Power Utility Consumption Capm in Uk Stock Markets

Pricing of Securities in Financial Markets 40141 – How well does the power utility consumption CAPM perform in UK Stock Returns? ******** 1 Hansen and Jagannathan (1991) LOP Volatility Bounds Volatility bounds were first derived by Shiller (1982) to help diagnose and test a particular set of asset pricing models. He found that to price a set of assets, the consumption model must have a high value for the risk aversion coefficient or have a high level of volatility.Hansen and Jagannathan (1991) expanded on Shiller’s paper to show the duality between mean-variance frontiers of asset portfolios and mean-variance frontier of stochastic discount factors. Law of one price volatility bounds are derived by calculating the minimum variance of a stochastic discount factor for a given value of E(m), subject to the law of one price restriction. The law of one price restriction states that E(mR) = 1, which means that the assets with identical payoffs must have the same price. For th is constraint to hold, the pricing equation must be true.Hansen and Jagannathan use an orthogonal decomposition to calculate the set of minimum variance discount factors that will price a set of assets. The equation m = x* + we* + n can be used to calculate discount factors that will price the assets subject to the LOP condition. Once x* and e* are calculated, the minimum variance discount factors that will price the assets can be found by changing the weights, w. Hansen and Jagannathan viewed the volatility bounds as a constraint imposed upon a set of discount factors that will price a set of assets.Therefore, when deriving the volatility bounds, we calculate the minimum variance stochastic discount factors that will price the set of assets. Discount factors that have a lower variance than these values will not price the assets correctly. Furthermore, Hansen and Jagannathan showed that to price a set of assets, we require discount factors with a high volatility and a mean close to 1. After deriving these bounds, we can use this constraint to test candidate asset pricing models.Models that produce a discount factor with a lower volatility than any discount factor on the LOP volatility can be rejected as they do not produce sufficient volatility. Hansen and Jagannathan find evidence that using LOP volatility bounds, we can reject a number of models such as the consumption model with a power function analysed in papers such as Dunn and Singleton (1986). 2 Methodology To test whether the power utility CCAPM prices the UK Treasury Bill (Rf) and value weighted market index returns, we first calculate the LOP volatility bounds.The volatility bound is derived by calculating the minimum variance discount factors that correctly price the two assets for given values of E (m). The standard deviations of the stochastic discount factors are then plotted on a graph to give the LOP volatility bound shown in figure one. Figure 1 here The CCAPM stochastic discount factors are then calculated for different levels of risk aversion. The mean and standard deviation of these discount factors are then plotted on the graph and compared to the LOP discount factor standard deviations.Pricing errors can then be calculated and analysed to see whether the assets are priced correctly by the candidate model. To accept the CCAPM model in pricing the assets, we expect the stochastic discount factors variance to be greater than the variance of the LOP volatility bounds. It is also expected that pricing errors and average pricing errors (RMSE) will be close to zero. These results will be analysed more closely in the later questions. 3 Power Utility CCAPM vs LOP Volatility Bounds In order for the power utility CCAPM to satisfy the Law of One Price volatility bound test at any level of risk aversion, the standard deviation f the CCAPM stochastic discount factor at that level of risk aversion must be above the Law of One Price standard deviation bound for the mean value of t he CCAPM stochastic discount factor at the same level of risk aversion. This is the null hypothesis and if it is accepted then the model satisfies the test. The alternative hypothesis is that it the standard deviation of the stochastic discount factor is below the Law of One Price standard deviation bound for the mean value of the stochastic discount factor.If the null hypothesis is rejected and the alternative hypothesis is accepted then the model does not satisfy the test. Table 1 here Figure 2 here Figure 2 shows LOP volatility bounds and the standard deviations and means of the CCAPM stochastic discount factors for levels of risk aversion between 1 and 20. It is obvious the standard deviations (Sigma(m)) of the CCAPM stochastic discounts factors are much lower than the LOP volatility bounds corresponding to the means (E(m)) of the CCAPM stochastic discount factors.This is true for any level of risk aversion, because the entire CCAPM (green) line lies below the LOP volatility bou nds (dark blue) line. Table 1 shows the standard deviations of the stochastic discount factors and the precise LOP volatility bound values, corresponding to the stochastic discount factor means so that the CCAPM can be formally tested. All of the standard deviations are lower than their respective volatility bound values. Therefore the null hypothesis is to be rejected and the alternative hypothesis is to be accepted for all levels of risk aversion between 1 and 20.Furthermore it would take a risk aversion of at least 54 to accept the null hypothesis. Therefore the power utility CCAPM stochastic discount factor does not satisfy the Law of One Price volatility bound test. These results are consistent with the equity premium puzzle study by Mehra and Prescott (1985). The study examines whether a consumption growth based model with a risk aversion value restricted to no more than 10 accurately prices equities. They have found that according to the model equity premiums should not excee d 0. 5% for values of risk aversion (? ) between 0 and 10 and values of the beta coefficient (? ) between 0 and 1. However the average observed equity premium based on the average real return on nearly riskless short-term securities and the S&P 500 for the period 1989-1978 was 6. 18%. This is clearly inconsistent with the predictions of the model. In particular if risk aversion is close to 0 and individuals are almost risk neutral, the model fails to explain why the sample’s average equity returns are so high.If risk aversion is significantly positive the model does not justify the low average risk-free rate of the sample. The results of Mehra and Prescott’s (2008) empirical study are consistent with our results, because the power utility CAPM did not satisfy our empirical tests. 4 Kan and Robotti (2007) Confidence Intervals The Law of One Price volatility bounds calculated in part 2 are subject to sampling variation. We have calculated point estimates of the volatilit y bounds, but we did not take into account that our results are based on a finite sample of Treasury Bill and market returns.To more accurately test whether the power utility CCAPM passes the LOP volatility bounds test, we need to identify the area in which the population volatility bound may lie. The area used is that between the upper and lower 95% confidence intervals for Hansen-Jagannathan volatility bounds obtained by Kan and Robotti (2007), shown in table 2. If the standard deviations of the CCAPM stochastic discount factors lie below that area for values of risk aversion between 1 and 20, then the power utility CCAPM model is to be rejected according to this test.Table 2 here Figure 3 here Figure 3 contains point estimates of the LOP volatility bounds, the standard deviations and means of the CCAPM stochastic discount factors for levels of risk aversion between 1 and 20 and the 95% confidence intervals for the volatility bounds. All of the standard deviations are below the ar ea in between the upper and lower confidence intervals for the volatility bounds. This indicates that at a 95% certainty the CCAPM does not satisfy the LOP volatility bound test even when sampling errors are taken into account. Performance of Power Utility CCAPM In recent academic literature on the subject of asset pricing models a common formal method of evaluating model performance is to calculate the pricing errors on a set of test assets. In this report the test assets are the Treasury Bill and Market Index quarterly returns from Q1 1963 to Q4 2009. The pricing error is calculated as [pic] Where [pic], [pic] Treasury Bill and Market Index returns, and [pic] is the pricing errors. Table 3 hereFor a model to correctly price an asset it would require that the pricing errors are as close to zero as possible since the pricing error is a measure of the distance between the model pricing kernel and the true pricing kernel. From Table 3 we can see that the pricing errors for the differe nt values of risk aversion are not close to zero and the size of the errors actually increases with the level of risk aversion. We can also see that the Route Mean Square Pricing Error (RSME) which measures the average distance from zero of the pricing errors is not as close to zero as we would hope and also increases with the level of risk aversion.If we note the case for a risk aversion level of 20 then the RSME is 6. 76%, since this is quarterly data this works out to an annual RSME of approximately 27%. With such large pricing errors we would not expect this model to perform strongly. Hansen and Jagannathan (1997) found that for different levels of risk aversion the pricing errors do not vary greatly. As noted above, this is not the case in our sample in which the error increases with the level of risk aversion, thus creating an ever wider dispersion of pricing errors.This is counterintuitive to what we would usually assume as with increased levels of risk aversion the consumer is only willing to accept a certain level of return for lower and lower levels of risk, therefore we would expect at some point that the mean variance level would pass the volatility bounds and therefore correctly price the assets. Conforming with this report Cochrane and Hansen (1992) found that in order to satisfy the levels of variance necessary to surpass the volatility bounds a risk aversion level of at least 40 was necessary.It should be noted that in reality this is quite unreasonable and also that for this level of variance to be attained the expected return might also have to drop below the level necessary to surpass the volatility bounds. Table 4 here From Hansen and Jagannathan (1991) we know that in order to price a set of assets correctly the stochastic discount factor (SDF) should be close to one and have high levels of volatility. Table 4 shows that SDF’s at low levels of risk aversion are relatively close to one but have very low levels of volatility.When the level of risk aversion increases the SDF’s get further and further away from one yet the volatility also increases. Therefore it seems reasonable to conclude that we would not expect any of these SDF’s to price the assets correctly. The results illustrated above are consistent with the earlier analysis and point to the conclusion that the power utility CCAPM does not do a good job in pricing the two test assets and thus does not perform well in UK stock returns. Cochrane and Hansen (1992) agree with this conclusion but Kan and Robotti (2007) find the opposite.The reason for this could be the use of sampling error in the Kan and Robotti paper and the different data used the in the analysis. This report illustrates that there exists not only an equity premium puzzle but also a risk free rate puzzle. This risk free rate puzzle as noted by Weil (1989) states that if consumers are extremely risk averse, a result of the equity premium puzzle, then why is the risk free rate s o low. Weil cites market imperfections and heterogeneity as the probable causes of this puzzle; however, this is not the explanation that Bansal and Yaron (2004) find.Using a model that accounts for investor reaction to news about growth rates and economic uncertainty they are able to go some way to resolving not only the risk free rate puzzle but also the equity risk premium puzzle. One method that could be used to improve the performance of the power utility CCAPM would be to construct the model using conditioning information; this would enlarge the possible payoff space available to investors. Kan and Robotti (2006) find that including conditioning information in models reduces the pricing errors by allowing the prices of volatility to move in line with the market.Although as Roussanov (2010) finds, conditioning information does not necessarily improve model performance and may actually exacerbate the problem. 6 Sampling Error in the Volatility Bounds When using the volatility bo unds as specified by Hansen and Jagannathan (1991) to test asset pricing models we must be wary of sampling error in the bounds. As noted previously if a model does not lie within the Hansen and Jagannathan volatility bounds then we can conclude that it does not price the test assets correctly.However, Gregory and Smith (1992) and Burnside (1994) first noted that this test does not take into account significant sampling variation and could therefore reject models that price assets correctly. Burnside (1994) uses Monte-Carlo simulation to illustrate that over repeated samples if sampling error is ignored the volatility bounds test performs poorly. Gregory and Smith (1992) state that the sampling error could be due to large variability in the estimated bounds or the use of sample data in the analysis.Kan and Robotti (2007) derive the finite sample distribution of the Hansen and Jagannathan bounds in order to take account of this sampling error. They argue that confidence intervals tha t take into account the variation can be constructed and used to test asset pricing models. The importance of this new method of testing cannot be underestimated as it could affect the decision to reject an asset pricing model or not, this is best illustrated with reference to examples. Kan and Robotti test the equity premium puzzle using data from Shiller (1989) to show the implications of taking into account sampling error.Through constructing the 95% confidence intervals for the Hansen and Jagannathan volatility bounds they are able to show that the time-separable power utility model being tested may not be rejected at low levels of risk aversion. This is in stark contrast to the findings when sampling error is not taken into account where the model is strongly rejected except for unfeasible levels of risk aversion. From Figure 3, as noted earlier, even when sampling error is taken into account for the model tested in this report it does not fall within the volatility bounds.Howe ver, it does decreases the distance between the model and the volatility bounds which is the major consequence of the Kan and Robotti paper. This new method goes some way to solving the problem noted by Cecchetti, Lam, and Mark (1994) who found using classical hypothesis tests that the Hansen and Jagannathan bounds without sampling error rejected true models too often. Again, an extension here could be to use conditioning information to improve the volatility bounds by using the methods of Ferson and Siegel (2003) and as a result hopefully reduce the sampling error in the bounds.References Bansal, R. and A. Yaron, 2004, Risks for the long run: A potential resolution of asset pricing puzzles, Journal of Finance, American Finance Association, vol. 59(4), pages 1481-1509, 08. Burnside, C. , 1994, Hansen-Jagannathan Bounds as Classical Tests of Asset-Pricing Models,† Journal of Business & Economic Statistics, American Statistical Association, vol. 12(1), pages 57-79 Cecchetti, S. G. , P. Lam, and N. C. Mark, 1994, Testing Volatility Restrictions on Intertemporal Marginal Rates of Substitution Implied by Euler Equations and Asset Returns, Journal of Finance, 49, 123–152.Cochrane, J. H. and L. P. Hansen, 1992, Asset Pricing Explorations for Macroeconomics, NBER Chapters, in: NBER Macroeconomics Annual 1992, Volume 7, pages 115-182 National Bureau of Economic Research, Inc. Dunn, K. , and K. Singleton, 1986, Modelling the term structure of interest rates under Non-separable utility and durability of goods, Journal of Financial Economics, 17, 1986, 27-55. Ferson, W. E. , and A. F. Siegel, 2003, Stochastic Discount Factor Bounds with Conditioning Information, Review of Financial studies, 16, 567–595. Gregory, A. W. and G. W Smith, 1992.Sampling variability in Hansen-Jagannathan bounds, Economics Letters, Elsevier, vol. 38(3), pages 263-267. Hansen, L. P. and R. Jagannathan, 1991, Implications of Security Market Data for Models of Dynamic Economies, Journal of Political Economy, Vol. 99, No. 2 (Apr. , 1991), pp. 225-262   Hansen, L. P. and R. Jagannathan, 1997. Assessing specification errors in stochastic discount factor models. Journal of Finance 52, 591-607. Kan, R. , and C. Robotti, 2007, The Exact Distribution of the Hansen-Jagannathan Bound. Working Paper, University of Toronto and Federal Reserve Bank of Atlanta. Mehra, R. , and E. C.Prescott, (1985), The equity premium: A puzzle, Journal of Monetary Economics 15, 145-161. Roussanov, N. , 2010, Composition of Wealth, Conditioning Information, and the Cross-Section of Stock Returns, NBER Working Papers 16073, National Bureau of Economic Research, Inc. Shiller, R. , 1982, Consumption, Asset Markets and Macroeconomic fluctuations, Carnegie–Rochester Conference Series on Public Policy, Vol. 17. North-Holland Publishing Co. , 1982, pp. 203–238. Shiller, R. J. , 1989, Market Volatility, MIT Press, Massachusetts. Journal of Economic Behavior & Organization, Elsev ier, vol. 16(3), pages 361-364.Weil, P. , 1989, The equity premium puzzle and the risk free rate puzzle, Journal of Monetary Economics 24. 401-422. Appendix [pic] Figure 1 LOP Volatility Bounds. The figure shows the LOP volatility bounds (dark blue line) which were found by using Treasury Bill and market returns as test assets. [pic] Figure 2 LOP Volatility Bounds with CCAPM.The figure shows the LOP volatility bounds (dark blue line) which were found by using Treasury Bill and market returns as test assets. It also shows the means and corresponding standard deviations of the CCAPM stochastic discount factors (green line) for values of risk aversion between 1 and 20. [pic] Figure 3 LOP Volatility Bounds with CCAPM and Confidence Intervals. The figure shows the LOP volatility bounds (dark blue line) which were found by using Treasury Bill and market returns as test assets.It also shows the means and corresponding standard deviations of the CCAPM stochastic discount factors (green line ) for values of risk aversion between 1 and 20. The figure contains the confidence intervals, with a 95% level of confidence, estimated by Kan and Robotti (2007) for E(m) between 0. 97 and 1. 0082 for the Law of One Price volatility bounds for their first set of test assets. The light blue line shows the upper bounds of the confidence intervals and the red line shows the lower bounds of the confidence intervals. Table 1 CCAPM stochastic discount factors’ means and standard deviations and corresponding LOP volatility bounds CCAPM |LOP volatility bounds |CCAPM | | |means | |st. dev. | | |0. 985121 |0. 82806186 |0. 011749 | |0. 980404 |1. 2067111 |0. 023503 | |0. 975849 |1. 57451579 |0. 035275 | |0. 971456 |1. 93015539 |0. 04708 | |0. 967223 |2. 27320637 |0. 58934 | |0. 963151 |2. 60350158 |0. 070853 | |0. 959239 |2. 92096535 |0. 082854 | |0. 955486 |3. 22555764 |0. 094953 | |0. 951893 |3. 5172513 |0. 107169 | |0. 94846 |3. 7960217 |0. 11952 | |0. 945187 |4. 06184126 |0. 132027 | |0. 942074 |4. 31467648 |0. 14471 | |0. 939121 |4. 5448604 |0. 15759 | |0. 93633 |4. 7812196 |0. 17069 | |0. 933701 |4. 99481688 |0. 184033 | |0. 931234 |5. 19520693 |0. 197645 | |0. 928931 |5. 38230757 |0. 211552 | |0. 926792 |5. 55602479 |0. 225781 | |0. 92482 |5. 71625225 |0. 240361 | |0. 923016 |5. 8628708 |0. 255322 |This table shows the means of the CCAPM stochastic discount factors for levels of risk aversion between 0 and 20, the corresponding LOP volatility bounds and the standard deviations of the CCAPM stochastic discount factors. Table 2 95% confidence intervals for E(m) between 0. 97 and 1. 0082 E(m) Lower Upper 0. 9700 3. 1823 5. 2069 0. 9710 2. 9385 4. 8383 0. 9719 2. 7038 4. 4830 0. 9729 2. 4781 4. 1411 0. 9738 2. 2617 3. 8125 0. 9748 2. 0544 3. 4974 0. 9757 1. 8565 3. 1959 0. 9767 1. 6680 2. 9080 0. 9776 1. 4890 2. 6337 0. 9786 1. 3195 2. 3731 0. 9795 1. 1597 2. 1262 0. 805 1. 0097 1. 8931 0. 9815 0. 8696 1. 6739 0. 9824 0. 7394 1. 4685 0. 9834 0. 6194 1. 2770 0. 9843 0. 5096 1. 0993 0. 9853 0. 4101 0. 9356 0. 9863 0. 3212 0. 7857 0. 9873 0. 2429 0. 6497 0. 9882 0. 1755 0. 5275 0. 9892 0. 1190 0. 4192 0. 9902 0. 0736 0. 3248 0. 9912 0. 0393 0. 2445 0. 9922 0. 0160 0. 1784 0. 9931 0. 0030 0. 1275 0. 9941 0 0. 0938 0. 9951 0 NaN 0. 9961 0 0. 0938 0. 9971 0. 0029 0. 1279 0. 9981 0. 0159 0. 1798 0. 9991 0. 0395 0. 2474 1. 0001 0. 0745 0. 3302 1. 0011 0. 1212 0. 280 1. 0021 0. 1796 0. 5408 1. 0031 0. 2498 0. 6689 1. 0041 0. 3317 0. 8123 1. 0051 0. 4255 0. 9714 1. 0061 0. 5309 1. 1461 1. 0072 0. 6481 1. 3368 1. 0082 0. 7769 1. 5437 This table shows the upper and lower bounds of the 95% confidence intervals Kan and Robotti (2007) calculated for the volatility bounds for their first set of test assets. The confidence intervals presented are for values of E(m) between 0. 97 and 1. 0082. Table 3 Pricing errors for the Treasury Bill (Rf) and the value weighted UK market index (Rm), and the Root Mean Square Pricing Error (RSME) for each level of risk av ersion Level of Risk Aversion |Error Rf |Error Rm |RSME | |1 |-0. 0104 |0. 0047 |0. 0080 | |2 |-0. 0152 |-0. 0001 |0. 0107 | |3 |-0. 0199 |-0. 0049 |0. 0144 | |4 |-0. 0244 |-0. 0094 |0. 0184 | |5 |-0. 287 |-0. 0138 |0. 0225 | |6 |-0. 0329 |-0. 0180 |0. 0265 | |7 |-0. 0369 |-0. 0221 |0. 0304 | |8 |-0. 0408 |-0. 0260 |0. 0342 | |9 |-0. 0445 |-0. 0297 |0. 0378 | |10 |-0. 0480 |-0. 0333 |0. 413 | |11 |-0. 0514 |-0. 0367 |0. 0446 | |12 |-0. 0546 |-0. 0399 |0. 0478 | |13 |-0. 0577 |-0. 0430 |0. 0508 | |14 |-0. 0606 |-0. 0459 |0. 0537 | |15 |-0. 0634 |-0. 0487 |0. 0564 | |16 |-0. 660 |-0. 0513 |0. 0590 | |17 |-0. 0684 |-0. 0537 |0. 0614 | |18 |-0. 0706 |-0. 0560 |0. 0636 | |19 |-0. 0727 |-0. 0580 |0. 0657 | |20 |-0. 0747 |-0. 0600 |0. 0676 | | | | | |The pricing errors above are calculated as [pic], where [pic], [pic] Treasury Bill and Market Index returns, and [pic] is the pricing errors. The RSME is simply the average pricing error of the stochastic discount factor for each level of risk aversion. Table 4 Summary Statistics for power utility CCAPM stochastic discount factor |Level of Risk Aversion |Average |St Dev |Min |Max | |1 |0. 9851 |0. 0117 |0. 9551 |1. 0436 | |2 |0. 804 |0. 0235 |0. 9214 |1. 1000 | |3 |0. 9758 |0. 0353 |0. 8889 |1. 1595 | |4 |0. 9715 |0. 0471 |0. 8575 |1. 2223 | |5 |0. 9672 |0. 0589 |0. 8273 |1. 2884 | |6 |0. 9632 |0. 0709 |0. 7981 |1. 3581 | |7 |0. 592 |0. 0829 |0. 7699 |1. 4316 | |8 |0. 9555 |0. 0950 |0. 7428 |1. 5090 | |9 |0. 9519 |0. 1072 |0. 7166 |1. 5906 | |10 |0. 9485 |0. 1195 |0. 6913 |1. 6767 | |11 |0. 9452 |0. 1320 |0. 6669 |1. 7674 | |12 |0. 421 |0. 1447 |0. 6434 |1. 8630 | |13 |0. 9391 |0. 1576 |0. 6207 |1. 9638 | |14 |0. 9363 |0. 1707 |0. 5988 |2. 0701 | |15 |0. 9337 |0. 1840 |0. 5777 |2. 1821 | |16 |0. 9312 |0. 1976 |0. 5573 |2. 3001 | |17 |0. 9289 |0. 116 |0. 5377 |2. 4245 | |18 |0. 9268 |0. 2258 |0. 5187 |2. 5557 | |19 |0. 9248 |0. 2404 |0. 5004 |2. 6940 | |20 |0. 9230 |0. 2553 |0. 4827 |2. 8397 | This table shows the average value, standard deviation, minimum and maximum for the stochastic discount factor at each level of risk aversion. ———————– 24th November 2011