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International Review of Financial Analysis

The effectiveness of the VaR-based portfolio insurance strategy: An empirical analysis☆ Chonghui Jiang a, Yongkai Ma a, Yunbi An b,⁎ a b

School of Management and Economics, University of Electronic Science and Technology of China, Chengdu, Sichuan, China, 610054 Odette School of Business, University of Windsor, Windsor, Ontario, Canada, N9G 3P4

a r t i c l e

i n f o

Article history: Received 11 October 2008 Received in revised form 7 April 2009 Accepted 11 April 2009 Available online 22 April 2009 JEL classiﬁcation: G11

a b s t r a c t This paper proposes an approach to constructing the insured portfolios under the VaR-based portfolio insurance strategy (VBPI) and provides a comprehensive analysis of its hedging effectiveness in comparison with the buy-and-hold (B&H) as well as the constant proportion portfolio insurance (CPPI) strategies in the context of the Chinese market. The results show that both of the insurance strategies are able to limit the downward returns while retaining certain upside returns, and their capabilities of reshaping the return distributions increase as the guarantee or the conﬁdence level rises. In general, the VBPI strategy tends to outperform the CPPI strategy in terms of both the degree of downside protection and the return performance. © 2009 Elsevier Inc. All rights reserved.

Keywords: Value-at-risk Portfolio insurance CPPI

1. Introduction The main objective of portfolio insurance is to hedge against the market downside risk inherently involved in ﬁnancial investments. The most well-known approaches to portfolio insurance include the option-based portfolio insurance (OBPI) method (Leland & Rubinstein, 1976) and the constant proportion portfolio insurance (CPPI) method (Black & Jones, 1987; Black & Perold, 1992). The OBPI strategy involves the investments in a risky asset and in a put option on that risky asset. As a result, the portfolio value will never fall below the strike of the put option regardless of the value of the asset at the terminal date. In the case that the put option is unavailable in a particular market, a synthetic put can be created by dynamically trading the risky and risk-free assets. The CPPI approach allocates assets dynamically over time based on its cushion, which is deﬁned as the difference between the portfolio value and a pre-speciﬁed ﬂoor. The exposure to the risky asset is the product of a predetermined multiple and the cushion; the rest of the fund is invested in the riskfree asset. The ﬂoor can be protected under this strategy, as the insured portfolio will be fully invested in the risk-free asset if the cushion reaches zero due to a signiﬁcant market down move. Such

☆ The authors thank the seminar participants at the 2008 China Finance Review International Symposium for their valuable comments. Jiang and Ma acknowledge the support from the Program for New Century Excellent Talents in Universities funded by the Ministry of Education in China (Contract #: NCET-05-0811). ⁎ Corresponding author. E-mail address: [email protected] (Y. An). 1057-5219/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.irfa.2009.04.001

portfolio insurance strategies appeal to investors because they not only limit the downside risk but also participate in upside markets. The subject of portfolio insurance has been widely investigated in the existing literature with a focus primarily on the classical OBPI and CPPI strategies and their relative performance. Hsuku (2009) analyzes an optimal dynamic OBPI strategy with stochastic volatility. Loria, Pham, and Sim (1991) assess the performance of a futures-based synthetic put strategy using Australian data, and ﬁnd that there is no perfect guarantee of loss prevention under any scenario. Boulier and Kanniganti (1995) evaluate the performance and risks of various modiﬁed CPPI strategies. Lee, Chiang, and Hsu (2008) study a variant of the CPPI strategy and ﬁnd that it yields better performance than the standard CPPI strategy at an expense of more complexity. Balder, Brandl, and Mahayni (2009) examine the impact of discrete-time trading on the performance of the standard CPPI strategy, and conﬁrm Loria et al.'s (1991) ﬁndings. The relative performance of the OBPI and CPPI strategies is documented by Annaert, Osselaer, and Verstraeter (2009), Bertrand and Prigent (2005), and Zhu and Kavee (1988), among others. Matsumoto (2007) focuses on a general portfolio insurance problem with liquidity risk. Some researchers, on the other hand, explore the beneﬁts of these insurance strategies compared with the simple buy-and-hold or stop-loss strategies (e.g., Garcia and Gould (1987), Cesari and Cremonini (2003), Do and Faff (2004), among others). The reported ﬁndings about the performance of portfolio insurance are mixed, but there is evidence that the protection is not perfect in practice due to market imperfections. The risk that the insured portfolio value falls below the protection level at maturity is known as

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gap risk. If the gap risk is substantially high, it may be even worse to employ such a strategy than no insurance at all given the explicit and implicit costs that portfolio insurance entails. Therefore, it is crucial for the insurers to understand the issues related to the gap risk, i.e., how high it is and how it can be controlled, etc. To quantify the risk that portfolio insurers face in a meaningful manner, researchers propose to apply the basic principles of value-at-risk (VaR) to the process of dynamically rebalancing the underlying insured portfolio (see, Chow and Kritzman (2001)). As per the VaR-based strategy, the insured portfolio is constructed and is frequently rebalanced such that the ﬁnal portfolio value exceeds the insured amount at a given conﬁdence level. In contrast with the standard OBPI and CPPI strategies, the VaR-based portfolio insurance (VBPI) approach clearly indicates the gap risk faced by the portfolio insurers. The design of the VBPI strategy is internally consistent with the performance measures for portfolio insurance, as VaR focuses on the downward tail of the return distribution. Furthermore, the method can be viewed as a generalized CPPI strategy and therefore, it provides insurers with an additional ﬂexibility to beneﬁt more from upward market movements while limiting potential downward returns. A profound understanding of its overall effectiveness relative to other standard strategies is of particular interest to practitioners and academics alike. The purpose of this paper is to provide a comprehensive analysis of the VBPI strategy. This is important because the gap risk is the major concern of portfolio insurers. The VBPI strategy not only provides a direct measure of the gap risk but also allows us to analyze how the portfolio should be constructed and rebalanced dynamically to control the gap risk. Previous studies on portfolio insurance focus on the standard OBPI and CPPI strategies and their variants, and the gap risk is mainly examined empirically. It is surprising to note that the VBPI strategy has received limited attention in the literature. The contribution of this paper is three-fold. First, we propose a method for the construction of the insured portfolio under the VBPI strategy in the Black-Scholes (1973) model framework. Some properties of this method are highlighted. Our method is essentially an application of the dynamic risk budgeting approach proposed by Strassberger (2006) to portfolio insurance. Other VaR-based methods (e.g. Zou, Qian, and Yu (2006)) allocate funds between the risk-free and risky assets, such that the return from the risk-free asset can cover the VaR for the risky asset. The VaR has to be estimated based on the assumption about the return distribution for the risky asset in order to build the underlying portfolio. In contrast, our method is only parameterized by the guaranteed value and the conﬁdence level; therefore, it is computationally simple and can be easily implemented in practice. Second, we empirically examine the overall effectiveness of the proposed VBPI approach as portfolio insurance by comparing it with the simple buy-and-hold strategy (B&H) and the standard CPPI strategy under various scenarios. The CPPI strategy serves as a benchmark, as it is simple and model independent. Additionally, CPPI is the only strategy employed by all portfolio insurers in the Chinese market, which is the focus of this analysis. In contrast with the existing studies, some new performance measures reﬂecting the ability to sustain the pre-speciﬁed insured value and to limit the downward returns while retaining upside returns are considered, in order to present a clear picture of the risk proﬁle for each strategy. Moreover, the Omega functions of the portfolio insurance strategies are plotted to further illustrate their relative performance. More importantly, the testing methodology adopted in this paper is designed to parallel the common market practice in the applications of the strategies, in order to capture the essence of gap risk in portfolio insurance. For example, the CPPI strategy examined in the test is the dynamic version of the standard CPPI strategy used by practitioners. Thus, our test procedure is more practical and relevant than those employed in existing literature. The underlying portfolio for each strategy is rebalanced frequently to reﬂect changes in the market

conditions. Apparently, rebalance frequency will have a great impact on the performance of the strategy under consideration. This is because rebalance frequency determines the degree of the strategy's participation in favourable markets and rebalancing entails transaction costs. The parameters chosen by fund managers that reﬂect their risk tolerances may also affect the hedging effectiveness. To demonstrate these effects, different parameter combinations and rebalance frequencies are considered. Third, this paper focuses on the Chinese market, which is one of the most fast-expanding emerging markets in the world. From an empirical perspective, emerging markets are of particular interest. Previous empirical work of portfolio insurance has mainly focused on developed markets rather than emerging markets. For example, Pressacco and Stucchi (1990) evaluate the OBPI strategy based on the Italian stock market, whereas the empirical analysis of Lee et al.'s (2008) research on CPPI uses the data from the U.S. market. On the other hand, Do and Faff (2004) investigate the relative performance of the OBPI and CPPI strategies in the Australia market. Due to various structural, institutional, and other problems, emerging markets may exhibit some unique behaviours and risk characteristics in comparison with mature markets, such as the U.S. and European markets. There is evidence that emerging markets are less liquid, less informationallyefﬁcient, and more volatile than developed markets (Domowitz, Glen, & Madhavan, 1998). Los and Yu (2008) examine the persistence characteristics of the Chinese stock markets and identify the lack of stationarity and ergodicity in the markets. Hammoudeh and Li (2008) investigate the volatility behaviour and persistence in the Gulf Arab stock markets and demonstrate that most of them are more sensitive to major global events than to local and regional factors. Santis and İmrohoroğlu (1997) explore the dynamics of expected stock return and risk in emerging markets including those of Mid-east, Asia, and Latin American, and show that these markets exhibit higher conditional volatility and conditional probability of large price changes than developed markets. Assaf (2009) also documents that it is crucial to characterize the fat tail behaviour in the emerging markets belonging to the Middle East and North African region. Consequently, emerging markets present themselves as an especially valuable case study for examining the effectiveness of our VaR-based portfolio insurance method. We ﬁnd that the VBPI strategy demonstrates a high degree of downside protection as long as the portfolios are not rebalanced too frequently, while the CPPI strategy is not capable of sustaining the ﬂoor value in the presence of transaction costs. Both are able to limit the negative returns while retaining certain high returns in comparison with the B&H strategy, and the return distribution of the insured portfolios is highly right-skewed for reasonable parameters. The degree of protection provided by these strategies increases as the protection level or the conﬁdence level rises. However, this comes at the expense of a decrease in their capability of capturing upward returns. We also ﬁnd that the VBPI strategy generally outperforms the CPPI strategy in terms of both the portfolio protection and the riskreturn trade-off. The remainder of this paper is organized as follows. Section 2 brieﬂy reviews the standard CPPI strategy and describes the proposed VBPI method. Section 3 discusses the datasets and the design of the empirical test. Section 4 reports and analyzes the empirical results, and Section 5 concludes this paper. 2. Portfolio insurance strategies 2.1. The CPPI strategy This section provides a review of the CPPI strategy proposed by Black and Jones (1987) and Black and Perold (1992), as it serves as a benchmark in our analysis. The approach is based on the assumption that the insured portfolio consists of a risky asset and a risk-free asset.

C. Jiang et al. / International Review of Financial Analysis 18 (2009) 185–197

To implement this strategy, the investor ﬁrst sets two parameters: an initial ﬂoor, F0, and a multiple, m. The ﬂoor is the guarantee or the insured value, which is assumed to grow at the risk-free rate, r, over time; namely, Ft = F0ert. The multiple is a constant and is assumed to be greater than one in order to get an option-like payoff function for the insured portfolio. According to this strategy, the amount invested in the risky asset, et, is determined by et = mCt = mðVt − Ft Þ;

ð1Þ

where Ct is the cushion at time t, which is the difference between the portfolio value, Vt, and the ﬂoor, Ft. The rest of the funds are invested in the risk-free asset. Following Do (2002), we impose short sale and credit constraints to make the strategy practical. In other words, the exposure to the risky asset is given by et = max fmin½mðVt − Ft Þ; Vt ; 0g:

ð2Þ

The portfolio is revised at each rebalancing time by shifting between the risky and risk-free assets to bring the exposure back to the same multiple of the cushion. When the market goes up, the cushion will increase, thereby calling for further investing in the risky asset. Conversely, when the market is down, funds will be switched to the risk-free asset due to a decrease in the cushion. If the cushion becomes zero, the portfolio will be fully invested in the risk-free asset and this will protect the portfolio value against falling below the ﬂoor. The exposure to the risky asset is larger if the multiple is higher or the ﬂoor is lower. In this case, the strategy will be more risky, and the investor can participate more in a sustained increase in the market returns at the cost of a lower degree of portfolio protection.

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where μ and σ are the instantaneous expected return and the volatility of the risky asset, respectively, and W is a standard Brownian motion. Moreover, µ N r. Denote the initial portfolio value as V0 and the insured amount as G at maturity T. Typically, the insured amount is assumed to be less than the terminal value of a pure risk-free investment, i.e., G b V0ert. If the proportion of the portfolio invested in the risk-free asset is w, then 1 −w will be the percentage in the risky asset. The units of the risk-free ð1 − wÞV0 0 and risky assets in the portfolio at time 0 are β = wV , B0 ; η = S0 respectively. Suppose that the portfolio is not rebalanced during the investment period [0,T], then the terminal value of the portfolio is given by VT = βBT + ηST

1 2 = wV0 exp ðrT Þ + ð1 − wÞV0 exp μ − σ T + σ WT : 2

ð5Þ

According to the VaR-based approach, we have −vp = −ðEðVT Þ − GÞ;

ð6Þ

where νp is the VaR for the portfolio at the pre-set conﬁdence level p. In this paper, the VaR horizon coincides with the investment period, which makes sense in the context of portfolio insurance. Plugging Eqs. (5) into (6) gives vp = wV0 expðrT Þ + ð1 − wÞV0 expðμT Þ − G:

ð7Þ

Therefore, w can be solved from Eq. (7) as follows w=

G + vp − V0 expðμT Þ : V0 ðexpðrT Þ − expðμT ÞÞ

ð8Þ

2.2. The VBPI strategy In theory, the CPPI strategy is perfect in the sense that the ﬁnal value of the insured portfolio will never fall below the protection level or ﬂoor.1 In practice, this is not always true due to the discrete portfolio adjustments and/or liquidity constraints of the assets. The standard portfolio insurance strategies do not explicitly indicate their abilities to meet the guarantee, one of the primary objectives of these investment strategies. To overcome this shortcoming, some researchers suggest converting optimal asset allocation into VaR assignments (see, Chow and Kritzman (2001)) and seeking to achieve the insurance objective at a given conﬁdence level. The VBPI strategy allocates funds between the risky and risk-free assets based on their return distributions and dynamically rebalances the insured portfolio such that its worst case loss relative to the insured value is equal to the VaR at a conﬁdence level. The VaR can be computed for different conﬁdence levels and insured values which are set by the investors reﬂecting their risk tolerances. Consequently, the ability to meet the protection level becomes a control factor in this strategy; this is its major advantage over the standard portfolio insurance strategies. To illustrate this approach, we assume that the strategy is implemented by trading only in a risky asset and a risk-free asset. The value of the risk-free asset Bt follows dBt = rBt dt;

B0 N 0;

ð3Þ

where r is the risk-free rate, and is assumed to be a constant. The risky asset St follows a geometric Brownian motion under the real probability measure: dSt = St ðμdt + σ dWt Þ;

S0 N 0;

ð4Þ

1 This is also true for the synthetic put OBPI strategy, which is not analyzed in this paper.

Eq. (8) indicates that the higher the investor's VaR, the lower the proportion invested in the risk-free asset and the higher the proportion in the risky asset. As the insured portfolio only consists of the risky and the risk-free assets, the VaR for the portfolio is equal to that for the risky asset. Namely, pﬃﬃﬃ 1 2 −vp = − ð1 − wÞV0 expðμT Þ −ð1 − wÞV0 exp μ − σ T − zp σ T ; 2

ð9Þ where zp is the p-fractile of the standard normal distribution, i.e., Φ(zp) = p. Φ(·)is the standard normal cumulative distribution function. Plugging Eqs. (9) into (8) gives pﬃﬃﬃ 2 G − V0 exp μ − 12 σ T − zp σ T w= pﬃﬃﬃ :

V0 expðrT Þ − V0 exp μ − 12 σ 2 T − zp σ T

ð10Þ

We can see from Eq. (10) that two parameters are needed to describe this strategy: an insured value, G and a conﬁdence level, p. The choice of these implementation variables entails important consequences for the strategy. In particular, a lower G / p implies a riskier strategy, as the exposure to the risky asset is higher in this case. Interestingly, w does not explicitly depend upon νp, indicating that this strategy only adopts the principles of value-at-risk and does not require the estimation of the potential loss of the investment when constructing the portfolio. If the portfolio is not rebalanced until the end of the investment period once it has been constructed, then the method based on Eq. (10) is called the static VBPI strategy. The dynamic strategy, on the other hand, frequently readjusts the portfolio mix based on either market movements or pre-speciﬁed time intervals, so that it can capture higher potential returns in a bullish market and can truncate negative

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returns in a bearish market. Note that the investor's VaR changes whenever the portfolio is rebalanced in such a dynamic strategy. Suppose that the dynamic strategy is employed and at time t = τ the portfolio needs to be rebalanced according to a particular rebalancing criterion. Then, the proportion of the risk-free asset should be revised as G − Vτ exp μ −

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðT − τ Þ − zp σ T − τ wτ = pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ :

Vτ expðr ðT − τ ÞÞ − Vτ exp μ − 12 σ 2 ðT − τÞ − zp σ T − τ 1 2 2σ

ð11Þ Eq. (11) shows that wτ is a decreasing function of the market value of the underlying portfolio, Vτ, in a relatively volatile market. This is quite intuitive, as Vτ reﬂects the market conditions. When the market goes up, a high Vτ implies less investment in the risk-free asset and more investment in the risky asset. When the Vτ value increases above a certain level, the strategy requires investing more funds in the risky assets than available by borrowing at the risk-free rate. In order to make the model more realistic, risk-free borrowing is not permitted in this paper; therefore, wτ is set at zero in the case when it is negative. Conversely, in a falling market, the optimal strategy requires an increase of wτ in order for the guarantee to be honoured at the given conﬁdence level. In contrast with the CPPI strategy, which is model independent, the VBPI strategy relies on a particular assumption about the stochastic process of the risky asset. Furthermore, the VBPI strategy needs the estimation of the expected return and volatility of the risky asset; therefore, its effectiveness depends on the estimation accuracy. From this point of view, the CPPI strategy is easier to implement and this is why it has become very popular with practitioners. However, the VBPI strategy clearly indicates the gap risk that investors face and allows them to dynamically budget the risk based on both the price of asset and the portfolio value. Both the VBPI and CPPI strategies have two user-deﬁned parameters and the choice of these parameters reﬂects the investor's degree of risk aversion. The VBPI is based on the choice of an insured value G and a conﬁdence level p, while the CPPI is based on an initial ﬂoor F0 and a multiple m. A more risk-averse investor will set a higher insured amount and a higher conﬁdence level in the VBPI strategy and higher ﬂoor and lower multiple levels in the CPPI strategy. The converse is true for an investor with a lower degree of risk aversion. The insured value G in the VBPI strategy plays the same role as the ﬂoor in the CPPI method. If the CPPI strategy allows the multiple to change over time, then a critical multiple mτ⁎ can be solved at each rebalancing time τ such that the strategy is equivalent to the VBPI strategy. In other words, the portfolio value under the discrete-time version of the CPPI strategy with a multiple mτ⁎ will stay above the ﬂoor at the conﬁdence level p. In this sense, the CPPI strategy is a special case of the VBPI strategy. 3. The test design and data 3.1. The strategies Three different strategies are investigated herein: the VBPI strategy, the CPPI strategy, and the B&H strategy. The VBPI and CPPI strategies are ﬁrst compared with the B&H strategy to gauge their effectiveness as portfolio insurance. Following Zhu and Kavee (1988), the B&H strategy is deﬁned as 100% of the investment in the risky asset until the end of the investment period. In contrast to the B&H strategy, the portfolio insurance strategies frequently adjust the amount invested in the risky asset based on the market movements. In particular, these approaches take on a risky character in a bullish market and a conservative one in a bearish market.

It is important to examine the advantages of the insurance strategies over the B&H strategy in terms of the downside risk hedge and the return performance. For this purpose, various insured amounts as well as conﬁdence levels are considered. This also enables us to investigate the impacts of these user-set parameters on the effectiveness of portfolio insurance strategies. Second, the VBPI strategy is compared with the standard CPPI strategy to see their relative hedging effectiveness. The VBPI strategy has different theoretical foundations than does the CPPI strategy, although both yield a similar payoff structure. Due to its simplicity, the CPPI strategy has been widely used among practitioners and it is the only method adopted by all portfolio insurers in the Chinese markets. Consequently, a comparison of the performance of the VBPI strategy with that of the CPPI strategy is particularly practical and relevant in the Chinese market context. To this end, the initial ﬂoor in the CPPI strategy is set equal to the present value of the insured amount in the VBPI strategy, so that both strategies provide the identical guarantee at the end of the insurance period. The multiple in the CPPI strategy should also be carefully chosen, as it has a great impact on the performance. The strategy with a higher multiple can generate higher expected returns but it provides a lower degree of ﬂoor protection. To make both strategies comparable in different scenarios, multiples in the CPPI strategy are chosen such that the strategy's initial percentage in the risky asset is the same as that under the corresponding VBPI strategy (see, Do (2002) for a similar treatment). 3.2. Portfolio rebalance disciplines Portfolio insurance induces dynamic management of the underlying portfolio. In general, the insured portfolios can be rebalanced based on three different disciplines: the time discipline, the market move discipline, and the portfolio mix discipline. The ﬁrst method requires a portfolio rebalance at predetermined time intervals, such as daily, weekly, or monthly, etc. The second method triggers a rebalance when the market has a pre-speciﬁed percentage move since the last rebalance, and the third one entails adjusting the portfolio whenever the difference between the required and the current portfolio mix exceeds a trigger point. It is well known that different rebalance disciplines may lead to different performances for a given portfolio insurance strategy (see, Etzioni (1986)). Typically, the strategy with higher frequency of portfolio rebalancing will provide better hedge against downside risk than that with lower rebalancing frequency but it will entail higher transaction costs. In this paper, to examine how the rebalancing frequency may affect the hedging effectiveness of different strategies, portfolios are rebalanced at three predetermined time intervals: daily, weekly and monthly.2 3.3. Performance measures As portfolio insurance limits downside losses while still retaining upward gains, the return distribution is asymmetric with a short left tail and a long right tail. For this reason, a risk-return analysis is inappropriate in the portfolio insurance context. To provide a profound understanding of the hedging effectiveness, each insurance strategy is assessed in terms of its ability to sustain the insured value and to reshape the return distribution. To this end, the following measures are considered: (1) Protection ratio. Following Do (2002), the protection ratio is deﬁned as the percentage of the realized portfolios that meet the insured value under a particular strategy. Similar to the protection level error used by Zhu and Kavee (1988), this measure shows the ability of the strategy in sustaining a pre2

In this paper, we deﬁne ﬁve trading days as a week and four weeks as a month.

C. Jiang et al. / International Review of Financial Analysis 18 (2009) 185–197

speciﬁed guarantee, the chief purpose of portfolio insurance. However, it does not present the asymmetric characteristics of the distribution created by the strategy. (2) 5% vintile (V5) and average value for the poorest-performed 5% portfolios (AV5). If we sort the realized portfolio values in ascending order and then divide them into 20 equal groups, AV5 is the average of the values in the ﬁrst group and V5 is the highest value in this group. Both measures focus on the downward tail of the frequency distribution, indicating the ability of the strategy to control the downside risk. From this viewpoint, they are conceptually similar to the VaR and expected shortfall (ES) at a conﬁdence level of 95%. Contrary to VaR and ES, higher V5 and AV5 values indicate that the strategy is more effective in limiting downward returns. (3) 75% quartile (Q75) and the average value for the bestperformed 25% portfolios (AQ75) are used to measure the ability of the strategy to capture upside returns. Apparently, the higher the 75% quartile, the more appealing the insurance strategy in the sense that it can retain higher returns in the upward markets. In addition to the above performance measures, we also analyze the traditional risk-return performance, the whole return distributions, and the Omega functions for each strategy in various cases. 3.4. Transaction costs Transaction costs will apparently affect the performance of these strategies. To investigate how big this effect is, we consider the cases with and without transaction costs. In China, the transaction costs include commission fees and taxes. The taxes imposed on security transactions may be re-adjusted from time to time. In January, 2005 the tax rate was reduced to as low as 0.1%, and commission fee was generally 0.08%; therefore, the total transaction cost of risky assets is set equal to 0.18% in our analysis. The transaction fees of government bonds, however, were no more than 0.1% in our sample period. 3.5. Test procedure Assume that the investment period is from time t = 0 to t = T and the initial investment is V0. The implementation procedure for the VBPI strategy can be described as follows: (1) At time t = 0, calculate the proportion w0 according to Eq. (10), and then construct the insured portfolio by investing (1 − w0) V0 in the risky asset and the rest in the risk-free asset. In the presence of transaction costs, the optimal amount invested in w0 ÞV0 ð1 − f1 Þ the risky asset becomes w0 ð1 −ð1 f− , where f1 and 2 Þ + ð1 − w0 Þð1 − f1 Þ f2 are the transaction costs of risky and risk-free assets, respectively. (2) On the next rebalancing date, record the portfolio value, which is the sum of the investments in the risk-free asset and in the risky asset adjusted for dividends. Then, reconstruct the portfolio based on the information available on this date. This step continues until the end of the investment period, t = T. (3) The above procedure tracks the portfolio values for one realization of the strategy considered. In order to perform an empirical analysis, the procedure is re-employed every trading day in the sample period and consequently we obtain a series of realizations with overlapped insurance periods in each scenario for each strategy. The CPPI strategy can be implemented in a similar fashion. Finally, performance measures are calculated based on these realizations and are used to evaluate the strategies under consideration. This test procedure is similar to the one adopted by An and Suo (2008, 2009).

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Fig. 1. The time series of CITIC-S&P 300 index and government bond index. Note: The daily data is from December 30, 1999 to December 28, 2007, 1926 observations for each series.

3.6. Data description The datasets used for this analysis include the China International Trust and Investment Corporation (CITIC) S&P 300 index and the CITIC-S&P government bond index from the Chinese market for the period from December 30, 1999 to December 28, 2007, obtained from CITIC-S&P service center website. The sample period contains both the bearish stock market from 2001 to early 2006 and the bullish market in 2007 in China. The CITIC-S&P 300 index is used as the risky asset and the CITIC government bond index is the proxy for the risk-free asset. There are 1926 observations in each data series. Fig. 1 displays the trends for the two series of indices. The daily return of the risky asset is deﬁned as rt = lnðSt = St − 1 Þ;

t = 1; 2; N ; n;

where rt is the return at date t, and St is the risky asset price at date t. The expected return and volatility are approximated by the annualized average return and standard deviation of the daily returns in the insurance period, respectively. Consequently, different estimates are used for different investment periods. 4. Empirical results In this analysis, the initial investment is $10,000 and the investment period is three months. Other implementation variables are set as follows: • Insured values/ﬂoors, G / FT: $9,800 (or 98%), and $10,000 (or 100%); • Conﬁdence levels, p: 90%, 95%, and 99%; As the procedure in this paper is re-employed on each trading day in the sample period, a total of 1866 different realizations are obtained in each scenario for each strategy and these trials are used to conduct our analysis. 4.1. The effectiveness of portfolio insurance strategies Table 1 reports the protection ratios for various strategies under different scenarios. From these results we can see that the degree of portfolio protection depends upon both the implementation variables in the insurance strategies and the portfolio rebalancing frequency. Not surprisingly, the protection ratio is higher when the conﬁdence level, p, is higher. This is because a higher conﬁdence level, which means a tighter VaR constraint, requires more investments in the riskfree asset in the VBPI portfolio, thereby resulting in higher protection ratios. A high p implies a low multiple in the CPPI strategy, which will increase the probability of the ﬂoor being met at maturity. However, it

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Table 1 Floor protection ratios for various strategies. B&H

Daily rebalancing

Weekly rebalancing

Monthly rebalancing

VBPI

VBPI

CPPI

VBPI

CPPI

Panel 1: G = 98%, without transaction costs p = 0.90 59.59% 50.13% 94.25% p = 0.95 59.59% 52.28% 97.31% p = 0.99 59.59% 70.88% 100.00%

CPPI

97.58% 98.39% 99.41%

84.58% 93.01% 99.25%

99.95% 100.00% 100.00%

67.06% 76.36% 92.69%

Panel 2: G = 98%,with p = 0.90 59.11% p = 0.95 59.11% p = 0.99 59.11%

90.65% 95.16% 98.93%

42.45% 50.78% 75.93%

99.84% 100.00% 100.00%

60.08% 69.96% 87.32%

Panel 3: G = 100%, without transaction costs p = 0.90 54.76% 47.13% 54.33% 96.56% p = 0.95 54.76% 45.67% 72.00% 97.80% p = 0.99 54.76% 54.70% 93.07% 98.76%

29.34% 39.01% 72.54%

99.95% 100.00% 100.00%

37.18% 39.66% 60.08%

Panel 4: G = 100%, with transaction costs p = 0.90 54.16% 18.75% 2.31% p = 0.95 54.16% 16.01% 1.61% p = 0.99 54.16% 7.20% 0.05%

18.11% 17.62% 17.89%

96.99% 99.68% 100.00%

34.55% 35.09% 43.90%

transaction costs 21.49% 9.30% 18.65% 8.06% 10.69% 5.64%

47.39% 39.39% 29.34%

Note: This table reports the protection ratios for various strategies in different cases. The guaranteed amount G is expressed in percentages of the initial investments, and p is the conﬁdence level.

is more difﬁcult to sustain a higher guarantee than a lower one for any given conﬁdence level, for both strategies. Contrary to intuition, the performance of the VBPI strategy improves as the rebalancing frequency decreases and this is true regardless of whether or not there are transaction costs. This is primarily because its hedging effectiveness depends greatly on the model assumption about the underlying stochastic process. Model errors may lead to an over/under protected portfolio, and this effect is larger when the portfolios are managed more strictly. In the absence of transaction costs, the CPPI strategy performs better when the rebalancing frequency is higher, which is in line with what the model predicts. The reason is that the composition of the CPPI portfolio is model independent. In general, the higher the rebalancing frequency, the more effective the strategy is. As we mentioned in Section 2, the CPPI strategy guarantees the ﬂoor value if the portfolio is rebalanced continuously. With transaction costs, however, both insurance strategies improve their performance when

the rebalancing frequency decreases, as a result of the cost savings. The extremely poor performance of both strategies in the case of daily rebalancing indicates that the transaction costs are unbearable when the portfolios are rebalanced too frequently. For example, only 10% of the total realizations under the VBPI strategy sustain the insured value when the conﬁdence level is 99%. In addition, the protection ratios of both the VBPI and the CPPI strategies with transaction costs are generally lower than those in the corresponding cases without transaction costs, and this is much more pronounced for the CPPI strategy. To see the extent to which the insurance strategies provide protection against downward risk relative to the non-insured investments, we similarly compute the so-called “protection ratio” for the B&H strategy. The results illustrate that both the insurance strategies generate higher protection ratios than the simple B&H strategy when the transaction costs are ignored. When there are transaction costs, the VBPI strategy outperforms the B&H strategy in the cases of weekly and monthly rebalancing, but under-performs the B&H strategy in the case of daily rebalancing. This implies that the VBPI strategy is better than the B&H strategy in terms of portfolio protection, so long as the portfolios are not rebalanced too frequently. The performance of the CPPI strategy is poorer than that of the B&H strategy for most of the scenarios when the transaction costs are considered. In this sense, the CPPI method belies its name, and is essentially worse than no insurance at all due to the unbearable transaction costs. This result is in line with the ﬁndings of Zhu and Kavee (1988). Table 2 presents the results of V5 and AV5 for various strategies. As pointed out in Section 3, these measure the ability of the insurance strategies to control the downside risk. The results show that when compared with the B&H strategy, both insurance strategies are indeed able to limit negative returns, as they have higher V5 and AV5 values for all the cases. The V5 and AV5 values for these insurance strategies also depend on the implementation variables, increasing with both the conﬁdence level and the insured/ﬂoor value. It follows that the insurance strategies are more effective in truncating downward returns when either G is higher or the VaR constraint is tighter. Intuitively, a higher G or p limits the potential room for risk budgeting, and leads to relatively higher investments in the risk-free asset in downside markets. An investor will choose a higher conﬁdence level or insured value for the insurance strategy to seek a higher degree of

Table 2 5% Vintile (V5) and the average values for the poorest-performed 5% vintile (AV5) realizations for various strategies. V5 B&H

AV5 Daily rebalancing

Weekly rebalancing

VBPI

VBPI

Panel 1: G = 98%, without transaction p = 0.90 84.45 93.77 p = 0.95 84.45 94.96 p = 0.99 84.45 96.21

CPPI

Monthly rebalancing CPPI

VBPI

B&H CPPI

Daily rebalancing

Weekly rebalancing

VBPI

CPPI

VBPI

Monthly rebalancing CPPI

VBPI

CPPI

costs 98.00 98.05 98.33

98.32 98.52 98.82

97.60 97.92 98.25

98.76 98.93 99.18

95.73 96.60 97.80

81.59 81.59 81.59

92.12 93.25 94.44

97.62 97.85 98.25

97.85 98.13 98.53

96.74 97.25 97.95

98.61 98.81 99.05

94.20 95.45 97.20

Panel 2: G = 98%, with transaction costs p = 0.90 84.30 89.22 93.14 p = 0.95 84.30 90.14 93.34 p = 0.99 84.30 91.84 93.69

97.87 98.01 98.21

96.73 97.13 97.58

98.55 98.72 98.94

95.58 96.45 97.60

81.45 81.45 81.45

87.42 88.5 89.96

92.69 92.96 93.45

97.38 97.58 97.94

95.87 96.35 97.14

98.43 98.61 98.84

94.08 95.29 97.02

Panel 3: G = 100%, without transaction costs p = 0.90 84.45 95.67 99.12 p = 0.95 84.45 96.51 99.47 p = 0.99 84.45 97.68 100.00

100.08 100.14 100.22

98.04 98.62 99.61

100.21 100.26 100.33

96.21 97.22 99.09

81.59 81.59 81.59

93.73 94.90 95.85

98.16 98.78 99.46

99.82 99.91 100.05

96.71 97.10 98.68

100.17 100.22 100.29

93.70 95.16 98.06

Panel 4: G = 100%, with transaction costs p = 0.90 84.30 89.99 94.09 p = 0.95 84.30 90.88 94.29 p = 0.99 84.30 92.21 94.54

99.31 99.33 99.34

97.42 97.89 98.70

100.02 100.06 100.11

96.15 97.22 98.92

81.45 81.45 81.45

88.20 89.24 90.48

93.35 93.64 94.36

99.11 99.16 99.22

95.95 96.32 97.87

99.97 100.03 100.08

93.65 95.25 97.98

Note: This table reports the V5 and AV5 values for various strategies in different cases. The guaranteed amount G and the results are expressed in percentages, and p is the conﬁdence level.

C. Jiang et al. / International Review of Financial Analysis 18 (2009) 185–197

191

Table 3 75% quartile (Q75) and the average values for the best-performed 25% quartile (AQ75) of realizations for various strategies. Q75 B&H

AQ75 Daily rebalancing

Weekly rebalancing

VBPI

CPPI

Monthly rebalancing

B&H

Daily rebalancing

Weekly rebalancing

Monthly rebalancing

VBPI

CPPI

VBPI

CPPI

VBPI

CPPI

VBPI

CPPI

VBPI

CPPI

Panel 1: G = 98%, without transaction costs p = 0.90 113.43 107.14 102.49 p = 0.95 113.43 101.42 102.41 p = 0.99 113.43 100.58 102.56

112.97 111.02 108.42

103.60 102.95 102.64

111.91 110.50 106.22

105.84 104.69 103.55

130.89 130.89 130.89

129.43 126.92 118.23

121.45 119.84 115.44

130.70 129.89 125.41

122.66 120.71 115.66

130.61 129.76 124.31

121.48 119.18 113.43

Panel 2: G = 98%, with transaction costs p = 0.90 113.23 95.83 94.97 p = 0.95 113.23 95.38 95.08 p = 0.99 113.23 95.27 95.19

110.78 108.88 105.55

100.87 100.60 100.74

111.51 109.91 105.66

105.07 104.10 102.99

130.66 130.66 130.66

115.54 113.45 106.42

103.49 101.90 98.38

127.94 127.12 122.52

118.85 116.83 111.86

129.92 129.06 123.61

120.68 118.36 112.67

Panel 3: G = 100%, without transaction costs p = 0.90 113.43 106.57 100.02 p = 0.95 113.43 101.02 100.12 p = 0.99 113.43 100.79 100.83

113.00 110.64 104.31

100.06 100.39 100.89

111.80 108.19 102.45

104.38 103.48 102.14

130.89 130.89 130.89

128.66 125.86 115.47

116.06 115.38 112.26

130.65 129.33 122.38

119.33 117.87 112.91

130.49 128.95 119.83

119.45 117.51 110.61

Panel 4: G = 100%, with transaction costs p = 0.90 113.23 95.33 94.91 p = 0.95 113.23 95.19 94.93 p = 0.99 113.23 95.02 94.96

111.17 108.55 100.66

99.31 99.41 99.69

111.2 107.37 101.86

103.46 102.58 101.66

130.66 130.66 130.66

114.54 111.88 104.63

97.88 96.97 95.30

127.95 126.64 119.08

114.75 113.11 107.69

129.8 128.24 119.03

118.55 116.48 109.61

Note: This table reports the Q75 and AQ75 values for various strategies in different cases. The guaranteed amount G and the results are expressed in percentages, and p is the conﬁdence level.

portfolio protection if he/she is more risk averse. The rebalancing frequency effects are different for the VBPI and CPPI strategies. In general, a higher rebalancing frequency leads to lower V5 and AV5 under the VBPI strategy. This is true even if the transaction costs are present and is also true for any conﬁdence levels and for different insured values. For the CPPI strategy, this still holds when there are transaction costs; however, the reverse is true in the absence of transaction costs. To investigate the capability of these strategies in retaining the upside returns, Q75 and AQ75 are reported in Table 3. It seems that the effects of portfolio rebalancing frequency on the Q75 and AQ75 values are similar to those found in Table 2. The strategies are more capable of capturing the upside returns when the portfolios are rebalanced weekly and monthly than when rebalanced daily. This is primarily because the transaction costs are higher in the case of daily rebalancing. Another explanation for this is that a low rebalancing frequency can avoid unnecessary switches from the risky asset to the

risk-free asset when the market moves down temporarily and will recover very soon. However, higher conﬁdence levels and insured values generally lead to lower Q75 and AQ75 results, indicating that their effects on the Q75 and AQ75 results are opposite to those on the V5 and AV5 values. Consequently, there is a trade-off between the strategy's capability of limiting downside returns and the capability of retaining upward returns. Furthermore, the results illustrate that the Q75 and AQ75 values under the VBPI and CPPI strategies are lower than those in the corresponding cases under the B&H strategy, and these differences reﬂect the implicit costs inherent in portfolio insurance. The above ﬁndings are not surprising, as insurance strategies keep part of the funds in the risk-free asset as “hedge”, and a higher degree of portfolio protection requires a larger size of “hedge”, which will surely affect the portfolio's capability of capturing the upside returns of the risky asset. Overall, the protection against the market losses can only be achieved by giving up some upward potential returns.

Fig. 2. Frequency distributions under the B&H strategy. Note: This ﬁgure plots the sample distributions for the ﬁnal portfolio values under the B&H strategy for the cases with and without transaction costs.

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To clearly illustrate the capability of the portfolio insurance strategies in reshaping the return distributions, the sample distributions for the ﬁnal portfolio values under different strategies are plotted. Fig. 2 is the plot for the B&H strategy and Fig. 3 is for the insurance strategies. For illustration purposes, only distributions for the insurance strategies with transaction costs in the case of weekly rebalancing are displayed.

Obviously, both the VBPI and the CPPI strategies have a desirable risk proﬁle with a short left tail in their distributions compared with the B&H strategy. Table 4 presents some statistics of the sample distributions in different cases for various strategies. For both of the insurance strategies, the results show that the effects of implementation variables and

Fig. 3. Frequency distributions under portfolio insurance strategies. Note: This ﬁgure plots the sample distributions for the ﬁnal portfolio values under the VBPI and CPPI insurance strategies for different combinations of the implementation variables. The portfolios are rebalanced weekly in the presence of transaction costs.

C. Jiang et al. / International Review of Financial Analysis 18 (2009) 185–197

193

Table 4 Statistics of sample distributions for various strategies. B&H

Daily rebalancing

Weekly rebalancing

Monthly rebalancing

VBPI

CPPI

VBPI

CPPI

VBPI

CPPI

Panel 1: G = 98%, without transaction costs p = 0.9 Return 0.0425 Stdev 0.1564 Skewness 0.8299 p = 0.95 Return 0.0425 Stdev 0.1564 Skewness 0.8299 p = 0.99 Return 0.0425 Stdev 0.1564 Skewness 0.8299

0.0445 0.1355 1.7035 0.0380 0.1308 1.9705 0.0252 0.1078 2.8214

0.0379 0.1067 2.3136 0.0364 0.1006 2.4170 0.0318 0.0821 2.7076

0.0770 0.1238 1.6298 0.0736 0.1223 1.7536 0.0619 0.1108 2.1736

0.0393 0.1090 2.1348 0.0374 0.1018 2.2498 0.0319 0.0820 2.6009

0.0777 0.1225 1.6835 0.0735 0.1214 1.8153 0.0590 0.1077 2.3176

0.0371 0.1044 1.8963 0.0351 0.0950 2.0641 0.0277 0.0698 2.3800

Panel 2: G = 98%,with transaction costs p = 0.9 Return Stdev Skewness p = 0.95 Return Stdev Skewness p = 0.99 Return Stdev Skewness

− 0.0198 0.1116 2.0124 − 0.0219 0.1063 2.2981 − 0.0343 0.0863 3.4698

− 0.0405 0.0697 3.7999 − 0.0428 0.0605 4.1796 − 0.0486 0.0332 5.2556

0.0621 0.1195 1.7165 0.0577 0.1185 1.8427 0.0454 0.1070 2.3159

0.0238 0.1019 2.3472 0.0215 0.0944 2.4915 0.0151 0.0737 2.9354

0.0735 0.1217 1.7024 0.0693 0.1206 1.8348 0.0549 0.1069 2.3462

0.0332 0.1030 1.9407 0.0310 0.0934 2.1109 0.0237 0.0682 2.4364

Panel 3: G = 100%, without transaction costs p = 0.9 Return 0.0425 Stdev 0.1564 Skewness 0.8299 p = 0.95 Return 0.0425 Stdev 0.1564 Skewness 0.8299 p = 0.99 Return 0.0425 Stdev 0.1564 Skewness 0.8299

0.0542 0.1277 1.8595 0.0480 0.1228 2.1242 0.0288 0.0987 3.3219

0.0333 0.0922 2.9341 0.0326 0.0889 3.0049 0.0280 0.0746 3.3139

0.0820 0.1204 1.7112 0.0744 0.1189 1.9001 0.0547 0.1055 2.6541

0.0379 0.0999 2.5129 0.0361 0.0942 2.6149 0.0288 0.0743 3.0859

0.0779 0.1208 1.7917 0.0703 0.1194 1.9973 0.0485 0.1003 2.9591

0.0360 0.0963 2.0638 0.0342 0.0884 2.2608 0.0245 0.0580 2.8636

Panel 4: G = 100%, with transaction costs p = 0.9 Return Stdev Skewness p = 0.95 Return Stdev Skewness p = 0.99 Return Stdev Skewness

− 0.0187 0.1086 2.1618 − 0.0219 0.1013 2.5280 − 0.0363 0.0812 3.9096

− 0.0484 0.0408 6.8635 − 0.0499 0.0324 8.2592 − 0.0526 0.0040 4.3832

0.0680 0.1164 1.7696 0.0593 0.1153 1.9670 0.0376 0.1023 2.8000

0.0213 0.0916 2.8465 0.0191 0.0851 3.0003 0.0104 0.0616 3.8558

0.0736 0.1201 1.8094 0.0662 0.1186 2.0150 0.0444 0.0996 2.9968

0.0320 0.0946 2.1246 0.0302 0.0862 2.3504 0.0205 0.0553 3.0027

0.0407 0.1564 0.8299 0.0407 0.1564 0.8299 0.0407 0.1564 0.8299

0.0407 0.1564 0.8299 0.0407 0.1564 0.8299 0.0407 0.1564 0.8299

Note: This table presents the mean, standard deviation, and skewness of the sample return distributions for various strategies under different scenarios. The guaranteed amount G is expressed in percentages, and p is the conﬁdence level.

rebalancing frequency on the average returns are generally similar to those on the Q75 and AQ75 values. However, the VBPI strategy generates higher average returns when the insured value is higher for any given conﬁdence level. The standard deviations for both strategies decrease while the conﬁdence level or insured value increases, conﬁrming the fact that higher conﬁdence levels and insured values imply a less risky insurance strategy. Compared with the B&H strategy, the VBPI strategy has lower standard deviations, but higher average returns when the portfolios are rebalanced weekly and monthly, and this is also true for the case of daily rebalancing in the absence of transaction costs. The CPPI strategy generally has lower standard deviations and lower average returns than the B&H strategy. As a result, the VBPI strategy dominates the B&H strategy in terms of risk-return performance, yet it is not clear whether the CPPI strategy or the B&H strategy should be preferred by the riskaverse investors. The high skews for both the VBPI and the CPPI strategies (in comparison with those for the B&H strategy) reconﬁrm their right-skewed asymmetric return distributions, observed in Fig. 3. 4.2. The VBPI strategy versus the CPPI strategy Now we analyze the relative performance of the VBPI and CPPI strategies under different scenarios. By comparing the protection

ratios for both insurance strategies in Table 1, we can see that the protection ratios of the VBPI strategy are higher than those of the CPPI strategy for most cases. The only exception is when the portfolios are rebalanced daily with no transaction costs. Therefore, in terms of the insured value protection, the VBPI strategy is generally better. The V5 and AV5 results in Table 2 show that the CPPI strategy is more capable of limiting the downside returns than the VBPI strategy in the case of daily rebalancing regardless of whether or not the transaction costs are present. However, when we move from daily rebalancing to weekly or monthly rebalancing, the VBPI strategy dominates the CPPI strategy. The differences in the V5 and AV5 values between both strategies are maximized when the conﬁdence level and rebalancing frequency are lower and the insured values are higher. Table 3 shows that the VBPI strategy generally generates higher Q75 and AQ75 values than the CPPI strategy in almost all of the cases, indicating that the former strategy outperforms the latter one in terms of their capabilities of capturing upward returns. The better performance of the VBPI strategy is generally expected, as it can be viewed as a generalized version of the CPPI strategy with a variable multiple. As a result, it provides the insurers with more ﬂexibility to obtain the beneﬁts from upward market movements than does the CPPI strategy. Omega (Keating & Shadwick, 2002) is a relatively new measure of portfolio performance, which is the ratio of the averages of the

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gains above a threshold to the averages of the losses below the same threshold. For any given threshold, the portfolio with a higher Omega fares better than the one with a lower Omega. As pointed out by Bacmann and Scholz (2003), Omega involves all the moments of the return distribution including skewness and kurtosis and therefore, it is an appropriate indicator of the effectiveness of insurance

strategies. To explore the relative performance of different strategies in terms of the Omega measure, we plot their Omega as a function of the threshold, L, under different scenarios in Figs. 4–6. The Omega functions show that when L is low the VBPI strategy is clearly superior to the CPPI strategy in the cases of weekly and monthly rebalancing. However, the Omega curve for the VBPI strategy lies

Fig. 4. Omega as a function of the threshold: daily rebalancing. Note: This ﬁgure plots the Omega functions of both the VBPI and the CPPI strategies for different cases when the portfolios are rebalanced daily. The ﬁrst six cases are those without transaction costs and the last six with transaction costs.

C. Jiang et al. / International Review of Financial Analysis 18 (2009) 185–197

195

Fig. 5. Omega as a function of the threshold: weekly rebalancing. Note: This ﬁgure plots the Omega functions of both the VBPI and the CPPI strategies for different cases when the portfolios are rebalanced weekly. The ﬁrst six cases are those without transaction costs and the last six with transaction costs.

below that for the CPPI strategy in the case when the portfolios are rebalanced daily. Their difference in the Omega values becomes relatively small as L increases. Bertrand and Prigent (2006) demonstrate that a lower threshold level implies that the investors are more concerned with risk control, and a higher level means that they worry more about the return performance of the portfolio. The results suggest that the VBPI strategy is especially preferable for the

more risk-averse investors and for the case of low rebalancing frequency. 5. Conclusions This paper proposes a method for constructing the insured portfolios under the VBPI strategy, assuming that the risky asset is driven by a

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Fig. 6. Omega as a function of the threshold: monthly rebalancing. Note: This ﬁgure plots the Omega functions of both the VBPI and the CPPI strategies for different cases when the portfolios are rebalanced monthly. The ﬁrst six cases are those without transaction costs and the last six with transaction costs.

geometric Brownian motion. The method is then compared with the CPPI strategy and the B&H strategy to gauge its effectiveness in terms of both the portfolio protection and the return performance. Using the Chinese market data, our analysis illustrates that the VBPI strategy demonstrates a high degree of portfolio protection as long as the portfolios are not rebalanced too frequently. However, the CPPI strategy performs poorly in sustaining the ﬂoor value when

transaction costs are present. Both are able to hedge against the downside risk in the sense that they indeed truncate the left tail of the return distribution. Moreover, their capability of reshaping the return distributions depends on the implementation variables, which are set by the portfolio insurers reﬂecting their degrees of risk aversion. In particular, the higher the insured value or the conﬁdence level, the more effective the strategy is in terms of downside return limitation.

C. Jiang et al. / International Review of Financial Analysis 18 (2009) 185–197

However, the hedge comes at the cost of low average upside returns compared with the B&H strategy. The conclusions are conﬁrmed by the frequency return distributions generated by these strategies. Regarding the relative performance of the VBPI and CPPI strategies, the VBPI strategy is generally more effective than the CPPI strategy in terms of portfolio protection. The VBPI strategy is less capable of limiting downside returns than the CPPI strategy when daily rebalancing is applied, but performs better in other cases. When measured in terms of the ability to retain upside returns, the VBPI strategy is always better. Additionally, the Omega functions display that the VBPI strategy is especially preferred by more risk-averse investors. The above ﬁndings are much more pronounced when there are transaction costs, indicating that the VBPI strategy is preferable to the CPPI strategy if investors face relatively high transaction costs. As with any other investment strategies relying on a particular assumption about the asset price dynamics, a problem associated with the VBPI method is the model risk, arising from the discrepancies between the real world and the one assumed in the model. This paper focuses on a simple analytically tractable case, where the price movements are continuous and the returns are normal. This explains in part the poor performance of the VBPI strategy measured by the V5 and AV5 values in the case of daily portfolio rebalancing. An interesting extension to this paper is to evaluate the VBPI strategy in the presence of occasional discontinuous price jumps, in particular the possibility of sharp falls in market prices. As the implementation of the VBPI strategy needs estimation for the market parameters such as the expected return and volatility of the risky asset, it is also worth examining how the accuracy of these estimates may affect its hedging effectiveness by conducting a simulation analysis. These issues are left for future research. References An, Y., & Suo, W. (2008). The compatibility of one-factor market models in caps and swaptions markets: evidence from their dynamic hedging performance. Journal of Futures Market, 28(2), 109−130. An, Y., & Suo, W., (forthcoming). An empirical comparison of option pricing models in hedging exotic options. Financial Management. Annaert, J., Osselaer, S. V., & Verstraete, B. (2009). Performance evaluation of portfolio insurance strategies using stochastic dominance criteria. Journal of Banking and Finance, 33(2), 272−280. Assaf, A., (forthcoming). Extreme observations and risk assessment in the equity markets of MENA regions: tail measures and value-at-risk. International Review of Financial Analysis. doi:10.1016/j.irfa.2009.03.007. Bacmann, J. F., & Scholz, S. (2003). Alternative performance measures for hedge funds. AIMA Journal, 1(1), 1−9. Balder, S., Brandl, M., & Mahayni, A. (2009). Effectiveness of CPPI strategies under discrete time trading. Journal of Economics and Control, 33(1), 204−220.

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