You are operating a gas turbine power station (GTPS). The risk factor driving cash inflows (revenues) is the electricity price. The risk factor driving cash outflows (expenses) is the gas price. At current prices, the yearly gas bill is EUR 50m (assume payment upfront) for producing a volume of electric power that can be sold for EUR 55m on the market (assume payment upfront).The electricity and gas price follow a multiplicative binomial distribution with the parameters u=1.8 and d =0.6 (electricity) and u=1.5 and d=0.8 (gas). Also for simplicity, we assume perfect correlation. The objective probability q for an upward movement is 0.5. The risk-free rate is equal 8.00% per yearWhat is the value of the GTPS closest to, if it lives for two years and one binomial step is equal one year? Conduct the valuation on a risk-neutral basis.

Question

BlackTom AI · Accepted Answer

We start by restating the setup in our own words to ensure clarity: a GTPS has revenues tied to electricity price and costs tied to gas price. At the baseline, the yearly gas bill is 50m EUR (paid upfront) and the revenue from selling electricity is 55m EUR (also treated as upfront in the problem). Prices evolve with a multiplicative binomial model with up = 1.8, down = 0.6 for electricity and up = 1.5, down = 0.8 for gas, and the two assets move in perfect correlation. The risk-free rate is 8% per year, and the project lasts two years with one-year steps. The risk-neutral valuation will be used.

Option-by-option analysis:

A. 15.00 million EUR
- To get a feel for the magnitude, imagine summing two years of payoffs under risk-neutral pricing. Under perfect correlation, in an up year electricity price multiplies by 1.8 and gas price by 1.5; in a down year they multiply by 0.6 and 0.8 respectively. Over two years, the UU state gives revenue 55 × 1.8^2 = 55 × 3.24 = 178.2, cost 50 × 1.5^2 = 50 × 2.25 = 112.5, net payoff 65.7. The DD state gives revenue 55 × 0.6^2 = 55 × 0.36 = 19.8, cost 50 × 0.8^2 = 50 × 0.64 = 32, net payoff -12.2. Discounting these back two years at (1.08)^2 = 1.1664 and weighting by the risk-neutral probabilities (p = (1.08 − 0.6)/(1.8 − 0.6) = 0.4 per year, so UU has probability 0.4^2 = 0.16 and DD has probability 0.6^2 = 0.36) gives an approximate present value around (0.16 × 65.7 + 0.36 × (−12.2)) / 1.1664 ≈ (10.512 − 4.392) / 1.1664 ≈ 6.120 / 1.1664 ≈ 5.25. This is far from 15.0, indicating option A is too high, as it implies a much larger risk-adjusted value than the computed figure.

B. 5.00 million EUR
- This matches the ballpark we just computed: a present value near 5.25m using the straightforward risk-neutral two-period binomial with perfect correlation. The key steps are: (i) compute the two-end-state payoffs UU and DD as shown above, (ii) use the year-1 risk-neutral up prob p = 0.4 (since (1.08 − 0.6)/(1.8 − 0.6) = 0.4 for electricity and the same computation holds for gas), so UU has probability p^2 = 0.16 and DD has probability (1 − p)^2 = 0.36, (iii) discount by (1.08)^2 = 1.1664. The resulting PV of the two-state payoff is around 5.25m, which is closest to 5.00m among the options. So B is plausible.

C. 16.23 million EUR
- This would require a substantially higher present value. Recalling the end-state payoffs (65.7 for UU and −12.2 for DD) and the risk-neutral weights (0.16 and 0.36) along with 2-year discounting, the PV tends to lie around the mid-single digits, not over 15m. The arithmetic shows the expected undiscounted payoff is 0.16×65.7 + 0.36×(−12.2) ≈ 6.12, and after discounting by 1.1664 it drops to roughly 5.25m. Therefore 16.23m is inconsistent with these numbers and would require an incorrect treatment of probabilities or payoffs.

D. 8.64 million EUR
- This option is somewhat higher than the computed ~5.25m, and to reach 8.64m you’d need a larger expected discounted payoff or different probabilities. Given p = 0.4 per year and the two-end-state framework with UU and DD, the weighted, discounted payoff sits around 5.0–5.5m. The 8.64m figure would imply either a mis-specified state payoff or a misapplied discounting. In short, this option overestimates the value in the specified model.

Summary observation:
- The core exercise yields a present value in the vicinity of about 5 million EUR when computed with the risk-neutral up probability p = 0.4, end-state payoffs UU = 65.7 and DD = −12.2, two-year discount factor 1.1664, and end-state probabilities 0.16 and 0.36. Among the choices, 5.00 million EUR is the value that aligns with that analysis, while the other options are inconsistent with the calculated risk-neutral valuation under the given parameters.

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