Oil Field Valuation – An Real Option Based Analysis

Oil Field Valuation – An Real Option Based Analysis

Final Report

Contemporary Concerns Study
To

Prof. V. Ravi Anshuman

On

August 28, 2007

Sumit Kumar (0611053)

There are various approaches available for valuation of oil fields, including the DCF approach but there are certain fundamental problems due to which one needs to look to another methodology for correct valuation. Option world provides such an alternative with real option valuation models constantly being developed for natural resources, and especially for oil. We have implemented one such model to gain a better understanding of effects of various changes of parameters on valuation and analyze recent trends in real-life valuations of oil fields.

The model splits the valuation of the oil field into three stages

1. 
   Production stage
   
2. 
   Development stage
   
3. 
   Pre development stage wherein one has an option to start development of the oil field within a fixed time frame
   

The model is used to analyze an illustrative example to understand the evolution of an oil field value through the three stages. Theoretical frameworks are used to understand this evolution. Analysis has also been done of the impact of varying two key parameters of the model on the valuation results.

In the last section, valuation of two actual oil fields, the Sakhalin-I oil field in Russia and the Atlantis oil field in USA, have been done. It is observed in both cases that the two oil field values have limited dependency on spot value of oil and the time left to the expiry of the development option. Moreover both have an extremely low critical spot price (Spot price above which it immediate exercise of the option is mandated) compared to the long term average oil price. All these aspects are identified to be a result of the relatively long operational life of the two oil fields which limits the impact of near time factors.

This model, in its current shape, can be used for the valuation of oil fields subject to the assumptions of a relatively rigid operational cash flow structure. It enables one to chart the evolution of the oil field value over the three stages and study the impact of changes in different parameter values.

In such a case, valuation of oil fields is expected to sky northwards. There are various ways of approaching valuation of an oil field (valuation is an art, not science), real option valuation is one such robust approach towards valuation.

It is projected that it will increase close to 50% in the next 25 years. Total fossil fuels, including coal, will continue to account for about 80% of this energy demand. Oil & Gas alone is about 60%. All other forms of energy account for 20%, including nuclear, hydro, biomass and other renewable forms of energy.

Solar and wind energy are growing the fastest of any of the fuel types – it is projected to grow by more than 10% per year - driven by government subsidies and related mandates. But, even with this growth rate, their share will only be about 1% of total energy in 2030, simply because the world energy market is so huge, and they are starting with such a small base.

Oil demand component is expected to grow about 1.4% per year from 2000 to 2030. Oil demand in the OECD countries of North America, Western Europe and Japan is expected to plateau in roughly 20 years. On the other hand, in the emerging markets of the non- OECD countries, oil demand is expected to grow at about 2.5% per year. At this rate, by 2030 non-OECD oil demand will be well in excess of that in the OECD countries.

The energy future which is being created is unsustainable. If continued as before, the energy supply to meet the needs of the world economy over the coming years will remain too vulnerable to failure arising from sudden supply interruption and will cause serious environmental problems. There are various key issues associated with this sensitive topic:

1. 
   The world is facing twin energy-related threats: that of not having adequate and secure supplies of energy at affordable prices and that of environmental harm caused by its use. A package of policies and measures that countries around the world are considering would, if implemented, significantly reduce the rate of increase in demand and emissions. Importantly, the economic cost of these policies would be more than outweighed by the economic benefits that would come from using and producing energy more efficiently.
   
2. 
   Oil demand grows by 1.3% per year through 2030 in the Reference Scenario, reaching 116 million barrels per day in 2030 – up from 84 million barrel/day in 2005.
   
3. 
   The transport sector absorbs most of the increase in global oil demand (63%). The lack of cost-effective substitutes for oil-based automotive fuels will make oil demand more rigid.
   
4. 
   Oil supply is increasingly dominated by a small number of major producers. The need for more transparent and comprehensive data on oil (and gas) reserves in all regions is a pressing concern.
   
5. 
   The oil industry needs to invest a total of $4.3 trillion (in year-2005 dollars) over the period 2005-2030, or $164 billion per year. The upstream sector accounts for the bulk of this. Almost 3/4^th of upstream investments will be required to maintain existing capacity.
   
6. 
   A critical uncertainty is whether the substantial investments needed in the oil production sector in key Middle East countries will, in fact, be forthcoming. These governments could choose deliberately to develop production capacity more slowly.
   

In contrast, the world’s proven reserves of oil (crude oil, natural gas liquids, condensates and non-conventional oil) amounted to 1,293 billion barrels at the end of 2005 – an increase of 14.8 billion barrels, or 1.2%, over the previous year. Proven reserves have grown steadily in recent years in volume terms, but have remained broadly flat as a percentage of production. Since 1986, the reserves-to-production, or R/P, ratio has fluctuated within a range of 39 to 43 years. A growing share of the additions to reserves has been coming from revisions to estimates of the reserves in fields already in production or undergoing appraisal, rather than from new discoveries. Some of these revisions have resulted from higher oil-price assumptions, allowing some oil that is known to exist to be reclassified as economically exploitable and, therefore, moved into the proven category.

According to traditional finance, valuation of oil fields should be done by NPV approach which means calculating the expected cash flows and discount them using some discount rate that takes into account risk premium (can be calculated by a model like CAPM). But there is a fundamental problem with this approach:

1. 
   High volatility of oil prices introduces some uncertainties which are difficult to handle using these DCF methods. For example, it becomes difficult to determine, when it is optimal to commit heavy investment resources to develop a marginal oil field and which oil price to use in valuing output.
   
2. 
   Oil price uncertainty translates into different risk premiums depending on the specific project operating leverage and its optimal investment & operating strategies.
   

The real option model for valuation of undeveloped oil field is a no-arbitrage model, as always has been the case with option valuation frameworks. The model assumes stochastic, but mean reverting, risk adjusted oil spot prices, and includes a timing investment option. The major advantage with real option valuation approach is that all possible real life decision possibilities can be incorporated in the model:

1. 
   The source of uncertainty is output price and firms can respond to this either by delaying production temporally or permanently, depending on price levels and volatility.
   
2. 
   The value is determined by using future prices and not by requiring predictions on spot prices, hence removing one of the main sources of error in natural resource investment valuations.
   
3. 
   Along with firm value, the optimal response of the firm is also determined (when to delay or resume production)
   

There are various variants to this basic model to take into account some specific characteristics of investment projects, like reduction of costs due to learning curve effects, considering investment rate as control variable instead of production level, changing source of uncertainty from commodity price to exchange rate or costs, etc. There are also customized models for valuation of copper mines, oil reserves, research and development, environmental technologies, and flexible production.

The model considers the valuation of an undeveloped oil field as a three stage process, in which starts by first calculating the value of the field in the third stage and works backwards to calculate the value of the field in first stage. This section seeks to give a brief idea of the concepts underlying this model¹.

The following are the major assumptions made in the development of this model.

1. 
   The project manager can decide optimally when to incur in a development investment which, once it is completed, initiates production.
   
2. 
   The option to delay investment expires and the firm must decide if it commits investment or gives up the concession.
   
3. 
   The oil field faces a competitive market for oil in which spot prices are uncertain, while firm characteristics like reserves, development investment requirements and production unit costs are considered deterministic.
   
4. 
   A risk-adjusted Brownian motion for oil prices is defined following the continuous time risk-neutral valuation approach standard in no-arbitrage finance models. Existing evidence of mean reversion in oil prices is incorporated into the model by using a mean convenience yield, which depends on the deviation of spot price to a long term average price S.
   

Where

1. 
   S – Spot unit price of oil
   
2. 
   Ŝ – Long term average unit price of oil
   
3. 
   β – Mean Reversion parameter
   
4. 
   r – risk free rate of interest (which is assumed to be constant)
   
5. 
   c – Mean convenience yield on holding one unit of oil
   
6. 
   σ – Instantaneous volatility of returns on holding one unit of oil
   
7. 
   dW – Increments to standard Gauss – Wiener process
   

1. 
   H(S, Ti) – Oil field value for Stage I
   
2. 
   U(S, Td) – Oil field value for Stage II
   
3. 
   V(S, Q) – Oil field value for Stage III
   
4. 
   Ti – Remaining time till expiration of development option
   
5. 
   Td – Remaining time till investment is completed in Stage II
   
6. 
   Q – Remaining reserves of oil
   
7. 
   t – time
   
8. 
   I – PV of development investment in Stage II
   
9. 
   qp – Annual production rate in Stage III
   
10. 
    a – Unit production cost in Stage III
    
11. 
    cc – Annual capital costs in Stage III
    
12. 
    dep – Depreciation
    
13. 
    R – Taxes (royalty rate)
    
14. 
    Corp – Taxes (corporate tax rate)
    
15. 
    λ – Country risk premium
    
16. 
    S*ci – Critical spot oil price
    

In order to estimate the values of parameters like the convenience yield associated with oil and the mean reversion parameter, one uses oil futures as a proxy and performs a regression analysis to arrive at their values.

Applying Ito’s Lemma to value of future:

dF(S,Τ) = F_s dS – F_t dt + ½ F_ss S² σ² dt

Returns on futures contract are perfectly correlated with those on the spot. So if there are no arbitrage opportunities in oil market, then an investor with a portfolio which is long in one unit of oil and short in (Fs)-1 units of future contract has hedged his price risk completely and hence will earn the risk-free interest rate:

dS + (c – β (Ŝ – S)) S dt – dF/F_S = r S dt

From the two equations above, we can obtain the following differential equation for the value of a futures contract:

½ Fss S² σ² + S F_S (r – c + β (Ŝ – S)) – F_t = 0

subject to boundary conditions

F(S,0) = S

F_ss(0, Τ) = 0

F_ss(∞, Τ) = 0

This equation is discretized using the numerical methods presented in the last part of this section and a regression analysis is done using historical data to arrive at the parameter values.

Value of an Oil Field

The oil field can be modeled as a real asset which could be in any of three different stages:

1. 
   Stage I is before committing to the development
   
2. 
   Stage II is during development
   
3. 
   Stage III is during production
   

The model is solved in a dynamic programming fashion by valuing the oil field in stage 3 and then work way backwards to solve for value of the oil field in stages 2 and 1.

Applying Ito’s Lemma to value of an oil field in stage 3, V(S, Q)

dV(S,Q) = V_S dS – q_p V_Q dt +1/2 V_SS S² σ² dt

The investor holding the oil field in stage 3 will receive a cash flow equal to M:

M = (q_p (S – a) – cc – t1 – t2)

t1 = R q_p S

t2 = Corp (q_p (S – a) – cc –dep – t1)

If an investor takes a long position in the oil field and Hs / Fs short position in futures contracts, then he hedges this risk and earns a return equal to the risk free interest plus a risk premium associated with the country where oil field is located

dV + M – (V_S/F_S) dF = (r + λ) V

Therefore, value of an oil field in stage 3 is given by

½ V_SS S² σ² dt - q_p V_Q + M + (r – c + β (Ŝ – S)) S V_S – (r + λ) V = 0

subject to boundary conditions

V(S,0) = 0; V_SS (0,Q) = 0; V_SS (∞,0) = 0)
Stage II

Using similar arguments, value of oil field in stage 2 is given by following differential equation:

½ U_SS S² σ² dt - U_Td + (r – c + β (Ŝ – S)) S U_S – (r + λ) U = 0

subject to boundary conditions

U(S,0) = V(S,Q); U_SS(0, T_d) = 0; U_SS(∞, T_d) = 0

Stage I

Finally the value of oil field must satisfy the following condition (if there is no arbitrage):

½ H_SS S² σ² dt - H_Ti + (r – c + β (Ŝ – S)) S H_S – (r + λ) H = 0

Subject to the boundary condition (when available time to initiate development is exhausted, the value of the undeveloped oil field will depend on whether it is optimal for the owner to make the development investment and obtain a stage 2 oil field, or to give up the concession):

H(S,0) = Max (U (S, T_i) – I, 0)

and other boundary conditions

H_SS(0, T_i) = 0; H_SS (∞, T_i) = 0

There is an optimal investment policy defined by a critical spot price (S), below which it is optimal to wait and above which it is optimal to develop:

H(S, Ti) = U(S, T_d ) – I if S > S*ci

By maximizing the left-hand side of above differential equation, subject to above conditions, the critical spot price can be determined. The value of the development option when it is optimal to wait as a function of the remaining time until expiration amounts to the difference between keeping the option alive and killing it. This development (or timing option) may be very valuable, depending on the characteristics of the oil field.

The model presented above does not have any analytical solution, hence must be solved by utilizing numerical methods. Finite difference methods are used to solve a discretized version of differential equations. For example, H(S,T_i) used in stage I solution is discretized as:

Fig 3.1 – Discretization of model

Following transformations are done to reach a numerical solution (Stage I)

S = i ΔS t = j ΔTⁱ

H_SS = (H_i+1,j + H_i-1,j – 2 H_i,j)/((ΔS)2)

H_S = (H_i+1,j – H_i-1,j )/(2 (ΔS))

To obtain the critical spot price, that value of ‘i’ is searched for which H(S, Tⁱ) is maximized, while satisfying the following equation:

U_i,m – I = H_i,j for j = 0,1…m

The model gives rise to following tridiagonal system of equations which must be solved to obtain the value of the oil field