Constant relative risk aversion utility and consumption CAPM: discount factors and risk aversions for Norway, Sweden, and the UK

Figures & data

Figure 1. CCAPM data: Index log returns (top) and consumption growth (bottom) for Norway, Sweden, and the UK, 1980-2022(q2).

Table 1. All Share Indices and Household Consumptions Characteristics for Norway, Sweden, and the UK, 1980-2022(q2).

Characteristics for All Share Indices and Consumption Growth 1981–2022(Q2)
Panel A	Norway OSEAX Index Returns
Mean (all)/	Median	Maximum /	Moment	Quantile	Quantile	Cramer-	Serial dependence		VaR
M (-drop)	Std.dev.	Minimum	Kurt/Skew	Kurt/Skew	Normal	von-Mises	Q(12)	Q^2(12)	(1; 2,5%)
0.00858	0.01689	0.11014	3.48554	0.09856	0.08058	0.55098	3.34892	5.10290	−0.187%
0.00901	0.05174	−0.23248	−1.37740	−0.02200	{0.9605}	{0.0000}	{0.7639}	{0.5307}	−0.138%
BDS-Z-statistic (ε = 1)					Phillips -	Augmented	ARCH	RESET	CVaR
m = 2	m = 3	m = 4	m = 5	m = 6	Perron test	DF-test	(12)	(12;6)	(1; 2.5%)
1.40569	1.77751	2.18022	2.06464	1.97461	−12.64848	−12.64610	4.33604	0.31011	−0.232%
{0.1598}	{0.0755}	{0.0292}	{0.0390}	{0.0483}	{0.0000}	{0.0000}	{0.6313}	{0.9581}	−0.176%
	Norway Consumption Growth
Mean (all)/	Median	Maximum /	Moment	Quantile	Quantile	Cramer-	Serial dependence		VaR
M (-drop)	Std.dev.	Minimum	Kurt/Skew	Kurt/Skew	Normal	von-Mises	Q(12)	Q^2(12)	(1; 2,5%)
0.00263	0.00899	0.04699	−0.67084	0.21112	3.21597	1.09000	269.148	160.678	−4.820%
0.00294	0.02355	−0.04821	−0.66382	−0.32419	{0.2003}	{0.0000}	{0.0000}	{0.0000}	−4.610%
BDS-Z-statistic (ε = 1)					Phillips -	Augmented	ARCH	RESET	CVaR
m = 2	m = 3	m = 4	m = 5	m = 6	Perron test	DF-test	(12)	(12;6)	(1; 2.5%)
3.11988	7.01564	16.24813	26.31176	38.34020	−41.26220	−4.10618	103.77979	8.72718	−4.821%
{0.0018}	{0.0000}	{0.0000}	{0.0000}	{0.0000}	{0.0001}	{0.0076}	{0.0000}	{0.0331}	−4.746%
Panel B.	Sweden OMXSPI Index Returns
Mean (all)/	Median	Maximum /	Moment	Quantile	Quantile	Cramer-	Serial dependence		VaR
M (-drop)	Std.dev.	Minimum	Kurt/Skew	Kurt/Skew	Normal	von-Mises	Q(12)	Q^2(12)	(1; 2,5%)
0.00882	0.02018	0.14814	1.21766	0.39526	3.63593	0.42380	8.03725	4.06777	−0.159%
0.00840	0.05143	−0.16569	−0.75585	−0.30391	{0.1624}	{0.0000}	{0.2354}	{0.6675}	−0.116%
BDS-Z-statistic (ε = 1)					Phillips -	Augmented	ARCH	RESET	CVaR
m = 2	m = 3	m = 4	m = 5	m = 6	Perron test	DF-test	(12)	(12;6)	(1; 2.5%)
3.15999	3.68117	4.04041	4.04766	4.44021	−11.82775	−11.81650	3.38698	1.07375	−0.166%
{0.0016}	{0.0002}	{0.0001}	{0.0001}	{0.0000}	{0.0000}	{0.0000}	{0.7589}	{0.7834}	−0.143%
	Sweden Consumption Growth
Mean (all)/	Median	Maximum /	Moment	Quantile	Quantile	Cramer-	Serial dependence		VaR
M (-drop)	Std.dev.	Minimum	Kurt/Skew	Kurt/Skew	Normal	von-Mises	Q(12)	Q^2(12)	(1; 2,5%)
0.00200	−0.00106	0.05743	−1.19116	−0.64676	3.35725	0.77638	603.616	190.199	−3.865%
0.00219	0.02539	−0.03883	0.19790	0.12951	{0.1866}	{0.0000}	{0.0000}	{0.0000}	−3.463%
BDS-Z-statistic (ε = 1)					Phillips -	Augmented	ARCH	RESET	CVaR
m = 2	m = 3	m = 4	m = 5	m = 6	Perron test	DF-test	(12)	(12;6)	(1; 2.5%)
123.126	173.147	237.368	346.256	516.416	−60.62088	−5.90918	117.653	21.3105	−3.883%
{0.0000}	{0.0000}	{0.0000}	{0.0000}	{0.0000}	{0.0001}	{0.0000}	{0.0000}	{0.0001}	−3.752%
Panel C.	UK FTSE All Shares Index Returns
Mean (all)/	Median	Maximum /	Moment	Quantile	Quantile	Cramer-	Serial dependence		VaR
M (-drop)	Std.dev.	Minimum	Kurt/Skew	Kurt/Skew	Normal	von-Mises	Q(12)	Q^2(12)	(1; 2,5%)
0.00337	0.00654	0.08564	2.00107	0.43585	1.35975	0.36511	5.75461	5.93282	−0.136%
0.00367	0.03832	−0.14592	−0.78800	−0.04069	{0.5067}	{0.0001}	{0.4512}	{0.4308}	−0.094%
BDS-Z-statistic (ε = 1)					Phillips -	Augmented	ARCH	RESET	CVaR
m = 2	m = 3	m = 4	m = 5	m = 6	Perron test	DF-test	(12)	(12;6)	(1; 2.5%)
1.70685	2.76085	3.63873	4.12829	4.88753	−14.15929	−13.98857	2.83387	2.34651	−0.146%
{0.0878}	{0.0058}	{0.0003}	{0.0000}	{0.0000}	{0.0000}	{0.0000}	{0.8294}	{0.5037}	−0.123%
	UK Consumption Growth
Mean (all)/	Median	Maximum /	Moment	Quantile	Quantile	Cramer-	Serial dependence		VaR
M (-drop)	Std.dev.	Minimum	Kurt/Skew	Kurt/Skew	Normal	von-Mises	Q(12)	Q^2(12)	(1; 2,5%)
0.00252	0.00994	0.07982	3.16068	0.09964	12.76860	2.09111	173.386	160.678	−7.438%
0.00261	0.02267	−0.10641	−1.06456	−0.67752	{0.0017}	{0.0000}	{0.0000}	{0.0000}	−4.937%
BDS-Z-statistic (ε = 1)					Phillips -	Augmented	ARCH	RESET	CVaR
m = 2	m = 3	m = 4	m = 5	m = 6	Perron test	DF-test	(12)	(12;6)	(1; 2.5%)
−1.2390	1.0714	7.4850	13.2827	19.4417	−22.5349	−3.08722	37.4212	54.1686	−10.641%
{0.2153}	{0.2840}	{0.0000}	{0.0000}	{0.0000}	{0.0000}	{0.1131}	{0.0000}	{0.0000}	−6.643%

*For all three bivariate series a breakpoint unit root test rejects unit root.

Table 3. CRRA Utility Model Parameters for Norway, Sweden, and the UK, 1980–2022(q2).

	Norway Utility Models Parameter Estimates (MCMC size 250 k)						Sweden Utility Models Parameter Estimates (MCMC size 250 k)						UK Utility Models Parameter Estimates (MCMC size 250 k)
	CRRA utility NPB		Posterior Mode =		−392.077		CRRA utility NPB		Posterior Mode =		−404.549		CRRA utility NPB		Posterior Mode =		−405.855
Parameters	Mean		Standard Deviations(VEC,VIC,VME,VMR)				Mean		Standard Deviations(VEC,VIC,VME,VMR)				Mean		Standard Deviations(VEC,VIC,VME,VMR)
	θ	θ	Extrinsic		Modified		θ	θ	Extrinsic		Modified		θ	θ	Extrinsic		Modified
Param	Extrinsic	Intrinsic	Extr Ctr	Intr Ctr	Extrinsic	Riemann	Extrinsic	Intrinsic	Extr Ctr	Intr Ctr	Extrinsic	Riemann	Extrinsic	Intrinsic	Extr Ctr	Intr Ctr	Extrinsic	Riemann
a0[1]	−0.30449	−0.33158	0.04263	0.05049	0.05048	0.05108	0.05025	0.07759	0.09723	0.11803	0.12514	0.12111	−0.84713	−0.86711	0.13393	0.13540	0.18744	0.13836
a0[2]	0.46780	0.48436	0.08117	0.08283	0.08284	0.08285	0.09389	0.09118	0.15629	0.14069	0.25176	0.14241	−0.44712	−0.47725	0.07367	0.07961	0.08746	0.08054
a0[3]	−0.10037	−0.03477	0.09273	0.11359	0.11360	0.11310	−0.05587	−0.00232	0.04293	0.06583	0.05533	0.06582	−0.09026	−0.07506	0.04962	0.05190	0.05967	0.05246
a0[4]	−0.10723	−0.12684	0.04215	0.04649	0.04650	0.04610	−0.00281	−0.03385	0.04297	0.05476	0.05665	0.05494	0.05566	0.00637	0.06411	0.08087	0.07099	0.08104
a0[5]	0.17072	0.19888	0.05981	0.06611	0.06615	0.06573	−0.09101	−0.07668	0.03462	0.04032	0.03894	0.04350	−0.39987	−0.42476	0.10171	0.10470	0.18350	0.10608
a0[6]	0.04560	0.09462	0.05950	0.07709	0.07710	0.07700	0.00653	−0.00016	0.04624	0.04575	0.05581	0.04656	−0.14415	−0.06954	0.06504	0.09897	0.10611	0.09962
a0[7]	−0.05537	−0.01592	0.03181	0.05067	0.05068	0.05051	−0.05801	−0.05451	0.02945	0.02654	0.03313	0.02756	−0.02860	0.01293	0.03704	0.05567	0.04867	0.05602
a0[8]	0.05853	0.02230	0.03258	0.04872	0.04873	0.04848	0.17233	0.19084	0.04301	0.04321	0.03192	0.04518	0.09084	0.07692	0.05882	0.06042	0.08415	0.06088
b0[1]	−0.31019	−0.37098	0.12186	0.13618	0.13623	0.13670	0.05408	0.06203	0.04346	0.04366	0.05618	0.04394	0.05064	0.01983	0.04479	0.05437	0.10784	0.05464
b0[2]	0.53814	0.46387	0.09854	0.12326	0.12326	0.12533	−0.03779	−0.03311	0.13378	0.14620	0.16643	0.15054	−0.11234	−0.12948	0.08187	0.08366	0.09554	0.08505
B(1,1)	0.01637	−0.02238	0.02601	0.04666	0.04669	0.04448	0.00166	0.00313	0.01331	0.03527	0.04601	0.01462	−0.12171	−0.13863	0.05970	0.06207	0.07208	0.06273
B(2,1)	−0.00749	−0.02880	0.02376	0.03192	0.03193	0.04039	−0.07041	−0.07377	0.05498	0.06876	0.08313	0.07208	−0.02315	−0.01997	0.02042	0.02067	0.02693	0.01626
B(1,2)	−0.04397	−0.10560	0.02784	0.06763	0.06761	0.05916	−0.02663	−0.05400	0.04820	0.05859	0.04283	0.06493	−0.15228	−0.13955	0.05675	0.05814	0.08195	0.05844
B(2,2)	−0.06757	−0.07498	0.04362	0.04424	0.04423	0.05394	−0.42947	−0.43576	0.05633	0.05359	0.06813	0.05462	−0.01480	−0.00696	0.01466	0.01662	0.02051	0.01540
R0[1]	0.91946	0.84989	0.04349	0.08205	0.08205	0.08223	0.28207	0.30575	0.04605	0.04236	0.07371	0.04417	0.30442	0.32573	0.03743	0.04306	0.04795	0.04332
R0[2]	−0.04968	−0.05934	0.02439	0.02624	0.02624	0.02637	−0.04762	−0.05642	0.02832	0.02699	0.03832	0.02763	−0.11864	−0.12175	0.02134	0.02156	0.04844	0.02171
R0[3]	0.64990	0.62037	0.04289	0.05206	0.05204	0.05075	0.37708	0.41703	0.05281	0.07087	0.06957	0.07164	0.30362	0.29537	0.03442	0.03540	0.04123	0.03590
P(1,1)	0.17758	0.20300	0.06963	0.07412	0.07415	0.07401	0.59268	0.66058	0.07781	0.08759	0.29275	0.08942	0.51001	0.53036	0.04658	0.05081	0.07311	0.05124
P(2,1)	−0.13339	−0.21189	0.05299	0.09472	0.09472	0.09378	−0.07804	−0.11742	0.06465	0.06906	0.08781	0.06908	−0.21319	−0.23617	0.03675	0.04336	0.05033	0.04360
Q(1,1)	–	–	–	–	–	–	0.76894	0.71946	0.05556	0.06253	0.15522	0.06148	0.83649	0.83620	0.02489	0.02491	0.03239	0.02513
beta (b)	0.99752	0.99827	0.00097	0.00123	0.00123	0.00671	0.99693	0.99640	0.00095	0.00105	0.00091	0.00141	0.99769	0.99868	0.00096	0.00138	0.00142	0.00243
gamma (γ)	3.27907	3.13085	2.49621	2.50065	2.50154	2.50050	0.19407	0.25813	0.18681	0.21928	0.34966	0.21006	6.17745	6.15927	1.55603	1.55592	2.64287	1.55846
β^4	0.99012	0.99310					0.98778	0.98568					0.99079	0.99473
1/(γ)	0.30496	0.31940					5.15278	3.87402					0.16188	0.16236
	Norway CRRA	Sweden CRRA	UK CRRA	Credibility regions CRRA Norway utility:			Norway Z-abcissae (quadrature rule):
Rejection rate	0.30094	0.45275	0.63594	Critical pnt	VIC	VMR	5%	−1.87822	⊂	−2.85697
Metropolis-Hastings reject average	0.25510	0.30516	0.40987	1.96	0.5239	0.5490	95%	1.33526	⊂	2.85697
Inequality failure average	0.04584	0.14758	0.22607	2.70	0.9708	0.9703	5%	−1.18963	⊂	−2.85697
Projection failure average	0.0	0.0	0.0				95%	1.92945	⊂	2.85697
Reverse projection failure average	0.0	0.0	0.0	Credibility regions CRRA Sweden utility:			Sweden Z-abcissae (quadrature rule):
Metropolis-Hastings accept average	0.69906	0.5	0.36406	Critical pnt	VIC	VMR	5%	−2.11780	⊂	−2.85697
Average |log f_x - log f_y|	0.33067	1.01420	0.86358	1.96	0.5714	0.3366	95%	1.28166	⊂	2.85697
Average |log p_x - log p_y|	0.70553	0.85210	1.85182	2.70	0.8975	0.6703	5%	−1.38673	⊂	−2.85697
Average |log p_v - log p_yp|	0.04955	0.07343	0.00713				95%	1.74788	⊂	2.85697
Average prior affect M-H decision	0.76233	0.79719	0.84686	Credibility regions CRRA UK utility:			UK Z-abcissae (quadrature rule):
Average proposal affect M-H decision	0.64409	0.69070	0.76446	Critical pnt	VIC	VMR	5%	−1.79326	⊂	−2.85697
Average Metropolis-Hastings ratio < 1	0.63241	0.67992	0.76386	1.96	0.6108	0.3863	95%	1.70977	⊂	2.85697
Normalisation constant	−438.753	−455.289	−450.701	2.70	0.9286	0.8564	5%	−1.26017	⊂	−2.85697
Standard deviation	{0.0185}	{0.0103}	{0.0177}				95%	1.71638	⊂	2.85697
Normalisation constant	−438.753	−455.289	−450.701	2.70	0.9286	0.8564	5%	−1.26017	⊂	−2.85697
Standard deviation	{0.0185}	{0.0103}	{0.0177}				95%	1.71638	⊂	2.85697

Figure 2. Trace plots $(50k)$ of Norwegian (top), Swedish (middle) and UK (bottom) CRRA log posterior paths. (a) Norway traces and parameter densities. (b) Sweden traces and parameter densities. (c) The UK traces and parameter densities.

Figure 4. Stochastic discount factor for a CRRA utility model Norway, Sweden, and the UK, 1980-2022(q2).

Table 2. Gamma and certainty equivalent for the CRRA utility function.

Value of γ	1	2	5	10	30
Value of WCE	70,711	66,667	58,566	53,991	51,209

Table 4. The mode, parameters for beta (β) and gamma (γ) and the normalization constant. Comparison Norway, Sweden, and the UK.

Comparing Mean Results for Norway, Sweden and the UK
		Period – 1980–2022 (q2)
		Norway	Sweden	the UK
Posterior Mode		−392.07741	−404.54867	−405.85519
Mean	Beta (β)	0.99752	0.99693	0.99769
discounting		{0.0010}	{0.0009}	{0.0010}
Relative	Gamma (γ)	3.27907	0.19407	6.17745
risk aversion		{2.4962}	{0.1868}	{1.5560}
Normalisation
constant		−438.7531	−455.2890	−450.7014
Standard dev		{0.01854}	{0.0103}	{0.0177}

Figure 5. NPB Simulation of 166 and 100k Index Returns and Consumption Growth. (a) Simulation 166 Index Returns and Consumption Growth observations from NPB model. (b) Simulation 100k Index Returns and Consumption Growth observations from NPB model.

Figure 6. Simulation 100k OSEAX returns and Consumption Movements Distributions.

Figure 7. Conditional Mean applying the original data-points.

Figure 8. Conditional Volatility using original data-points.

Figure 9. Conditional Covariance/Correlation using original data-points. (a) Norway’s CRRA Mean One-Step-Ahead Density 1980-2022(q2). (b) Sweden’s CRRA Mean One-Step-Ahead Density 1980-2022(q2). (c) UK’s CRRA Mean One-Step-Ahead Density 1980-2022(q2).

Figure 10. One-Step Ahead Mean Surface Densities NPB estimations 1980-2022(Q2) for Norway, Sweden, and UK.