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1 EC907 - ECONOMICS OF FINANCIAL MARKETS Note 4 Professor SHERI MARKOSE Optimal Portfolio Selection with Dynamic Regime Switching Weights Abstract In a number of classic papers by Hamilton (1989), Engel and Hamilton (1990), the significance of regime switching (RS) in economic time series was established. The behaviour of equity markets is characterised by bull and bear markets and there is an asymmetry in the mean, volatility and correlation in asset returns during these respective market regimes. Typically when the average market returns are positive there is lower volatility and correlation while during bear market conditions when the average returns are negative, there is higher volatility and correlation. In Ang and Bekaert (2002), it was noted that the asset allocation research appears to have overlooked these implications of RS on asset returns for portfolio optimization. In particular, the asymmetry of volatility and asset correlation behaviour is not captured in the determination of portfolio weights. In the case of diversification, due to the effect of asymmetric correlations, the benefits of diversification may be severely distorted and found to be lacking precisely when it is needed most during major market down turns. Also, in the case of active portfolio management, the optimal portfolio in bear markets would be substantially different from the one in bull markets. Early work in the area has focussed on portfolios composed of international stock indexes. This paper gives a comprehensive analysis of how to dynamically rebalance an equity portfolio with borrowing and lending using optimal regime switching portfolio weights. The results are remarkable in that the RS approach produces substantial improvements on a non-RS static case with the superiority of the former arising from a greater capacity for market timing. Keywords: Regime Switching, Dynamic Portfolio Optimization, Asymmetric Correlation, market Timing 2 Part I: Theory 1Introduction 1.1 Background Traditional static mean-variance portfolio choice model relies on the sample mean and variance-covariance to determine the optimal portfolio weights. For a sample size T, the estimate for mean return is simply the average and the estimate for the variance is likewise the sample variance. To rebalance portfolio weights on say a monthly basis, the sample data and the sample statistics for mean and variance-covariance is updated monthly. How good are sample statistics as estimates for the two moments of returns relevant for the next holding period ? Sample statistics are very crude inputs for determining the optimal portfolio weights. Ceteris paribus, the latter directs more money into those assets which have a higher sample mean and a lower sample variance. Sample statistics are not very responsive to sudden market movements and hence portfolios weights of stocks with high sample means when returns turn negative continue to have these stocks overrepresented in the portfolio. Hence, traditional portfolio optimization framework has been criticised (see, Michaud, 1989) as one which maximizes estimation error. To address this problem, whilst retaining the mean variance quadratic programming framework, there have been two strands of work. The first is to input the portfolio optimization problem with ‘better’ estimates of mean and variance covariance of asset returns and the other is the development of robust optimization which explicitly accounts for errors in point estimates. 1.2 Regime Switching Models A number of asset prices have clear cut periods of up movements and down movements. These are called regimes. For example, the behaviour of equity markets is characterised by bull and bear markets. Likewise, exchange rates are known to have long runs of appreciation and depreciation. There are lengthy periods of high interest rates and then periods of low interest rates. In a number of classic papers by Hamilton (Econometrica, 1989), Engel and Hamilton (American Economic Review,1990), the significance of regime shifts in economic time series was established. What drives regime switches ? It is generally accepted that the business cycle (measured by GDP growth) drives the regimes in the stock market as a whole with the latter represented by all share indexes. In other words, macro business cycle movements impact the stock market as a whole and vice versa rather than any one company sector specific risk characteristics. Hence, the justification that in the case of equity markets that it is the national or international stock indexes that best proxy for the business cycle related regime switches. 1.3 Stylized Facts about asset returns during bull and bear regimes: (i) There exists a stable, less-risky regime with high returns and a volatile, unstable regime producing low returns. That is, in bull markets when market 3 returns are on average positive, there is lower volatility and in bear markets when the market returns are on average negative there is high volatility. There is an inverse relationship between boom markets and stock returns volatility. During a boom, volatility is low and during a bear market when stock prices are falling there is high volatility. So it is now recognized that during a boom when the seeds of the bust are sowed, the low volatility of the stock market masks the growing threat. This is called the paradox of volatility . In fact, at the height of the boom, the low volatility regime was called the Great Moderation by likes of Ben Bernanke. It lulled policy makers into a state of false confidence about the economy. “Paradox of stability” : Stock Index and Volatility Index Paradox of Volatility (Borio and Drehman(2009); Minsky (1982)) V-FTSE is the volatility index on the FTSE-100. VIX is the volatility index on the S&P 500 : Note below as S&P 500 is booming, volatility is low when SP500 is falling as in recent period VIX is spiking at close to 80% in 2008 24 Oct. 4 (ii) In the recent stock market collapse, it was widely noted that portfolios did poorly as in the market downturn all assets fell together substantially. The benefits of diversification seem to be diminished. The feature that asset correlations are less strong in bull markets rather than bear markets is called asymmetric correlation. In the case of international equity return, this phenomenon of asymmetric correlation is a well-known fact1. Longin and Solnik (2001) showed that asymmetric correlation is statistically significant. Ang and Bekaert (2002) pointed out that the asymmetric correlation is not restricted to international portfolios. Recently, Ang and Bekaert (2002, 2004) and Tu (2007) identified the asymmetric correlation phenomenon correspondent to the bull/bear regimes, with higher correlation between assets in bear regime and lower correlation in bull regime. Their works are directed to the portfolio decision based on bull/bear regime and asymmetric correlations and provided inspiring investment insights. They also stated that regime-switching models of Hamilton (1989) do well at replicating the degree of asymmetric correlations in the equity 1See Erb, Harvey and Viskanta (1994), Campbell, Koedijk and Kofman (2002) and Ang and Bekaert (2003). 5 returns data. 1.4 Implications for Regime Switching Weights for Portfolios Note the main difference between RS and non RS analysis is that in the latter, there is no regime sensitivity of the statistics for mean and variance-covariance of asset returns. In particular the asymmetry of volatility and asset correlation behaviour is not captured in the determination of portfolio weights. Asymmetric correlations between assets could have profound effects on investment decisions such as portfolio selection and asset allocation. Firstly, in the case of diversification, it is worthwhile to take into account the effect of asymmetric correlations as the correlations tend to be higher in the bear market than bull market, the benefits of diversification may be severely distorted. In fact when you need the benefits of diversification most, they appear to work the least if the portfolio is insensitive to the fact that correlations between assets are higher during market downturns. Also, in the case of active portfolio management and asset allocation, the optimal portfolio in bear markets would be substantially different with the one in bull markets. Ang and Bekaert (2002) claim that the asset allocation research appears to have overlooked these implications of regime switches on asset returns for portfolio optimization. The work of Markose and Yang (2008) reported here provides an equity portfolio selection model based on Hamilton’s (1989) regime-switching model, which focuses on domestic (UK) market rather than international equities used by Ang and Bekaert (2002). We found the asymmetric correlations exist in the UK equity market and can be utilized to produce regime-dependent portfolios. For ease of understanding, a 3 asset portfolio is constructed with and without Regime Switching weights. The portfolio is rebalanced monthly at the end of each month after a in sample period. The stylized facts for the RS statistics for the FTSE-100 index are given. In a RS model, at the point when the portfolio is being rebalanced, the portfolio manager must determine which regime it is and then form conditional expectations for the mean and variance for the market index and also for each of the assets for the next period. With the stock returns estimated from single factor CAPM models, the asymmetry of correlation in the assets in the portfolio are shown. The cumulative return from a 1GBP invested in a portfolio with non-RS and RS portfolio weights is compared for their out of sample performance. 2 An Equity Market Model with Regime-Switching 2.1 The Single-factor CAPM Market Model Suppose the equity return during a holding period (one month in this case) is related to the return on a market index (such as FTSE100) during the same holding period. Then the equity return can be captured by the CAPM-inspired single-factor market model. rit i i rmt it (1) 6 where ri = return on security i for a given period rm = excess return on market index for a given period = intercept term i i = slope term i i = idiosyncratic standard deviation of security i = random error term from N 0,1, iid According to this model, the total risk of security i, i2 , measured by its variance equals the following: 2 i2 i2 m i2 (2) where m2 denotes the variance of returns on the market index. Thus, the total risk of security i consists of two parts: (1) market (or systematic) risk i2 m2 and (2) specific (or unsystematic) risk i2 . In the first formulation of the model, the alpha and beta coefficients for each asset are not assumed to be sensitive to the regimes, but are given by the in sample OLS regression estimation of asset returns on the excess market returns. 2.2 The Regime-Switching CAPM Model The key change to the CAPM model in Eq. (1) is that the term rmt , i.e. the excess return on market index, is now subject to regime-switching. Suppose the expected return and volatility (as measured by standard deviation) on the market index can have two values, depending on which regime is realized for the market. Then the regime-switching equity market model is given by the following equations: rit i i rmt i it (3) 7 Here, rmt , the return on the stock market index such as the FTSE-100 is assumed to be driven by two regimes: rmt m (st ) m (st ) mt where st (4) denotes the regime variable whose value depends on regime realization (1 or 2 in m (st ) this case) of market index, on market index and ( st ) m denotes the regime-dependent mean of excess return denotes the regime-dependent conditional volatility, measured by standard deviation. The switches between the two regimes are governed by the Markov chain which can be characterized by the following two transition probabilities: P p(st 1 st 1 1) Q p(st 2 st 1 2) Figure 1 (5) 8 The States of Regime-switching • State1: High Return, Low Volatility • State2: Low Return, High Volatility • Governed by Transition Probabilities: P and Q St 1 St 2 St 1 1 P (1-P) St 1 2 (1-Q) Q In this model, the excess return of any security rit is subject to regime switching only through its relation with the market index, that is, its i . Therefore, there is no asset-specific regime in this model. The time-varying means, volatilities and correlations of asset returns are all driven by the regime switches of the market index. Simple Introduction to Parameter Estimation of Hamilton Regime Switching for the Stock Index :The parameters to be estimated are = {m1, m2,m1, m2, P, Q} In the two state regime switching model we assume that the time series for rmt is drawn from two normal distributions with their respective means and variances being driven by the two hidden states st. Then the conditional density function for rmt(st), st =1,2, 1 f(rmt| st, Rmt-1) = 2 2 st 1 (rmt st ) 2 . exp st 2 (6) 9 Here Rmt-1 is the past observation on stock index returns. Statistical inference, especially for the estimation for parameters , rely on the conditional density function based on past observations of the random variable and the likelihood function being the product of the conditional density function for each t. However, as in the above density function the regime states are unobservable and themselves random, we have to factor out the unobservable st from the conditional density function f(rmt| st, Rmt-1) in (6). We specify a conditional likelihood P(st| Rmt-1) and multiply it to f(rmt| st, Rmt-1) to yield the joint density function. As we have assumed that the conditional likelihood of each state P(st| Rmt-1) follows the Markov structure in (5), ie P and Q denoted as st , the likelihood function is L({m1,m2,m1,m2,P,Q}= T st 2 2 2 st t 1 s 1 1 (rmt st ) 2 . exp st 2 (7) The Maximum Likelihood Estimator (denoted by asterisk) of st* is the average of the so called filter probabilities of the state at time t obtained after observing rmt . Start with initial values for ({m1, m2,m1, m2, P, Q}, then in the filter probability function (8) we obtain the number of times the series was in state 1 relative to the number of total number of times in both states : t = 1 f rmt | st 1, Rt 1 1 f (rmt | st 1, Rt 1 ) 2 f (rmt | st 2 , Rt 1 ) (8) The Maximum Likelihood Estimate MLE (denoted by asterisk) for the transition probabilities P is the average of the filter probabilities for state 1 at time t obtained after observing rm at t-1: P1* = 1 T T t 1 (9) 1t The MLE for the mean for regime 1 m1 The MLE for the variance for regime 1 *= 1 T 1t T m1*= t 1 * 1 rmt 1 T 1t (rmt m1*) 2 T t 1 1* (10) (11) 10 Likewise, the conditional likelihood of state 2 is obtained by: t = Q 1 f rmt | st 2, Rt 1 1 f (rmt | st 1, Rt 1 ) 2 f (rmt | st 2 , Rt 1 ) * = 1 T T t 1 (12) 2t The MLE for the mean for regime 2 m2*= The MLE for the variance for regime 2 1 T m2 2t T *= t 1 * rmt (13) 2 1 T 2t (r m 2 *) 2 * mt T t 1 2 (14) We iterate the process in (8), (9), (10), (11) and (12) till the values for 1,2* , m1,2* and m1,2 stabilize. Equations (10,13) and (11,14) for the RS means and variances are simply the weighted sample mean and variance weighted by the ratio of the filtered probability of being in that state in each t with the estimated sample transition probability of that state. 3.1 Equity Portfolio Selection with Regime-switching Now let us investigate the portfolio selection problem with Regime-switching from a portfolio manager’s perspective. According to Markowitz’s approach of portfolio selection, a portfolio manager should view the return associated with portfolios as random variables, whose probability distribution can be described by their moments, two of which are expected mean and standard deviation. Under regime-switching, the expected mean and standard deviation (variance) vary through time because the random variable could be drawn from two different probability distributions associated with two different regimes. If one were to use the unconditional portfolio efficiency frontiers for the 2 regimes and also of that for the non-RS, we have following type of graph. 11 Mean-Variance Efficient Frontiers of Regime-switching Equity Model Regime2 Frontier Regime1 Frontier Unconditional Frontier Expected Return 0.02 0.015 SR=0.3 SR=0.15 0.01 SR=0.06 Risk-free Rate 0.005 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Standard Deviation Figure 2 : Mean-variance efficient frontiers of unconditional regime-switching equity model compared with that for the standard M-V efficiency frontier The dashed curve near the top represents the mean-variance efficient frontier in the normal regime1, while the dashed curve near the bottom represents mean-variance efficient frontier in regime2. The solid curve in the middle is the unconditional frontier implied from regime-switching model, which is obtained by averaging the moments in the two regimes. Of the three frontiers, the one for reigme1 has the best risk-return trade-off. Intuitively, this is because investor takes into account the likelihood of a bear market regime at next time period is small as regimes are persistent. The Sharp Ratio along the capital allocation line (the line emanating from the risk-free rate on the vertical axis tangent to the frontier) is 0.3. In the volatile regime2, the risk-return trade-off worsens substantially and can only realize a Sharp Ratio of 0.06. The investor simply has a very different portfolio in the Regime2, correspondent to the regime-dependent means and covariance. As for the unconditional frontier, not surprisingly, the Sharp Ratio (0.15) here is between those from Regime1 and Regime2. Wikipedia : The Sharpe ratio or Sharpe index or Sharpe measure or reward-to-variability ratio is a measure of the excess return (or risk premium) per unit of deviation in an investment asset or a trading strategy, typically referred to as risk (and is a deviation risk measure), named after William Forsyth Sharpe. Since its revision by the original author in 1994, it is defined as: 12 where R is the asset return, Rf is the return on a benchmark asset, such as the risk free rate of return, E[R − Rf] is the expected value of the excess of the asset return over the benchmark return, and σ is the standard deviation of the excess of the asset return. 3.2 Dynamic updating of Regime Switching Portfolio weights Step 1: Divide Data into In Sample and Out of Sample (for Example Whole sample1986-2006; In sample 1986-1996; out of sample 1996-2006) Load all the in sample data and do OLS Regression of each security with respect to the market excess return and obtain the and coefficients. Step 2: Do Whole Sample and In Sample Statistics for Mean Variance and Covariance of the Stock Returns and the mean and variance for the Market Excess Return Step 3 : Whole sample and in sample data of the stock index, respectively into Regime Switching Estimator and record the in sample and whole sample unconditional parameters ({m1, m2,m1, m2, P, Q} Step 4: Take the filter probabilities at the end of the in sample period and iteratively update at the end of each month to get the inference of which state it is. Then calculate the conditional RS mean, variance and covariance of the CAPM beta based returns of assets using equations in (15) and (16) at each t. Step 5: Use the inputs from Step 4 to the M-V quadratic programming problem at the end of each month over the out of sample period Step 6: Calculate the cumulative return on £1 invested in the RS and non RS portfolios over the out of sample period. Suppose the portfolio manager knows which regime is realized at each point of time, but he/she dose not know which regime will be realized next time point. If the market is currently in regime1, the probability of remaining in regime1 at next time point is P, and the probability of transitioning to regime2 is (1 - P). Similarly, if the market is currently in regime2, the probability of remaining in regime2 at next time point is Q, and the probability of transitioning to regime1 is (1 - Q). With constant transition probability P and Q, he/she would have same expectations every time when regime1 (regime2) is realized. The Markov process and the estimation of conditional probabilities is illustrated in Figure3. Figure 3: The Transition of Market Regimes: Conditional Mean and Variance 13 Time t Time t+1 P Regime 1 Regime Realization (shaded) Regime 1 Q Regime 2 e m (1) : Regime 2 e m (2) : P m (1) (1 P) m (2) Conditional Expected Return Regime 1 1-Q 1-P Regime 2 Time t+2 (1 Q) m (1) Q m (2) (15) m (1) : m (2) : Conditional Expected P[ m (1)] 2 (1 P)[ m (2)] 2 Variance P(1 P)[ m (1) m (2)]2 (1 Q)[ m (1)]2 Q[ m (2)]2 Q(1 Q)[ m (1) m (2)]2 (16) The shaded box denotes the regime which is realized at the time point. In this example, the market is in regime1 at time t, regime2 at time t+1 and regime1 again at time t+2. 3.3 Conditional Expected Return and Variance/Covariance for Securities To implement the steps in portfolio optimization problem, the expected return and variance/covariance of securities are required. They can be derived from the market model expressed in Eq. (3) and (4). Since the market is switching between two regimes, the securities are regime-dependent through their relation with market. Let us denote the regime-dependent expected returns of security i as ri ( st ) , then ri (st ) i im(st ) s t 1,2 (17) 1 1 Let and . The regime-dependent expected mean vector for N N N 14 securities is given by: R(st ) um(st ) st 1,2 Then the conditional expected return vectors for N securities given that either Regime1 or Regime2 is indicated to be the current state: e(1) PR(1) (1 P) R(2) (18) e(2) QR (2) (1 Q) R(1) The variance/covariance matrix has three components. First, the conditional variance of individual assets depends on the asset’s exposure to systematic risk through its beta with respect to the market. Therefore, the differences in systematic risk across the different assets and the correlations are completely driven by the variance of the market. However, because the market variance next time point depends on the realization of the regime, we have two possible variance matrices for the unexpected returns next time period. Second, each asset has an idiosyncratic volatility term V2 unrelated to its systematic exposure (Note V2 is the standard error obtained from the OLS regression of betas) . Therefore, the regime-dependent variance for any security will be: i2 i2 m2 ( st ) i2 (19) 1 Let V , then the regime-dependent covariance matrix for N securities is given by: N ( st ) ( )[ m ( st )] 2 V 2 , st 1,2 (20) It is straightforward to see that this model implies that the correlation given by (1) will be lower that the correlation given by ( 2) . Finally, the actual covariance matrix today takes into account the regime structure, in that it depends on the realization of the current regime and it adds a jump component to the conditional variance matrix, which arises because the conditional means change from one regime to the other. As a consequence, the conditional expected covariance matrix for N securities in regime1 and regime2 can be written as: K (1) P(1) (1 P)(2) P(1 P)[ (1) (2)] 2 15 K (2) (1 Q)(1) Q(2) Q(1 Q)[ R(1) R(2)]2 (21) Now, using Eq. (17), (18), (19), (20) and (21), the portfolio optimization steps can proceed. The quadratic optimisation problem is: N N Min i 1 j 1 i j ij N Subject to R i i 1 i 0 r f ER p Empirical Results: Part II of the Regime Switching Lecture Note STEP 1: Table 1 below provides the estimation results of the market model using the time series data of FTST 100 index excess return from 1986-2005. Regime 1 Regime 2 Transition Probability m(1) m (1) m(2) m(2) P Q Estimates 0.69 4.03 -4.92 9.58 0.9793 0.6703 Std Err 0.33 0.32 4.47 3.04 0.0207 0.2340 Table 1: Estimation results of regime-switching market model on the full-sample, i.e. FTSE 100 from 1986-2005. All the parameters are monthly and reported in percentage except the transition probabilities. The two regimes inferred by the data show significantly different features. Regime 1 is the normal, stable regime, in which the market index is expected to yield 0.69% per month (8.28% per annum), with a volatility of 4.03% per month (13.96% per annum). Regime 2 is 16 the abnormal, volatile regime, in which the market index is expected to yield -4.92% per month (-59.04% per annum), with a volatility of 9.58% per month (33.19% per annum). The transition probabilities P and Q demonstrate the persistence of the two regimes. According to the estimates, Regime 1 is much more persistent than Regime 2. In fact, there is 98% of chance that the market will be in Regime 1 next time period given that it is currently in Regime 1. The persistence of Regime 2 is much lower than Regime 1, with only 67% of chance of staying in Regime 2 next time period given that the process is already in Regime 2 today. The results are generally in line with the stylized facts established in the literature that equity returns are drawn from several different regimes (distributions), in this case two, one being less volatile with high means, the other being more volatile with lower means. Step 2 The three securities we choose are the stocks of Barclays British Petroleum, and BT Group, which are from different major sectors of the FTSE 100 Index. Figure 4.2 below plots the prices of the three asst along with the FTSE 100 Index for the whole sample period from 1986 to 2005 and Table 2 below reports the statistics of these assets. Figure 4: Prices of Three Stocks versus FTSE 100 Index for whole sample 17 18 Sample Observations Whole Sample : Jan. 1986 -Dec. 2008 276 British Petroleum (BP) British Telecom (BT) FTSE -0.067016197 0.00535888 0.073204371 3.764961394 -0.074422979 0.007213523 0.084932464 0.951038296 -0.069030508 0.003112354 0.055788472 4.990296385 -0.958451333 -0.413066478 -1.447136822 0.015269134 1.235309055 0.073927272 0.506546439 -0.004086538 0.91162098 0.057226223 0.48266247 -0.004515555 1.01270332 0.068923948 0.442492544 0.009376072 0.00266363 0.00410899 0.003844719 0.00266363 0.00410899 0.003844719 0.00535888 0.002513598 0.002837287 0.002513598 0.007213523 0.003151891 0.002837287 0.003151891 0.003112354 1 0.37577369 0.499632735 0.711720759 0.37577369 0.499632735 0.711720759 1 0.404282696 0.694739138 0.404282696 1 0.66520113 0.694739138 0.66520113 1 Barclays Descriptive Statistics Mean Variance Std Deviation Excess Kurtosis Skewness OLS Regression Intercept Slope Standard Error Slope R Square Covariance Matrix Barclays British Petroleum (BP) British Telecom (BT) FTSE Correlation Matrix Barclays British Petroleum (BP) British Telecom (BT) FTSE -0.07000487 7 0.009376072 0.096830118 9.621022894 -1.69771866 1 Table 2: Statistical Description of Three Assets As we can see from table 2 that BT shows the lowest average return and high volatility making it unpopular choice for portfolio. Comparatively Barclays has the largest excess return which may be explained by its high beta value of 1.23. Implying than when the average market returns are high the returns from Barclays will be on average 1.23 times higher than the market return. .Further, evidence of asymmetric correlations has been shown for our 3 assets in relation to the FTSE (Table 3). The bear sample (regime 2) has higher correlations compared to the more stable regime1. This is consistent with previous studies, Ang and Bekaert (2002; 2004). 19 Observations: 36 BULL Sample: Jan.2004 -Dec.2006 BEAR Sample: Jan.2001 -Dec.2003 FTSE FTSE Descriptive Statistics Correlation Matrix Barclays British Petroleum (BP) British Telecom (BT) 0.595065 0.805085183 0.609471 0.121986 0.644156137 0.724764658 Table 3: Evidence of Asymmetric Correlation (Bear Sample) Step 4 : Regime Indicator The essence of regime switching portfolio model is the ability of making the inferences about regime and constructing the portfolio accordingly. Therefore, the accuracy of these inferences is crucial to the performance of RS strategy. Let us now investigate this issue first. As introduced in the previous section, the regime ‘switches’ in the model is only on the market, which is proxied by the FTSE 100 Index in this practice. The time-varying nature of the stock return and correlation are captured by their association with the market. Figure 4.6 below depicts the updated inferences of regimes (Regime Indicator) through time versus the ex-post smoothed probabilities for the FTSE 100 index. The former is the regimes that have been used by the portfolio model in the RS strategy and the latter represents the ‘true’ regime the data has been in. 1 Updated Regime Indicator ex-post Smoothed Prob 0.9 0.8 Probability of Regime 1 0.7 0.6 0.5 0.4 0.3 0.2 Wrong Bets 0.1 0 1996 1998 2000 2002 Time 2004 2006 20 Figure 5 : Updated Regime Inference vs ex-post Smoothed Probability of Regime 1 on FTSE 100 Index 1996 – 2005. We can see that the updated Regime Indicator is more volatile than the ex-post smoothed probabilities. In general it gives higher probability of the data being a certain regime. In several cases, it made the ‘wrong bet’ of regime judging by the ‘0.5 rule’. However, in large, the updated Regime Indicator is a reasonable inference of the states of the data series. STEP 5 Cumulated Wealth With Risk-free Borrowing and Lending In addition to the portfolio of risky assets, we can also include the risk-free asset (the Libor) in to the model, which offers the investor the opportunity to freely borrow/lend money at the risk-free rate. In another word, the investor has the choice to investment in cash or lends cash and put it into his/her risky portfolio, all depending on the investor’s portfolio selection strategy and risk aversion. The end-of-period return of a portfolio of risky and risk-free assets is given by: R 0 Rf (1 0) Rp N Rp iRi i 1 N with 1, i i i 1 N and 0 (1 0) i 1, 0 i 1 where 0 is the fraction of risk-free asset in the whole portfolio, Rp is the returns of portfolio is risky assets, Rf is the return of risk-free asset, i is the weight of risky asset i in the risky portfolio and Ri is the return of risky asset i. Note that when 0 0 , the investor is putting non-negative amount of money into cash investment while when 0 0 , the investor is borrowing money at the risk-free and enhance the allocation to the risky portfolio. If we put a constraint of i 0 on the model, then the short-selling of stocks is not permitted. In the practice, the risk-free rate is the one-month Libor available at time t. Figure 4.13 below show the accumulated wealth of the three strategies includes the risk-free asset but without short-selling of risky asset. 21 4 3.5 RS Strategy non-RS Strategy Mkt Cap Strategy ex-post Smoothed Prob of Regime 1 Accumulated Portfolio Wealth 3 2.5 2 1.5 1 0.5 0 1996 1998 2000 2002 2004 2006 Time Figure 6: Accumulated wealth of portfolio including the risk-free asset without short-selling The R-S dependent strategy outperforms the other two strategies for the most part of the out-of-sample period, especially in the last 5 years. The end-of-period wealth is 3.56 GBP, 1.80 GBP and 1.99 GBP for the RS strategy, non-RS strategy and Market Cap strategy, respectively. The Sharp Ratios are 0.175, 0.036 and 0.051 for the RS strategy, non-RS strategy and Market Cap strategy, respectively. Comparing with the case of no risk-free asset, the RS strategy improves the both end-of period wealth (3.05 GBP to 3.56 GBP) and the Sharp Ratio (0.11 to 0.175). The improvement comes from the market-timing ability of the RS strategy. References Ang, A., G. Bekaert, 2002, ‘International asset allocation with regime switching’, Review of Financial Studies, Vol.15, No.4, pp.1137-1187 Ang, A., G. Bekaert, 2004, ‘How do regimes affect asset allocation’, Financial Analysts Journal, 60, pp. 86-99 Campbell, R., K. Koedijk, and P. Kofman, 2002, ‘Increased Correlation in Bear Market’, Financial Analysts Journal, Jan-Feb: 87-94 Erb, C.B., C.R. Harvey, and T.E. Viskanta, 1994, ‘Forecasting International Equity Correlations’, Financial Analysts Journal, Nov-Dec: 32-45 22 Gibbons, M. R., 1982, ‘Multivariate tests of financial models’, Journal of Financial Economics, Vol.10, pp.3-27 Hamilton, J.D, (1989), “A New Approach to the Economic Analysis of Non Stationary Time Series and Business Cycle”, Econometrica, Vol. 57, No. 2, pp. 357-384. Huang, H. C., 2000, ‘Test of regimes-switching CAPM’, Applied Financial Economics, Vol.10, pp. 573-578 Huang, H. C., 2003, ‘Test of regimes-switching CAPM under price limits’, International Review of Economics and Finance, 12(2003) 305-326 Longin, F., and B. Solnik, 2001, ‘Correlation Structure of International Equity Market During Extremely Volatile Periods’, Journal of Finance, vol. 56, no. 2 (April): 649-676 Tu, J., 2007, ‘Is Regime Switching in Stock Returns Important in Asset Allocations?’, http://ssm.com/abstract=1028445