Can we still beat “buy-and-hold” for individual stocks?

Physica A xx (xxxx) xxx–xxx

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Physica A journal homepage: www.elsevier.com/locate/physa

Can we still beat ‘‘buy-and-hold’’ for individual stocks? Q1

Eddie C.M. Hui ∗ , Ka Kwan Kevin Chan Department of Building and Real Estate, The Hong Kong Polytechnic University, Hong Kong

highlights • • • • •

We tests our 2 strategies on HSI and HSP indices and 12 constituent stocks of HSI. Our strategies are less effective on individual stocks than on stock indices. Our strategies are more effective on property stocks than on non-property stocks. Our strategies work better on stocks whose Shiryaev–Zhou indices fluctuate less. Our strategies work better during ‘‘bad times’’ than during ‘‘good times’’.

article

info

Article history: Received 13 March 2014 Received in revised form 19 May 2014 Available online xxxx Keywords: Shiryaev–Zhou index Individual stocks ‘‘Buy-and-hold’’ Short-selling Transaction costs

abstract Many investors seek for a trading strategy to beat the ‘‘buy-and-hold’’ strategy. In light of this, Hui and Yam (2014) and Hui et al. (2014) derived a trading strategy from the Shiryaev–Zhou index, and found that the resulting strategy outperformed the ‘‘buy-andhold’’ strategy for western and Asian securitized real estate indices respectively. However, whether the trading strategy works on individual stocks or not is still unknown. This is the first study to test whether the trading strategy can beat the ‘‘buy-and-hold’’ strategy on individual stocks. We construct two trading strategies and compare the resulting profits with the profits arising from the ‘‘buy-and-hold’’ strategy on Hang Seng Index (HSI), Hang Seng Property (HSP) Index and 12 constituent stocks of HSI during the period December 29, 1995– December 31, 2013. The second strategy (Strategy 2) is a new strategy which incorporates short-selling, and has the effect of multiplying the profit. The results show that our trading strategies are less effective on individual stocks than on stock indices, and are more effective on property stocks than on non-property stocks. Moreover, our strategies outperform ‘‘buy-and-hold’’ by a larger extent on stocks of which the Shiryaev–Zhou indices fluctuate less frequently. Furthermore, by tracking the resulting profits of the three strategies at different times along the whole period of observation, our strategies work better during ‘‘bad times’’ than during ‘‘good times’’. This reflects that our trading strategies are especially useful in protecting investors from substantial loss during market downturns. © 2014 Elsevier B.V. All rights reserved.

1. Introduction

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Investors always seek for a strategy to maximize their profits. One well known trading strategy is the ‘‘buy-and-hold’’ strategy based on the efficient market hypothesis (EMH). According to the EMH, at any time, security prices fully reflect all

∗ Correspondence to: ZN 744, Department of Building and Real Estate, The Hong Kong Polytechnic University, Hong Kong. Tel.: +852 2766 5881; fax: +852 27645131. E-mail addresses: [email protected] (E.C.M. Hui), [email protected] (K.K. Kevin Chan). http://dx.doi.org/10.1016/j.physa.2014.05.061 0378-4371/© 2014 Elsevier B.V. All rights reserved.

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available information [1]. The EMH was supported by a number of studies like Malkiel and Fama [1] and Malkiel [2,3]. Barber and Odean [4] showed that households trading stocks more frequently were beaten by the market by a larger extent. This result also supports the ‘‘buy-and-hold’’ strategy and hence the EMH. However, some other studies gave evidence contrary to the EMH. For example, Joel-Carbonell and Rottke [5] found REIT market irregularities during the period 1991–2008, which was contrary to underlying rational human behaviour. Hence someone tries to explore a strategy to outperform the ‘‘buyand-hold’’ strategy. This lays out the background of our study. The above said problem motivates scholars to study portfolio profit optimization. The first to work on this topic was Markowitz [6], who introduced the mean–variance modern portfolio theory (MPT). The MPT was incorporated with fuzzy set theory by Hui et al. [7] to study portfolio optimization in direct real estate investment. Many dynamic models, e.g. the Merton portfolio [8], [9] and the continuous-time Markowitz model [10], lead to continuously rebalancing optimal portfolios. They do not result in the pure ‘‘buy-and-hold’’ strategy. Other recent works on optimal trading strategies are described as follows. Liehr and Pawelzik [11] derived an optimal trading strategy with variable investment for minimizing the risk to profit ratio. They tested their trading strategy on DAX and S&P indices using different types of prediction models in comparison. Krystalogianni and Tsolacos [12] developed a Markov switching strategy to investigate the structure of yields between broad asset classes. Their resulting Markov switching model was superior to simple buy-and-hold strategies. Applying cointegration methods, Gallo et al. [13] constructed globally diversified real estate portfolios which beat the mean–variance optimized portfolio by almost 600 basis points each year. The method this study applies is the Shiryaev–Zhou index, which is named in honour of its two founders, A. Shiryaev and X.Y. Zhou. Hui and Yam [14] and Hui et al. [14] derived a trading strategy from the Shiryaev–Zhou index, and tested the strategy on western and Asian securitized real estate markets respectively. Both of them found that their derived trading strategy outperformed the ‘‘buy-and-hold’’ strategy generally. However, both studies used data of securitized real estate indices. Whether the trading strategy works on individual stocks or not is still unknown. In order to solve this myth, we use data of Hang Seng Index (HSI), Hang Seng Property (HSP) Index and 12 constituent stocks of HSI. We divide the 12 stocks into 6 property stocks and 6 non-property stocks to compare the performance of our strategies on different types of stocks. Secondly, for Hui and Yam [14] and Hui et al.’s [14] strategy, when the estimated value of Shiryaev–Zhou index is negative, the strategy is to hold entire cash. This strategy is based on the rationale that a stock/stock index is usually falling when its Shiryaev–Zhou index is negative, but with the assumption that short-selling is not allowed. However, in reality, short-selling is allowed in some markets. In this study, we will construct a new trading strategy so that we short-sell the stock/stock index when its estimated value of Shiryaev–Zhou index is negative, so that we can take advantage of the adverse movement of stocks by short-selling. Furthermore, we will track the performance of our strategy and the ‘‘buy-and-hold’’ strategy along the whole timeline to compare the performances of the strategies at different times during the period of observation. This study derives two trading strategies from the Shiryaev–Zhou index, and tests the strategies on HSP Index and six of its constituent stocks over the period December 29, 1995–December 31, 2013. We compare the resulting profits with the profit derived from the ‘‘buy-and-hold’’ strategy, and consider three scenarios: (i) no transaction costs, (ii) 0.1% transaction costs, and (iii) 0.2% transaction costs. The paper proceeds as follows: Section 2 reviews previous studies related to the Shiryaev–Zhou index. Section 3 lays out the formula of the Shiryaev–Zhou index and its statistical estimation. Section 4 describes the trading strategy derived by Hui and Yam [14] and Hui et al. [24], and constructs a new trading strategy which incorporates short-selling. Section 5 explains the data source. Section 6 analyses the trends of the Shiryaev–Zhou indices of HSP Index and the 12 constituent stocks chosen. Section 7 tests our two trading strategies and the ‘‘buy-and-hold’’ strategy on HSP Index and the 12 constituent stocks, and compares the resulting profits. Finally, we draw a conclusion in Section 8.

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2. Literature review

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The Shiryaev–Zhou index is derived from the problem of finding the optimal selling time to minimize the expected relative error between the selling price of a stock and its maximum price. The first attempt to solve this problem was done by Graversen et al. [15], who solved the problem of stopping a Brownian motion in order to minimize the square error deviation from the maximum. Shiryaev et al. [16] developed a ‘‘goodness index’’ γ of a stock to determine the optimal selling time of the stock. Adopting the probabilistic approach, they showed that the optimal selling time t was determined by t = T (T is the end of the period [0, T ]) if γ ≥ 21 , and t = 0 if γ ≤ 0. For the case 0 < γ < 21 , Shiryaev et al. [16] claimed that the result is same as the case γ ≤ 0, and referred to Dai et al.’s [17] PDE approach for this result. Du Toit and Peskir [18] provided another probabilistic proof of Shiryaev et al.’s [16] result. Applying the techniques in solving the secretary problem, Yam et al. [19–21] resolved the same problem in the binomial tree setting, deriving and generalizing the Shiryaev–Zhou index over the corresponding framework. Note that Shiryaev et al.’s [16] ‘‘goodness index’’ is larger than Yam et al.’s [19–21] Shiryaev–Zhou index by a magnitude of 1/2. The optimal trading strategies derived from Shiryaev et al. [16], Du Toit and Peskir [18] and Yam et al. [19–21] are called ‘‘bang–bang’’ strategies, of which the practicality was empirically investigated by Wong et al. [22], and hence a dynamic bang–bang strategy allowing for parameters of the return distribution to vary over time was derived. Provided that λ was estimated using recent returns, Wong et al.’s [22] dynamic bang–bang strategy beat the ‘‘buy-and-hold’’ strategy on the CRSP, FTSE 100 and Hang Seng indices. The sign of λ is the same as the sign of the Shiryaev–Zhou index [22] and determines when one should buy or sell an asset. Therefore, we can combine Wong et al.’s [22] dynamic bang–bang strategy with the Shiryaev–Zhou index, building up the theoretical and conceptual framework of our study.

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Hui et al. [23] were the first to implement a practical application of the Shiryaev–Zhou index. They listed the latest selling dates of each stock of Hong Kong listed property companies, but did not calculate the resulting profit. Hui and Yam [14] derived a trading strategy from the Shiryaev–Zhou index, and applied the strategy on securitized real estate indices of four European and North American countries. Their resulting strategy outperformed the ‘‘buy-and-hold’’ strategy in general. Hui et al. [24] applied the same strategy on securitized real estate indices of six Asian economies. They also found that their trading strategy generally beat the ‘‘buy-and-hold’’ strategy. However, both Hui and Yam [14] and Hui et al. [14] investigated securitized real estate indices instead of individual stocks. This study fills in the gap by testing the trading strategy on individual stocks. Moreover, Hui and Yam [14] and Hui et al. [24] assumed that short-selling is disallowed. This study constructs a new trading strategy which incorporates short-selling. Finally, we track the performance of the three strategies along the whole timeline to compare the performances of the strategies at different times during the period of observation. This is omitted in [23,24] and [14].

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3. The Shiryaev–Zhou index and its statistical estimation

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The Shiryaev–Zhou index is derived from the problem of minimizing the time between the selling price and the maximum price of the stock:

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



Sτ (3.1) ST where T > 0, E denotes expectation, and ST = max0≤s≤t Ss is the maximum stock price during the period [0, t ]. As mentioned in Section 2, Shiryaev et al. [16] showed that the solution to the problem (3.1) is V ∗ = T when γ ≥ 21 , and ∗ V = 0 otherwise, where γ is the ‘‘goodness index’’ of the stock. Yam et al. [19–21] resolved problem (3.1) in the binomial tree setting and derived that V ∗ = T when µ ≥ 0, and V ∗ = 0 otherwise, where µ is the Shiryaev–Zhou index defined by Yam et al. [19–21], Hui et al. [23,24], Hui and Yam [14], and Hui et al. [24]: V ∗ = max E

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0≤τ ≤T

µ = (α − 0.5σ )/σ = α/σ − 0.5, 2

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2

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(3.2)

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where α is the annual growth rate of the stock and σ is its annual volatility (α, σ are constants). The trading strategy is determined as follows: hold the stock until the end of the period if µ ≥ 0, and sell it immediately otherwise [19–21,23,14,24]. As Shiryaev et al.’s [16] ‘‘goodness index’’ γ relates to the Shiryaev–Zhou index µ by the relationship γ = µ + 12 , the trading strategies derived by Shiryaev et al. [16], and Yam et al. [19–21] are, in fact, the same. In the formula (3.2), the drift α and the volatility σ are constants. However, in reality, these parameters are always varying. More importantly, their exact values are normally not known. Hence we adopt the moving window approach to estimate their values: we use the closing stock prices from day i − n + 1 to day i to estimate the values of α and σ on day i, and hence obtain the estimated value of the Shiryaev–Zhou index on day i. Here we set n = 130. The estimator of the Shiryaev–Zhou index µ on day i (i > n) is:

µ ˆi =

αˆ i − 0.5σˆ , σˆ i2 2 i

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(3.3)

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where αˆ is the estimator of α on day i, and σˆ is the estimator of σ on day i. For details of derivation of the formula (3.3), refer to [14,24].

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4. Our trading strategies Here we use the Shiryaev–Zhou index to construct two trading strategies for trading stocks or stock indices. The Shiryaev–Zhou index of a stock on each day is estimated by the formula (3.3). We make the following two assumptions: (1) The transaction price (buying and selling price) of a stock is its closing price of that day. (2) The amount of cash we hold in the beginning is sufficient to cover all transactions during the period. Before deriving our two strategies, we introduce the ‘‘buy-and-hold’’ strategy first. ‘‘Buy-and-hold’’ normally refers to buying a stock and holding it for a long time. However, since the period of observation here is December 29, 1995–December 31, 2013, the ‘‘buy-and-hold’’ strategy in this study is defined as follows: buying one unit of the stock on December 29, 1995, holding it throughout the whole period, and selling the entire one unit of the stock on December 31, 2013. Our first trading strategy was first derived by Hui and Yam [14], and then applied by Hui et al. [14]. We modify their strategy a bit as follows (we call the following strategy ‘‘Strategy 1’’ in this paper): 1. On Day 1 (December 29, 1995), if µ ˆ 1 ≥ 0, buy one unit of the stock. Otherwise, do not take any action. 2. From Day 2 to the second last day of the period, trade the stock according to Table 1. 3. On the last day of the period, sell the entire one unit of the stock if one is still holding one unit of the stock. Otherwise, do not take any action. Note that in the strategy derived by Hui and Yam [14] and Hui et al. [14], one does not take any action in the first 130 days. However, in this study, the HSP Index and the six stocks we choose are listed more than 130 days before the beginning of the timeline (December 29, 1995), so we modify the strategy as above.

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E.C.M. Hui, K.K. Kevin Chan / Physica A xx (xxxx) xxx–xxx Table 1 Our trading strategy from Day 2 to the second last day according to Strategy 1.

µ ˆ i−1

µ ˆi

Action

≥0 ≥0 <0 <0

≥0 <0 ≥0 <0

No action (keep holding one unit of the stock) Sell the entire one unit of the stock we hold Buy one unit of the stock No action (keep holding entire cash)

Table 2 Our trading strategy from Day 2 to the second last day according to Strategy 2.

µ ˆ i−1

µ ˆi

Action

≥0 ≥0 <0 <0

≥0 <0 ≥0 <0

No action (keep holding one unit of the stock) Sell the entire one unit of the stock we hold, and then short-sell one unit of the stock Buy back one unit of the stock originally short-selling, and then buy one unit of the stock No action (keep short-selling one unit of the stock)

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Strategy 1 is constructed under the assumption that short-selling is not allowed. However, in reality, short-selling is allowed in some markets. Therefore, we construct a new trading strategy (called ‘‘Strategy 2’’ in this paper) as follows:

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1. On Day 1, if µ ˆ 1 ≥ 0, buy one unit of the stock. Otherwise, short-sell one unit of the stock.

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2. From Day 1 to the second last day of the period, trade the stock according to Table 2. On the last day of the period, sell the entire one unit of the stock if one is still holding one unit of the stock. Otherwise (i.e. one is short-selling one unit of the stock), buy back the one unit of stock originally short-selling. We consider the following three scenarios:

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(1) No transaction costs exist.

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(2) Both buying and selling (including short-selling) costs are equal to 0.1% of the transaction price (i.e. the closing price).

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(3) Both buying and selling (including short-selling) costs are equal to 0.2% of the transaction price (i.e. the closing price). Case (2) is closest to the real life situation as for Hong Kong Stock Exchange (HKSE), the transaction cost consists almost entirely of stamp duty, which is 0.1% of the transaction price. Let ai , bi and ci be the numerical profits on day i (i ≥ 2) by applying the ‘‘buy-and-hold’’ strategy, Strategy 1 and Strategy 2 respectively, without transaction costs. If µ ˆ i−1 ≥ 0, then for all the three strategies, we buy one unit of the stock at the closing price on day i − 1, so we hold one unit of the stock on day i. Hence ai = bi = ci . If µ ˆ i−1 < 0, then for the ‘‘buy-andhold’’ strategy, we hold one unit of the stock on day i. For Strategy 1, we hold nothing but entire cash on day i. For Strategy 2, we are in a position of short-selling one unit of the stock on day i. Hence the following relationship holds:

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ci − bi = bi − ai .

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Q4 In fact, (7.1) also holds when µ ˆ i−1

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≥ 0, with both sides of the equality equal to zero. Therefore, this equality holds no matter the sign of µ ˆ i−1 is. 4698 4698 4698 Let a = i = 2 ai , b = i=2 bi and c = i=2 ci be the total numerical profits by applying the ‘‘buy-and-hold’’ strategy, Strategy 1 and Strategy 2 respectively (the whole period December 29, 1995–December 31, 2013 contains a total of 4698 observations). Taking summation on both sides of Eq. (4.1) from i = 2 to i = 4698 results in the following equality: c − b = b − a,

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or, by rearranging the variables,

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c − a = 2 (b − a) .

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(4.1)

(4.2)

(4.3)

When transaction costs exist, the equality signs in (4.1) and (4.2) become approximation due to the initial and final buying/selling of the stock. From Eq. (4.3), we can see that Strategy 2, which incorporates short-selling, has the effect of multiplying the profit. If Strategy 1 outperforms the ‘‘buy-and-hold’’ strategy, Strategy 2 will outperform the ‘‘buy-and-hold’’ strategy by a larger extent. If Strategy 2 underperforms the ‘‘buy-and-hold’’ strategy, Strategy 2 will underperform the ‘‘buy-and-hold’’ strategy even more. This is our rationale behind Strategy 2. In [19–21]’s strategy, one should sell the stock immediately if µ < 0. Therefore, in our strategies using the estimator µ ˆ i of the Shiryaev–Zhou index, we assume that the stock price usually falls when µ ˆ i < 0. In Strategy 1, short-selling is not allowed, so we hold entire cash. However, short-selling is allowed in some financial markets. In order to model this scenario, in Strategy 2, we short-sell the stock when µ ˆ i < 0, aiming to take advantage of the adverse stock price movement to earn more profits. Hence we can beat the ‘‘buy-and-hold’’ strategy by an even larger extent. This is the potential advantage of Strategy 2. However, if the stock price turns out to be rising most of the time when µ ˆ i < 0, then both Strategies 1 and 2 would underperform the ‘‘buy-and-hold’’ strategy, and Strategy 2

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Table 3 The constituent stocks we choose. Property stocks Name Cheung Kong Holding Limited (Cheung Kong) Henderson Land Development Company Limited (Henderson) Sun Hung Kai Properties Limited (SHK) New World Development Company Limited (New World) Sino Land Company Limited (Sino) Hang Lung Properties Limited (Hang Lung)

Stock code Period of being a constituent stock of HSI

Period of being a constituent stock of HSP

1

December 29, 1995–December 31, 2013

December 29, 1995–December 31, 2013

12

December 29, 1995–December 31, 2013

December 29, 1995–December 31, 2013

16

December 29, 1995–December 31, 2013

December 29, 1995–December 31, 2013

17

December 29, 1995–June 8, 2003, June 6, 2005–December 31, 2013 December 29, 1995–June 8, 2003, June 6, 2005–December 31, 2013 December 29, 1995–December 31, 2013

December 29, 1995–June 8, 2003, September 10, 2012–December 31, 2013 December 29, 1995–June 8, 2003, June 6, 2005–December 31, 2013 December 29, 1995–December 31, 2013

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Non-property stocks Name The Hong Kong and China Gas Company Limited (Towngas) The Wharf (Holdings) Limited (Wharf) HSBC Holdings Limited (HSBC) Hang Seng Bank Limited (HSB) Hutchison Whampoa Limited (Hutchison) Swire Pacific Limited A (Swire A)

Stock code Period of being a constituent stock of HSI

Sector of HSI of which the stock belongs to

3

December 29, 1995–December 31, 2013

Utilities

4

December 29, 1995–December 31, 2013

5 11 13

December 29, 1995–December 31, 2013 December 29, 1995–December 31, 2013 December 29, 1995–December 31, 2013

Commerce & Industry (re-classified to HSP Index since September 10, 2012) Finance Finance Commerce & Industry

19

December 29, 1995–December 31, 2013

Commerce & Industry

would underperform ‘‘buy-and-hold’’ even more than Strategy 1 does. This highlights the potential drawback of Strategy 2. Therefore, Strategy 2 is riskier than Strategy 1. In this study, we test Strategies 1 and 2 on HSI and HSP Indices and the 12 stocks selected to find out whether the strategies can outperform the ‘‘buy-and-hold’’ strategy. We consider 3 scenarios: no transaction costs, 0.1% transaction costs and 0.2% transactions costs. Before proceeding to our results, we briefly describe our data source first.

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5. Data source

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The period of observation is December 29, 1995–December 31, 2013, a total of 4698 observations. However, from Section 3, in order to obtain the estimated value of Shiryaev–Zhou index µ ˆ i on day i, the stock price on day i − 130 must be known, so we have to extend the timeline back by 130 days, i.e. back to June 30, 1995. We obtain the values of HSI and HSP Indices and 12 constituent stocks during the period June 30, 1995–December 31, 2013 from Bloomberg. The 12 constituent stocks are divided into 6 property stocks and 6 non-property stocks. We select the constituent stocks according to the following criteria:

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1. All stocks must be listed on Hong Kong Stock Exchange (HKSE) during the whole period June 30, 1995–December 31, 2013.

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2. All stocks must be constituent stocks of Hang Seng Index (HSI) for at least 12 years during the period of observation December 29, 1995– December 31, 2013.

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3. Each of the 6 property stocks must be a constituent stock of HSP Index for at least 8 years during the period December 29, 1995–December 31, 2013. 4. The 6 non-property stocks are chosen so that there is at least one stock from each of the three sectors of HSI (except the property sector)—Finance, Utilities, Commerce and Industry. Only constituent stocks of HSI are chosen because they are the most frequently traded stocks in Hong Kong. They have much larger market values and transaction volumes than other stocks. Hence they can reflect the overall performance of Hong Kong’s market, i.e. they are representative. Since HSI is divided into 4 sectors—Finance, Utilities, Properties, Commerce and Industry, we select the non-property stocks so that each of the three sectors, Finance, Utilities, Commerce and Industry, contains at least one stock so as to achieve a balanced composition of stocks. According to the above criteria, the stocks listed in Table 3 are selected. With the data chosen, we use the formula (3.3) to estimate the Shiryaev–Zhou indices of HSI and HSP Indices and the 12 selected stocks on each day during the period of observation. Then we apply the ‘‘buy-and-hold’’ strategy, Strategy 1 and Strategy 2 as described in Section 4 on HSI and HSP Indices and the 12 selected stocks during the period of observation under the 3 scenarios: no transaction costs, 0.1% transaction costs and 0.2% transactions costs, and compare their resulting profits. Furthermore, we track the resulting profits of the three strategies at different times along the whole period (without transaction costs). The results are shown in Sections 6 and 7 respectively.

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Fig. 1. HSI and its Shiryaev–Zhou index.

Fig. 2. HSP and its Shiryaev–Zhou index.

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6. Overview of Shiryaev–Zhou index of the two indices and each stock For HSI and HSP Indices and each constituent stock chosen in Section 5, we use the formula (3.3) to estimate the Shiryaev–Zhou index on each day in the period of observation (December 29, 1995–December 31, 2013). The following figures show the trends and the corresponding Shiryaev–Zhou indices of HSP Index and the 12 constituent stocks over the period of observation. Note that in Figs. 1–14, ‘‘Holding period’’ and ‘‘Non-holding period’’ correspond to the periods when µ ˆ i ≥ 0 and µ ˆi < 0 respectively. They are related to our trading strategy described in Section 3. We will explain this further in Section 7. From the above figures, for HSI and HSP Indices and each of the constituent stocks, the Shiryaev–Zhou index is normally positive when the stock price is rising and vice versa. Hence the Shiryaev–Zhou index can act as an indicator of the performance of its corresponding stock. Secondly, the Shiryaev–Zhou indices of HSI and HSP Indices and the 12 constituent stocks followed a similar pattern over a certain time period. For example, the Shiryaev–Zhou indices of HSI and HSP Indices and most of the 12 stocks remained positive in 1996, when Hong Kong experienced an economic boom. On the other hand, there are several long periods when the Shiryaev–Zhou indices of the two indices and most of the stocks stayed negative: late 1997–late 1998 (the Asian financial crisis), mid 2002–mid 2003 (the SARS outbreak), early 2008–early 2009 (the global

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Fig. 3. Cheung Kong’s stock price and its Shiryaev–Zhou index.

Fig. 4. Henderson’s stock price and its Shiryaev–Zhou index.

financial crisis), and early 2011–early 2012 (the European sovereign debt crisis). Within those periods, either financial crisis broke out or Hong Kong’s economy underwent a downturn. Hence the Shiryaev–Zhou index can act as an indicator of Hong Kong’s economic condition to some extent.

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7. The results

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7.1. The resulting profits of the three strategies

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7.1.1. No transaction costs Here we apply the ‘‘buy-and-hold’’ strategy, Strategy 1 and Strategy 2 as described in Section 4 on HSI and HSP Indices and the 12 constituent stocks during the period of observation December 29, 1995 –December 31, 2013, and compare their resulting profits. Firstly, we consider the case without transaction costs. The resulting profits are shown in Table 4 (the figures in the brackets indicate the percentage profit. Note that for our trading strategy, the base for calculating the percentage profit is the initial cost).

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Fig. 5. SHK’s stock price and its Shiryaev–Zhou index.

Fig. 6. New World’s stock price and its Shiryaev–Zhou index.

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From Table 4, we can see that Eqs. (4.2) and (4.3) hold for both indices and all the 12 stocks. Strategy 2 really has a multiplying effect. For both HSI and HSP Indices, both Strategies 1 and 2 outperform the ‘‘buy-and-hold’’ strategy, and we can even earn more profit by adopting Strategy 2. The two strategies also beat the ‘‘buy-and-hold’’ strategy for five of the six property stocks. In particular, our strategies work better for New World, Sino and Hang Lung, where Strategies 1 and 2 outperform the ‘‘buy-and-hold’’ strategy by a larger extent. For New World, the ‘‘buy-and-hold’’ strategy yields a 65% loss. By adhering to Strategy 1, the loss is reduced to 7%. If short-selling is allowed, we can even earn a 52% profit by adopting Strategy 2. The only exception is Cheung Kong, where both Strategies 1 and 2 are outperformed by the ‘‘buy-and-hold’’ strategy, and Strategy 2 underperforms the ‘‘buy-and-hold’’ strategy by an even larger extent. However, for the non-property stocks, Strategies 1 and 2 outperform the ‘‘buy-and-hold’’ strategy for Wharf and Hutchison only, but underperform the ‘‘buy-andhold’’ strategy for other four stocks. We can look back to Figs. 1–14 to explain this result. ‘‘Holding period’’ corresponds to the period when µ ˆ i ≥ 0, in which according to Strategies 1 and 2 we should keep holding the stock. From the figures, we can see that during ‘‘Holding period’’, HSI and HSP Indices and the 12 stocks are rising most of the time, so we can earn a profit. Whilst during ‘‘Non-holding period’’, the two indices and the majority of the 12 stocks are falling for the majority of the time, so we can avoid a loss by not holding the stocks. If short-selling is allowed, we can short-sell the stocks to earn more profit. Hence Strategies 1 and 2 can outperform the ‘‘buy-and-hold’’ strategy. In particular, New World suffered a 65% fall

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Fig. 7. Sino’s stock price and its Shiryaev–Zhou index.

Fig. 8. Hang Lung’s stock price and its Shiryaev–Zhou index.

during the period of observation (see Table 4). It has long non-holding periods during which the stock plunged drastically: late 1997–late 1998 (the Asian financial crisis), mid 2002–mid 2003 (the SARS outbreak), and early 2008–early 2009 (the global financial crisis). Therefore, the ‘‘buy-and-hold’’ strategy yields a substantial loss. However, for Strategies 1 and 2, we need not hold the stock during the non-holding periods when the stock is falling most of the time, so the loss is much smaller for Strategy 1. For Strategy 2, we can take advantage of the adverse movement of the stock by short-selling, so we can even earn a profit. This reflects that Strategies 1 and 2 are especially useful in protecting investors during market downturns. Even when the market keeps falling, we may still earn a positive return, which was thought to be impossible in the past. However, for Cheung Kong, Towngas, HSBC, HSB and Swire A, Strategies 1 and 2 are outperformed by the ‘‘buy-and-hold’’ strategy. For these five stocks, during the non-holding periods, the time during which the stock price is rising is more than the time when the stock price is falling, so both Strategies 1 and 2 underperform the ‘‘buy-and-hold’’ strategy, and Strategy 2 performs even worse due to short-selling during the non-holding periods. The most extreme case is HSBC. From Table 4, the ‘‘buy-and-hold’’ strategy yields a 133% profit for this stock, but the profit falls significantly to 22% when we apply Strategy 1, and we even suffer an 89% loss if we adopt Strategy 2. Fig. 11 shows that there are many non-holding periods in which the stock price of HSBC is rising most of the time (e.g. late 1997–mid-1998, mid 1998–early 1999, and mid-2003).

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Fig. 9. Towngas’ stock price and its Shiryaev–Zhou index.

Fig. 10. Wharf’s stock price and its Shiryaev–Zhou index.

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7.1.2. 0.1% transaction costs Real life stock trading includes transaction costs. How would this affect our results? In the second scenario, the buying or selling cost is equal to 0.1% of the transaction price. The resulting profits are shown in Table 5. Comparing Table 5 with Table 4, when 0.1% transaction costs exist, the profits arising from Strategies 1 and 2 reduce by a larger extent than the profits from using the ‘‘buy-and-hold’’ strategy do. The reason is that Strategies 1 and 2 require investors to trade the stock/stock index a number of times. Therefore, the transaction costs build up and may even offset the gain. By applying Strategy 2, each time when the Shiryaev–Zhou index turns from positive to negative, one has to change his position from long one unit of stock to short one unit of stock. Hence he has to sell two units of stock. Similarly, when the Shiryaev–Zhou index turns from negative to positive, one has to buy two units of stock. Therefore, the total amount of transaction costs arising from Strategy 2 is approximately double of the amount of transaction costs arising from Strategy 1. As a result, the profits arising from Strategy 2 diminish by an even larger extent when there are 0.1% transaction costs. Strategies 1 and 2 still outperform the ‘‘buy-and-hold’’ strategy for HSI and HSP Indices when 0.1% transaction costs exist, but the performance of the two strategies vary for individual stocks. For SHK, Strategies 1 and 2 outperform the ‘‘buy-and-hold’’ strategy slightly when transaction costs do not exist, but are beaten by the ‘‘buy-and-hold’’ strategy with the presence of 0.1% transaction costs. For the other six stocks in which Strategies 1 and 2 outperform ‘‘buy-and-hold’’ with no transaction costs, the two strategies are still superior with 0.1% transaction costs, but the two strategies beat ‘‘buy-and-hold’’ by a smaller

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Fig. 11. HSBC’s stock price and its Shiryaev–Zhou index.

Fig. 12. HSB’s stock price and its Shiryaev–Zhou index.

extent. For the remaining five stocks in which Strategies 1 and 2 underperform ‘‘buy-and-hold’’ even with no transaction costs, the two strategies underperform ‘‘buy-and-hold’’ by an even larger extent with 0.1% transaction costs. In particular, for HSBC, ‘‘buy-and-hold’’ yields a 132% profit, but both Strategies 1 and 2 incur a loss. 7.1.3. 0.2% transaction costs In the third case, both buying and selling costs are 0.2% of the transaction price. Table 6 displays the result. From Table 6, as transaction costs double, the profits arising from Strategies 1 and 2 diminish further. Strategies 1 and 2 still outperform the ‘‘buy-and-hold’’ strategy for HSI and HSP Indices when 0.2% transaction costs exist, but the two strategies perform worse for individual stocks in overall. Comparing the results in Tables 5 and 6, Strategies 1 and 2 beat ‘‘buy-andhold’’ for Henderson with 0.1% transaction costs, but are outperformed by ‘‘buy-and-hold’’ when transaction costs increase to 0.2%. For the other five stocks in which Strategies 1 and 2 outperform ‘‘buy-and-hold’’ with 0.1% transaction costs, the two strategies are still superior with 0.2% transaction costs, but the two strategies beat ‘‘buy-and-hold’’ by a smaller extent. For the remaining six stocks in which Strategies 1 and 2 underperform ‘‘buy-and-hold’’ with 0.1% transaction costs, the two strategies underperform ‘‘buy-and-hold’’ by an even larger extent with 0.2% transaction costs. In particular, Strategies 1 and 2 perform the worst on HSBC, yielding 81% and 293% losses respectively, while ‘‘buy-and-hold’’ would earn a 132% profit on the same stock. In contrast, applying the ‘‘buy-and-hold’’ strategy incurs a 65% loss on New World, but Strategy 2 still yields an 18% profit even with 0.2% transaction costs present.

1 2 3

4

Q6

5 6 7 8 9 10 11 12 13 14 15 16

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Fig. 13. Hutchison’s stock price and its Shiryaev–Zhou index.

Fig. 14. Swire A’s stock price and its Shiryaev–Zhou index.

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Comparing with Hui et al.’s [14] results, the only case which Hui et al.’s [14] strategy underperforms the ‘‘buy-and-hold’’ is the case of Hong Kong’s securitized real estate index with 0.2% transaction costs. However, in this study, our two strategies underperform the ‘‘buy-and-hold’’ strategy for five stocks even without transaction costs. When there are 0.2% transaction costs, the ‘‘buy-and-hold’’ strategy beats our strategies for seven stocks. Our strategies seem to be less effective on individual stocks than on stock indices, especially when transaction costs are present. What is the reason behind this? One possible reason is that individual stocks usually have a larger volatility than the stock indices they belong to, hence their stock prices fluctuate more frequently than the stock indices do. From Section 3, the estimated value of Shiryaev–Zhou index µ ˆ i on day i is calculated using stock prices from day i–130 to day i, so µ ˆ i in fact lags behind the stock price. When the stock price fluctuates more frequently, its Shiryaev–Zhou index also fluctuates more frequently. Each ‘‘Holding period’’ (period when µ ˆ i ≥ 0) and ‘‘Non-holding period’’ (period when µ ˆ i < 0) is shorter, so there is a higher chance that the stock price is rising when µ ˆi is negative, so Strategies 1 and 2 would underperform the ‘‘buy-and-hold’’ strategy in this case. A stock index comprises many stocks and hence is less volatile. It has a smoothing effect, eliminating fluctuations. Hence each ‘‘Non-holding period’’ is longer with the stock price falling in the majority of time. In this case, Strategies 1 and 2 would outperform the ‘‘buyand-hold’’ strategy. Moreover, the Shiryaev–Zhou index fluctuates more frequently means that it changes sign for a greater number of times. This means we have to trade the stock more frequently, so the transaction costs increase (if transaction costs exist), making Strategies 1 and 2 less effective when transaction costs exist.

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Table 4 The profit of the three strategies on HSI and HSP Indices and the 12 constituent stocks without transaction costs. Indices Index

HSI

HSP

Buy-and-hold Strategy 1 Strategy 2

132 33.00 (131.37%) 177 49.06 (176.20%) 222 65.12 (221.03%)

109 32.68 (62.22%) 200 69.64 (114.21%) 292 06.60 (166.21%)

Property stocks (unit: $) Stock code

1

12

16

17

83

101

Buy-and-hold

75.30 (159.87%) 61.20 (129.94%) 47.10 (100.00%)

1.89 (4.46%) 14.24 (33.62%) 26.59 (62.78%)

35.10 (55.49%) 35.35 (55.89%) 35.60 (56.28%)

−18.29 (−65.13%) −1.89 (−6.72%) 14.51 (51.69%)

5.15 (94.32%) 12.09 (221.56%) 19.03 (348.80%)

16.80 (218.18%) 26.89 (349.22%) 36.98 (480.26%)

Strategy 1 Strategy 2

Non-property stocks (unit: $) Stock code

3

4

5

11

13

19

Buy-and-hold

14.75 (486.80%) 10.49 (346.04%) 6.22 (205.28%)

35.69 (151.15%) 56.57 (239.58%) 77.45 (328.01%)

48.04 (133.01%) 7.92 (21.93%) −32.20 (−89.16%)

56.45 (81.52%) 27.85 (40.22%) −0.75 (−1.08%)

62.58 (146.16%) 98.74 (230.60%) 134.89 (315.04%)

38.45 (73.31%) 35.49 (67.67%) 32.53 (62.02%)

Strategy 1 Strategy 2

Table 5 The profit of the three strategies on HSI and HSP Indices and the 12 constituent stocks with 0.1% transaction costs. Indices Index

HSI

HSP

Buy-and-hold Strategy 1 Strategy 2

131 99.62 (130.90%) 156 50.27 (155.21%) 181 00.92 (179.51%)

108 86.60 (61.89%) 168 22.85 (95.64%) 227 02.08 (129.06%)

Property stocks (unit: $) Stock code

1

12

16

17

83

101

Buy-and-hold

75.13 (159.35%) 46.40 (98.41%) 17.67 (37.48%)

1.80 (4.25%) 7.34 (17.31%) 12.78 (30.15%)

34.94 (55.18%) 19.94 (31.50%) 4.75 (7.51%)

−18.33 (−65.20%) −4.26 (−15.17%) 9.78 (34.80%)

5.13 (93.93%) 10.85 (198.68%) 16.55 (303.04%)

16.77 (217.55%) 24.55 (318.47%) 32.28 (418.76%)

Strategy 1 Strategy 2

Non-property stocks (unit: $) Stock code

3

4

5

11

13

19

Buy-and-hold

14.73 (485.63%) 8.62 (284.15%) 2.47 (81.50%)

35.61 (150.65%) 51.49 (217.88%) 67.26 (284.60%)

47.92 (132.55%) −10.64 (−29.43%) −69.19 (−191.40%)

56.26 (81.15%) 9.15 (13.20%) −37.95 (−54.75%)

62.43 (145.67%) 88.51 (206.50%) 114.58 (267.33%)

38.31 (72.96%) 25.63 (48.81%) 12.76 (24.31%)

Strategy 1 Strategy 2

Table 7 shows the number of times the Shiryaev–Zhou index changes sign during the period December 29, 1995– December 30, 2013 (the final day, 31 December, 2013, is excluded because no matter the sign of the Shiryaev–Zhou index is, we have to return to the position of holding entire cash) for HSI and HSP Indices and the 12 constituent stocks. Comparing the results in Table 7 with those in Tables 4–6, we can see the following relationship: in general, when the Shiryaev–Zhou index fluctuates more frequently, Strategies 1 and 2 would outperform the ‘‘buy-and-hold’’ strategy by a smaller extent. This effect is more significant when transaction costs exist. Table 7 shows that the Shiryaev–Zhou indices of HSI and HSP Indices change sign least frequently (121 and 154 times respectively). Table 6 shows that Strategies 1 and 2 outperform ‘‘buy-and-hold’’ even with the presence of 0.2% transaction. For the five stocks in which the two strategies beat ‘‘buy-and-hold’’ under 0.2% transaction costs, Table 7 reveals that the Shiryaev–Zhou indices of these five stocks change sign less frequently than the Shiryaev–Zhou indices of the other seven stocks do. In particular, the Shiryaev–Zhou index of HSBC changes sign most frequently (211 times), while Tables 4–6 reflects that Strategies 1 and 2 underperform ‘‘buy-and-hold’’ by the largest extent for this stock. We can also see that in general, the Shiryaev–Zhou indices of property stocks fluctuate less

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Table 6 The profit of the three strategies on HSI and HSP Indices and the 12 constituent stocks with 0.2% transaction costs. Indices Index

HSI

HSP

Buy-and-hold Strategy 1 Strategy 2

131 66.24 (130.44%) 135 51.48 (134.26%) 139 36.73 (138.08%)

108 40.53 (61.57%) 135 76.05 (77.10%) 161 97.56 (91.99%)

Property stocks (unit: $) Stock code

1

12

16

17

83

101

Buy-and-hold

74.96 (158.84%) 31.60 (66.96%) −11.76 (−24.92%)

1.72 (4.04%) 0.43 (1.02%) −1.02 (−2.41%)

34.78 (54.87%) 4.54 (7.16%) −26.10 (−41.17%)

−18.36 (−65.27%) −6.64 (−23.60%)

5.11 (93.54%) 9.61 (175.84%) 14.07 (257.36%)

16.74 (216.91%) 22.20 (287.79%) 27.57 (357.39%)

Strategy 1 Strategy 2

5.05 (17.94%)

Non-property stocks (unit: $) Stock code

3

4

5

11

13

19

Buy-and-hold

14.71 (484.46%) 6.75 (222.39%) −1.28 (−42.02%)

35.52 (150.15%) 46.42 (196.22%) 57.08 (241.28%)

47.80 (132.08%) −29.19 (−80.67%) −106.18 (−293.43%)

56.06 (80.79%) −9.55 (−13.76%) −75.16 (−108.31%)

62.29 (145.18%) 78.28 (182.45%) 94.27 (219.71%)

38.16 (72.62%) 15.76 (29.99%) −7.00 (−13.33%)

Strategy 1 Strategy 2

Table 7 No of times the Shiryaev–Zhou index changes sign for HSI and HSP Indices and the 12 constituent stocks. Indices Stock code

HSI

HSP

No. of times the Shiryaev–Zhou index changes sign

121

154

Property stocks Stock code

1

12

16

17

83

101

No. of times the Shiryaev–Zhou index changes sign

165

168

172

156

144

162

Stock code

3

4

5

11

13

19

No. of times the Shiryaev–Zhou index changes sign

190

164

211

185

141

170

Non-property stocks

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frequently than the Shiryaev–Zhou indices of non-property stocks do, causing the two strategies to work better on property stocks. In order to further illustrate this relationship, we plot a graph of difference in percentage profit between Strategy 1 and ‘‘buy-and-hold’’ under 0.1% transaction costs (variable y) against the number of times the Shiryaev–Zhou index changes sign (variable x). Since stock indices behave different from individual stocks, we include all the 12 constituent stocks, but exclude HSI and HSP Indices. We use linear regression to find the best-fit line. The result is shown in the following figure: From Fig. 15, the coefficient of determination R2 is about 0.73, which is fairly high, showing that the linear model fits the data quite well. Hence we see that when the Shiryaev–Zhou index fluctuates less, Strategy 1 (and hence Strategy 2) beats the ‘‘buy-and-hold’’ strategy by a larger extent, and the increase in the profit gap (in percentage) is proportional to the decrease in number of times the Shiryaev–Zhou index changes sign. In addition, the x-intercept x0 of the line in Fig. 15 represents the threshold. If the Shiryaev–Zhou index changes sign less than this threshold, Strategy 1 (and hence Strategy 2) would outperform the ‘‘buy-and-hold’’ strategy. Otherwise, the two strategies would underperform ‘‘buy-and-hold’’. From 2286 = 166.25. This means that if the Shiryaev–Zhou index changes sign the equation of the line, we can deduce that x0 = 70..0435 for less than or equal to 166 times, then we should adopt Strategy 1 (Strategy 2 if short-selling is allowed). Otherwise, we should adhere to the ‘‘buy-and-hold’’ strategy.

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7.2. The resulting profits of the three strategies at different times throughout the period

1 2 3 4 5 6 7 8 9 10 11 12 13 14

17 18 19

In the second part, we compare the resulting profits of the ‘‘buy-and-hold’’ strategy and Strategies 1 and 2 on HSI and HSP Indices and the 12 constituent stocks at different times along the whole period of observation. For the sake of convenience, we only consider the case without transaction costs. The results are shown in the following figures.

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Fig. 15. The graph of difference in percentage profit between Strategy 1 and ‘‘buy-and-hold’’ under 0.1% transaction costs against the number of times the Shiryaev–Zhou index changes sign.

Fig. 16. Profit of the three strategies on HSI Index (without transaction costs).

Comparing Figs. 16–29 with Figs. 1–14, during the ‘‘Holding period’’ (when the Shiryaev–Zhou index is non-negative), the three strategies move in the same direction. During the ‘‘Non-holding period’’ (when the Shiryaev–Zhou index is negative), Strategy 1 moves horizontally, while Strategy 2 and the ‘‘buy-and-hold’’ strategy move in opposite directions. So it is the movement of stock price during the ‘‘Non-holding period’’ which makes the difference in profits between the three strategies. From Figs. 16 to 29, we can see that Strategies 1 and 2 outperform the ‘‘buy-and-hold’’ strategy during some times, but underperform during other times. Overall Strategies 1 and 2 beat the ‘‘buy-and-hold’’ strategy for the majority of the time for HSI and HSP Indices and the majority of the 12 stocks. In particular, for the two indices and most of the stocks, Strategies 1 and 2 outperform the ‘‘buy-and-hold’’ strategy the most in mid-1998 and from late 2008 to early 2009. If we look back to Figs. 1–14 we can see that these two periods correspond to the long non-holding periods when the Shiryaev–Zhou

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Fig. 17. Profit of the three strategies on HSP Index (without transaction costs).

Fig. 18. Profit of the three strategies on Cheung Kong (without transaction costs).

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index stays negative for a long time and the stock/stock index keeps falling. Hence Strategies 1 and 2 beat the ‘‘buy-and-hold’’ strategy, while Strategy 2 is even superior as we can take advantage of the adverse movement of the stock by short-selling. In fact, these two periods represent the time when the Asian financial crisis and the global financial crisis were most severe when many stock markets attained a trough. Our two strategies are most effective around the trough of an economic cycle. On the other hand, during mid-1997 and from late 2007 to early 2008, Strategies 1 and 2 underperform the ‘‘buy-and-hold’’ strategy for HSI and HSP Indices and most of the 12 stocks (except for New World and Hutchison, where Strategies 1 and 2 still beat the ‘‘buy-and-hold’’ strategy in late 2007–early 2008). Referring to Figs. 1–14, HSI and HSP Indices and the prices of the 12 stocks attain a peak within those periods. This is because the stock price movements are not perfectly smooth. During a long period of stock price boom, the stock price seldom keeps on rising continuously. There are always short periods when the stock price falls. Hence its Shiryaev–Zhou index remains positive most of the time, but there are short periods when the Shiryaev–Zhou index turns negative. However, since the estimated value of Shiryaev–Zhou index lags behind the stock price, the stock price is often still rising during those non-holding periods. This is reflected in Figs. 1–14 that during the period 2004–2007, HSI and HSP Indices and the 12 stocks are on a rising trend in general. Most of the time in this period belongs to ‘‘Holding period’’, but there are several short non-holding periods, during which the stock prices are still rising for the majority of the time. Therefore, we can see from Figs. 16–29 that Strategies 1 and 2 outperform the ‘‘buy-and-hold’’ strategy in 2004, but the gap becomes narrower gradually, and in late 2007, Strategies 1 and 2 even underperform the

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Fig. 19. Profit of the three strategies on Henderson (without transaction costs).

Fig. 20. Profit of the three strategies on SHK (without transaction costs).

‘‘buy-and-hold’’ strategy except for New World and Hutchison. Our two strategies are least effective around the peak of an economic cycle. Among HSI and HSP Indices and the 12 stocks, Strategies 1 and 2 work best on New World, as seen from Fig. 21 that the two strategies have beat the ‘‘buy-and-hold’’ strategy since 2000. It is the only stock of which Strategies 1 and 2 beat the ‘‘buy-and-hold’’ strategy all the time during the period 2000–2013. From Fig. 6, the stock price of New World suffers a sharp plunge from nearly $50 in mid-1997 to a trough of about $2 in early 2003, creating several long non-holding periods during which its stock price keeps on falling. Hence Strategies 1 and 2 beat the ‘‘buy-and-hold’’ strategy by a large extent. This can be seen from Fig. 21 that the gap between ‘‘buy-and-hold’’ and ‘‘Strategy 1’’ (and also between ‘‘Strategy 1 and ‘‘Strategy 2’’) becomes wider and wider from 2000 to 2003. Although its share price gradually rises to about $28 in late 2007, its stock price is still much lower than the peak of nearly $50 in 1997, so the gap just narrows from 2004 to 2007. Strategies 1 and 2 are still superior. In 2008, the global financial crisis broke out, so there is a long non-holding period from early 2008 to early 2009, during which the share price drops rapidly, so the gap widens again. Our two strategies are particularly effective on adverse-performing stocks. On the other hand, HSBC is the worst performing stock under Strategies 1 and 2. From Fig. 26, Strategies 1 and 2 underperform the ‘‘buy-and-hold’’ strategy almost all the time since 1998. During two non-holding periods late 1997–mid-1998 and mid-1998–early 1999, its stock price is still rising for the majority of time, so the gap between ‘‘buy-and-hold’’ and ‘‘Strategy 1’’ (and also between ‘‘Strategy 1 and ‘‘Strategy 2’’) widens, indicating that ‘‘buy-and-hold’’ outperform the two strategies by a larger extent. Since then, HSBC’s stock price has risen gradually to a peak of about $140 in late 2007. However, during this period, its Shiryaev–Zhou index fluctuates a lot (from Table 7, HSBC’s Shiryaev–Zhou index fluctuates most frequently, changing sign for 211 times), creating more non-holding periods in

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Fig. 21. Profit of the three strategies on New World (without transaction costs).

Fig. 22. Profit of the three strategies on Sino (without transaction costs).

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which the stock price is still rising, so the gap widens further. The gap becomes the widest in late 2007, when the ‘‘buy-andhold’’ strategy yields a profit of over $100, but adopting Strategy 2 would suffer a loss of over $60. Although the stock price falls sharply during the global financial crisis, Strategies 1 and 2 beat ‘‘buy-and-hold’’ only within a short period in early 2009, when the stock price attained a trough of about $30, but soon the position reverses again as the stock price rebounds although the Shiryaev–Zhou index remains negative.

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8. Conclusion

1 2 3 4

7 8 9 10 11 12 13 14 15

This study tests our two trading strategies and the ‘‘buy-and-hold’’ strategy on HSI and HSP Indices and 12 constituent stocks during the period December 29, 1995–December 31, 2013, and compares the resulting profits of the three strategies. The results are summarized as follows: (1) Strategy 2 has the effect of multiplying the profit. If Strategy 1 outperforms the ‘‘buy-and-hold’’ strategy, Strategy 2 will outperform the ‘‘buy-and-hold’’ strategy by a larger extent. If Strategy 1 underperforms the ‘‘buy-and-hold’’ strategy, Q7 Strategy 2 will underperform the ‘‘buy-and-hold’’ strategy even more. (2) For HSI and HSP Indices, Strategies 1 and 2 outperform the ‘‘buy-and-hold’’ strategy even with 0.2% transaction costs present. However, the two strategies underperform the ‘‘buy-and-hold’’ strategy for five stocks even without transaction costs. With 0.2% transaction costs, the ‘‘buy-and-hold’’ strategy is superior for seven stocks.

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Fig. 23. Profit of the three strategies on Hang Lung (without transaction costs).

Fig. 24. Profit of the three strategies on Towngas (without transaction costs).

(3) Overall, our two strategies are more effective on property stocks than on non-property stocks. (4) Our two strategies outperform ‘‘buy-and-hold’’ by a larger extent on stocks of which the Shiryaev–Zhou index fluctuates less frequently. (5) By tracking the resulting profits of the three strategies at different times along the whole period of observation, we can see that our two strategies work better around troughs than around peaks. (6) Our two strategies work best on New World, of which the ‘‘buy-and-hold’’ strategy yields a 65% loss, but Strategy 2 still yields a profit. Moreover, it is the only stock of which Strategies 1 and 2 beat the ‘‘buy-and-hold’’ strategy all the time during the period 2000–2013 without transaction costs. On the other hand, our two strategies work worst on HSBC, of which the ‘‘buy-and-hold’’ strategy yields a profit of over 130%, but Strategy 2 yields a huge loss. In fact, HSBC is the stock of which the Shiryaev–Zhou index fluctuates most frequently during the period. This study shed light on better strategies of investment compared with previous studies. Hui and Yam [14] and Hui et al. [24] found that their trading strategies beat the ‘‘buy-and-hold’’ strategy in general for western and Asian securitized real estate indices respectively. Wong et al.’s [22] dynamic bang–bang strategy also beat the ‘‘buy-and-hold’’ strategy for the CRSP, FTSE 100 and Hang Seng indices. However, this study shows that our two trading strategies would have a higher chance of underperforming the ‘‘buy-and-hold’’ strategy for individual stocks than for stock indices, especially when transaction costs exist, so our strategies are less effective on individual stocks. On the other hand, our results have a similarity with

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Fig. 25. Profit of the three strategies on Wharf (without transaction costs).

Fig. 26. Profit of the three strategies on HSBC (without transaction costs).

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Hui and Yam [14] and Hui et al.’s [24] results: Hui et al.’s [24] strategy outperforms ‘‘buy-and-hold’’ the most for Canada’s securitized real estate index, which fell by more than 50% during the period. Hui et al.’s [14] strategy outperforms ‘‘buyand-hold’’ the most for Thailand’s securitized real estate index, which fell by nearly 80% during the period. Our strategies outperform ‘‘buy-and-hold’’ the most for New World, which fell by 65% during the period. Thus the trading strategies derived from the Shiryaev–Zhou index beat the ‘‘buy-and-hold’’ strategy by the largest extent on adverse-performing stocks/stock indices. These results are useful for investors in real life investment. By the EMH, no strategies can beat the ‘‘buy-and-hold’’ strategy (see Section 1). However, the two strategies derived in this study outperform the ‘‘buy-and-hold’’ strategy for HSI and HSP Indices and certain stocks. Does this imply that the EMH no longer holds? The answer to this is uncertain. The time period we cover is December 29, 1995–December 31, 2013, which includes several major financial crises which affect Hong Kong seriously: the Asian financial crisis in 1997–1998, the SARS outbreak in 2003, and the global financial crisis in 2008–2009. In particular, the global financial crisis in 2008–2009 is the most severe financial crisis since the Great Depression in the 1930s. It caused heavy criticism of the EMH. The hypothesis was attacked by a number of scholars such as Jeremy Grantham, Roger Lowenstein and former Federal Reserve chairman Paul Volcker (Wikipedia [27]). They accused the EMH of causing people to underestimate the burst of asset bubbles and over-believe in rational expectations and market efficiency. Hence EMH may not work. From Section 7.2, which tracks the resulting profits of the three strategies at different times along the whole period of observation, we find that Strategies 1 and 2 outperforms the ‘‘buy-and-hold’’ strategy the most during the 1997–98 Asian financial crisis and the 2008–09 global

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Fig. 27. Profit of the three strategies on HSB (without transaction costs).

Fig. 28. Profit of the three strategies on Hutchison (without transaction costs).

financial crisis. The market irrationality during those crises caused people to cast doubt on the EMH. The EMH might have failed during those periods so that our strategies can beat ‘‘buy-and-hold’’. Another possible reason is that Hong Kong’s market has become more closely related to China’s market in recent years. Malkiel [26] warned that Shanghai and Shenzhen’s markets exhibited considerable serial correlation, non-random walk, and evidence of manipulation, and hence were not empirically efficient. Our results may be an indicator showing that Hong Kong’s market may not be efficient, especially during crises, but more evidence is needed to prove this. Regarding future research, one can attempt to find the optimal moving-window size n which maximizes the profit of Strategies 1 and 2. Increasing the moving-window size n would result in a ‘‘smoothing effect’’, reducing the fluctuation of the Shiryaev–Zhou index. By the analysis in Section 7.1, our two strategies would be more profitable, especially when transaction costs exist. However, if n is too large, the estimated value of Shiryaev–Zhou index µ ˆ i would lag behind the stock price even more, so the chance that the stock price is rising when µ ˆ i is negative increases (see Section 7.1), reducing the profits of our two strategies. Another possible scope of future study is to consider whether our trading strategies can beat the ‘‘buy-and-hold’’ strategy for other individual stocks listed in Hong Kong or overseas.

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Fig. 29. Profit of the three strategies on Swire A (without transaction costs).

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Acknowledgement We are grateful for the financial support from the PolyU Internal Research Grants (Project #G-YH96 and 4-ZZC8). References [1] [2] [3] [4] [5] [6] [7]

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