Computing the optimal rebalance interval and the optimal boundary for different investment horizons

Computing the optimal rebalance interval and the optimal

boundary for different investment horizons

JEL classifications: C15, G11, G17

Keywords: rebalance portfolio; transaction costs; moving blocks bootstrap

Author: Roelof Waalkens Msc. thesis Finance (EBM866B20)

University of Groningen, faculty of Economics and Business Student number: 1793624

Supervisor: A. Plantinga Date: 26-6-2014

Abstract

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1. Introduction

Since 2002, Dutch financial institutions have to inform investors about the risks of the financial instruments they offer. This information has to be published in the so called “financial information leaflet”, a simplified version of a prospectus. The financial information leaflet was implemented in 2002 by the Dutch Authority Financial Markets. The financial information leaflet is used by financial institutions to inform their clients about risks of the product, which they have to publish on their website. Furthermore, they have to send the information leaflet costless to clients if they ask for it. Nowadays, financial institutions have to indicate the risk profile of the instruments they offer. Usually, there are five risk classes ranging from very defensive to very offensive. The portfolios differ in allocation to stocks, bonds, cash and alternative investments.

Over time, as a portfolio’s different investments produce different returns, the portfolio drifts away from its target asset allocation. The new allocation acquires risk and return characteristics, which could be inconsistent with the investor’s goals and preferences. Therefore, a rebalancing strategy is needed to control the portfolio risk and return characteristics (Ritter and Chopra, 1989).

Comparable with the selection of a portfolio allocation, a rebalancing strategy involves a trade-off between risk and return. Rebalancing results in lower risk, but also in higher costs and therefore, lower returns. The contradiction between portfolio allocation and rebalancing strategy is, in portfolio allocation risk and return are absolutes. Where risk and return are measured relative to the performance of the target asset allocation in choosing a rebalancing strategy (Pliska and Suzuki, 2004). Result of the trade-off between risk and return is that there is an optimal rebalancing strategy for every investor, depending on their risk tolerance and investment horizon. This paper will investigate this optimal strategy.

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In practice, professionals are not unanimous. According to van Zadelhoff, director Privat Banking at Triodos Bank, Triodos Bank rebalances when in advance set boundaries are exceeded, so they use the threshold rebalancing strategy. The rebalancing policy at van Landschot Bank is not determined. They give portfolio managers freedom to rebalance when they think it is needed. The asset allocation strategy used in this paper is the constant mix strategy. The constant mix maintains the exposure of financial assets to a constant proportion of wealth, due to rebalancing (Perold and Sharpe, 1988). Constant mix strategies benefit from mean reversion in which the buy and hold strategy benefit from momentum. Because financial institutions offer their clients a portfolio with a risk profile, they are restricted to use a constant mix strategy because otherwise the portfolio allocation will drift away from their initial allocation to an allocation of another risk profile.

Furthermore, this paper will indicate the importance of the investment horizon. According to Conrad and Kaul (1998), different investment horizons have different optimal strategies. Momentum strategies show significant better results at medium horizons, which is 3 to 12-months. Mean reversion strategies show better results at long horizons, which is 3 years or longer. So when an investor has a medium horizon, investing in a portfolio with a risk profile could have a negative effect on their utility, because risk profile based portfolios have to use a constant mix strategy to maintain the asset allocation in its risk profile. However, Kritzman (1994) state that under some assumptions investment horizon doesn’t matter when making decisions about asset allocation. These assumptions are; your risk aversion is invariant to changes in your wealth, you believe that risky returns are random, and your future wealth depends only on investment results. Besides these assumptions, he did not take transaction cost into account.

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This study his contribution to the current literature is that it uses the bootstrap method to evaluate the optimal rebalance strategy. Other methods are used to find the optimal rebalance strategy like Monte Carlo simulations but the bootstrap method is never used.

The data contains weekly returns of stocks, bonds and cash between 1983 and 2013 and real estate returns between 1997 and 2013. The remainder of this thesis starts with an overview of the relevant literature. Thereafter, it gives an overview of the data used. Then the methodology is explained. Thereafter, the results will be discussed and finally an interpretation and conclusion of the results will be given.

2. Literature review

Markowitz (1952) derives a method to maximize the utility of portfolios of individual securities. Utility formalizes the trade-off between risk and return and is different for every investor. The first decades after Markowitz, research was mainly qualitative research. Arnott and Lovell (1993) investigate that rebalancing increases return. The authors use historical bond- and stock returns between 1968 and 1991. Their research consists of three rebalancing strategies. The first strategy is calendar rebalancing, which is rebalancing after a given period of time. Drawback of this strategy is that market behaviour is neglected. You always have to rebalance even if the portfolio is close to optimal (Sun, Fan, Chen and Schouwenaars, 2006). The second strategy is rebalancing to allowed range. This strategy sets boundaries and rebalances when these boundaries are exceeded. It rebalances the asset which exceeded the boundary to the boundary. The final strategy is threshold rebalancing. Just like rebalancing to allowed range it rebalances when a boundary is exceed. The difference with the rebalancing to allowed range strategy is that the threshold rebalancing strategy rebalances all assets to its initial distribution instead of one asset to the boundary. The authors conclude that the calendar rebalancing strategy is superior to the rebalancing to allowed range strategy and the threshold strategy. Furthermore, Sun et al. conclude that monthly rebalancing is the optimal rebalancing interval.

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future returns. The author concludes that every market has its own optimal strategy. Furthermore, the author describes the importance of the relation between investment horizons, trading costs and rebalancing. A portfolio with a short investment horizon is less likely to need rebalancing because there is less time for the portfolio to drift from the target asset allocation. In addition, such a portfolio is less likely to benefit from rebalancing because there is less time to benefit from the trading and the trading cost which are made. Similar research is done by Das, Kaznachey and Goyal (2011), who include trading cost and find a formula to maximize the investors utility for any given investment horizon. The authors use a log-optimal strategy, this is appropriate when investors be devoted to a logarithm based utility function. The authors assume that the portfolio asset returns follow a Geometric Brownian motion. Their results show that when investors implement their designed strategy, they gain significantly when comparing with the calendar rebalancing strategy. Kuhn and Luenberger (2008), find that calendar rebalancing only slightly outperforms threshold rebalancing if there are no transaction costs and if the rebalancing intervals are shorter than about one year.

Woodside-Oriakhi, Lucas and Beasley (2012) study the influence of investment horizons and transaction costs on portfolio rebalancing. They split up transaction costs into two types, fixed and variable costs. Fixed transaction costs are paid irrespective of the amount traded in which variable transaction costs are related to the amount traded. They model the portfolio rebalancing problem as a mixed- integer quadratic program with an explicit constraint on the amount that can be paid in transaction cost.

Perold and Sharpe (1988) describe the constant mix strategy. The constant mix strategy maintains the exposure of every financial asset to a constant proportion of wealth, due to rebalancing. So the risk of the portfolio stays constant and is not related to wealth. When the value of a financial instrument relative increase, you sell it, and when the value of a financial instrument relative decrease, you buy it. In other words, you buy low and sell high.

In theory, higher risk should lead to higher returns. The Sharpe ratio measures this theory. According to Sortino and van der Meer (1991), the Sharpe ratio uses the capital market line as a benchmark to evaluate the performance of the portfolio. A higher Sharpe ratio implicates a higher return per unit of risk.

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When comparing the calendar rebalancing strategy and the threshold rebalancing strategy, I expect that threshold rebalancing gives better results because it only rebalances when the portfolio is out of its boundary. I especially expect threshold rebalancing to outperform calendar rebalancing when transaction costs are taken into consideration.

3. Methodology and hypothesis

In this study the moving blocks bootstrap method is used to calculate the optimal rebalancing strategy. Bootstrap methods involve estimating a model many times using simulated data. Quantities computed from the simulated data are then used to make inferences from the actual data (Mc Kinnon, 2006). The block bootstrap method divides the quantities that are being resampled into blocks. Künsch (1989) improved the bootstrap method by adding the block-bootstrap method. The block bootstrap method takes blocks out of the sample instead of single values. The main advantage of this method is that it retains serial correlation, also known as momentum. According to Ayadi and Kryzanowski (2011), three advantages of the block bootstrap method are: it accounts for possible violations of the normality assumption underlying the performance tests and inferences; it can accommodate possible nonlinearities in fund returns without requiring the estimation of the complex joint distribution of performance across all mutual funds and it can deal with time-series dependencies in the data through basic setup extensions. Politis and White (2004), review the different block bootstrap methods for time series, and present them in a unified framework. Furthermore, they come up with useful estimators of the optimal block size for the aforementioned block bootstrap methods. According to Lahiri (1999), the best block bootstrap approach is the moving blocks bootstrap. The moving blocks bootstrap allows overlapping blocks and block size is fixed. Example 1 gives an example of the moving block bootstrap method.

Example 1: We have historical returns X, which is a vector with values [0,1,2,3,4,5,6]. The number of elements (L) in a segment is 4 and shift size (M) is 1. We have 3 possible bootstrapped samples (L-M=3). These samples are [1,2,3,4], [2,3,4,5] and [3,4,5,6].

In this study the length of X is 886 for real estate, and 1565 for stocks, bonds and cash. L is 11, which is in agreement with Lahiri (1999). According to the author, the optimal block length = . The average number of observations of the four assets is 1395.

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This study uses two rebalancing strategies, calendar rebalancing which is rebalancing to target weights at a consistent time interval and threshold rebalancing which is rebalancing to target weights whenever any asset class moves beyond some predetermined boundary (Sun et al., 2006). The rebalance intervals used for calendar rebalancing are ranging from weekly to two years. The boundaries used for threshold rebalancing are 10%, 25% and 50%. Rebalancing can be done to the initial distribution, to the boundary or somewhere between the initial distribution and the boundary; in this study we rebalance to the initial distribution. Furthermore, we rebalance all assets when a time period is expired or when an asset exceeds its boundary. To calculate the transaction costs the turnover is multiplied by transaction costs of 0%, 1% and 2.5%, which is in agreement with Chen, Fabozzi and Huang (2011). This means that transaction cost is variable, so the amount of transaction cost depends on how much is bought or sold. The transaction costs will be subtracted from the return.

For calendar rebalancing the expected return is:

(1)

where (2)

(3)

And v=value, w=weight in portfolio, t=time, R=bootstrapped return1_{, i= stocks, bonds, cash or real}

estate, t=time, c=transaction costs. Transactions costs are:

Where Total turnover= (3)

(4)

And P=variable transactions cost of 0%, 1% or 2.5%.

For threshold rebalancing the expected return is:

(1)

Where (2)

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(5)

And Q is a dummy variable, Q=1 if:

(6) And Q=0 if (6) does not hold. bdr= boundary of 0.1, 0.25 or 0.5.

Where Total turnover= (3)

(4)

The Sharpe ratio (Sortino and van der Meer, 1991) is used as a performance measure to compare the different strategies. According to Sortino and van der Meer (1991), the Sharpe ratio is:

(7)

Where =expected return, =risk-free rate, which is the average return on cash between 1983 and 2013 (4,4%) and = the standard deviation of the expected return.

4. Data and data description

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assets are weekly returns and adjusted, so dividends or coupons are included. The returns of stocks, bonds and cash are for the period between 1983 and 2013. The real estate returns are for the period between 1997 and 2013. This differs because there is no MSCI US REIT data available before 1997. Because I resample the data using the bootstrap method, this is no disputed point. However, it could be a problem if the returns of the assets differ between the period before 1997 and the period after 1997. The risk-free rate is the average return on cash over the data sample. Table 1 gives an overview of the descriptive statistics of the assets. Stocks have the highest expected return and the highest standard deviation. The average return is important for portfolio rebalancing, because significant differences between asset returns increase the risk of significant deviation from the target allocation. Therefore, it increases the need to rebalance (Tokat, 2007). There are substantial differences between the returns of the assets. The return on stocks is almost twice as high as the return on cash. Furthermore, the volatility is important for portfolio rebalancing. High volatility increases the fluctuation of the asset class weightings around the target allocation. Therefore, higher volatility implies a greater need to rebalance (Tokat, 2007). The volatility is measured by the standard deviation of the expected returns. Stocks and real estate have a substantial higher standard deviations compared with bonds and cash. All asset returns are skewed to the left, which means that the outliers are low returns. The kurtosis outcome shows that the distribution of stocks and real estate are leptokurtic whereas the distribution of bonds and cash are platykurtic. The Jarque Bera test is a goodness of fit test, to test if we can assume that the data is normally distributed. The outcome shows that at a 1% significance level we can reject the null-hypothesis that the data is normally distributed. In other words, the data is not normally distributed. Since this paper does no test which assumes normality, this is not a problem.

Table 1. Table 1 presents the descriptive statistics of the assets stocks, bonds, cash and real estate.

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Stocks Bonds Cash Real estate

Average return (%) 8.34 6.54 4.43 6.66

Standard deviation year (%) _{16.75 0.31 0.39 24.78}

Serial correlation (k=1) -0.06 1.00 1.00 -0.03 Serial correlation (k=2) 0.06 1.00 1.00 0.00 Skewness -0.82 0.74 0.09 -0.48 Kurtosis 6.50 0.61 -0.85 9.55 Jarque Bera 978 517 967 1616 JB 1% sign no no no no Median weekly (%) 0.31 0.12 0.10 0.25

Max return weekly (%) 11.36 0.26 0.22 21.71

Min return weekly (%) -20.08 0.04 0.00 -21.28

Number of observations 1565 1565 1565 886

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Stocks Bonds Cash Real estate

Stocks 1 0.01 0.02 0.02

Bonds 1 0.87 0.01

Cash 1 -0.01

Real estate 1

The portfolio used in this paper is based on the average distribution implemented by multiple Dutch financial institutions. Table 3 shows the average distribution of the assets in the different risk profiles. Alternative investments differ for every bank. Some banks call it alternative investments, others real estate or commodities. In this paper real estate returns will be used as measurement for alternative investments, real estate and commodities. The distribution is based on risk profiles of the ABN Amro Bank, Delta Lloyd, ING Bank, van Lanschot, Rabobank and Triodos Bank. The portfolio set up of all the financial institutions can be found in the appendix. This paper focuses on the portfolio of the neutral risk profile.

Table 3: Table 3 shows de asset distribution of different risk profiles. The distributions are the average distributions of Dutch banks. Real estate is an aggregation of real estate, commodities and alternative investments.

Risk Profiles Very defensive Defensive Neutral Offensive Very offensive

Stocks (%) 0.06 0.24 0.41 0.62 0.82

Bonds (%) 0.77 0.60 0.47 0.27 0.06

Real estate (%) 0.03 0.05 0.07 0.08 0.09

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5. Results

The results consist of three sections with outcomes. The results of calendar rebalancing, threshold rebalancing and buy and hold are shown for every level of transaction costs. The levels of transaction costs are 0%, 1% and 2.5%, which is consistent with Chen et al. (2011). For calendar rebalancing, this study uses rebalance intervals ranging from weekly to every two year. In this context, monthly rebalancing means rebalancing every four weeks (not every = 4.33 weeks). To compare the results of the rebalancing strategies, I include the buy and hold strategy as a benchmark. With the buy and hold strategy you buy the initial portfolio and do nothing. Therefore, there are no transaction costs.

5.1 The results of the strategies without transaction costs are present in Table 4. The table shows that for a 10 year investment horizon, threshold rebalancing with a 0.5 boundary, results in the highest expected return. 3 month or semi-annually rebalancing results in the lowest risk. If we combine risk and return, monthly rebalancing results in the highest Sharpe ratio, which means that monthly rebalancing gives the best return for its risk. For a 20 year investment horizon, the expected return of the strategies does not differ much but there are differences in risk. We can see that the standard deviation is lowest for 3 month calendar rebalancing. When comparing the Sharpe ratios we see that 3 month- and semi-annual- calendar rebalancing have the highest Sharpe ratio. If the investment horizon is 30 years we see differences between the returns of calendar rebalancing and threshold rebalancing. Every return of threshold rebalancing is higher than the highest return of calendar rebalancing. When comparing the standard deviations we see that the month- and 3 month calendar rebalancing have the lowest standard deviation. When comparing the Sharpe ratios we see that threshold rebalancing with a boundary of 0.25 has the highest Sharpe ratio and therefore, is optimal.

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Table 4: Table 4 shows the outcomes of calendar rebalancing, threshold rebalancing and the buy and hold strategy, without transaction cost. The first column shows the investment horizons. In the table Panel A stands for calendar rebalancing strategy, Panels B stands for threshold rebalancing strategy, E(r) stands for total expected return and Stdev stands for the standard deviation of the expected return. Sharpe stands for Sharpe ratio.

10 year 20 year 30 year

Panel A

time E(r) StDev Sharpe E(r) StDev Sharpe E(r) StDev Sharpe

Week _{6.9% 7.1% 0.35 6.8% 7.1% 0.34 7.0% 7.1% 0.37} Month _{6.9% 6.9% 0.36 6.8% 7.2% 0.33 6.8% 6.9% 0.35} 3 Month _{6.7% 6.8% 0.34 6.9% 6.8% 0.36 6.9% 6.8% 0.37} Semi-annual _{6.8% 6.8% 0.35 6.9% 7.0% 0.36 6.9% 7.0% 0.36} Annual _{6.5% 7.4% 0.27 6.9% 7.4% 0.34 7.0% 7.4% 0.35} 2 Year _{6.0% 8.6% 0.19 6.7% 8.5% 0.26 6.8% 8.5% 0.28} Panel B boundary 0.10 _{7.0% 7.1% 0.36 6.9% 7.1% 0.34 7.1% 7.1% 0.38} 0.25 _{7.0% 7.3% 0.35 6.9% 7.2% 0.34 7.2% 7.1% 0.39} 0.50 _{7.1% 7.8% 0.35 6.7% 7.7% 0.29 7.1% 7.6% 0.34}

Buy and hold strategy

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rebalancing interval results in a higher turnover and therefore, a lower expected return. Furthermore, a tighter the boundary results in a higher turnover and therefore, a lower expected return. We can see this when comparing table 5 with table 4. The buy and hold strategy gives the highest expected return for every time horizon. However, the standard deviation is high for every investment horizon and therefore the Sharpe ratios of the buy and hold strategy are low compared with calendar- and threshold rebalancing. When comparing calendar- and threshold rebalancing, we find evidence that for a 10 year investment horizon, the expected return is highest with threshold rebalancing with a boundary of 0.5. The standard deviation is the lowest for 3 month and semi-annual rebalancing. The Sharpe ratio is highest for semi-annual rebalancing and rebalancing with a 0.5 boundary. For 20 year investment rebalancing, the expected return is highest for semi-annual or semi-annual rebalancing. The standard deviation is lowest for 3 month rebalancing. The Sharpe ratio is highest for semi-annual rebalancing. When comparing the results for 30 a year investment horizon, the expected return of annual rebalancing or a 0.5 boundary rebalancing is highest. The standard deviation of 3 monthly rebalancing is lowest. The Sharpe ratio of semi-annual or annual rebalancing is highest.

To conclude, with 1% variable transaction cost calendar- and threshold rebalancing outperform the buy and hold strategy. Furthermore, we can conclude that semi-annual rebalancing is optimal for every investment horizon. There is barely any difference between the optimal calendar rebalancing interval and the optimal threshold rebalancing boundary. However, there are differences within the strategies. For calendar rebalancing, we see that the optimal interval moves from around 3 months to semi-annual for a 10 year investment horizon, to semi-annual to annual for a 30 years investment horizon. This is consistent with the hypothesis, that the longer the investment horizon, the larger the rebalance interval. For threshold rebalancing, we see that the optimal boundary moves from a boundary of 0.5 for a 10 year investment horizon to a boundary of 0.25 for a 20- and 30 year investment horizon. This is not conforming to the hypothesis, which is that a wider boundary is better for a longer investment horizon.

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10 year 20 year 30 year

Panel A

time E(r) StDev Sharpe E(r) StDev Sharpe E(r) StDev Sharpe

Week _{5.5% 7.1% 0.16 5.3% 7.1% 0.13 5.4% 7.1% 0.13} Month _{6.2% 6.9% 0.25 6.0% 7.2% 0.22 6.0% 6.9% 0.23} 3 Month _{6.3% 6.8% 0.27 6.4% 6.8% 0.29 6.4% 6.8% 0.29} Semi-annual _{6.5% 6.8% 0.30 6.5% 7.0% 0.30 6.5% 7.0% 0.30} Annual _{6.1% 7.4% 0.23 6.5% 7.4% 0.29 6.6% 7.4% 0.30} 2 Year _{5.8% 8.6% 0.15 6.4% 8.5% 0.23 6.5% 8.5% 0.25} Panel B boundary 0.10 _{6.0% 7.1% 0.22 5.8% 7.1% 0.19 6.0% 7.1% 0.22} 0.25 _{6.4% 7.3% 0.27 6.2% 7.2% 0.25 6.5% 7.1% 0.29} 0.50 _{6.7% 7.8% 0.30 6.2% 7.7% 0.23 6.6% 7.6% 0.28}

Buy and Hold

6.8% 17.8% 0.13 6.8% 18.3% 0.13 7.0% 19.1% 0.14

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week rebalancing is remarkable. The outcome is negative because the risk-free rate (4.4%) is higher than the expected return. If we compare the results of a 20 year investment horizon, the expected return of annual rebalancing is optimal. The standard deviation of 3 month rebalancing is lowest. When looking at the Sharpe ratios we see that semi-annual or annual rebalancing is optimal. For a 30 year investment horizon, the expected return of annual or 2 year rebalancing is highest. The standard deviation of 3 month rebalancing is lowest. The Sharpe ratio of annual rebalancing is highest.

To conclude, with 2.5% transaction costs rebalancing is better than the buy and hold strategy. There is barely any difference between the optimal calendar rebalancing interval and the optimal threshold boundary. However, there are differences within the strategies. For calendar rebalancing, the optimal interval moves from semi-annual for a 10 year investment horizon to semi-annual or annual for a 30 year investment horizon. Regarding the threshold rebalancing strategy, it is optimal to use a boundary of 0.5 for every investment horizon.

Table 6: Table 6 shows the outcomes of calendar rebalancing, threshold rebalancing and the buy and hold strategy, with 2.5% transaction cost. The first column shows the investment horizons. In the table Panel A stands for calendar rebalancing strategy, Panels B stands for threshold rebalancing strategy, E(r) stands for total expected return and Stdev stands for the standard deviation of the expected return. Sharpe stands for Sharpe ratio.

10 year 20 year 30 year

Panel A

time E(r) StDev Sharpe E(r) StDev Sharpe E(r) StDev Sharpe

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Buy and Hold

6.8% 17.8% 0.13 6.8% 18.3% 0.13 7.0% 19.1% 0.14

Table 7, gives an overview of the optimal strategy for every investment horizon and transaction costs. In the first row the investment horizons are shown, In the first column the transaction costs are shown. CR means calendar rebalancing and is followed by the optimal rebalancing interval, TR means threshold rebalancing and is followed by the optimal rebalance boundary.

10 year 20 year 30 year

0% CR, month CR 3 month or CR semi-annual TR 0.25

1% CR semi-annual, TR 0.5 CR semi-annual CR semi-annual, CR annual 2.50% TR 0.5 CR semi-annual or CR annual CR annual

6. Conclusion

In this paper we have considered the problem of how to rebalance a portfolio, when transaction costs are involved with different investment horizons. The initial portfolio is based on the average Dutch portfolio. Financial institutions want their portfolio to stay in its risk profile because otherwise the characteristics of the portfolio do not match with the characteristics they publicise in the information leaflet. Therefore, they use a constant mix strategy. This paper uses two methods of rebalancing, calendar rebalancing and threshold rebalancing. The buy and hold strategy is used as a benchmark and the block bootstrap method is used to resample the data.

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implementing the calendar rebalancing strategy it is better to use longer rebalance intervals if the investment horizon increases. When implementing the threshold rebalancing strategy, the optimal boundary depends on the investment horizon. When transaction costs are taken into account, we see that wider boundaries and longer rebalance intervals perform better. This is because these strategies rebalance less often and therefore, their transaction costs are lower.

Furthermore, we can conclude that the investment horizon influence the optimal rebalance interval. However, results provide not consistent support for the hypothesis that a longer investment horizon results in a higher rebalance interval. In all cases, it is optimal to rebalance between every quarter of a year and every year. When comparing these results with the existing literature these results are consistent with part of the literature. Previously research (see e.g., Arnott and Lovell, 1993, Kuhn and Luenberger, 2008), showe an optimal rebalancing interval between a month and a year.

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7. References

Arnott, R., Lovell, R.M., 1993. Rebalancing: Why?, When? How often?. The Journal of Investing 2 .1, 5-10.

Ayadi, M., Kryzanowski, L., 2011. Fixed-income fund performance: Role of luck and ability in tail membership. Journal of Empirical Finance 18.3, 379–392.

Chen A., Fabozzi F., Huang D., 2012. Portfolio revision under mean-variance and mean-CVaR with transaction costs. Review of Quantative Finance and Accounting 39, 509–526.

Conrad, J., Kaul, G., 1998. An anatomy of trading strategies, Review of Financial Studies 11. 489– 519.

Das, S., Kaznachey, D., Goyal, M., 2011. Computing optimal rebalance frequency for log-optimal portfolio. Quantative Finance, ahead of print, 1-14.

Kuhn, D., Luenberger, D.G., 2010. Analysis of the rebalancing frequency in log-optimal portfolio selection. Quantitive Finance 10, 221–234.

Künsch, H.R., 1989. The jackknife and the bootstrap for general stationary observations. The Annals of Statistics 17, 1217–1241.

Kritzman, M., 1994. What practitioners need to know about time diversification. Financial Analysts Journal 50.1, 14-18.

Lahiri, S. N., 1999. Theoretical comparisons of block bootstrap methods. The Annals of Statistics 27, 386–404.

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Pliska, S., Kiyoshi S., 2004. Optimal tracking for asset allocation with fixed and proportional transaction costs. Quantitative Finance 4.2, 233-243.

Politis, D. N., White, H., 2004. Automatic block-length selection for the dependent bootstrap. Econometric Reviews 23.1, 53–70.

Ritter, J. R., Chopra, N., 1989. Portfolio rebalancing and the turn-of-the-year effect. The Journal of Finance 44, 149–166.

Sortino, F. A., van der Meer, R., 1991. Downside risk. The Journal of Portfolio Management. 17.4, 27-31.

Sun, W., Fan, A., Chen, L., Schouwenaars, T., Albota, M., 2006. Optimal rebalancing for institutional portfolios. Journal of Portfolio Management 32.2, 33-43.

Tokat, Y., Nelson W., 2007. Portfolio rebalancing in theory and practice. Journal of Investing 16, 52-59.

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Appendix

Investment distribution of the financial institutions.

ABN Amro

Very