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Exhaustibility and Risk as Asset Class Dimensions: A Social Investor Approach to Capital-Resource Economies

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Abstract

This paper develops a framework in which asset class dimensions are extended to include both risk and exhaustibility for explaining the evolution of shadow prices of marginal units of exhaustible natural resources in capital-resource economies. It is shown that the pricing kernel function required for socially valuing marginal units of exhaustible resource, hereafter called the Exhaustion-Stochastic Discount Factor, combines a factor that discounts for risk and another factor that discounts for resource exhaustion over time. The social rate of return on the marginal unit of resource stock adds to the risk-premium an exhaustion premium that accounts for the resource depletion over time. In this setting, the principle of no-arbitrage holds by extending asset-class dimensions to include not only a risk dimension but also an exhaustibility dimension.

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Notes

  1. The structure of social preferences needs not be intertemporally separable (Browning 1991). With a time-additive separable utility, the marginal substitution rate of current for future utility is constant and therefore is independent of future utility levels. The standard time additive separable framework is too rigid to describe important aspects of choice over time as well as risk attitudes (Samuelson 1937; Epstein 1987; Skiadas 2007). Several empirical studies show that individuals’ discount rate adjusts to outcomes that arrive in the distant future, suggesting a weakness of the constant discounted time-additive utility approach.

  2. The recursive utility framework pushes beyond the constant discounted time-additive separable utility by allowing a flexible marginal rate of substitution between current-period utility and the utility to be derived from all future periods.

  3. A comprehensive measure of a country’s wealth includes below-ground exhaustible natural resources such as subsoil oil and gas reserves (Dasgupta 2009, 2011; Diaz and Harchaoui 1997). The non-negligible role played by states over the management of natural resources indicates that there is a public interest in understanding the social valuation of exhaustible natural resources (Marchak 1998).

  4. Since sustainability issues often have long-term financial impacts, it becomes a matter of financial prudence to consider them. For instance, Calvert Investments has recently launched a Green Bond Fund Fund in order to support the financing of environmentally sustainable projects, while Morgan Stanley has established an Institute for Sustainable Investing. In Norway, the Government Pension Fund Global is required to act on behalf of the Norwegian people as a responsible investor in a way that takes into account intergenerational issues as well as environmental and social terms.

  5. The United Nations Principles for Responsible Investment is a voluntary framework which investors can use to incorporate responsible investment into their decision-making and ownership practices.

  6. The pricing kernel generally refers to a compact functional form that incorporates all relevant information on fundamental economic determinants of a valuation model. Pricing kernels are often used as valuation operators for comparing valuation models.

  7. As mentioned by Solow (1974a), the divergence between the private discount rate used by firms and the social discount rate is a critical issue to be considered when analyzing exhaustible natural resources. The literature on agency problems shows that the relationship between ownership and control is not self-evident (Jensen and Meckling 1976; Fama and Jensen 1983). Discounting issues can pertain to agency problems. For instance, there is empirical evidence that the discount rate used by the owners have no necessary implications for those used by the corporate executives (Chirinko and Schalle 2004; Stein 1996; Dow and Gorton 1997).

  8. Then, notice that the cost of extraction is \(r(t)K_{x}(t)=r(t)\upgamma ({\uptheta } _{2}(t))x(t)\), \(r(t)\) being the opportunity cost of capital.

  9. This is a detailed way to say that \({\upsigma }_i{\upxi }_i \sqrt{dt}\) is normally distributed with expectation zero and variance \(t\) - a standard Wiener process.

  10. In the case where \(\mu _2\) does not depend on the cumulative extraction, the resource becomes a homogenous Hotelling resource. Thank you to an anonymous referee for pointing this out.

  11. \({\mathcal {D}}L(.)\) is expressed as:

    $$\begin{aligned} {\mathcal {D}}L(p(t),q(t),{\uptheta }_{1}(t))&= \frac{1}{dt}E_{t}d L(p(t),q(t),{\uptheta }_{1}(t))\nonumber \\&= L_{1}\mu _{p}+L_{q}\mu _{q}+L_{1}\mu _{1}+\frac{1}{2}\Big [L_{11}{\upsigma }_{11}^{2}+L_{pp}{\upsigma }_{p}^{2} +L_{qq}{\upsigma }_{q}^{2}\nonumber \\&\quad +2L_{pq}{\upsigma }_{pq}+2L_{p1}{\upsigma }_{1p}+2L_{q1}{\upsigma }_{1q} \Big ], \end{aligned}$$
    (6)

    where \(L_{i}\) denotes the derivative of \(L\) with respect to \(i=p,q,{\uptheta }_{1}\) and \(L_{ij}\) denotes the cross-derivative with respect to \(i\) and \(j=p,q,{\uptheta }_{1}\), \(\mu _{i}\) is the drift of \(i\), \({\upsigma }_{i}\) is the volatility of \(i\), and \({\upsigma }_{ij}\) is the conditional covariance between \(i\) and \(j\).

  12. As it will be shown latter, in the general equilibrium, optimality of the consumption requires that the marginal utility of consumption equals the shadow value of the marginal unit of wealth/income [Eq. (24)]. The marginal utility of consumption \(q(t)=f_c(c(t),V(t))\) is the demand price (in utils) of the composite good. Since \(\lambda (t)=p(t)-r(t)\upgamma ({\uptheta }_2(t))\) is the marginal profit expressed in terms of the composite good, it follows that \(q(t)\lambda (t)=q(t)[p(t)-r(t)\upgamma ({\uptheta }_2(t))]\) is the marginal profit expressed in utils at time \(t\).

  13. \({\mathcal {D}}H(.)\) is expressed as:

    $$\begin{aligned} {\mathcal {D}}H(p(t),q(t),{\uptheta }_{2}(t))&= \frac{1}{dt}E_{t}d H(p(t),q(t),{\uptheta }_{2}(t))\nonumber \\ \nonumber&= -x H_{S}+H_{1}\mu _{p}+H_{q}\mu _{q}+H_{1}\mu _{2}+\frac{1}{2}\Big [H_{22}{\upsigma }_{22}^{2}+H_{pp}{\upsigma }_{p}^{2} +H_{qq}{\upsigma }_{q}^{2}\\&\quad + 2H_{pq}{\upsigma }_{pq}+2H_{p2}{\upsigma }_{1p}+2H_{q2}{\upsigma }_{2q} \Big ], \end{aligned}$$
    (11)

    where \(H_{i}\) denotes the derivative of \(H\) with respect to \(i=p,q,{\uptheta }_{1}\) and \(H_{ij}\) denotes the cross-derivative with respect to \(i\) and \(j=p,q,{\uptheta }_{2}\), \(\mu _{i}\) is the drift of \(i\), \( {\upsigma }_{i}\) is the volatility of \(i\), \({\upsigma }_{ij}\) is the conditional covariance between \(i\) and \(j\).

  14. That is the price of the marginal unit of the resource on the flow market \(p(t),\) net of the cost of taking it out of the ground \(r(t)\upgamma ({\uptheta }_2(t)).\)

  15. In addition, the transversality condition should be satisfied. That is,

    $$\begin{aligned} \lim _{t\rightarrow \infty }\lambda (t)S(t)=0. \end{aligned}$$
    (15)

  16. It is worth mentioning that recursive utility was first introduced under certainty by Koopmans (1960), and extended in a continuous-time setting under certainty by Epstein (1987), Epstein and Allan (1983), and in a continuous-time setting under uncertainty by Duffie and Epstein (1992b). An additional feature of the recursive utility in a stochastic setting is that it provides a way to disentangle intertemporal substitution from risk aversion.

  17. The recursive representation of the time-additive utility function is given by the following aggregator:

    $$\begin{aligned} f(c,V)=U(c)-{\uprho } V. \end{aligned}$$
    (18)

    The solution to the integral Eq. (16), with the aggregator (18), is given by the following future utility \(V(t)=E_{t}\left[ \int _{t}^{\infty }U(c(s))e^{-{\uprho } (s-t)}ds\right] \). From Eq. (18), it is readily seen that the marginal rate of substitution of current for future utility, \(-f_{V}(c,V)={\uprho },\) is constant. Therefore, the marginal rate of substitution of current for future utility is independent of the levels of current and future utility.

  18. The production and the extraction functions depend upon stochastic productivity indices, therefore the exhaustible resource stock and the physical capital are two risky assets. The rate of return on the physical capital is endogenously determined [see Eq. (7)]. The rate of return on the bond is assumed to be exogenously given and risk-free.

  19. Interactions among markets can be reexpressed as follows: Taking \(\lambda \) and \(r\) as given, the social investor invests a fraction of her wealth in the nonrenewable natural resource risky asset, with return \(d\lambda /\lambda \), and the rest in the accumulated composite good, with return \(r\). Her consumption and portfolio decisions send price signals in utils \(q(\lambda )\) to the resource firms and the firms producing the composite good. Taking the demand prices in utils \(q\) as given, the firms make their extraction and production decisions, which in turn generate the value \(\lambda (q)\) and the asset return \(\frac{d\lambda (q)}{\lambda (q)}\) that the social investor takes as given when establishing her opportunity set, as determined by her wealth constraints. In the general equilibrium, the prices and asset returns are taken to be those that simultaneously equilibrate all the markets.

  20. In other words,

    $$\begin{aligned} {\mathcal {D}}V({\uptheta }_{1}(t),{\uptheta }_{2}(t),W(t))&= \frac{1}{dt}E_{t}dV({\uptheta }_{1}(t),{\uptheta }_{2}(t),W(t))\nonumber \\&= V_{1}\mu _{1}+V_{2}\mu _{2} +V_{W}\Big [W(1-\upomega )[\mu _{S} -r]+Wr-c\Big ]+\frac{1}{2}\Big [V_{11}{\upsigma }_{11}^{2} +V_{22}{\upsigma }_{22}^{2} \nonumber \\&\quad + 2V_{12}{\upsigma }_{12}+2W(1- \upomega )(V_{1W}{\upsigma }_{1S}+V_{2W}{\upsigma }_{2S})+V_{WW}(1 -\upomega )^{2}W^{2}{\upsigma }_{S}^{2}\Big ], \end{aligned}$$
    (23)

    where \(V_{i}\) denotes the derivative of \(V\) with respect to \({\uptheta }_{i}\) and \(V_{ij}\) denotes the cross-derivative with respect to \({\uptheta }_{i}\) and \({\uptheta }_{j}\), \(\mu _{i}\) is the drift of \(i\), \( {\upsigma }_{i}\) is the volatility of \(i\), and \({\upsigma }_{ij}\) is the conditional covariance between \(i\) and \(j\).

  21. The shadow price \(V_{W}\) is the increase in social wellbeing that would be enjoyed if a unit more of the wealth were made available, when behaving optimally. As pointed out by Cochrane (2005, p. 240), the shadow price of the wealth \(V_{W}\) answers the question: “How much happier would you be if you found a dollar on the street ?” It measures “hunger”- marginal utility, not total utility.

  22. As in Skiadas (2007, p. 812), on the optimal consumption path, \(q(t)=V_W(t)=f_{c}(c(t),V(t))\) represents the shadow price of the time-\(t\) wealth constraint. That is, the first-order utility increment (per unit of wealth) as a result of slightly increasing time-\(t\) wealth (Schroder and Skiadas 2003, p. 169). See also Skiadas (2008, p. 211 ).

  23. With a recursive social utility function, the social discount rate is implicit to the structure of the aggregator, and measured by the term \(-f_{V}(c(t),V(t))\).

  24. As noted by Cochrane (2005), the covariances of returns with the utility index captures shocks to the investor’s future prospects.

  25. The utility gradient is computed as the Gateaux derivative of a dynamic utility. For more details, see Duffie and Skiadas (1994, p. 20) and Schroder and Skiadas (1999, p. 76).

  26. Note also that this paper’s approach is broader than economic models in which the dynamic of natural resource windfalls is exogenously given. Assuming that natural resource assets are a fixed, indestructible factor of production is questionable since Nature consists of degradable resources (Dasgupta 2010).

  27. As argued by Kiernan (2009), it is hard to conjure up any compelling arguments against acquiring additional information for investing in a sustainable world.

  28. A generic risky security, or more precisely, a unit of an asset, is a title to receive payoffs in amounts that depend on which state of the world occurs.

  29. Recall that \(\lambda (t)=p(t)-r(t)\upgamma ({\uptheta }_2(t))\) is the price of the extracted resource minus the marginal cost of extraction, expressed in terms of the composite good.

  30. The no-arbitrage asset pricing interpretation of the Hotelling (1931) rule is extensively discussed by Solow (1974a).

  31. There is some evidence that mining assets consistently trade at market values that differ from values obtained using traditional discounted cash flow techniques relating only to risk factors (Moyen et al. 1996). It is also of some interest to mention that commodities have gained traction among asset managers in recent years as a separate asset class beyond traditional asset class labels.

  32. Asset classification is not cast in iron or some platonic universe, but it is meant to serve a purpose (Dasgupta 2011). In addressing sustainability issues, it makes sense to consider a broad notion of asset classification that accounts for the twin presence of risk and exhaustibility in capital-resource economies.

  33. Incorporating sustainability concerns into the investment process is about long-term investing. In Norway, the view that not only present but also future generations should benefit from the oil and gas activities has led to the creation of a trustee fund for future generations that takes into account the estimation of the size of the “remaining petroleum wealth in the ground” (Van den Noord and Vourc’h 1999, p. 11). The exhaustion premium gives a rationale for the inclusion of subsoil oil and gas reserves in analyzing the strategic portfolio allocation of a sovereign wealth fund that takes into account intergenerational issues generated by resource exhaustion.

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  1. Department of Economics, Morehouse College, 830 Westview Dr. SW, Atlanta, GA, 30314, USA

    Johnson Kakeu

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Appendices

Appendix 1: The Derivation of Eq. (27)

To derive Eq. (27), differentiate the maximized Hamilton–Jacobi–Bellman Eq. (22) with respect to \(W\) and apply the envelope theorem to obtain:

$$\begin{aligned} \small f_{V}V_{W}=-\frac{\partial }{\partial W}\Big ({\mathcal {D}}\,V\Big ). \end{aligned}$$
(26)

From (23), the right hand side of (26) becomes:

$$\begin{aligned} - \frac{\partial }{\partial W}\Big ({\mathcal {D}}\,V\Big )\!= & {} \!-({\mathcal {D}} V_{W}+V_{W}r)\nonumber \\&- (1-\upomega )\Bigg [ V_{W}(\mu _{S}-r)\!+\!V_{WW}{\upsigma }_{S}^{2}(1\!-\!\upomega )W\!+\!(V_{1W}W{\upsigma }_{1S}\!+\!V_{2W}W{\upsigma }_{2S})\Bigg ].\nonumber \\ \end{aligned}$$
(41)

Using the first-order condition with respect to \((1-\upomega ),\) that is Eq. (25), it is readily seen that the term in the large brackets, in Eq. (41), vanishes. Therefore Eq. (26) becomes:

$$\begin{aligned} f_{V}V_{W}=-({\mathcal {D}} V_{W}+V_{W}r). \end{aligned}$$
(42)

Equation (27) follows straightforwardly from substituting the first-order condition with respect to consumption (24), that is Eq. (24), into Eq. (42).

Appendix 2: The Derivation of Eq. (30)

To derive Eq. (30), use the Itô product rule to obtain

$$\begin{aligned} \frac{d(f_{c}(c,V)\lambda )}{f_{c}(c,V)\lambda }= \frac{d\lambda }{\lambda }+ \frac{df_{c}(c,V)}{f_{c}(c,V)}+\frac{df_{c}(c,V)d\lambda }{f_{c}(c,V)\lambda }, \end{aligned}$$
(43)

where the quadratic variation is given by

$$\begin{aligned} \frac{df_{c}(c,V)d\lambda }{f_{c}(c,V)\lambda } =\left[ \frac{cf_{cc}}{f_{c}}{\upsigma }_{c}{\upsigma }_{S}+\frac{Vf_{cV}}{f_{c}}{\upsigma }_{V}{\upsigma }_{S}\right] dt+ o(dt). \end{aligned}$$
(44)

Now apply the operator \(\frac{1}{dt}E_{t}\) to both sides of Eq. (43) and use Eq. (28) to obtain (30).

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Kakeu, J. Exhaustibility and Risk as Asset Class Dimensions: A Social Investor Approach to Capital-Resource Economies. Environ Resource Econ 65, 677–695 (2016). https://doi.org/10.1007/s10640-015-9917-x

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