A methodology using option pricing to determine a suitable discount rate in environmental management

With the present analysis the authors propose an approach for determining an adequate discount rate in environmental management problems, more specifically radioactive waste management. It is shown that the classical Black-Scholes pricing formula can be used for determining the adequate present funding to be set aside for the future. The average funding is equal to the net present value (NPV) of the future costs, including technical-scenario uncertainties. For taking into account the financial uncertainties, the NPV is identified with the strike price of a European put option, and the asset value in the managed fund is identified with the current price. The discount rate to be determined is identified with the risk-free interest rate. The paper shows that the adequate present funding can be determined for given multi-generational risk levels and an asset allocation by fixing the discount rate and adding a premium to the NPV of future costs. 1. SCOPE OF THE ANALYSIS The choice of a discount rate for comparing environmental costs distributed over time is a difficult task, because of the multi-generational impacts of longterm externalities. The literature on this topic is vast, but a universal approach reconciling in any situation the economics with the multi-generational aspects does not seem to exist. With the present analysis the authors propose an approach for determining an adequate discount rate in the specific case where funding in the present supports the future generations in removing the long-term externalities. This methodology has been used by ONDRAF/NIRAS the Belgian agency for radioactive waste management for determining the right level of funding. 2. BASIC ASSUMPTIONS OF THE MODEL In this note we consider a general nuclear-wastemanagement project to be properly funded at the inception time, i.e., just before the project starts. It is assumed that the overall budget of the project, including uncertainty margins, is completely known. The problem is to evaluate the discount rate (d.r.) for calculating the Net Present Value of the total cost distributed over an extended time horizon, which is also assumed to be known. The funding is to be set aside in a fund, managed according to a specific strategic asset allocation (AA). To make things simple in this methodological paper, we will compare two AA’s in the following, using constant currency units at the date t=0, and thus real rates of return: A low-risk portfolio with 100% bonds. The couple (return=2,30%; standard deviation=3,90% ) is estimated from historical data series; A medium-risk portfolio with 50% equities/50% bonds. The couple (return=5%; standard deviation=10%) is estimated from historical data series. (Note that a an environment agency being often the property of the State would generally not be willing to implement a high-risk AA with 75% to 100% equities). Of course many other assumptions can be used. Further the assumption is made for the asset evolution of brownian motion, so that the resulting distribution of the asset price S is lognormally distributed at t=T. T is the portfolio liquidation date. We consider several T-values, and we examine in each case the funding ratio. We will be using the values T= 5, 15, 25, 50, 100 years, which are typical for radioactive-waste management. More formally, ln S(T) is normally distributed (see Hull 2000; Beninga 2000): 2 0 ln ( / 2) , S T T μ σ σ   Φ + −   (1) , where 0 ln S represents the value of the asset in t=0; ( , ) m s Φ represents the cumulated normal distribution of mean value m, and standard deviation s. Under those conditions, we can use the theory of financial options, i.e., here simply the Black-Scholes formula, to calculate the probability that the asset S(T) will be superior (inferior) to the liability covered by the funding. 3. EQUILIBRIUM IN THE FUNDING AT THE EXERCISE TIME T In the following we assimilate the spot price 0 S in t=0 to the assets placed in the fund at the inception date in t=0. In the same way we assimilate the strike price X in t=T (exercise time) to the future value of the total liability of the fund at T. Remember that in this paper we consider only financial uncertainties, so we assume that all technical uncertainties have been taken into account when calculating the X-value. Calling a the discount rate (d.r.), the funding at start in t=0 to have the equality asset = liability will be: 0 aT S Xe = (2a) (‘initial-funding’ equation) The available assets in t=T are represented by the lognormal distribution defined in (1). Adequate funding situations in T will impose: ( ) S T X ≥ (2b) (‘adequate-funding in T’ equation) Because S is a stochastic variable, lognormally distributed according to (1), this equation will be verified for a given confidence level. Taking for example a confidence level 95% means that, with a probability of 95%, 0 S must be larger than X, i.e.: 2 0 1 ln ( ) ln 2 (95%) 1.645 S T k T X k σ μ σ