Optimal asset allocation for commodity sovereign wealth funds

Abstract

This paper studies the dynamic asset allocation problem faced by an infinitively lived commodity-based sovereign wealth fund under incomplete markets. Assuming that the fund receives a non-tradable stream of commodity revenues until a predetermined date, the optimal consumption and investment strategies are state and time-dependent. Using data from the Norwegian Petroleum Fund, we find that the optimal demand for equity should decrease gradually from 60% to 40% over the next 60 years. However, the solution is particularly sensitive to the correlation between oil and stock price changes. We also estimate wealth-equivalent welfare losses, relative to the optimal rule, when following alternative suboptimal investment rules.

Keywords:

• Optimal asset allocation
• Sovereign wealth fund
• Commodities
• Income risk
• Suboptimal investments

JEL Classifications:

• E21
• G11
• G23
• Q32

Acknowledgements

We appreciate the comments and suggestions from two anonymous referees, as well as from Juan Pablo Nicolini, Knut Einar Rosendahl, Olvar Bergland, Camilo Castillo, Olesia Verchenco, and seminar participants at Norges Bank, Norwegian University of Life Sciences, Central Bank of Chile and the Ministry of Finance of Norway. We also thank Yacine Ait-Sahalia for providing us with Matlab code to perform the maximum-likelihood estimation.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 A description of the investment behavior of SWFs in equity markets and their allocation strategies across industries and sectors is provided by Miceli (2013).

2 Using monthly data for the period 1997–2008, the author does not find any significant correlation between the nominal returns on the MSCI World index and the nominal price of crude oil. Therefore, he concludes that global equities do not provide a hedge against fluctuations in oil prices. In the face of a similar insignificant correlation, Døskeland (2007) proposed to use a cointegration approach to identify the long-term relationship between financial assets and non-tradable assets. When applied to the Norwegian case, he finds a similar result: the government should increase its current (initial) holdings in equity and reduce it over time.

3 According to Statistics Norway, mainland Norway refers to all the domestic production activity with the exception of exploration of crude oil and natural gas, transport via pipelines and ocean transport.

4 Notice that the commodity converges to zero asymptotically but never gets fully depleted in finite time.

5 Note that the fund could also choose derivatives instruments, e.g. futures and forward contracts, to hedge unspanned oil income risk. This is an interesting avenue to explore; however, we abstain from doing it for several reasons. First, we do not observe in practice NBIM taking positions in futures contracts on oil on behalf of the Norwegian GPFG. Marking to market of daily profit and losses for a large number of contracts in oil may lead to liquidity constraints, a point made in van den Bremer et al. (2016). Second, the institutional mandate of the Norwegian GPFG requires NBIM to exclusively focus on the risk of the portfolio (see Norges Bank Investment Management 2022). Third, it may be that because of the sheer size of the portfolio that NBIM manages, the futures market is not able to provide sufficient liquidity.

6 The stochastic optimal control problem in (9(9) $J_0=\underset{(C,\alpha )\in {\mathcal{A}}_t}{max}{\mathbb{E}}_0\left[{\int }_0^{\hat{T}}f\right(C_t,J_t)\mathrm{d}t+J\left(W_{\hat{T}}\right)],$(9) ) is equivalent to a finite horizon dynamic asset allocation problem with terminal utility $J\left(W_{\hat{T}}\right)=A(W_{\hat{T}}^{1-\gamma }/1-\gamma )$. The constant A, which measures the relative utility weight of wealth at instant $\hat{T}$ and intermediate consumption, is then chosen to be consistent with the assumption that the fund manager optimizes the intertemporal utility of consumption even after the flow of revenues terminates.

7 To bring down the exposure to income volatility one could follow the ideas of Bick et al. (2009, 2013) and introduce an additional traded asset (e.g. derivative) to ‘artificially’ complete the market (see footnote 5). The resulting hedging strategy provides additional exposure to oil income risk compared with a strategy where the fund only invests in one risky asset. Extensive research has shown that derivative securities can be used to complete financial markets and improve welfare (see Liu and Pan 2003 and Hsuku 2007 and the references therein.).

8 The multivariate diffusion in (24(24) $\mathrm{d}{\overrightarrow{X}}_t=\mu ({\overrightarrow{X}}_t;{\mathit{\theta }}_1)\mathrm{d}t+\sigma ({\overrightarrow{X}}_t;{\mathit{\theta }}_1)\mathrm{d}{\overrightarrow{Z}}_t,$(24) ) is reducible in the sense that it is possible to transform the diffusion process $\overrightarrow{X}$ into a diffusion process $\overrightarrow{Y}$ with diffusion matrix equal to the identity matrix (see Ait-Sahalia 2008).

9 The corresponding median trajectories from 10,000 simulations of the model together with their 15th and 85th percentiles are reported in Appendix 3.

10 Using quarterly data on U.S. aggregate income from the National Income and Product Accounts (NIPA) for the period 1951–2003, Munk and Sørensen (2010) estimate a volatility of disposable labor income of 2.08%, and a correlation coefficient between income and equity price changes of 16.73%. Their estimate of the volatility of the S&P500 index is 16.13%. When using aggregate income data from the Panel Study of Income Dynamics (PSID) survey, their estimate of the labor income volatility is 1.64%. On the other hand, and consistent with the evidence reported in Carroll and Samwick (1997) and Chamberlain and Hirano (1999) using PSID data, Viceira (2001) uses a volatility of labor income of 10% in his benchmark calibration. A similar value is estimated in Cocco et al. (2005), who additionally estimate a correlation coefficient with equity returns between 0% and −1.75%.

11 We also perform a sensitivity check to explore the role of different initial values of ratio $W_0/Y_0$ on the optimal allocation. We find that initial values have moderate effects on the optimal equity holding, whereas the impact on the consumption-wealth ratio is more sizable. However, the effects are not quantitatively important. The results are available upon request.