Download V-FTSE is the volatility index on the FTSE-100.

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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
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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.
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(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).
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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)
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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)
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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:
P1* =
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.
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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:
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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
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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  im(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