CAPM with various utility functions: Theoretical developments and application to international data

Figures & data

Table 1. Estimated values of the constant coefficient of relative risk aversion

Study	CRRA range
Arrow (1971)	1
Friend and Blume (1975)	2
Hansen and Singleton (1982,1984)	0–1
Mehra and Prescott (1985)	55
Ferson and Constantinides (1991)	0–12
Epstein and Zin (1991)	0–12
Cochrane and Hansen (1992)	40–50
Jorion and Giovannini (1993)	5.4–11.9
Normandin and Saint-Amour (1998)	$<3$
Sophie Coutant (1999)	0–11.404
Aït-Sahalia and Lo (2000)	12.7
Guo and Whitelaw (2001)	3.52
Bliss and Panigirtzoglou (2004)	0.37–15.97

Table 2. The sample descriptive statistics

Indexes	Mean	Median	Standard deviation	Skewness	Kurtosis	Min	Max
CAC40	$-0.00012$	0.00021	0.02350	$-26.65$	1,136.70	-1	0.11180
DAX	0.00017	0.00067	0.02340	$-27.08$	1,160.60	$-1$	0.11400
BEL20	$-0.00012$	0.00022	0.02260	$-29.96$	1,326.60	$-1$	0.09780
SP500	$-0.00002$	0.00048	0.02240	$-30.94$	1,382.00	$-1$	0.11580
Tunindex	0.00018	0.00010	0.01950	$-47.33$	2,432.70	$-1$	0.04190
MSCI Europe	0.00000	0.00056	0.02370	$-26.27$	1,113.60	$-1$	0.11290
MSCI Emerging Markets	0.00014	0.00130	0.02470	$-23.09$	932.44	$-1$	0.13680
MSCI South Africa	0.00022	0.00062	0.02240	$-30.99$	1,384.00	$-1$	0.06140
MSCI World	$-0.00004$	0.00078	0.02150	$-35.14$	1,635.10	$-1$	0.09520

Table 3. Calculation of the traditional CAPM beta

Indexes	$\mathit{\beta }$
CAC40	1.023
DAX	1.017
BEL20	0.978
SP500	1.007
Tunindex	0.758
MSCI Europe	1.044
MSCI Emerging Markets	1.022
MSCI South Africa	0.907

Table 4. Calculation of the CAPM beta extracted from the negative exponential utility function ${10}^{-2}$

$\mathit{\beta }$	0.5	1	2	10	20
CAC40	$-49.90$	$-49.87$	$-49.67$	$-47.35$	$-40.41$
DAX	$-49.9$	$-49.87$	$-49.76$	$-45.35$	$-40.40$
BEL20	$-49.91$	$-49.87$	$-49.76$	$-47.35$	$-40.41$
SP500	$-49.9$	$-49.87$	$-49.76$	$-47.35$	$-40.41$
Tunindex	$-49.92$	$-49.88$	$-49.77$	$-47.36$	$-40.42$
MSCI Europe	$-49.9$	$-49.87$	$-49.76$	$-47.34$	$-40.40$
MSCI Emerging Markets	$-49.9$	$-49.87$	$-49.76$	$-47.34$	$-40.40$
MSCI South Africa	$-49.91$	$-49.87$	$-49.77$	$-47.35$	$-40.41$

Table 5. Calculation of the CAPM beta extracted from the power utility function ${10}^4$

$\mathit{\gamma }$	0.5	2	3	4	10	0
CAC40	2.9301	9.3767	15.628	23.442	103.15	1.5626
DAX	4.8733	15.5594	25.99	38.984	171.53	2.5991
BEL20	3.1618	10.118	16.864	25.296	111.31	1.6862
SP500	6.9744	22.319	37.199	55.798	245.52	3.7195
Tunindex	4.6299	14.815	24.692	37.037	162.96	2.4693
MSCI Europe	$-6,161.31$	$-19,716$	$-32,860$	$-49,290$	$-216,880$	$-3,286$
MSCI Emerging Markets	7.8865	25.236	42.06	63.09	277.59	4.2062
MSCI South Africa	4.6299	14.815	24.692	37.037	162.96	2.4693

Table 6. Calculation of the CAPM beta extracted from the hyperbolic utility function ${10}^{-2}$

Indexes	$\mathit{\gamma }=-10;\mathit{\beta }=2$		$\mathit{\gamma }=-3;\mathit{\alpha }=1$		$\mathit{\gamma }=10;\mathit{\beta }=2$
	$\mathit{\alpha }=1$	$\mathit{\alpha }=10$	$\mathit{\beta }=1$	$\mathit{\beta }=2$	$\mathit{\alpha }=1$	$\mathit{\alpha }=10$
CAC40	$-49.91$	$-49.36$	$-49.91$	$-49.9$	$-49.9$	$-48.99$
DAX	$-49.91$	$-49.35$	$-49.92$	$-49.9$	$-49.9$	$-48.99$
BEL20	$-49.91$	$-49.36$	$-49.92$	$-49.9$	$-49.9$	$-48.99$
SP500	$-49.91$	$-49.36$	$-49.92$	$-49.9$	$-49.9$	$-48.99$
Tunindex	$-49.92$	$-49.37$	$-49.93$	$-49.91$	$-49.91$	$-49$
MSCI Europe	$-49.91$	$-49.35$	$-49.91$	$-49.9$	$-49.9$	$-48.99$
MSCI Emerging Markets	$-49.91$	$-49.35$	$-49.91$	$-49.9$	$-49.9$	$-48.99$
MSCI South Africa	$-49.91$	$-49.36$	$-49.92$	$-49.91$	$-49.91$	$-48.99$

Table 7. Regression results of the CAPM equation

Index	Constant term		ICAPM $\mathit{\beta }$		$R^2$
	Value	p-value	Value	p-value
CAC40	$9.3010\times {10}^{-5}$	0.5539	0.9983	0	0.9991
DAX	$-1.91339.3010\times {10}^{-4}$	0.223	0.9971	0	0.9991
BEL20	$7.1362\times {10}^{-5}$	0.6503	1.0005	0	0.9991
SP500	$-1.5909\times {10}^{-5}$	0.8842	0.9989	0	0.9996
Tunindex	$-2.2774\times {10}^{-4}$	0.3038	1.0016	0	0.9982
MSCI Europe	$-2.6437\times {10}^{-5}$	0.8516	0.9981	0	0.993
MSCI Emerging Markets	$-1.7314\times {10}^{-4}$	0.4155	0.9984	0	0.9984
MSCI South Africa	$-2.5250\times {10}^{-4}$	0.2296	0.9992	0	0.9984

Arrow, K. J. (1971). Essays in the theory of risk bearing. Chicago, IL: Markham Publishing.

 Google Scholar

Friend, I., & Blume, M. E. (1975). The demand for risky assets. American Economic Review, 65, 900–922.

 Web of Science ®Google Scholar

Hansen, L. P., & Singleton, K. J. (1982). Generalized instrumental variables estimation of nonlinear rational expectations models. Econometrica, 50, 1269–1286.

 Web of Science ®Google Scholar

Hansen, L. P., & Singleton, K. J. (1984). Errata. Econometrica, 52, 267–268.

 Web of Science ®Google Scholar

Mehra, Rajnish, & Prescott, E. C. (1985). The equity premium: A puzzle. Journal of Monetary Economics, 15, 145–61.

 Web of Science ®Google Scholar

Ferson, W. E., & Constantinides, G. M. (1991). Habit persistence and durability in aggregate consumption: Empirical tests. Journal of Financial Economics, 29, 199–240.

 Web of Science ®Google Scholar

Epstein, L. G., & Zin, S. E. (1991). Substitution, risk aversion, and the temporal behavior of consumption and asset returns: An empirical analysis. The Journal of Political Economy, 99, 263–286.

 Web of Science ®Google Scholar

Cochrane, J. H., & Hansen, L. P. (1992). Asset pricing explorations for macroeconomics. L. Fischer & O. J. Blanchard (Eds), NBER macroeconomics annual. Cambridge: MIT Press.

 Google Scholar

Normandin, M., & St-Amour, P. (1998). Substitution, risk aversion, taste shocks and equity premia. Journal of Applied Econometrics, 13, 265–281.

 Web of Science ®Google Scholar

Coutant, S. (1999). Implied risk aversion in option prices using hermite polynomials (CEREG Working Paper No 9908).

 Google Scholar

Aït-Sahalia, Y., & Lo, A.W. (2000). Nonparametric risk management and implied risk aversion. Journal of Econometrics, 94, 9–51.

 Web of Science ®Google Scholar

Guo, H., & Whitelaw, R. F. (2001). Uncovering the risk-return relation in the stock market (Working Paper, No 2002-001A). Federal Reserve Bank of St. Louis.

 Google Scholar

Bliss, R. R., & Panigirtzoglou, N. (2004). Option implied risk aversion estimates. The Journal of Finance, 1, 407–446.

 Web of Science ®Google Scholar