Uncertain Demand,” European Journal of Operational Research 117: 415–428.
[4] D´ecamps, J-P, T Mariotti, and S Villeneuve (2006), “Irreversible Investment in Alternative
Projects,” Economic Theory 28: 425–448.
[5] Deng, S-J (2005), “Valuation of Investment and Opportunity-to-Invest in Power Generation Assets with Spikes in Electricity Price,” Managerial Finance 31(6): 94–114.
[6] Deng, S-J, B Johnson, and A Sogomonian (2001), “Exotic Electricity Options and the Valuation of
Electricity Generation and Transmission Assets,” Decision Support Systems 30: 383–392.
[7] Dixit, AK (1993), “Choosing Among Alternative Discrete Investment Projects under Uncertainty,”
Economics Letters 41: 265–268.
[8] Dixit, AK and RS Pindyck (1994), Investment under Uncertainty , Princeton University Press,
Princeton, NJ, USA.
[9] Eydeland, A and K Wolyniec (2003), Energy and Power Risk Management: New Developments in
Modeling, Pricing, and Hedging , John Wiley & Sons, Hoboken, NJ, USA.

Transmission Investment Timing and Sizing under Uncertainty
20 x 10
7
Value of Option to Invest in Transmission Capacity
(limited liability,
σ
= 2.96)
X
2
X
6
X
7
15
10
5
V
1
(X) − I
1
V
2
(X) − I
2
V
3
(X) − I
3
F(X), X < X
2
F(X), X
6
< X < X
7
0
16
−5
20 21 22 23 24
X (US$/MWh)
25 26 27 28
Figure 6: NPV and Value of the Option to Invest under Discrete Capacity Choice: Limited Liability with Low Volatility
[10] Huisman, R and R Mahieu (2003), “Regime Jumps in Electricity Prices,” Energy Economics 25:
425–434.
[11] Joskow, P and J Tirole (2005), “Merchant Transmission Investment,” Journal of Industrial Economics 53(2): 233–264.
[12] Kristiansen, T and J Rosell´on (2006), “A Merchant Mechanism for Electricity Transmission Expansion,” Journal of Regulatory Economics 29(2): 167–193.
[13] N¨as¨akk¨al¨a, E and Fleten, S-E (2005), “Flexibility and Technology Choice in Gas Fired Power Plant
Investments,” Review of Financial Economics 14(3–4): 371–393.
[14] Perez-Arriaga, IJ, FJ Rubio, JF Puerta, J Arceluz, and J Marin (1995), “Marginal Pricing of
Transmission Services: an Analysis of Cost Recovery,” IEEE Transactions on Power Systems 10(1):
546–553.
[15] Pindyck, R (1989), “Irreversible Investment, Capacity Choice, and the Value of the Firm,” The
American Economic Review 78(5): 969–985.
[16] Saphores, J-D, E Gravel, and J-T Bernard (2004), “Regulation and Investment under Uncertainty: an Application to Power Grid Interconnection,” Journal of Regulatory Economics 25(2): 169–186.
[17] Skantze, P and M Ilic (2000), “The Joint Dynamics of Electricity Spot and Forward Markets:
Implications on Formulating Dynamic Hedging Strategies,” working paper, Massachusetts Institute of Technology Energy Laboratory, Cambridge, MA, USA.

Transmission Investment Timing and Sizing under Uncertainty
2
1
0
−1
−10
5
4
3
7
6
8 x 10
8
0
Value of Option to Invest in Transmission Capacity
(limited liability,
σ
= 9.43)
X
3 V
1
(X) − I
1
, X
≥
c
1
V
1
(X) − I
1
, X < c
1
V
2
(X) − I
2
, X
≥
c
2
V
2
(X) − I
2
, X < c
2
V
3
(X) − I
3
, X
≥
c
3
V
3
(X) − I
3
, X < c
3
F(X) c
3
10 20 30
X (US$/MWh)
40 50 60
17
Figure 7: NPV and Value of the Option to Invest under Discrete Capacity Choice: Limited Liability with High Volatility
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Power Plant under Regulation of Electricity Price,” Decision Support Systems 37: 449–456.
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Osaka, Japan (5–7 August 2002).
The value of the option to invest is trichotomous:


A 0
1 e β
1
X
F ( X ) =


C 0 e
G 0
β
1 e β
1
X
X
+ D 0
+ H 0 e β
2
X e β
2
X if X < X 0
1 if X 0
4
< X < X 0
5 if X 0
6
< X < X 0
7
(A-1)
Here, X 0
1
< X 0
4
< 0
12
0
12
< X 0
5
< 0
23
, X 0
5
< X 0
6
< 0
23
, and X 0
7
>
0
12
0
23 are congestion rent thresholds
0
23
. In order to find the four endogenous constants, C 0 , D 0 , G 0 , and H 0 , along with the indifference investment thresholds, X 0
4
,
X 0
5
, X 0
6
, and X 0
7
, we write the following value-matching and smooth-pasting conditions:
⇒ C 0 e β
1
X
0
4
+
F
D
( X 0
4
0 e β
2
) =
X
0
4
V
1
( X 0
4
= K
1 h
) − I
1 h
X 0
4
η + µ
+ d
1 i
− I
1
(A-2)

Transmission Investment Timing and Sizing under Uncertainty 18
30
25
20
15
40
35
10
5
0
2
Investment Thresholds
Invest in 1060 MW line
3 4
Wait
Invest in 796 MW line
5 6
σ
7
X
2
X
~
12
X
6
X
7
(X
3
for
σ
> 2.96)
X
~
23
8 9 10
Figure 8: Investment Thresholds under Discrete Capacity Choice with Limited Liability
F 0 ( X 0
4
) = V
1
0 ( X 0
4
)
⇒ β
1
C 0 e β
1
X
0
4
+ β
2
D 0 e β
2
X
0
4
= K
1 h
η + µ
(A-3)
⇒ C 0 e β
1
X
0
5
+
F
D
( X 0
5
0 e β
2
) =
X
0
5
V
2
( X 0
5
= K
2 h
) − I
2 h
X 0
5
η + µ
+ d
2 i
− I
2
F 0 ( X 0
5
) = V
2
0 ( X 0
5
)
⇒ β
1
C 0 e β
1
X
0
5
+ β
2
D 0 e β
2
X
0
5
= K
2 h
η + µ
(A-4)
(A-5)
⇒ G 0 e β
1
X
0
6
+
F
H
(
0
X e
0
6
) =
β
2
X
0
6
V
=
2
( X 0
6
K
2 h
) − I
2 h
X
6
η + µ
+ d
2 i
− I
2
F 0 ( X 0
6
) = V
2
0 ( X 0
6
)
⇒ β
1
G 0 e β
1
X
0
6
+ β
2
H 0 e β
2
X
0
6
=
K
2 h
η + µ
⇒ G 0 e β
1
X
0
7
+
F
H
(
0
X e
0
7
) =
β
2
X
0
7
V
=
3
( X 0
7
K
3 h
) − I
3 h
X
0
7
η + µ
+ d
3 i
− I
3
F 0 ( X 0
7
) = V
3
0 ( X 0
7
)
⇒ β
1
G 0 e β
1
X
0
7
+ β
2
H 0 e β
2
X
0
7
= K
3 h
η + µ
(A-6)
(A-7)
(A-8)
(A-9)

Transmission Investment Timing and Sizing under Uncertainty 19
15 x 10
6 Option Value of Limited Liability
10
5
0
2 3 4 5 6
σ
7 8 9 10
Figure 9: Option Value of Limited Liability