On the impact of feeding cost risk in aquaculture valuation and decision making

Figures & data

Table 1. Fish farm parameters.

Name	Symbol	Value	Unit
Bertalanffy growth factor	a	1.113	–
Bertalanffy growth factor	b	1.097	–
Bertalanffy growth factor	c	1.43	–
Asymptotic weight	${\mathrm{w}}_{\mathrm{\infty }}$	6	kg
Mortality rate	$\mathrm{m}$	10	%
Conversion rate	$\mathrm{c}$	1.1	kg feed/ kg fish
Number of recruits	${\mathrm{n}}_0$	10 000	fish
Time horizon	T	3	years
Harvesting possibilities	N	72	–
Salmon spot price	${\hat{S}}_0^1$	see table 2	NOK/ kg
Production costs	$\mathrm{PC}$	$0.5{\hat{S}}_0^1$	NOK/ kg
Harvesting costs	${\mathrm{H}}_0$	$0.1\mathrm{PC}$	NOK / kg
Feeding costs	${\mathrm{F}}_0$	$0.25\mathrm{PC}$	NOK / kg year
Salmon initial value	$S_0^1$	${\hat{S}}_0^1-\mathrm{PC}+{\mathrm{H}}_0+{\mathrm{F}}_0$	NOK/ kg
Soy initial value	$S_0^2$	1	–

Table 2. Parameters for different scenarios of salmon models.

Scenario	${\sigma }_1^1$	${\sigma }_2^1$	${\kappa }^1$	${\alpha }^1$	${\lambda }^1$	${\rho }^1$	${\delta }_0^1$	${\hat{S}}_0^1$
Down, down	0.23	0.75	2.6	0.02	0.01	0.9	0.57	95
Down, up	0.23	0.75	2.6	0.02	0.2	0.9	0.57	95
Up, up	0.23	0.75	2.6	0.02	0.6	0.9	0.57	95

Figure 1. Interpolated (nearest neighbor) market future surfaces for salmon (left) and soy (right) from 01/01/2006 till 01/01/2023. On the x-axis are dates identified by integers, on the y-axis the time-to-maturity and on the z-axis the corresponding price. Deep blue associates a low and yellow a high price.

Figure 2. Close spot and future prices for different model parameters.

Figure 3. Mean relative changes of salmon (left) and soy (right) prices for the different scenarios in tables 2 and 3.

Table 3. Parameters for different scenarios of soybean models.

Scenario	${\sigma }_1^2$	${\sigma }_2^2$	${\kappa }^2$	${\alpha }^2$	${\lambda }^2$	${\rho }^2$	${\delta }_0^2$	${\hat{S}}_0^2$
Low volatility	0.5	0.4	1.2	0.06	0.14	0.44	0	1500
Medium volatility	1	0.4	1.2	0.06	0.14	0.44	0	1500
High volatility	2	0.4	1.2	0.06	0.14	0.44	0	1500

Table 4. Relative improvements using stochastic feeding costs. Mean with confidence intervals using $m=10000$ trials each with $M=10000$ simulations for LSMC.

	Salmon
Soy	Down, down	Down, up	Up, up
Low vol	${\mathrm{RI}}^{\tau }=1.0045$	${\mathrm{RI}}^{\tau }=1.0027$	${\mathrm{RI}}^{\tau }=1.0004$
	$(1.0040,1.0051)$	$(1.0020,1.0034)$	$(0,9999,1.0010)$
Medium vol	${\mathrm{RI}}^{\tau }=1.0301$	${\mathrm{RI}}^{\tau }=1.01940$	${\mathrm{RI}}^{\tau }=1.0052$
	$(1.0296,1.0307)$	$(1.0187,1.0200)$	$(1.0046,1.0058)$
High vol	${\mathrm{RI}}^{\tau }=1.1203$	${\mathrm{RI}}^{\tau }=1.0926$	${\mathrm{RI}}^{\tau }=1.0478$
	$(1.1195,1.1211)$	$(1.0918,1.0934)$	$(1.0472,1.0485)$

Table 5. Value of (4(4) $\begin{array}{rl}{V}_0^z\left(x\right)& :={\mathbb{E}}^{\mathbb{Q}}[\exp (-r{\tau }^z)\left(S_{{\tau }^z}^1\mathrm{B}\right({\tau }^z)-\mathrm{CH}({\tau }^z\left)\right)\\ & -{\mathrm{CF}}^{\mathrm{stoch}}\left({\tau }^z\right)|X_0^{\mathrm{stoch}}=x]\end{array}$(4) ) using stochastic feeding costs with different stopping rules.

	Salmon
Soy	Down, down	Down, up	Up, up
Low vol	$V_0^{\tau ,\mathrm{determ}}=1922265$	$V_0^{\tau ,\mathrm{determ}}=2397588$	$V_0^{\tau ,\mathrm{determ}}=3897333$
	$V_0^{\tau ,\mathrm{stoch}}=1930654$	$V_0^{\tau ,\mathrm{stoch}}=2403022$	$V_0^{\tau ,\mathrm{stoch}}=3898161$
Medium vol	$V_0^{\tau ,\mathrm{determ}}=1951032$	$V_0^{\tau ,\mathrm{determ}}=2429512$	$V_0^{\tau ,\mathrm{determ}}=3933336$
	$V_0^{\tau ,\mathrm{stoch}}=2009449$	$V_0^{\tau ,\mathrm{stoch}}=2475496$	$V_0^{\tau ,\mathrm{stoch}}=3952845$
High vol	$V_0^{\tau ,\mathrm{determ}}=2004857$	$V_0^{\tau ,\mathrm{determ}}=2488747$	$V_0^{\tau ,\mathrm{determ}}=3998265$
	$V_0^{\tau ,\mathrm{stoch}}=2244406$	$V_0^{\tau ,\mathrm{stoch}}=2717344$	$V_0^{\tau ,\mathrm{stoch}}=4187842$

Table A1. Correlation matrix inferred from filtered log-spot prices and convenience yield using two separate Kalman filters for salmon and soybean.

Panel	${\rho }_{12}$	${\rho }_{13}$	${\rho }_{14}$	${\rho }_{23}$	${\rho }_{24}$	${\rho }_{34}$
$[1,2,3,4,5,6]$	0.9560	0.0225	−0.0107	0.0130	−0.0244	0.4957
$[1,2,6,7,11,12]$	0.9334	0.0133	0.0058	−0.0151	−0.0261	0.7849
$[6,7,11,12,15,16]$	0.9833	−0.0596	−0.0382	−0.0679	−0.0436	0.9232

Table A2. Correlation matrix deduced by a joint Kalman filter for salmon and soybean.

Panel	${\rho }_{12}$	${\rho }_{13}$	${\rho }_{14}$	${\rho }_{23}$	${\rho }_{24}$	${\rho }_{34}$
$[1,2,3,4,5,6]$	0.944	0.011004	0.13451	0.052189	0.1236	0.19658
$[1,2,6,7,11,12]$	0.93374	0.030779	0.11054	0.058464	0.09726	0.52233
$[6,7,11,12,15,16]$	0.99687	−0.16528	−0.10078	−0.16908	−0.10595	0.87066

Table A3. Correlation matrix inferred from filtered log-spot prices and convenience yield using a joint Kalman filter for salmon and soybean.

Panel	${\rho }_{12}$	${\rho }_{13}$	${\rho }_{14}$	${\rho }_{23}$	${\rho }_{24}$	${\rho }_{34}$
$[1,2,3,4,5,6]$	0.9501	0.0637	0.1328	0.0826	0.1310	0.1259
$[1,2,6,7,11,12]$	0.9500	0.0612	0.0910	0.0727	0.0751	0.5784
$[6,7,11,12,15,16]$	0.9963	−0.1886	−0.1406	−0.1932	−0.1467	0.8852

Table A4. Relative improvements using stochastic feeding costs with full correlation matrix.

	Salmon
Soy	Down, down	Down, up	Up, up
Low vol	${\mathrm{RI}}^{\tau }=1.0045$	${\mathrm{RI}}^{\tau }=1.0027$	${\mathrm{RI}}^{\tau }=1.0004$
	$(1.0040,1.0050)$	$(1.0020,1.0034)$	$(0.9998,1.0010)$
	${\mathrm{RI}}^{+}=1.0049$	${\mathrm{RI}}^{+}=1.0023$	${\mathrm{RI}}^{+}=1.0004$
	$(1.0045,1.0053)$	$(1.0017,1.0029)$	$(0.9998,1.0010)$
	${\mathrm{RI}}^{-}=1.0045$	${\mathrm{RI}}^{-}=1.0018$	${\mathrm{RI}}^{-}=1.0004$
	$(1.0041,1.0049)$	$(1.0011,1.0019)$	$(0.9998,1.0010)$
Medium vol	${\mathrm{RI}}^{\tau }=1.0301$	${\mathrm{RI}}^{\tau }=1.0194$	${\mathrm{RI}}^{\tau }=1.0052$
	$(1.0306,1.0296)$	$(1.0188,1.0200)$	$(1.0046,1.0058)$
	${\mathrm{RI}}^{+}=1.0293$	${\mathrm{RI}}^{+}=1.0194$	${\mathrm{RI}}^{+}=1.0058$
	$(1.0289,1.0297)$	$(1.0189,1.0199$	$(1.0053,1.0063)$
	${\mathrm{RI}}^{-}=1.0299$	${\mathrm{RI}}^{-}=1.0179$	${\mathrm{RI}}^{-}=1.0043$
	$(1.0295,1.0303)$	$(1.0173,1.0185)$	$(1.0036,1.0050)$
High vol	${\mathrm{RI}}^{\tau }=1.1203$	${\mathrm{RI}}^{\tau }=1.0926$	${\mathrm{RI}}^{\tau }=1.0478$
	$(1.1195,1.1211)$	$(1.0934,1.0918)$	$(1.0472,1.0484)$
	${\mathrm{RI}}^{+}=1.1110$	${\mathrm{RI}}^{+}=1.0883$	${\mathrm{RI}}^{+}=1.0478$
	$(1.1104,1.1116)$	$(1.0876,1.0890)$	$(1.0471,1.0485)$
	${\mathrm{RI}}^{-}=1.1252$	${\mathrm{RI}}^{-}=1.0941$	${\mathrm{RI}}^{-}=1.0471$
	$(1.1245,1.1259)$	$(1.0933,1.0949)$	$(1.0464,1.0478)$

Table A5. Farm values using stochastic feeding costs with full correlation matrix.

	Salmon
Soy	Down, down	Down, up	Up, up
Low vol	$\rho =0$: 1 930 654	$\rho =0$: 2 403 022	$\rho =0$: 3 898 161
	$\rho =-0.2$: 1 925 131	$\rho =-0.2$: 2 397 483	$\rho =-0.2$: 3 896 229
	$\rho =0.2$: 1 937 654	$\rho =0.2$: 2 407 189	$\rho =0.2$: 3 899 199
Medium vol	$\rho =0$: 2 009 449,	$\rho =0$: 2 475 496	$\rho =0$: 3 952 845
	$\rho =-0.2$: 1 996 878	$\rho =-0.2$: 2 464 371	$\rho =-0.2$: 3 946 433
	$\rho =0.2$: 2 022 187	$\rho =0.2$: 2 486 452	$\rho =0.2$: 3 958 609
High vol	$\rho =0$: 2 244 406	$\rho =0$: 2 717 344	$\rho =0$: 4 187 842
	$\rho =-0.2$: 2 233 547	$\rho =-0.2$: 2 708 055	$\rho =-0.2$: 4 181 273
	$\rho =0.2$: 2 255 045	$\rho =0.2$: 2 728 003	$\rho =0.2$: 4 196 131