# 1 INTRODUCTION

According to the efficient market hypothesis, particularly the semi-strong form efficiency, market prices reflect all publicly available information including information contained in historical data and information contained in current financial statements. As such, any attempts to generate abnormal returns by using fundamental analysis technique such as examining financial statements, industry and lifecycles will more often than not prove to be futile. Notwithstanding, in many situations where institutions seek funding for their activities, there is a cyclical process where each cycle begins by funders investing in the institutions and ends by evolution of their performance. The funding handed out in the next cycle is based upon how well each institution did in the evaluation.

The goal of this project will be to investigate the possible consequences of errors in the evaluation process by investigating the relationship between the past and present performance of the UK real estate sector over the period between 1981 and 1996. In particular, the data will be analysed in a bid to identify variations in investment performance classification. The main mathematical tool will be the concept of stationery states in Markov chains.

A Markov process is a stochastic process where only the present value of all the variables is important for predicting the future, which makes the past history of the variable irrelevant (Hull, 2009). Stock prices are considered to follow such a process because future stock prices are influenced by their prices in the past. The only relevant information in a stock that can be used to predict its future price is its current price. Consequently, the prediction of future stock prices is uncertain and follows probability distributions.

The Markov property suggests that the probability distributions of future stock prices are not influenced by the price history of the stock and is therefore consistent with the weak form of market efficiency. The latter states that the current price of a stock summarises all the information obtained from historical prices, which renders is impossible to beat the market by studying historical stock prices. The weak form market efficiency is supported by the fact that there is a very large number of traders in the market, who always trying to take advantage of market movements. As such, at any given time there will always be many traders contemplating to take advantage of historical price patterns, which prevents such patterns from reoccurring because they are cancelled by market competition.

Despite the difficulty associated with using past information about the behaviour of stock and other asset prices to predict their future performance, most investors are still compelled to invest in institutions that that have performed well in the past and withdraw their investment from those that have performed poorly. However, in their defence, there is actual empirical evidence supporting the fact that security returns may be predictable over short horizons. Additionally, real estate returns have been found to be more persistent compared to bonds and stocks, which are more liquid and traded more efficient markets.

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# 2 LITERATURE REVIEW

Several authors have over the years explored the persistence of the performance of various investment projects including real estate properties, real estate investment trusts (REITS) and mutual funds. Young & Graff (1996) explored the serial dependence of total annual returns of assets in the National Council of Real Estate investment Fiduciaries (NCREIF) database which revealed a statistically significant persistence in real estate assets in the first and fourth quartiles of disaggregated data between 1978 and 1994. More precisely, they observed that assets which exhibited superior performance generally continued to exhibit superior performance while those that exhibited inferior performance generally continued to exhibit inferior performance. To the contrary, there was virtually no evidence to support serial dependence in the second or third quartiles, neither combined nor taken separately. The serial independence among real estate returns was empirically rejected thereby casting doubt on the conclusions of research studies founded on models that incorporate the assumption of serial independence.

Graff & Young (1997) also investigated the annual and monthly real estate investment trust (REIT) returns, which both exhibited statistically significant serial persistence. There was however a qualitative difference between the two types of persistence behaviour. On the other hand, quarterly REIT returns did not exhibit any serial persistence, which strongly implied that linear multifactor market models cannot suffice to describe REIT investment behaviour. Annual REIT returns failed to reflect corresponding persistence behaviour in underlying real estate returns precisely when the REITs are large enough to attract institutional investor interest. Lack of persistence was occasioned by the fact that institutional investors alternated between of large-capitalization REITs in ways that had a negative impact investment returns.

Myer, et al. (1997) analysed the performance of non-securitized real estate investments in relation to two phenomena – the impact of asset management fees and the persistence of the returns. This was achieved by evaluating the cumulative returns and Jensen’s alpha are used for 72 CREFs using data from the Townsend Real Estate Universe on CREFs. These returns were benchmarked against the NCREIF Index. Results of the analysis indicated that asset management fees do not have a significant influence on performance rankings but they are instead influenced by the choice of a risk-adjusted performance measure or a non-risk-adjusted performance measure. The results therefore implied that practicing real estate professionals should not dwell on ranking funds based on net or gross returns because the results will more often than not be quite similar.

Graff, et al. (1999) investigated the serial persistence of total annual returns for all the properties in the Property Council of Australia database and illustrated that serial persistence was statistically significant in all quartiles of disaggregated returns between 1985 and 1997. As aforementioned, this implied that performance in a particular quartile was generally followed by continued performance in the same quartile. However, when the data was grouped by property type, there were persistence differences. The empirical evidence of serial persistence among real estate returns in the Property Council of Australia database again cast doubts on the conclusions of research based upon models that incorporate the assumption of serial independence.

In a research study that sought to evaluate the presence of both price continuation and price reversals in international real estate securities, Stevenson (2002) identified evidence of performance persistence in international markets over short and medium term horizons and relatively less compelling evidence on price reversals. Variance Ratio and Augmented Dickey-Fuller tests were utilized to carry out empirical analysis tests for mean reversion. In both cases, there was no consistent evidence of mean reversion in international real estate securities. The author further applied portfolio switching tests, which reveal some evidence of performance reversals. It was however established that while under-performing markets has the capacity to outperform over longer horizons, evidence of their superior performance was not statistically significant.

Lin & Yung (2004) analysed the performance of real estate mutual funds between 1993 and 2001. The results of this study suggested that on average, real estate mutual funds do not provide positive abnormal performance. It was established the performance of real estate mutual funds is influenced a great deal by the performance of the real estate sector as a whole. The influence of risk factors such as book-to-market ratio, size, and market momentum became immaterial after the inclusion of the real estate market index in the evaluation model. The results also showed that fund performance only persists in the short run. Additionally, it was established that risk-adjusted real estate fund returns were influenced by fund size, but the was no apparent correlation between these returns and the expense ratio, management tenure and turnover.

Devaney, et al., (2007) examined the individual level property returns to determine whether there was evidence of persistence in performance, i.e. if the probability that well (or poorly) performing properties would continue to perform well (or poorly) in subsequent periods was greater than expected. A similar methodology to the one that had originally been adopted by Young & Graff (1996, 1997, 1999) was applied, facilitating a direct comparison of the results that were obtained with those for the US and Australian markets. However, the authors used a relatively larger database covering all commercial property data relating to the UK available in the Investment Property Databank (IPD) ranging from 1981 to 2002, which entailed at least 216,758 individual property returns. While the findings of this research were more or less similar to those obtained by Young & Graff (1996, 1997, 1999) in the US and Australia – greater persistence in the extreme first and fourth quartiles – there was also evidence of persistence in the moderate second and third quartiles, a notable deviation from previous studies. Similarly, patterns across property type, time, location, and holding period were remarkably similar. These findings suggested that performance persistence is not a feature unique to particular markets, but instead may characterize most advanced real estate investment markets. Apart from being a geographical extension of previous research this paper also explored possible reasons for such persistence, consideration of which led the authors to conclude that behaviours of institutional-grade commercial real estate investment managers were deeply rooted and persistent, and perhaps influenced by agency effects.

Apart from real estate related assets, several authors have in the recent past made deliberate attempts to examine performance persistence in various financial institutions such as mutual funds. Eser, (2007) pointed that this line of research has produced several stylized facts. For instance, it has been established that actively managed funds rarely outperform passively managed diversified portfolios. Additionally, both individual and institutional investors have found it increasingly difficult to identify, ex ante, funds that have the capacity to outperform broad market indicators after adjusting for risk. As such, most investors continue to rely on past returns as predictors of future performance. This strategy compensates winners handsomely in terms of net fund flows and does not penalize losers as much, except for the extreme worst performers.

In line with previous research findings, Eser (2007) concurred that prior performance has a significant influence on future performance. The author further pointed out that most findings are inclined towards the fact that poor performance persists to a much larger degree than superior performance and that persistence of superior performance can only be explained entirely by taking stock market momentum into account. However, even after taking the stock market momentum into account, persistence of poor performance is left unexplained.

Despite ackowledging the contributions made by other scholars, Eser (2007) identified two important shortcomings in literature. First, the author argued that there was a possibility that the documented persistence could have been due to calendar-related distortions of fund returns as opposed to the individual competencies of the fund manager. Secondly, he added that Carhart’s passive momentum conjecture had not been formally tested in persistence literature. Apart from that, the Eser (2007) also explored the possibility that Carhart’s 4-factor model of performance attribution might not adequately explain the returns of prior performance-ranked portfolios of mutual funds.

The author utilized performance attribution models that employed in previous literature, to illustrate that once the calendar year-end noise was masked, measures of persistence were strikingly smaller, regardless of the period under investigation. He further argued that the variation in persistence results induced by varying the timing of portfolio formation is robust to weighting schemes, survival requirements, fund size, and sample periods. We also find that there is very little persistence in performance of “robot” funds that are formed by randomly selecting stocks. We also find that using a longer-term momentum factor results in a much better performance attribution model compared to using Carhart’s short-term momentum factor. Once the longer-term momentum factor replaces the short-term momentum factor, persistence infund performance disappear even when the portfolios are formed at the end of the calendar years.

(Liow, 2009) While the long memory property is examined in the literature for the US REIT returns, this paper extends the analysis to international securitized real estate markets with the hope of finding answers or confirming prior stock market evidence regarding the presence (or absence) of long memory volatilities for 40 weekly real estate indices (original and hedged). Using a battery of five econometric tests on three alternative risk measures; weekly observed absolute and squared mean deviations and conditional variances, we find statistically significant evidence of long memory in the volatility structure of most securitized real estate markets studied. Volatilitypersistence is particularly strong in Asia, but is not consistent throughout the period of study.

(Vasques, et al., 2009) Real Estate Investment Funds (REIFs) inPortugal are a recent and fast growing industry. Despite the scarcity of research addressing this type of investment vehicle, there are strong arguments leading to the assumption that the present fund regulatory framework conditions the determination of their returns, influencing distribution features and inducing predictability. This study develops a detailed persistence analysis using contingency tables to investigate the issue of predictability. Relevant and robust evidence of both short- and long-term performance persistence within the overall property fund industry and for the restricted universe of open-ended funds is found.

(Bond & Mitchell, 2010) This paper investigates whether fund managers investing in the direct real estate market can systematically and persistently deliver superior risk-adjusted returns. The research that has been published has typically focused on the performance of managers trading public real estate securities. Our study draws on a unique data set of commercial real estate funds collated by the Investment Property Databank (IPD) in the United Kingdom, covering up to 280 funds over the period 1981 to 2006. The widespread finding is that very few managers appear to be able to generate excess risk-adjusted returns. Furthermore, there is little evidence of performance persistence in either fund returnsor risk-adjusted fund returns.

In this paper we investigate whether fund managers investingin the direct real estate market can consistently maintain their performance. The question of whether the performance of fund managers persists over time has been the focus of a long line of research in financial economics. Surprisingly, despite its importance to property investors and fund managers, and the widely held view that real estate markets are “inefficient”, there has been comparatively little research on the extent to whichreal estate fund managers can systematically and persistently deliver superior risk-adjusted returns. The research that has been published has tended to focus on the performance of managers trading public real estate securities. Our study draws on a unique data set of commercial real estate funds collated by the Investment Property Databank (IPD) in the United Kingdom. The widespread finding is that very few managers appear to be able to maintain consistency in their performance rankings.

(NewsRX, 2010) This study examines thereturn patterns of hotel real estate stocks in the U.S. during the period from 1990 to 2007.We find that the magnitude andpersistence of future mean returns of hotel real estate stocks can be predicted based on past returns, past earnings surprise, trading volume, firm size, and holding period.

(Benos & Jochec, 2011) Using daily return data from 448 actively managed mutual funds over a recent 9-year period, we look for persistence, over two consecutive quarters, in the ability of funds to select individual stocks and time the market. That is, we decompose overall fund performance into excess returns resulting from stock selection and timing abilities and we separately test for persistence in each ability. We find persistence in the ability to time the market only among well performing funds and in the ability to select stocks only among the very best and worst performers. The existing literature patterns appear only when funds are ranked by their overall performance, which includes stock selection, market timing and fees. With respect to overall performance, there ispersistence among most poorly performing and only the top well performing funds. Furthermore, the profitability of a winner-picking strategy depends on the rebalancing frequency and potentially the size of the investment. Small investors cannot profit, whereas large investors can take advantage of the class-A share fee structure and realize positive abnormal returns by annually rebalancing their portfolios.

(Sing, et al., 2013) Extreme shocks if occur will have significant and permanent impact on the risk premiums of the stock markets. Modeling these events in a conditional variance framework assuming that the stock market will mean-revert in a short time could produce spurious results. Using the Markov-switching autoregressive conditional heteroskedasticity (MS-GARCH) model to filter out the high volatility states from the low and medium volatility states, we found that the volatility persistence (“large news”) increases the returns of the equity real estate investment trust (EREIT). However, when the volatility persistence is interacted with negative shocks, it cause the EREIT returns to decline. The negative volatility persistence effects fit the story of inter-temporal asset substitution, which explain why risk-averse REIT investors substitute risky REIT assets by risk-less assets inperiods of prolong negative shocks.

(Dawe, et al., 2014) evaluates the performance persistence of equity and blended mutual funds in Kenya for the period 2006 to 2009. The objective of the study was to establish persistence of funds’ performance. The target population was seven mutual funds for which net asset values were available over the period from 1st January 2006 to 31st December 2009. The data was collected from the funds database and annual reports available in the business daily newspapers and in some cases from fund managers’ themselves. The data included mutual funds dailyreturns and annual reports for the period 2005 to 2009. The data was used to calculate the performance persistence of mutual funds in Kenya. Performance persistence of mutual funds was analyzed using regression equation developed by Grinblatt and Titman (1993). The general finding was that for both equity and blended fund, there was evidence of performance differences which tend to persist over time. This implies that there is significant performance persistence over the research period and therefore investors can successfully use the measures of past performance as a decision tool for fund selection.

(Schindler, 2014) This paper extends the analysis of predictability and persistenceof inflation-adjusted house price movements in the UK housing market both on a regional level across 13 regions and on a nationwide level. Applying a univariate time series approach, the results from the quarterly transaction-based Nationwide Building Society indices from 1974 to 2009 provide empirical evidence for a high persistence of house price movements. In addition to conducting parametric and non-parametric tests, we provide technical trading strategies as a robustness check to compare predictability across markets and to test whether or not the detected persistence can also be used for detecting turning points in the market. The empirical findings from the technical trading strategies support the results from the statistical tests. Moving average-based trading strategies perform extremely wellin the southern regions, while trading strategies are less profitable for the northern regions and Wales. Thus, from an investors’ perspective, there are excess real returns from moving average-based strategies compared to a buy-and-hold strategy for most regional markets. From a household perspective, the findings support the importance of derivative markets where households could hedge their risk exposure from being homeowner.

# 3 METHODOLOGY

## 3.1 Markov Chains

The following conventions will adopted for purposes of this research: random variables will be denoted by capital letters and, while realizations of random variables will be denoted by corresponding lowercase letters and. A stochastic process will be a sequence of random variables, say and,, while a sample path will be a sequence of realizations and . Roughly speaking, a stochastic process has the Markov property if the probability distribution

for any. A Markov chain is a stochastic process with this property

and takes values in a finite set. A Markov chain is characterized by a

triple of three objects: a state space identified with an *n*-vector *x*, an *n*-by-*n*

transition matrix *P*, and an initial distribution, an *n*-vector.

Let

Then the transition matrix has elements with the interpretation

So, if a row *i *is fixed then the elements in each of the *j *columns give the conditional probabilities of transiting from state to. In order for these to be well-defined probabilities, the following is required

For example, ifand

then the conditional probability of moving from state to is while the conditional probability of moving from state to is only . It should be noted that the diagonal elements of *P* give a sense of the persistence of the chain, because the elements give the probability of staying in a given state.

Similarly the initial distribution has the elements with the interpretation

and in order for these to be well-defined probabilities, the following is required

## 3.2 Higher order transitions

The main advantage of the Markov chain model is the ease with which it can be manipulated using ordinary linear algebra. For example,

whereis the-thelement of the matrixMore generally,The Markov Chain gives a law of motion for probability distributions over a finite set of possible state values. The sequence of unconditional probability distributions is where eachis an *n*-vector. Let be a vector whose elements have the interpretation

Then the sequence of probability distributions is computed using

where the prime denotes the transpose. In short, probability distributions evolve according to the linear homogeneous system of difference equations

The conditional and unconditional moments of the Markov chain can be calculated using similar reasoning. For example,

while conditional expectations are

a function with typical element on the state . Then

## 3.3 Stationary Distribution

A steady state or stationary probability distribution is a vector such that

or, equivalently

is obtained by solving this matrix equation. Clearly, for a non-trivial, detis needed, which will always be satisfied for the problems in this research. If *P* is regular, i.e., for each *i, j*, then the non-trivial be unique. This will however not be proven in this paper

### 3.3.1 Computing Stationary Distributions By Hand

Consider a two-state Markov chain with transition matrix

for. Then this Markov chain has the unique invariant distribution which can be solved for as follows

or

Carrying out the calculations, it is found that

The se two equations only provide one piece of information, particularly

But it is also known that the elements must satisfy

So these two equations can be solved in two unknowns to get

### 3.3.2 Computing Stationary Distributions in MATLAB

On method of computing stationary distribution using a computer is by using brute

force. If for instance *P* is the transition matrix, can simply be computed for some large T.

Then a matrix with identical rows that are equal to the chain’s stationary distribution will be obtained. For example, if

Then,

and the stationary distribution is 0, which is what would be obtained from the “by-hand” method. This is not very efficient, and as such, the best approach is to use MATLAB to compute the eigenvalues and eigenvectors of P.

In order to appreciate why eigenvalues and eigenvectors are used, it is important to understand the equation that describes the steady state

It should be noted that solving this equation for amounts to solving for the eigenvector of corresponding to an eigenvalue equal to 1. We can be sure that will always have an eigenvalue equal to 1 because of the property that, and. We can also be sure that it will have only one eigenvalue equal to 1, so that the eigenvector corresponding to it will be unique, provided is regular. These claims will also not be proven in this paper. The requirement that is a normalization of the

eigenvector.

It should be noted that the example above, P does not satisfy the regularity property. So the convergence is obtained because the regularity property is only sufficient for uniqueness. In MATLAB, the first step would be to compute matrix of eigenvalues and eigenvectors of

This yields a matrix of eigenvectors V and a diagonal matrix D whose entries are the eigenvalues of *P*.

Secondly, since P is a transition matrix, one of the eigenvalues is 1, the column of V associated with the eigenvalue 1 is picked. With the matrix P given above, this will be the second column, which translates to

Finally, the eigenvector is normalized to sum to one, that is

## 3.4 Simulating Markov Chains in MATLAB

While simulating Markov Chains in MATLAB an *n*-state Markov chain ) for time periods, will have to be simulated. Bold **x **is used to distinguish the vector of possible state values from sample realizations from the chain. Iterating on the Markov chain will produce a sample path where for each . In the exposition below, it is assumed that n = 2 for simplicity so that the transition matrix can be written

The first step is to set the values for each). Secondly, the initial state will be determined. This will be achieved by drawing a random variable from a uniform distribution on [0, 1]. That can be called realization. In MATLAB. This can be done with the rand() command. If the number, set . Otherwise, set .

Finally, vector of length *T* is drawn of independent random variables from a uniform distribution on [0, 1]. A typical realization will be called. Again, in MATLAB, this can be done with the command rand (T,1). Now the current state is it is the checked if . If so, the state transits to with. Otherwise, the state remains at *i* and . Iterating in this manner builds up an entire simulation.

## 3.5 Application of Stationary Markov Chains in Property Valuation

According to (Lee & Ward, 2001), the movement of properties or funds between quartiles is characterized by a Markov process, particularly the Markov Chain model, which is perhaps the most suitable model for describing and predicting property performance rankings over time. In this regard, the Markov process is based on the assumption that “if any population of properties can be classified into various groups or ‘states’, movements of properties between states

over time can be regarded as a stochastic process. ” (Lee & Ward, 2001, p. 283). As such, given a set of states (), it is possible to estimate the probabilities () of properties moving from to.

The Markov process is particularly based on the assumption that the probabilities at that any state will occur in a time sequence at t relies on the state of the system at and independent of all other states. Consequently, the transition between the state of the system at and the state of the system at can be described mathematically as set of probabilities arrayed in matrix form. The quartile distribution of properties of a given year say 1990 is a function of their distribution during the previous year, 1989, modified by a change component that covers the intervening period.

The Markov chain conforms to a similar set of assumptions only it also characterised by the stationarity assumption, which stipulates that any state S occurring in a particular time t in a sequence t is does not depend on its position in that sequence. This implies that the matrix of transitional probabilities does not change between periods. For this reason for a Markov process to qualify as a Markov chain it should be tested for stationarity.

To test for stationarity, these probabilities of movements are expressed in the form of s square transition matrix as illustrated below.

The value of represents the probability that a property in the second quartile ranking at a given time to will remain the same quartile ranking at. On the other hand, represents the likelihood that a property in the first quartile ranking at time period will move to the second quartile in period . In essence, probabilities to the right of the leading diagonal indicate the probability of a property moving to a lower quartile ranking while probabilities to the left indicate the likelihood of moving to a higher quartile. The test for the stationarity assumption is based on a study by Anderson & Goodman (1957) in which they illustrated that the maximum likelihood of estimates of stationary transition probabilities are:

Where is the transition probability that the property will move from state to state in time period estimated by calculating the number of movements () of properties from state into state in time period and comparing this with its quartile rank at time period . Accordingly, change in property ranking is considered as one movement. The movements between quartile ranks and are then summed up, aggregated and divided by the total sum of movements across all the quartile ranks. Given the estimates, , the null hypothesis can be tested to determine whether the transition probabilities are stationary. To achieve this, the following test statistics is used.

This statistic is distributed as Chi-Square statistic with degrees of freedom. To carry out this test the sample set have to be divided in to sub-periods. It should however be noted that the number of subdivisions is arrived at when there is suitable trade-off between the number of observations in each subdivision and the fitness of the test. For instance, if each period is used as a subsample there will be several estimates, but the confidence level of the estimated transition matrices will be very minimal. A smaller number of sub periods will on the other hand generally translate to greater confidence in the resultant transition matrices, the stability test will be less sensitive. For this research, the data will divided into four sub periods in order to test for stability.

After the stability of the data sets has been confirmed, the next step will be to test for persistence. Without any persistence or pattern from one state to another, the probability of a property staying in a specific states would not be any larger than its probability of changing into another state, that is, . On the other end of the spectrum, persistence would imply that the elements in the leading diagonal are greater than the off diagonal elements. The Chi-squared test can be used to test this assumption in which case, the null hypothesis will state that there is not persistence and that the elements in the leading diagonal are not any larger than the off diagonal elements.

## 3.6 Data

Due to time and resource constraints, the data for this research was obtained from a previous study by Lee & Ward (2001), which sought to investigate the persistence of real estate returns in the UK. The data for the study was mainly derived from the *Local Markets Report, *which contained a total of 13,721 properties with an aggregate value of £65.4 billion by the end of 1997. The sample data consist of the total returns on properties in three different sectors, Retail, Office and Industrial at a number of locations in the UK over the period between 1981 and 1996, which yields a total of 392 asset possibilities. For confidentiality purposes, data for locations containing fewer than four properties in any year was not published.

Year | All | Retail | Office | Industrial | London | South-East | Rest of UK |

1981 | 8,990 | 4,572 | 2,746 | 1,672 | 2,930 | 2,247 | 3,813 |

1982 | 9,953 | 4,993 | 3,047 | 1,913 | 3,199 | 2,531 | 4,223 |

1983 | 9,958 | 4,909 | 3,102 | 1,947 | 3,119 | 2,581 | 4,258 |

1984 | 10,173 | 5,036 | 3,167 | 1,970 | 3,087 | 2,697 | 4,389 |

1985 | 10,307 | 5,168 | 3,221 | 1,918 | 3,029 | 2,811 | 4,467 |

1986 | 10,529 | 5,337 | 3,303 | 1,889 | 2,990 | 2,980 | 4,559 |

1987 | 10,130 | 5,319 | 3,117 | 1,694 | 2,794 | 2,901 | 4,435 |

1988 | 9,837 | 5,339 | 2,949 | 1,549 | 2,661 | 2,952 | 4,224 |

1989 | 9,967 | 5,465 | 2,971 | 1,531 | 2,643 | 3,106 | 4,218 |

1990 | 10,328 | 5,591 | 3,108 | 1,629 | 2,641 | 3,286 | 4,401 |

1991 | 10,652 | 5,680 | 3,268 | 1,704 | 2,627 | 3,482 | 4,543 |

1992 | 10,955 | 5,777 | 3,318 | 1,860 | 2,641 | 3,640 | 4,674 |

1993 | 10,623 | 5,578 | 3,218 | 1,827 | 2,539 | 3,513 | 4,571 |

1994 | 10,383 | 5,503 | 3,099 | 1,781 | 2,469 | 3,353 | 4,561 |

1995 | 11,393 | 6,093 | 3,337 | 1,963 | 2,515 | 3,697 | 5,181 |

1996 | 10960 | 5938 | 3148 | 1874 | 2363 | 3537 | 5060 |

Total | 165138 | 86298 | 50119 | 28721 | 44247 | 49314 | 71577 |

## 3.7 Estimation of the transitional matrix

In order to estimate the transition matrix, data describing the movements of each individual property in the database over time was required. Using this data, the transition probabilities were estimated by obtaining the average trend in these movements over time. Different methods were used to estimate the transition matrices in a bid to determine whether the results for the overall dataset were consistent across time, sectors, property types, size and risk, and the presence of any systematic differences associated across these subgroups which were not reflected in total returns.

First, the 16 years of annual total returns data for 392 locations were partitioned into four non-overlapping four/one-year intervals to determine whether the performance relationship was consistent over time or period specific. Secondly, the overall data set was partitioned into two volatility based groups (High and low) to find out whether volatility had any significant influence on the returns, which would essentially imply that some return/risk trade-off exists. Finally, the data set was examined to determine whether the performance relationship was consistent across sectors, size and regions.

## 3.8 The overall transition matrix

The operation of a Markov chain model for the performance of the properties relies on the definition of a set of mutually exclusive and comprehensive states, which are perceived to be the quartile rankings of the return performance of the properties. Two approaches to classifying the quartiles were adopted. Consistent with Young and Graff (1997), the first approach was simply to rank the properties in each and every year. This approach, however, has a major shortcoming when used in a Markov chain analysis in which there are no entrants or leavers, since it always implies that the long-run performance will be consistent with non-persistent performance, that is, the probability of being in any quartile ranking must be 25%. In order to overcome this problem and to provide a stronger test of persistence in performance, another ranking procedure was adopted. In this approach, the returns in any one year were expressed as the deviation from the average of the sample of properties in that year. The resultant values in any one year could then be considered as abnormal or excess returns for that year. The excess returns for each year were then combined and the total 6,272 (392×16) data points placed into quartile ranks. The subsequent data was then reassigned to the individual years. Using this approach, it was possible to obtain more data in some quartile ranks and less data in others in any one year if a property or group of properties displayed persistence in performance over the total 16-year period. The rationale for this approach is motivated by the fact that the variation in performance across real estate portfolios within periods is unusually high (Morrell, 1997). The transformation used in this study facilitated the modelling of persistence in superior/inferior performance that would be achieved if the portfolios were bought and held for successive periods.

The values of the transition matrix, which is the basis of the Markov chain model were derived based on the movements of the 392 ‘properties’, between the quartile rankings in each and every year. This was done for the entire 16 years of data and for the four sub-periods, in order to test for stationarity. The transition matrix is shown in Table 1 below for the entire period.

Table 1: Transition matrix for the whole period (Lee & Ward, 2001)

From/to | First quartile | Second quartile | Third quartile | Fourth quartile |

First quartile | 0.444 | 0.284 | 0.180 | 0.093 |

Second quartile | 0.243 | 0.347 | 0.261 | 0.148 |

Third quartile | 0.175 | 0.252 | 0.299 | 0.274 |

Fourth quartile | 0.128 | 0.144 | 0.264 | 0.463 |

After testing the Markov system for stationarity, the Chi-squared statistic was found to be 0.572, which is insignificantly different from zero at the 99.99% level. As a consequence the null hypothesis of stationarity was accepted.

The principal diagonal contains fairly large coefficients, which would indicate stability of performance ranking, as suggested by Young & Graff (1997). For example there is a 44% chance that a ‘property’ that was ranked in the first quartile in time period *t *will remain in that quartile in the next period. Similarly, a property in the lowest quartile rank in one period has a 46% chance of remaining in that rank in the following period. The corresponding figures for the second and third quartile are 35% and 30% respectively. These probabilities indicate that property in the highest and lowest rankings have a high probability of staying where they are. The test of persistence confirmed this with a Chi squared value of 864.41. Consequently, the null of equality was rejected. However, the existence of a high degree of persistence in total return performance rankings between different time periods cannot be taken as an indicator of market inefficiency. As discussed above, such persistence in return performance could be the result of appraisal or valuation smoothing on the part of valuers. In addition to which, any persistence in performance is difficult if not impossible to exploit by real estate fund managers owing to the high search and transaction costs exhibited in direct real estate markets.

The off-diagonal elements of the transition matrix also provide some worthwhile information not readily available from other types of estimations. For instance, the coefficients in row 1 of Table 2 indicate that approximately 28% of properties ranked in the first quartile in time period *t *will enter the second quartile. Similarly, less than 18% of the properties in the third performance rank in period *t *will move up to the first rank in. Upper off-diagonal probabilities of the matrix are typically higher than the lower off-diagonal probabilities, although only marginally. This suggests that there is more likelihood of performance for the best performing assets to decline more rapidly than that a property to improve its performance.

Collectively, the results in the transition matrix above indicate that there is a 73% chance that a property in the top quartile in one year will achieve above-average performance in the next year. On the other hand, a property in the lowest quartile has a 73% chance of exhibiting below-average returns in the next period. In contrast, properties that appeared in the second quartile in one period have only a 59% chance of achieving above-average returns in the next year. Similarly, for properties in the third tier in one year have a 57% chance of exhibiting below-average returns in the next period.

The evidence suggests two important rules-of-thumb for property fund managers who wish to maximise their performance. First, they should avoid properties with below-average performance. Secondly, they should invest in properties in the upper quartile of performance in one year as they have a higher-than-average chance of achieving above-average returns next year. In other words, a fund manager would be advised to stay with the best and avoid the worst.

## 3.9 Sub-division transition matrices

The data was subsequently divided up in a number of ways to test whether the results were consistent across sectors, regions, size and volatility. For example, Brown *et. al*. (1992), Hendricks *et al*. (1993) and Malkiel (1995) contend that, for mutual funds, persistence is inversely related to total return volatility. Under this interpretation, high-risk mutual funds that survive represent instances of mutual fund managers taking a high-risk strategy and succeeding, whereas high-risk funds that took a high-risk strategy and failed, no longer exist. In this context, the remaining mutual funds would be biased towards persistence in performance through time. In other words, investors who purchase the best investments on the basis of past performance would in fact be buying volatility.

In an attempt to test for this potential impact, the real estate assets are ranked by their variance of returns over the 16-year period, and then partitioned into either a high-risk or low-risk group, respectively. In other words, the properties with a variance above the median are categorised as high-risk assets, while median and below are categorised as low-risk assets. In a similar fashion, the properties were classified into large and small local markets, based on their average market value in 1981. Finally, the data was sub-divided into the three sectors, Retail, Office and Industrial and into the four regions, London, the South East, the South West and the North. The results of the analysis suggest that consistency in performance

High/low-risk matrices

*Table **2**: Low-risk transition matrix (Lee & Ward, 2001)*

0.4261 | 0.2810 | 0.1784 | 0.1145 |

0.2486 | 0.3076 | 0.2795 | 0.1643 |

0.1867 | 0.2545 | 0.2835 | 0.2752 |

0.1180 | 0.1790 | 0.2878 | 0.4151 |

Stationary test 1.67 and Persistence test 370.38.

Table 3: High-risk transition matrix (Lee & Ward, 2001)

0.4368 | 0.2855 | 0.1750 | 0.1026 |

0.2525 | 0.3529 | 0.2468 | 0.1478 |

0.1964 | 0.2437 | 0.2939 | 0.2660 |

0.1137 | 0.1503 | 0.2758 | 0.4601 |

Stationary test 3.31 and Persistence test 490.82.

Large value/small value matrices

Table 4: Large value towns transition matrix (Lee & Ward, 2001)

0.4280 | 0.2773 | 0.1773 | 0.1173 |

0.2504 | 0.3281 | 0.2574 | 0.1641 |

0.1906 | 0.2707 | 0.2831 | 0.2555 |

0.1331 | 0.1331 | 0.2754 | 0.4585 |

Stationary test 2.39 and Persistence test 435.92.

Table 5: Small value towns transition matrix (Lee & Ward, 2001)

0.3876 | 0.2698 | 0.1971 | 0.1455 |

0.2496 | 0.3189 | 0.2655 | 0.1659 |

0.1965 | 0.2578 | 0.2892 | 0.2565 |

0.1451 | 0.1834 | 0.2731 | 0.3984 |

Stationary test 1.86 and Persistence test 261.75.

Sector transition matrices

Table 6: Retail sector transition matrix (Lee & Ward, 2001)

0.3790 | 0.2738 | 0.2028 | 0.1445 |

0.2470 | 0.3063 | 0.2591 | 0.1876 |

0.2078 | 0.2471 | 0.2993 | 0.2458 |

0.1447 | 0.2038 | 0.2629 | 0.3887 |

Stationary test 1.17 and Persistence test 234.41.

Table 7: Office sector transition matrix (Lee & Ward, 2001)

0.4185 | 0.2837 | 0.1517 | 0.1461 |

0.2515 | 0.3402 | 0.2456 | 0.1627 |

0.1919 | 0.2326 | 0.3169 | 0.2587 |

0.1345 | 0.1401 | 0.2941 | 0.4314 |

Stationary test 2.22 and Persistence test 189.96.

Table 8: Industrial sector transition matrix (Lee & Ward, 2001)

0.3906 | 0.2299 | 0.1773 | 0.2022 |

0.2145 | 0.2931 | 0.2568 | 0.2356 |

0.2006 | 0.2537 | 0.2861 | 0.2596 |

0.1923 | 0.2225 | 0.2967 | 0.2885 |

Stationary test 2.62 and Persistence test 61.00.

Regional transition matrices

Table 9: London region transition matrix (Lee & Ward, 2001)

0.4751 | 0.2805 | 0.1176 | 0.1267 |

0.2609 | 0.3623 | 0.2512 | 0.1256 |

0.1845 | 0.2184 | 0.3447 | 0.2524 |

0.0950 | 0.1403 | 0.2851 | 0.4796 |

Stationary test 4.60 and Persistence test 189.23.

Table 10: South East region transition matrix (Lee & Ward, 2001)

0.4131 | 0.2685 | 0.1910 | 0.1274 |

0.2547 | 0.3302 | 0.2491 | 0.1660 |

0.1800 | 0.2852 | 0.2799 | 0.2549 |

0.1453 | 0.1367 | 0.2907 | 0.4273 |

Stationary test 3.86 and Persistence test 281.63.

Table 11: South and West region transition matrix (Lee & Ward, 2001)

0.4677 | 0.2612 | 0.1891 | 0.0821 |

0.2348 | 0.3398 | 0.3011 | 0.1243 |

0.1618 | 0.2732 | 0.2812 | 0.2838 |

0.1089 | 0.1708 | 0.2500 | 0.4703 |

Stationary test 3.65 and Persistence test 302.81.

Table 12: Northern region transition matrix (Lee & Ward, 2001)

0.4441 | 0.2875 | 0.1725 | 0.0958 |

0.2567 | 0.3333 | 0.2567 | 0.1533 |

0.2007 | 0.2274 | 0.3244 | 0.2475 |

0.0912 | 0.1792 | 0.2516 | 0.4780 |

# 4 REFERENCES

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Bond, S. A. & Mitchell, P., 2010. Alpha and Persistence in Real Estate Fund Performance. *Journal of Real Estate Finance and Economics, *41(1), pp. 53-97.

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