(28)
A Numerical Case
The Data
Taiwan High Speed Rail BOT Project is now used as a numerical case to apply the foregoing pricing formulas. The Project’s detailed financial data and projections can be obtained from the authors. As summarized in Table 1, the Project’s total concession period is 36 years, the present value of the total capital investment cost is NT$ 242,422 M, and the present value of the operating costs is NT$ 101,429 M. The variation of the Project’s revenues is small, so the Generalized Wiener process can be applied. The average growth rate is thus estimated at 6.02%, and we use 6% as an approximation. The volatility of Taiwan’s stock market price varies from 0.2 to 0.4, and we use 0.3 as the volatility of the growth rate. Finally, the Project’s discount rate is estimated at 12% by the financial consultant.
For the purpose of this application, we suppose the Project has an MRG, and the annual MRG levels are equal to the corresponding annual operating costs. In addition, although the construction commencement date is not specified in the concession contract, suppose the pre-construction period of the project is one year after the contract signing date. Table 1 includes a pre-construction cost and its present value is denoted as J0. The cost is required to keep the Project alive until t
= tB. It will reduce the Project’s NPV, or NPV = P0 – I0 – J0, and affect the value of the investment option. Therefore, the price formula for F should be adjusted.
When the MRG is not available, the price formula is as the following:
0 ( )1 0 ( ) 2
F =P N k −I N k − J0 (29)
When the MRG is available, the price formula for FM is adjusted as:
0 0 0 0
1 1 2 2 2
1
0 1
( , ; ) ( , ; ) ( ) ( )
n tn tn n tn tn n n tn
i
m
M t B n t B n t t
i m
t i
F R B a b t t M B a b t t D M N a J m0
F J
=
=
⎡ ⎤
= ⎣ − − − ⎦
= −
∑
∑
−
(30)
Results and Interpretations
The results of this numerical case are shown in Table 2, from which we have the following observations:
(a) The Project’s NPV is NT$139,508 M, so the Project is financially feasible.
(b) The option to abandon creates values. The value of the option, or f, is NT$7,436 M, calculated by either equation (5) or (28). By equation (5), for example, the shortfall rate is calculated at 6% from the growth rate and the discount rate. N(k1) = 0.9047, N(k2) = 0.8434, and F = NT$ 146,944 M. F is greater than the Project’s NPV, and the amount of f is obtained by subtracting the NPV from it.
(c) The MRG also creates values. The option value of the MRG, or Q, is
$7,716 M, calculated by (9).
(d) When the options are combined, the foregoing valuations change. The value of the option to invest with the MRG, or FM, is NT$152,897 M, calculated by the adjusted compound option pricing formula in (29). The MRG value or Qf is NT$5,953 M, calculated by subtracting F from FM by (19). And the value of the option to abandon is $5,673 M, calculated by subtracting the NPV and the value of the MRG from FM by (21).
(e) As a result, when the MRG and the option to abandon are combined, their values are reduced. The values disappeared are called joint value, which is equal to NT$1,763 M in this numerical case. This result is consistent with Trigeorgis’
observation. When options intended to control down-side risks are exercised at the same time, they will counteract each other, and thus reduce their own values (Trigeorgis, 1993). There are two counteracting forces in our compound option.
On one hand, the value of the MRG cannot be realized if the option to abandon is exercised at tB. So long as there is a chance that the project will be abandoned, the value of the MRG will not be fully realized. On the other, the presence of the MRG will increase the value of the underlying assets of the option to abandon, and thus reduce the value of the option itself.
(f) Although the option values are reduced, the total value created by the compound option is still substantial. The value is NT$13,389 M, calculated by subtracting the joint value from the original values of the MRG and the option to abandon in the single option models.
Policy Implications
Both the MRG and the option to abandon are valuable policy tools as we have shown, but the government should check if they will produce intended policy effects when combined in the same BOT package. For example, Table 3 shows if the pre-specified level of the MRG is increased by 250%, then the value of the option to abandon will decrease to NT$10M, which is no longer substantial compared to the Project’s NPV.
In general, the investment value of BOT projects increases with the MRG level, but the increase of the MRG will decrease the value of the option to abandon.
In terms of risk allocation, the higher the MRG level is, the more the down-side risk is transferred to the government, and the less likely the concessionaire will exercise the option to abandon. When both the MRG and the option to abandon are proposed at the same time, the government should check if the proposed MRG level will render the option to abandon worthless.
In practice, the MRG may be valuable from an investor’s perspective, but it requires substantial budgetary commitments. The benefits and costs of the commitments should be justified, and the pricing formulas can be used as valuation tools. When the project is not bankable due to high revenue risks, the lender may ask the government to provide MRG for a minimum level of debt coverage. In this case, MRG can also be justified by credit enhancement benefits to the concessionaire, such as reduced financing costs. However, to know how much the financing costs can be saved, MRG should be valued from lender’s perspective. This appears to be another open issue.
Whenever there are budgetary constraints, the option to abandon is a preferable, more easily justified policy choice; especially it can only be used in the pre-construction phase. If the option can be used during construction and operation,
abandonment will cause greater disruptions and disturbances, and this will make the justification more difficult.
Conclusions
Two single option pricing models were first developed for the valuation of the MRG and the option to abandon in the pre-construction phase. The MRG was further combined with the option to abandon to develop a compound option model.
Taiwan High-Speed Rail Project was used as a numerical case to apply the derived option pricing formulas. The results indicated that both MRG and the option to abandon could create substantial values. When the MRG and the option to abandon were combined, they counteracted each other and their values were reduced. If the level of the MRG were high enough, the option to abandon would be rendered valueless.
MRG involves substantial budgetary commitments, and its benefits and costs should be carefully justified. The option to abandon is a preferable policy choice under budgetary constraints, and its justification is more straightforward.
Overall, the application of the real option approach in BOT project evaluation looks promising. The option pricing formulas developed in this paper can be used as valuation tools for the foregoing justifications as well as extensions of traditional project evaluation approaches, such as the discounted cash flow model.
But the presented formulas are limited in scope. They do not consider the options to abandon during construction and operation. In addition, if the MRG is to be used as a credit enhancement tool, it should also be valued from the lender’s perspective. Future researches are required to tackle these problems.
Acknowledgements
The authors wish to thank the National Science Council for research grant under contract number NSC 93-2211-E-009-043.
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Table 1 Data and assumptions.
Pre-construction 1 Expected growth
rate α 0.06 Construction 5 Growth rate
volatility σ 0.3 The
concession period
(yrs) Operation 30
Estimated parameters
Discount rate µ 0.12 Pre-construction
cost J0 : 3,400 Construction
cost B0 : 242,422 Investment
costs (NT$
million)
Operating cost
C0 : 101,429
Annual level of MRG,
ti
M , i=1~30
The corresponding annual operating cost,
ti
C
Table 2 Results.
Unit: million of NT$
Year (i) Rti Cti Mti N(-d2) N(-d1) Qti
1 75,879 12,740 12,740 0.0321 0.0041 43 2 81,125 13,586 13,586 0.0463 0.0057 64 3 86,744 14,630 14,630 0.0633 0.0076 91 4 92,685 15,568 15,568 0.0798 0.0093 116 5 99,155 16,537 16,537 0.0964 0.0108 140 6 105,999 17,353 17,353 0.1109 0.0119 158 7 113,303 18,582 18,582 0.1291 0.0135 183 8 120,457 19,942 19,942 0.1489 0.0153 211 9 128,184 21,144 21,144 0.166 0.0165 230 10 136,347 22,229 22,229 0.181 0.0174 243 11 145,027 23,599 23,599 0.1978 0.0185 259 12 154,256 24,969 24,969 0.2135 0.0194 270 13 164,069 26,540 26,540 0.2299 0.0203 283 14 174,501 28,763 28,763 0.2505 0.0220 306 15 185,594 30,434 30,434 0.2651 0.0226 311 16 197,386 32,090 32,090 0.2785 0.0230 313 17 209,924 34,503 34,503 0.2961 0.0242 326 18 222,578 36,379 36,379 0.3094 0.0246 325 19 235,987 38,600 38,600 0.3238 0.0252 327 20 250,196 41,005 41,005 0.3382 0.0258 328 21 265,253 44,198 44,198 0.3556 0.0269 337 22 281,206 46,728 46,728 0.3681 0.0272 332 23 298,110 49,493 49,493 0.3807 0.0275 328 24 316,018 52,620 52,620 0.3938 0.0279 325 25 334,993 56,671 56,671 0.4096 0.0288 329 26 355,096 59,865 59,865 0.4205 0.0289 321 27 376,393 63,756 63,756 0.4331 0.0292 316 28 398,341 67,404 67,404 0.4439 0.0294 308 29 416,926 71,081 71,081 0.4565 0.0298 300 30 434,757 75,295 75,295 0.4705 0.0305 295
Table 2 Continued.
Year (i) *0
ti
R D t0i B(a2,b2) B(a1,b1) N(a2) Fti
1 22,433 22,516 0.8473 0.9185 0.8669 10,675 2 21,272 21,392 0.8367 0.9172 0.8669 10,103 3 20,174 20,337 0.8236 0.9157 0.8669 9,560 4 19,118 19,319 0.8104 0.9143 0.8669 9,042 5 18,140 18,375 0.7970 0.9130 0.8669 8,564 6 17,199 17,457 0.7849 0.9121 0.8669 8,109 7 16,305 16,597 0.7698 0.9107 0.8669 7,674 8 15,374 15,702 0.7532 0.9092 0.8669 7,221 9 14,511 14,860 0.7388 0.9080 0.8669 6,805 10 13,689 14,050 0.7260 0.9073 0.8669 6,414 11 12,914 13,292 0.7116 0.9063 0.8669 6,043 12 12,183 12,571 0.6981 0.9054 0.8669 5,695 13 11,493 11,892 0.6840 0.9045 0.8669 5,367 14 10,841 11,266 0.6663 0.9031 0.8669 5,053 15 10,226 10,653 0.6537 0.9025 0.8669 4,764 16 9,646 10,070 0.6420 0.9021 0.8669 4,491 17 9,099 9,534 0.6267 0.9010 0.8669 4,231 18 8,557 8,986 0.6150 0.9006 0.8669 3,977 19 8,046 8,473 0.6024 0.9001 0.8669 3,738 20 7,566 7,991 0.5899 0.8995 0.8669 3,512 21 7,114 7,546 0.5746 0.8985 0.8669 3,299 22 6,689 7,112 0.5636 0.8982 0.8669 3,101 23 6,289 6,703 0.5526 0.8979 0.8669 2,914 24 5,913 6,320 0.5410 0.8975 0.8669 2,739 25 5,560 5,968 0.5272 0.8967 0.8669 2,573 26 5,227 5,622 0.5175 0.8966 0.8669 2,418 27 4,914 5,301 0.5064 0.8962 0.8669 2,273 28 4,612 4,987 0.4968 0.8961 0.8669 2,133 29 4,282 4,645 0.4858 0.8957 0.8669 1,979 30 3,960 4,314 0.4734 0.8951 0.8669 1,829
Table 3 Comparative static.
Change of the
MRG Level NPV FM Qf fM
Joint Value 0% (no MRG) 139,508 146,944 (= F ) 0 7,436 (= f ) 0
50% 139,508 148,267 1,323 6,992 444 100% 139,508 152,897 5,953 5,673 1,763 150% 139,508 161,233 14,290 4,132 3,304 250% 139,508 187,708 40,715 883 6,602 350% 139,508 227,037 80,093 10 7,426
to tm=T
t = 0 t = tB
Pre-construction Construction Operation Transfer
Figure 1 Typical BOT project life-cycle after contract signing.
Payoff
tn n R
Mt
Payoff
tn
M
tn
R
Payoff
tn n R
M
tFigure 4 The nth period revenue payoff function from MRG.
Figure 3 The nth period revenue payoff function from R.
Figure 2 The nth period revenue payoff function.
Payoff Payoff
tn n R
M
t tnn R Mt tn
M
Figure 6 A European style call option.
Figure 5 A constant cash flow