10.7 Silver and the gold/silver ratio

10.7.1Dual demand#

About half or more of silver demand is industrial (electronics, solar photovoltaics, electric vehicles, brazing alloys); the rest is investment and jewellery. Silver's market is much smaller than gold's, its above-ground stock is less concentrated in vaults, and its price is more cyclical and more volatile.

Silver Institute, World Silver Survey 2026 (April 2026; https://www.silverinstitute.org/), as reported by Crux Investor, FXStreet and SMM: a sixth consecutive market deficit forecast for 2026 of about 46.3 Moz (2025: about 40.3 Moz; an earlier 67 Moz projection was superseded); about 762 Moz drawn from above-ground stocks since 2021; mine supply roughly flat; 2026 industrial demand forecast down about 3% as solar manufacturers thrift silver at high prices (one source says PV demand −19%). Read the primary report before quoting these figures.

The storage theory of 10.2 applies directly: a run of deficits draws down inventories and moves silver toward the steep part of the storage curve, where spreads and spot volatility rise. Thrifting is the demand-side response that Hotelling-type reasoning ignores: at high prices, users engineer the metal out.

Levels Silver reached an all-time high of about $121.64/oz in January 2026 and was about $51.9 on 6 Nov 2026 (secondary spot quotes; the LBMA Silver Price is the benchmark but its history is licence-restricted), down about 57% from the peak.

10.7.2Is the ratio mean-reverting? Testing it properly#

Model Take $x$ to be an Ornstein–Uhlenbeck process, $dx = \kappa(\theta - x)\,dt + \sigma\,dW$. Its exact solution over a step $\Delta$ is

so the sampled series is an AR(1): $x_{t+1} = a + \varphi x_t + e_{t+1}$ with $\varphi = e^{-\kappa\Delta}$, $a = \theta(1 - \varphi)$. (The derivation of (10.18) by Itô's lemma, maximum-likelihood estimation of $(\kappa,\theta,\sigma)$ and the precision of $\hat\kappa$ are in M13A · 13A.7, the course's canonical treatment of the OU spread; here only the result is used.)

Proof

Write $z_t = x_t - \mu$; then $z_{t+1} = \varphi z_t + e_{t+1}$ because $a = \mu(1 - \varphi)$. Iterating, $z_{t+h} = \varphi^h z_t + \sum_{j=0}^{h-1}\varphi^j e_{t+h-j}$, and the sum has conditional mean zero. Setting $\varphi^{h} = 1/2$ gives $h\ln\varphi = -\ln 2$, i.e. (10.19). For the OU process $\ln\varphi = -\kappa\Delta$, so $h_{1/2}\Delta = \ln 2/\kappa$.∎

A common approximation regresses $\Delta x_t$ on $x_{t-1}$, gets slope $\gamma = \varphi - 1$, and uses $h \approx -\ln 2/\gamma$. Since $\ln\varphi \approx \varphi - 1$ for $\varphi$ near 1, the two agree for slow mean reversion.

The ADF test Write the AR(1) as $\Delta x_t = a + \gamma x_{t-1} + \sum_{j=1}^p \delta_j \Delta x_{t-j} + e_t$ with $\gamma = \varphi - 1$ and lagged differences to absorb serial correlation. $H_0$: $\gamma = 0$ (unit root, no mean reversion); $H_1$: $\gamma < 0$. The OLS $t$-statistic on $\gamma$ does not have a Student-$t$ distribution under $H_0$, because $x_{t-1}$ is nonstationary: Dickey and Fuller (1979) showed it converges to a functional of Brownian motion. With a constant in the regression the 1%, 5% and 10% critical values are about −3.43, −2.86 and −2.57 (MacKinnon's tables, used by statsmodels.adfuller); using the normal value −1.645 would reject far too often. The limiting distribution, its derivation and the full table of critical values are in M13A · 13A.2.2, the course's canonical treatment of unit-root tests (proof also in Hamilton, Time Series Analysis, 1994, chapter 17).

Two small-sample problems

1. Bias. OLS $\hat\varphi$ is biased downward in finite samples. With an intercept, Kendall's (1954) approximation is $\mathbb{E}[\hat\varphi] - \varphi \approx -(1 + 3\varphi)/n$. So estimated half-lives are too short: the ratio looks more mean-reverting than it is. A first-order correction is $\tilde\varphi = (n\hat\varphi + 1)/(n - 3)$. The test test_kendall_correction_reduces_bias checks this by simulation.
2. Uncertainty. $h_{1/2}$ is a convex function of $\varphi$ that blows up as $\varphi \to 1$. With $\text{se}(\hat\varphi) \approx \sqrt{(1 - \varphi^2)/n}$, a confidence interval for $\varphi$ that reaches 1 gives a half-life interval that reaches infinity. Report the interval, not the point estimate.

What the data say (computed for this course, 6 Nov 2026) Using Yahoo Finance continuous front-month COMEX futures (GC=F, SI=F), daily closes from 2 Jan 2001 to 5 Nov 2026, with the script logic of m10_ratio_meanrev.py:

Sample	Frequency	$n$	ADF stat	$p$-value	$\hat\varphi$	Raw half-life
2001–2026	month-end	311	−2.83	0.055	0.9503	13.6 months
2001–2019	month-end	228	−2.27	0.18	0.9514	13.9 months
2010–2026	month-end	203	−2.16	0.22	0.9543	14.8 months
2001–2026	daily	6,486	−2.85	0.052	0.99732	258 trading days

The ratio ranged from 32.0 (25 Apr 2011, when silver neared $49) to 125.9 (18 Mar 2020, the COVID liquidation) on daily closes and stood at 79.0 on 5 Nov 2026. Reading: over the full sample, the unit root is rejected at 10% but not at 5%. In both sub-samples it is not rejected at 10%. Kendall's correction moves the full-sample monthly half-life from 13.6 to about 18 months, and a 95% interval for $\varphi$ of roughly $[0.916, 0.985]$ maps to half-lives of about 8 to 46 months. With three samples tried, the borderline full-sample result also carries a small multiple-testing penalty (M04). The honest conclusion: there is weak evidence of very slow mean reversion, with a half-life measured in years and an interval too wide to trade on alone. Continuous front-month series also contain roll jumps; a careful study would use spot fixes or back-adjusted series.