Summary of key results · M10 Commodities

Result	Formula
Cost of carry (M07)	$F = S e^{(r+u-y)T}$ (10.1)
Cash-and-carry bound	$F \le S e^{(r+u)T}$ , or $F \le (S + U)e^{rT}$ (10.2)
Inventory-holder / lease bound	$F \ge S e^{(r+u-\ell)T}$ ; no lower bound without inventory (10.3)
Implied convenience yield	$y = r + u - \ln(F/S)/T$ (10.4)
Competitive storage	$I_t\,[\mathbb{E}_tS_{t+1}/(1+r) - S_t - k] = 0$ , bracket $\le 0$ (10.5)
Supply of storage	$\ln(F/S)/T = r + u - y(I)$ , increasing, concave in $I$ (10.6)
Gold carry	$F \approx S e^{(r-\ell)T}$ (10.7)
Excess return decomposition	$\ln(F_{t'}/F_t) = \ln(S_{t'}/S_t) + \Delta\ln(F/S)$ (10.8)
Held to expiry	$\ln(S_T/F_t) = \ln(S_T/S_t) - c_t(T-t)$ (10.9)
Rolled long	cumulative excess $= \ln(S_N/S_0) - \Delta\sum c_k$ (10.10); total $= $ spot $+\Delta\sum(y_k - u_k)$ (10.11)
Curve roll yield	$-\ln(F_2/F_1)/(T_2 - T_1)$
Hotelling	$p_t - c = (p_0 - c)e^{rt}$ (10.13)
Durable asset (Barsky–Summers)	$p_g = \delta/(\rho - g)$ , $\partial\ln p_g/\partial\rho = -1/(\rho - g)$ (10.14)
Import parity	$P_{10g} = (P_{oz} + q)/31.1034768 \times 10 \times X \times (1+d)$ ; retail $\times 1.03$ (10.15)
INR gold return	$\Delta\ln P^{\text{INR}} = \Delta\ln P^{\text{USD}} + \Delta\ln X + \Delta\ln(1+d)$ (10.16)
OU → AR(1)	$\varphi = e^{-\kappa\Delta}$ , $a = \theta(1-\varphi)$ (10.18)
Half-life	$h_{1/2} = -\ln 2/\ln\varphi$ ; OU: $\ln 2/\kappa$ (10.19)
Kendall bias	$\mathbb{E}\hat\varphi - \varphi \approx -(1 + 3\varphi)/n$
ADF critical values (constant)	−3.43 (1%), −2.86 (5%), −2.57 (10%)
3-2-1 crack	$(2P_{\text{gas}} + P_{\text{dist}} - 3P_{\text{crude}})/3$ (10.20)
Black-76	(10.21)