Gold's long-term expected return

Appendix A: Data

The data used in our analyses comes from various sources.

• For gold price data, we use the LBMA London PM spot price from January 1971, sourced from Bloomberg
• Nominal global GDP is sourced from the Federal Reserve Bank of St Louis FRED database
• Annual world equity market capitalisation from 1975 to 2022 is sourced from the World Federation of Exchanges and is backdated to 1971 using Wilshire 5000 index returns from Bloomberg
• Annual bond market capitalisation, represented by total global non-financial debt outstanding, is sourced from the BIS
• Data on the supply and demand categories is sourced from Metals Focus, and prior to 2010, from Refinitiv GFMS
• The stock of gold is sourced from Goldhub, via Metals Focus, and historical values are generated by subtracting from respective categories
• Forecasts of global nominal GDP in US dollars come from Oxford Economics, and equity and bond returns forecasts come from the J.P. Morgan Long-Term Capital Market Assumptions 2024 (LTCMA, 28th Edition). Additional building blocks sourced from Morgan Stanley’s LTCMA 2023 for illustration.

Appendix B: Robustness tests of OLS regressions

The initial econometric specification for estimating gold’s long-run expected return is presented in Table 1. In Table 3 we show some alternative specifications and discuss them relative to Model (2) above, which is our preferred model.

Regardless of specification GDP always has a positive coefficient. The coefficient on global portfolio is positive if GDP is not included in the equation. This switching of the sign is a marginal effect as discussed above and is consistent with GDP being the stronger effect on gold returns, as is sometimes seen in microeconomic studies where the income effect dominates the substitution effect, causing the sign on substitutes to flip when a measure of income is included.

Using alternatives for the financial component, such as equity or debt market caps, gives the same intuition as for global market cap where the sign is consistently negative when paired in a regression with GDP as the other independent variable. The marginal negative coefficient for debt in particular may reflect that issuance often must be absorbed regardless of yield, as we saw in Europe after the Global Financial Crisis, which might crowd out investments in alternatives such as gold.

While some specifications do show evidence of cointegration when only growth or financial factors are included in univariate regressions, it is clear from the Philips-Perron tests that the best examples of cointegration, and therefore long-run equilibrium systems, are found when both are included.

But there are two challenges with these specifications. The first is the presence of multicollinearity among the independent variables. Multicollinearity exists when there is a strong correlation among the independent variables that can give rise to several issues:

• Unreliable coefficient estimates from the standard errors are inflated making them less precise 
• Instability in coefficient estimates with small changes have potentially large impacts on parameter estimates
• Model overfitting, which can lead to fitting more noise than the actual theoretical relationships. 

Multicollinearity can be tested by the variance inflation factor (VIF). The VIF assesses how much the variance of an estimated regression coefficient is “inflated” by the presence of multicollinearity. When VIF values are high, it indicates that a predictor variable can be accurately predicted by other variables in the model, suggesting redundancy or high correlation. VIF values above 10 are often considered a concern, indicating potentially problematic multicollinearity. The VIF values for the estimated OLS equation are shown in Table 4.

A common method of dealing with multicollinearity would be to remove one or more of the independent variables. For this study, it is critical to include all three variables. Ridge regression is an extension of OLS that is designed to address multicollinearity.

Ridge regression modifies the ordinary least squares by adding a penalty term or shrinkage parameter to the regression equation. This penalty term is based on the sum of the squares of the coefficients (also known as L2 regularisation), effectively constraining the coefficients and preventing them from reaching extreme values. It does so by shrinking the coefficients towards zero, particularly those of highly correlated predictors, without eliminating them entirely. This helps to reduce the variance of the coefficient estimates, making them more reliable and less sensitive to small changes in the data.

The model was estimated using ridge regression and the results are shown in Table 5.

The estimated coefficients from the ridge regression are smaller in absolute value than the OLS model, but well within a range that allows one to conclude that the underlying theoretical relationships have not changed meaningfully. This approach helps to address the presence of highly correlated independent variables and would help reduce the variance in the model but it does not address the second challenge with the original OLS model: attempting to estimate a cointegrating relationship among the variables.

Cointegration is a statistical concept that describes a long-term equilibrium relationship between two or more non-stationary time series variables. In simpler terms, cointegration reflects a situation where multiple variables are linked in such a way that even though, individually, they might wander away from each other in the short run, they tend to move together in the long run.

This study was designed to better understand the long-run expected return of nominal gold prices. The independent variables were chosen based on their theoretical relationship with gold over the long run. Therefore, estimating the most precise cointegrating equation possible is a stronger concern than dealing with multicollinearity.

OLS estimation of a single equation cointegrating model has been widely used since Engle and Granger introduced the two-step procedure in 1987. OLS is commonly used in this framework due to its computational efficiency and ease of interpretation. There are, however, some disadvantages:

• Inefficient parameter estimates resulting from the violation of the OLS assumption of strictly exogenous independent variables 
• The presence of serial correlation potentially leads to biased standard errors and/or incorrect inference 
• The lack of an error-correction mechanism means that for both short and long term effects are estimated.

Phillips and Hansen introduced fully modified least squares (FM-OLS) to address these issues and improve the coefficient estimates in a cointegrating framework. The advantages of FM-OLS over OLS include:

• Correcting for endogeneity leads to less biased and more consistent coefficient estimates
• Correcting for the possible presence of serial correlation leads to possibly more efficient estimates
• Some of the stricter OLS assumptions can be relaxed.

The model was estimated using FM-OLS. The results are shown in Table 6.

The model fit in both the FM-OLS and OLS is similar, with a 91% adjusted R². The coefficient estimates are also similar in size and magnitude.

This Appendix addresses the estimation challenges of both multicollinearity and cointegration. There is no clear method that allows both issues to be addressed simultaneously and there is a trade-off when addressing one over the other. The ridge regression was estimated to demonstrate the effect on coefficient estimates to address multicollinearity. The FM-OLS model was estimated to address cointegration. Both additional models resulted in similar coefficient estimates, providing support to the original OLS coefficient estimates and the theoretical relationships discussed in this report.

Appendix C: Why 1971?

Our analysis starts in 1971, rather than 1968 or any other important turning points in the gold market that have been used elsewhere.

If we used a point for our analysis that started before the failure of the gold pool in 1968,¹  we would need to include data where the price did not adjust to market pressures: at that time it was set by central banks and governments. There were brief periods of free gold prices on the London market in the 1920s, and again in the 1930s, but these were seen as interim periods between the officially desired Gold Standard, rather than permanent changes.² Additionally, gold was a different asset during the Bretton Woods era, acting as money rather than a financial asset. The same logic applies to start dates in the 1800s.

April 1968 is often used as a starting point for analysis as this was the time when a free-floating gold price re-emerged in the London gold market.³ But there remained an “official” market for gold running in parallel until 1971, as it was widely expected that a form of Gold Standard would make an imminent return.

When the gold window was closed in 1971, suspending convertibility of gold into US dollars at a fixed price, only a free market for gold remained. Price movements, while at times driven by the possibility of a return to a Gold Standard, were not constrained by official actions to limit price and gold became more of an investment asset, remaining so to the present day.

Another often used date is the end of 1974, when US citizens were legally allowed to buy gold for the first time since 1933. However, a number of reasons make this a less important date for a change in the gold market than the idea might suggest. Gold prices reached a peak the day before the liberalisation of the market, in expectation of a surge of pent-up US demand. These record prices resulted in a lack of demand in the US for newly available gold futures or physical gold products. Demand was also lower than expected as, despite restrictions, some Americans already owned gold, which they held abroad, and others had only memories of gold investment that involved the confiscation of their holdings in 1933 – a further deterrent.⁴

Appendix D: GDP as a driver of demand

Here we replicate and update some of the results from the Goldman Sachs document ‘Precious Metals Primer: Fear and Wealth’ (2017), in particular Exhibit 13.

We explore the drivers of demand in world, emerging and developed markets, in Table 8.

In Panel A we can see that jewellery’s income elasticity of demand – measured through GDP – is greater than 1 in all cases. A 1% rise in GDP sees a 1% average increase in demand across the globe, and a reaction twice as large in EM markets – indicating the importance of growth for physical gold markets.

Panel A shows a clear negative price elasticity for jewellery demand regardless of whether we look at world, DM or EM. This reflects the sensitivity of jewellery buyers to the price of gold, with a 1% price rise in EM markets resulting in a nearly 2% fall in tonnage demand for jewellery.

Equally, retail bar and coin demand is significantly impacted by a rise in savings – a more concentrated proxy for wealth – but fear (investment flows into bonds less equities) has dominated over the last decade and a half, likely driven by the impact of the Global Financial Crisis.

Appendix F: References

Barsky, R., Epstein, C., Lafont-Mueller A. and Yoo, Y. (2021) ‘What drives gold prices?’, Chicago Fed Letter, 464, pp. 1-6.

Baur, D. and Lucey, B. (2010) ‘Is gold a hedge or a safe haven? An analysis of stocks, bonds and gold’, Financial Review, 45, no. 2, pp. 217-29.

He, Z., O’Connor, F. and Thijssen, J. (2022) ‘Identifying proxies for risk-free assets: Evidence from the zero-beta capital asset pricing model’, Research in International Business and Finance, 63(101775).

Hotelling H. (1931) 'The economics of exhaustible resources', Journal of Political Economy, 39, pp. 137-75.

Levin, E., Abhyankar, A. and Ghosh, D. (1994) ‘Does the Gold Market Reveal Real Interest Rates?’, Manchester School, 62, pp.93–103.

O'Connor, F., Lucey, B., Batten, J. and Baur, D. (2015). 'The financial economics of gold - A survey', International Review of Financial Analysis, 41, pp.186-205.

O’Connor, F., Lucey, B., and Baur, D. (2016) 'Do gold prices cause production costs? International evidence from country and company data', Journal of International Financial Markets, Institutions and Money, 40, pp.186-196.

O’Connor, F. and Lucey, B. (2024) 'The Efficiency of the London Gold Fixing: From Gold Standard to Hoarded Commodity 1919-68', Financial History Review (forthcoming).