Risk Management Performance

Predictability in covariance between two assets’ returns, as measured by traditional criteria that focus on the size of the forecast error, does not necessarily imply that an investor can make profits or reduce risk from a trading strategy based on such forecasts.

Therefore, we also use the other category of performance measure which is based on the views of risk managers, and two types of criteria are used. One is to calculate the value at risk (VaR) as an evaluation of the estimator. For a two-asset portfolio with δ invested in the first asset and

(

^1−δ

)

in the second asset, the one-step-ahead VaR at time t and atα , assuming normality, is %

( ) (

ˆ1,

( )

ˆ2,

)

²ˆ11,

( )

² ˆ22,

( )

ˆ12,

VaR_t α = −^⎡⎢⎣ δr_t+ −1 δ r _t −z_α⋅ δ h _t + −1 δ h _t +2δ 1−δ h _t^⎤⎥⎦, (3.24)

where z_a is the right quantile at α . To compare and evaluate model performances on % several different model specifications, we choose the following criteria. By definition, the failure rate (FR) is the proportion of returns (in absolute value) exceed the forecasted VaR, i.e. FR

( )

1 ^T1

(

1,_t

(

1

)

2,_t VaR_t

( ) )

T t₌ I δr δ r α

=

∑

+ − < − ^{, where I}

^{( )}

^⋅ is the indicator

function. Hence, if the VaR model is correctly specified, the failure rate should be equal to the prespecified VaR level. We also use Kupiec LR test (1995) to examine if the model is correctly specified. To test H :f₀ =α against H₁: f ≠α , the LR statistic is

( )

( ) (

^{( )}

⁽

^{( )}

⁾ )

2 ln ^{T N} 1 ^N 2 ln ^{T N} 1 ^N

LR= − α ⁻ −α + N T ⁻ − N T , where N is the number of VaR violations, T is the total number of observations and f is the theoretical failure rate. Under the null hypothesis, the LR test statistic is asymptotically distributed as^χ2

( )

^{1 .}

Figure 3.3 plots one period ahead 5% VaR forecast comparison for a hedged portfolio with weights (1,-1) and three weighted portfolios with weights (0.5, 0.5), (0.3, 0.7), and (0.7, 0.3). The fractions of VaR violations and p-values of the Kupiec (1995) failure rate test are reported in Table 3.8. For in-sample data, the p-values for the null hypothesis of the

hedged portfolio are all smaller than 0.05 when the linear model is considered. The threshold model performs very well as there are no p-values smaller than 0.05. Thus the switch from the linear model to threshold model yields a significant improvement in the VaR performance in the hedged portfolio. For the weighted portfolios consisting of S&P and Nasdaq100 spot markets, the failure rates are very close to the prespecified VaR level and p-values are larger than 0.05 for in-sample predictions in all linear and threshold models. For the out-of-sample VaR performance comparison, we find that there are no p-values smaller than 0.05 no matter what kinds of portfolios or models which we consider here. But, the fractions of VaR violation based on the threshold model are closer to the prespecified VaR level than those based on the linear model except the weighted portfolio with weights (0.7, 0.3). In view of in-sample and out-of-sample empirical results, we find that the linear model may sometimes overestimate the VaR, so using the threshold model to calculate the portfolio’s VaR may be more appropriate.

A. 5% VaR for S&P500 Futures-Spot

200001 200007 200101 200107 200201 200207 200301 200307 200401 200407 200501

Ret(S-F) VaR(S-F)_M1 VaR(S-F)_M4 VaR(S-F)_M7

B. 5% VaR for 0.5*S&P500+0.5*Nasdaq100

-10

200001 200007 200101 200107 200201 200207 200301 200307 200401 200407 200501

Ret(Portfolio) VaR_M1 VaR_M4 VaR_M7

C. 5% VaR for 0.3*S&P500+0.7*Nasdaq100

-15

200001 200007 200101 200107 200201 200207 200301 200307 200401 200407 200501

Ret(Portfolio) VaR_M1 VaR_M4 VaR_M7

D. 5% VaR for 0.7*S&P500+0.3*Nasdaq100

-10

200001 200007 200101 200107 200201 200207 200301 200307 200401 200407 200501

Ret(Portfolio) VaR_M1 VaR_M4 VaR_M7

FIGURE 3.3－One Period Ahead 5% VaR Forecast Comparison for Various Portfolios, 2000/1-2005/3.

TABLE 3.8 － In- and Out-of-sample 5% VaR Failure Rate Results for the S&P500 Futures-Spot and S&P500-Nasdaq100 Spot Markets

A. Fraction of VaR Violation

In sample Out of sample S&P500

Fut-Spot S&P500-Nasdaq100 S&P500

Fut-Spot S&P500-Nasdaq100

B. P-Value of Kupiec LR Test

In sample Out of sample

S&P500

Fut-Spot S&P500-Nasdaq100 S&P500

Fut-Spot S&P500-Nasdaq100

Note:This table shows the 5% VaR forecast results of four different portfolios for alternative models. Panel A is the fraction of VaR violations, and the results of Kupiec LM test (1995) are showed in Panel B. The data used are daily S&P500 index futures, S&P500 and Nasdaq 100 index prices. The in-sample data period is from January 1, 2000 to December 31, 2004 and out-of-sample data period is from January 1, 2005 to March 31, 2005. M1,M2,…,M9 denote the same models in Tables 3.2.

While futures contracts are popular among investors as a class of speculative assets, they are important in the financial markets due to their use as a hedging instrument.

Furthermore, hedging with futures contracts may be the simplest method to manage market risk resulting from adverse movements in the price of various assets. In this section, we assume the hedger attempts to minimize the conditional variance of the spot-futures portfolio. It is well known that the optimal hedge ratio (OHR) is the ratio of the conditional covariance between spot and futures returns over the conditional variance of the futures return. So, the one-step-ahead forecasts of optimal hedge ratios can then be calculated as

( ) ( )

*

1 , 1

t t t s f t t f

HR =Cov₊ r r Var₊ r , (3.25)

where r_s and r_f are spot and futures returns, respectively. The variance of the estimated optimal hedged portfolio can be characterized as

(

^{s t, t* f t,}

)

Var r −HR r⋅ .

To evaluate hedging performance, the typical criterion is based on the percentage variance reduction (PVR) of the hedged portfolio relative to the unhedged position. It can be calculated as

When the futures contract completely eliminates risk, PVR=100 is obtained, otherwise PVR=0 is obtained when hedging with the futures contract does not reduce risk. Hence, a larger PVR indicates better hedging performance.

The in- and out-of-sample hedged portfolio variances and hedging effectiveness of alternative models for the S&P500 futures contract are presented in Table 3.9. The variances of hedged portfolio returns are calculated under the following eleven

alternative models: three linear VECM-DCC-GARCH models with different lag parameter L (M₁,M₂,M₃), six threshold VECM-DCC-GARCH models with different lag parameter L and the number of regime G (M₄,---,M₉), hedging with a constant OHRs estimate using regression methods of returns and the naïve hedge with hedge ratio of 1 at all times. The results show that the three-regime threshold VECM(3)-DCC-GARCH model has the lowest in-sample hedged portfolio variance, with a 93.037% in-sample variance reduction compared to the variance of the unhedged position. In addition, the in-sample hedging performance of the linear model is even worse than that of OLS or naïve strategy.

Table 3.9. In- and Out-of-sample Hedging Effectiveness of Alternative Models for S&P500 Spot and Futures Markets

In Sample Out of Sample

Model (Mi) Variance Variance

Reduction Rank Variance Variance

Reduction Rank

Note: The table reports the variance of the hedged portfolio, percentage variance reductions, and the ranks of hedging effectiveness for several different models. The data used are daily index futures and spot price. The in-sample data period is from January 1, 2000 to December 31, 2004 and out-of-sample data period is from January 1, 2005 to March 31, 2005. M1,M2,…,M9 denote the same models in Tables 3.2.

However, active hedgers are likely to be more concerned about future hedging

performance. Therefore, the comparison of out-of-sample performance is a better way to evaluate our hedging strategy. We find that the ranking of out-of-sample hedging effectiveness is the same as that of in-sample hedging effectiveness. The two-regime threshold VECM(3)-DCC-GARCH model has a 97.065% out-of-sample variance reduction and outperforms all linear dynamic and static hedging models we considered.

Overall, the dynamic hedge with threshold model has a better hedging performance than that with linear VECM-DCC-GARCH model. Both naïve and OLS strategies tend to outperform the linear VECM-DCC-GARCH model. This fact may indicate that the threshold model has superior ability to forecast the optimal hedge ratios.

6. CONCLUSIONS

We proposed a robust multivariate VAR-DCC-GARCH model that extends existing approaches by admitting multivariate thresholds in conditional means, conditional volatilities and conditional correlations. In addition, such threshold variables are defined by a weighted average of endogenous variables and the weights are estimated from the data. This threshold setting can not only enhance the robustness of the model but also has some economic meanings or values. Moreover, the Markov chain Monte Carlo method is implemented for the Bayesian inference. We studied the performance of our model in an application to two data sets consisting of daily S&P500 futures and spot prices and S&P500 and Nasdaq100 spot prices.

We develop a Bayesian testing scheme for model selection among several competing models and select the model with a higher posterior probability. We also adopt several criteria, which are based on the views of statistical loss and risk managers, to evaluate the prediction performance of the conditional covariance matrix.

In our real data application we find that estimated conditional volatilities are strongly characterized by both GARCH and multivariate threshold effects. Dynamic correlations are still apparent between S&P500 and Nasdaq100 spot returns, while slighter between S&P500 futures and spot markets. In addition, the estimation results suggest that S&P500 futures market is price leader between S&P500 futures and spot markets and S&P500 spot market is price leader between S&P500 and Nasdaq100 spot markets. For the comparison in covariance matrix forecasting performance, the threshold model has a better in-sample and out-of-sample forecasting performance relative to the linear model across most measure criteria.

Chapter 4. Summary and Conclusions

There are more and more academics shows that financial time series, such as stock returns, exchange rates series and etc., exhibit strong signs of nonlinearity. For this reason, some of traditional financial models should be modified appropriately. In this dissertation, we focus on the threshold model and use the Bayesian approach to settle the difficulty in the maximum likelihood estimation and inference. In addition, the applications in several important issues in financial markets are discussed, including the mutual fund performance evaluation and the forecasting in conditional covariance matrix.

The first essay in this dissertation uses three-regime Bayesian unconditional and conditional threshold four-factor models to study how fund managers react to change in market conditions. Our empirical analyses show that there are more apparent differences between unconditional and conditional three-regime threshold models instead of two-regime threshold models. In addition, we find that most managers’ market timing ability comes from the skills to forecast the downside market. The three-regime threshold models have more power to detect significant timing activity when lagged public information is taken into account.

In addition, for the relationship between fund performances and various characteristics, we find that investors prefer to select funds with better past selectivity performance and upside market timing ability instead of downside market timing skill.

Moreover, fund clients favor large size funds and funds with lower turnover, total load charges, and expenses. High turnover funds tend to have worse (better) selectivity performances in the downside (upside) market. We also find that contemporaneous net cash flows are negatively associated with downside market timing ability, but are positively correlated to upside market timing skills. In addition, funds with higher

expenses have alert sensitivities to discover downside and upside markets.

In the second essay, we proposed a robust multivariate VAR-DCC-GARCH model that extends existing approaches by admitting multivariate thresholds in conditional means, conditional volatilities and conditional correlations. In addition, such threshold variables are defined by a weighted average of endogenous variables and the weights are estimated from the data. We studied the performance of our model in an application to two data sets consisting of daily S&P500 futures and spot prices and S&P500 and Nasdaq100 spot prices.

Our empirical analyses find that estimated conditional volatilities are strongly characterized by both GARCH and multivariate threshold effects. Dynamic correlations are still apparent between S&P500 and Nasdaq100 spot returns, while slighter between S&P500 futures and spot markets. In addition, the estimation results suggest that S&P500 futures market is price leader between S&P500 futures and spot markets and S&P500 spot market is price leader between S&P500 and Nasdaq100 spot markets. For the comparison in covariance matrix forecasting performance, the threshold model has a better in-sample and out-of-sample forecasting performance relative to the linear model across most measure criteria.

References

Andersen T., Bollerslev T., Diebold F., Ebens H., 2001. The distribution of realized stock return volatility. Journal of Financial Economics 61: 43-76.

Bauwen L., Lubrano M., Richard J.F., 1999. Bayesian Inference in Dynamic Econometric Models. Oxford University Press: New York.

Bollen N.P.B., Busse, J.A., 2001. On the timing ability of mutual fund managers. Journal of Finance 56: 1075-1094.

Bollerslev T., 1990. Modeling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH model. Review of Economics and Statistics 72:

498-505.

Bollerslev T., Engle R., Wooldridge J.M., 1988. A capital asset pricing model with time varying covariances. Journal of Political Economy 96: 116-131.

Brooks C., 2001. A double-threshold GARCH model for the French Franc/Deutschmark exchange rate. Journal of Forecasting 20: 135-143.

Brown S.J., Goetzmann W.N., Ibbotson R.G., Ross S.A., 1992. Survivorship bias in performance studies. Review of Financial Studies 5: 553–580.

Brown S.J., Goetzman W.N., 1995. Performance persistence. Journal of Finance 50: 679-698.

Busse J.A., 1999. Volatility timing in mutual funds: Evidence from daily returns. Review of Financial Studies 12: 1009-1041.

Caner M., Hansen B.E., 2001. Threshold autoregression with a unit root. Econometrica 69:

1555-1596.

Carhart M.M., 1997. On persistence in mutual fund performance. Journal of Finance 52:

57-82.

Chang E.C., Lewellen W.G., 1984. Market timing and mutual fund investment performance.

Journal of Business 57: 57-72.

Chen C.W.S., So M.K.P., 2006. On a threshold heteroscedastic model. International Journal of Forecasting 22: 73-89.

Chen J., Huang H., Hong M., Kubik J.D., 2004. Does fund size erode mutual fund performance? The role of liquidity and organization. American Economic Review 94:

1276-1302.

Chevalier J., Ellison G., 1997. Risk taking by mutual funds as a response to incentives.

Journal of Political Economy 105: 1167-1200.

Chib S., Greenberg E., 1995. Understanding the Metropolis-Hastings algorithm. American Statistician 49, 327-335.

Dellva W.L., Gerard T.O., 1998. The relationship between mutual fund fees and expenses and their effects on performance. The Financial Review 33: 85-104.

Doan T.A., 1990. Regression Analysis of Time Series, Evanston, IL: VAR Econometrics.

Edelen R.M., 1999. Investor flows and the assessed performance of open-end mutual funds.

Journal of Financial Economics 53: 439-466.

Elton E.J., Gruber M.J., Das S., Hlavka M., 1993. Efficiency with costly information: a reinterpretation of evidence from managed portfolios. Review of Financial Studies 6:1-22.

Fama E.F, French K.R., 1993. Common risk factors in the returns on stocks and bonds.

Journal of Financial Economics 33: 3-56.

Engle R.F., Yoo B., 1987. Forecasting and testing in cointegrated systems. Journal of Econometrics 35: 143-159.

Engle R.F., Ng V., Rothschild M., 1990. Asset pricing with a factor-ARCH covariance structure: empirical estimates for Treasury bills. Journal of Econometrics 45: 213-238.

Engle R.F., Kroner K., 1995. Multivariate simultaneous GARCH. Econometric Theory 11:

122-150.

Engle R.F., 2002. Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business and Economic Statistics 20: 339-350.

Ferson W.E., Schadt R., 1996. Measuring fund strategy and performance in changing economic conditions. Journal of Finance 51: 425-462.

French K.R., Schwert G.W., Stambaugh R.F., 1987. Expected stock returns and volatility.

Journal of Financial Economics 19: 3-29.

Geweke J., 1992. Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. Bayesian Statistics 4: 169-193.

Geweke J., 1996. Bayesian reduced rank regression in econometrics. Journal of Econometrics 75: 121-146.

Gilks W.R., Richardson S., Spiegelhalter D.J., 1996. Markov chain Monte Carlo in practice.

Chapman & Hall.

Glosten L.R., Jagannathan R., 1994. A contingent claim approach to performance evaluation.

Journal of Empirical Finance 1: 133-160.

Goetzmann W.N., Ingersoll J., Ivkovic Z., 2000. Monthly measurement of daily timers.

Journal of Financial and Quantitative Analysis 35: 257-290.

Grinblatt M., Titman S., 1989. Mutual fund performance: An analysis of quarterly portfolio holdings. Journal of Business 62: 393-416.

Grinblatt M., Titman S., 1994. A study of monthly mutual fund returns and performance evaluation techniques. Journal of Financial and Quantitative Analysis 29: 419-444.

Gruber M., 1996. Another puzzle: the growth in activity managed and mutual funds. Journal

of Finance 51: 783-810.

Hammersley J.M., Clifford P., 1968. Markov fields of finite graphs and lattices. Preprint, Univ.

of Calif.-Berkeley.

Hansen B.E., 1996. Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64: 413-430.

Hansen B.E., 2000. Sample splitting and threshold estimation. Econometrica 68: 575-603.

Hansen B.E, Seo B., 2002. Testing for two-regime threshold cointegration in vector error-correction models. Journal of Econometrics 110: 293-318.

Hastings W.K., 1970. Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57: 97-109.

Henriksson R.D., 1984. Market timing and mutual fund performance: An empirical investigation. Journal of Business 57: 73-96.

Hsieh D., 1989. Testing for Nonlinearity in Daily Foreign Exchange Rate Changes. Journal of Business 62: 339-368.

Hsieh D., 1991. Chaos and Nonlinear Dynamics: Application to Financial markets. Journal of Finance 46: 1839-1877.

Hsieh D., 1993. Implication of nonlinear dynamics for financial risk management. Journal of Financial and Quantitative Analysis 28: 41-64.

Ippolito R., 1989. Efficiency with costly information：A study of mutual fund performance 1965-1984. The Quarterly Journal of Economic 104: 1-23.

Jain P.C., Wu J.S., 2000. Truth in mutual fund advertising: Evidence on future performance and fund flows. Journal of Finance 55: 937-958.

Jensen M., 1968. The performance of mutual funds in the period 1945-1964. Journal of Finance 23: 389-416.

Jones Christopher S., Shanken Jay, 2005. Mutual fund performance with learning across funds. Journal of Financial Economics 78: 507-552.

Kinal T, Ratner J., 1986. A VAR forecasting model of a regional economy: Its construction and comparative accuracy. International Regional Science Review 10: 113-26.

Koop, G., Potter, S.M., 1999. Bayes factors and nonlinearity: Evidence from economic time series. Journal of Econometrics 88: 251–281.

Koop, G., Potter, S.M., 2003. Bayesinan analysis of endogenous delay threshold models.

Journal of Business & Economic Statistics 21: 93–103.

Kupiec P., 1995. Techniques for verifying the accuracy of risk management models. Journal of Derivatives 3: 73-84.

Lamoureux C.G., Lastrapes W.D., 1990. Persistence in variance, structural change, and the GARCH model. Journal of Busines & Economic Statistics 8: 225-234.

Li C.W., Li W.K., 1996. On a double-threshold autoregressive heteroscedastic time series model. Journal of Applied Econometrics 11: 253-274.

Litterman R.B., 1980. Techniques for forecasting with vector autoregressions. Working paper, Massachusetts Institute of Technology.

Litterman R.B., 1986. Forecasting with Bayesian vector autoregressive-five years of experience. Journal of Business & Economic Statistics 4: 25-38.

Malkiel B., 1995. Returns from investing in equity mutual funds 1971 to 1991. Journal of Finance 50: 549-572.

Merton R.C., Henriksson R.D., 1981. On the market timing and investment performance of managed portfolios II – statistical procedures for evaluating forecasting skills. Journal of Business 54: 513-533.

Metropolis N., Rosenbluth A.W., Rosenbluth M.N., Teller A.H., Teller E., 1953. Equation of

state calculations by fast computing machines. Journal of Chemical Physics 21,:1087-1092.

Mikosch T., Starica C., 2004. Nonstationarities in Financial Time Series, the Long Range Dependence and the IGARCH Effects. Review of Economics and Statistics 86:

378-390.

Ni S, Sun D., 2003. Noninformative priors and frequentist risks of Bayesian estimators of vector-autoregressive models. Journal of Econometrics 115: 159-197

Ni S, Sun D., 2005. Bayesian estimates for vector autoregressive models. Journal of Business

& Economic Statistics 23: 105-117.

Osiewalski J., Pipień M., 2004. Bayesian comparison of bivariate ARCH-type models for the main exchange rates in Poland. Journal of Econometrics 123: 371-391.

Pastor L., Stambaugh Robert F., 2000. Comparing Asset Pricing Models: An Investment Perspective. Journal of Financial Economics 56: 335–381

Pastor L., Stambaugh Robert F., 2002. Investing in Equity Mutual Funds. Journal of Financial Economics 63: 351–380.

Rabemananjara R., Zakoian J.M, 1993. Threshold ARCH models and asymmetries in volatility. Journal of Applied Econometrics 8: 31-49.

Sharpe W.F., 1964. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance 19: 425-442.

Sirri E., Tufano P., 1998. Costly search and mutual fund flows. Journal of Finance 53:

1589-1622.

Sims C.A., 1980. Macroeconomics and reality. Econometrica 48: 1-48.

So M.K.P., Chen C.W.S., Chen M.T., 2005. A Bayesian threshold nonlinearity test for financial time series. Journal of Forecasting 24: 61-75.

Tong H., 1983. Threshold Models in Non-linear Time Series Analysis: Lecture Notes in Statistics 21. Berlin: Springer-Verlag.

Tong, H., 1990. Non-linear time series: A dynamical system approach. Oxford University Press: Oxford.

Treynor J.L., Mazuy, K., 1966. Can mutual funds outguess the market? Harvard Business Review 44: 131-136.

Tsay R.S., 1989. Testing and modeling threshold autoregressive process. Journal of the American Statistical Association 84: 231- 240.

Tsay R.S., 1998. Testing and modeling multivariate threshold models. Journal of the American Statistical Association 93: 1188-1202.

Villani M., 2001. Bayesian prediction with cointegrated vector autoregressions. International Journal of Forecasting 17: 585-605.

Vrontos D, Dellaportas P, Politis D.N., 2003. Inference for some multivariate ARCH and GARCH models. Journal of Forecasting 22: 427-446.

Warther V.A., 1995. Aggregate mutual fund flows and security returns. Journal of Financial Economics 39: 209-235.

Wermers R., 2000. Mutual fund performance: An empirical decomposition into stock-picking talent, style, transactions costs, and expenses. Journal of Finance 4:

1655-1703.

Zellner A., 1971. An Introduction to Bayesian Inference in Econometrics. New York: Wiley.

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