Out-of-sample testing of several portfolio
strategies in the US stock market
JEL classifications: G1, G11.
Author: Harm Hommes MSc Finance Thesis
University of Groningen, faculty of Economics and Business Subject: Portfolio theory
Date: June 2013 Supervisor: Lammertjan Dam
Abstract:
2 Introduction
People mainly invest because they want to generate more cash in the future than they possess know. The main objective, which depends on the investment philosophy, has to be determined before the investment (Minahan, 2006). Concretely, the investor has to seek for an optimum between return, volatility and downside protection. In the past decades several strategies are invented to meet the different objectives. Markowitz (1959) started the basis of portfolio analysis with comparing the expected return and volatility of the stocks to compute a portfolio with the lowest volatility for a given return. These analyses are still very common in modern portfolio theory. Interestingly, other academics gave evidence that less sophisticated strategies have the same or higher return and are easier to implement (Jobson and Korkie, 1980). Downside protection of the portfolio is also not included in the analysis of Markowitz. Nowadays there is a lot of discussion in the field of portfolio management about the assumptions underlying investment strategies and their performance.
This research investigates several characteristics of some commonly used investment strategies. Theory provides information about how the strategies should behave. The empirical evidence, presented in this research, shows whether the theory corresponds with practice and shows the advantages and disadvantages of several strategies. Some of the prior research is done within the sample of the research or with Monte Carlo simulations. In this empirical research the comparison will be made of several strategies and they will be tested out-of-sample. The strategies that will be considered are the buy-and-hold strategy, constant-mix strategy, equally-weighted portfolio, minimum variance portfolio, momentum strategy, CPPI strategy and TIPP strategy.
3
performance of the several investment strategies. Next to these performance measures, the four factor model of Cahart (1997) is used to identify what the effect of a size premium, value premium or momentum is on the returns. The stocks from the S&P 500, a tracker of the S&P 500 and the 3-months Treasury Bill are used to create the portfolios. The analysis of the portfolios starts in 1989 and ends in 2013.
In contrast to the abundance of theoretical results on portfolio strategies, the empirical validation of these strategies is scarce. We aim to fill this gap in current literature through empirical testing of commonly used investment strategies. Within sample testing implies that estimating parameters and testing strategies is done on the same data. Jegadeesh and Titman (1993) mention this problem in their research. An investor invests with the history in his mind and the future in front. He will never have the possibility to invest in the past. Therefore, within sample testing gives a too positive view on the advantages of strategies.
Monte Carlo simulations require estimates of the return and the standard deviation to project all future paths. Simulations are based on theory and will adjust the results to the estimates. This method will therefore increase the probability that the conceived theory will be found. Empirical out-of-sample testing does not incur these problems. Hence, we will use out-of-sample evaluation of the different portfolio strategies, which, to our knowledge, has never been done before.
Moreover, theory does not give an overview and comparison of the several strategies. It only provides information about the advantages and disadvantages of a separate strategy in a specific context. This research investigates the strategies on the same stocks in the same time frame. A comparison of the strategies, thus, adds relevant information for investors in practice who have to choose the best strategy with respect to the objective.
4
managing strategy is preferable, because actively managed funds are charged with a higher fee than passive managed funds. The trade-off between the positive and negative aspects of actively and passive managed funds will be included in the discussion.
5 Theoretical background
In this section the theoretical framework of portfolio construction is considered first. Next, short selling is explained. Theory behind the portfolios and path dependency is communicated thereafter. Then, the portfolios are divided in tactical asset allocation and stock picking strategies. Afterwards, mean reverting, momentum and payoff structures will be analysed. The last part of this section considers the implications of the strategies.
Theoretical framework of portfolio construction
The theoretical optimal portfolios in line with the EMH including stocks are shown in Figure I. In this kinds of frameworks financial economists assume information efficiency and rationality of investors and employ equilibrium asset pricing models, like the capital asset pricing model (CAPM) (Fama and MacBeth, 1973). However, active portfolio managers assume inefficiently priced securities. They search for under- and overvalued stocks or signals that predict returns (Clarke, Da Silva and Thorley, 2006). Grosmann and Stiglitz (1980) even advocate that equilibria cannot exist. DeBondt and Thaler (1995) find evidence that behavioural factors are overlooked by academics and explain part of the difference in theory and practice on financial markets.
6
the tangency portfolio and the riskless asset is never optimal. They find evidence that estimation errors cause a very bad out-of-sample performance.
Figure I.
Figure I shows the opportunity set of all possible portfolios with stocks. The MVP point represents the portfolio with the lowest volatility possible for a portfolio consisting only stocks. The efficient frontier is that part of the opportunity set where the portfolio has the highest risk for a given level of volatility. The capital market line (CML) represents the highest return possible with respect to a certain amount of risk or vice versa including the tangency, or market, portfolio (TP) and the risk-free rate (Rf).
Short selling
If an investor has a long position in a stock it means that he buys the stock and holds it. The investor anticipates on an increase of the stock price. When an investor goes short, sells short or has a short position, he is anticipating on a decrease in the value of the stock. The investor borrows the stock from the broker’s inventory and sells it. The price of the share is directly available on the bank account. At a certain point the investor has to buy the stock in order to give it back to the broker. When the price decreased, the investor is able to buy the stock at a lower price then he sold it at the beginning and vice versa.
7
all kind of constructions with financial instruments including short selling. Politicians wanted to ban short selling from financial markets and liquidity even further decreased (Berber and Pagano, 2010). The MVP and CPPI use short selling in their portfolios. Due to the reasons mentioned above, unrestricted short selling might cause a problem in the practical application of the strategies.
Buy-and-hold
The buy-and-hold is based on the EMH. When prices reflect all the available information in the appropriate way, there is no need to change the portfolio. After a move the stock price still reflects the most appropriate value, because it represents the equilibrium in supply and demand. Market timing is not possible in the view of this strategy as well. Since stock values reflect the true value it is not possible to time when markets go up or down. Changing the portfolio, because an investor wants to pick the right stock or time the market, only leads to higher transaction costs and not to higher return. Because of that, the initial portfolio will be held for the entire investment period. The buy-and-hold is a ‘do-nothing’ strategy and doesn’t aim to beat the market. Because prices are efficient an investor on average will have a return equal to the market. The most extreme buy-and-hold advocates state that one should never sell a stock unless one needs the money.
Some of the main characteristics of the buy-and-hold strategy are explained in the paper of Perold and Sharpe (1988). The portfolio value increases proportional in relation to the stock market since the portfolio reflects the market. When the stock market declines, normally, the buy-and-hold portfolio does as well.
Equally weighted portfolio
8
what every investor wants. Negative returns in the EWP occur when stocks default. When the price of a stock decreases, the strategy prescribes to buy more of this stock. Afterwards increasing stock prices will compensate the losses. At the point the stock defaults, the initial investment plus the extra investments will be evaporated.
A point of discussion is the degree of diversification in the EWP portfolios (Eun and Resnick, 1988). Because all the assets are weighted equally the correlation within the portfolio might increase. When an investor establishes the weights by himself, he can take care of the optimal allocation for diversification.
Momentum
Richard Driehaus is seen as the father of momentum investing and his philosophy is “buy high and sell even higher”, which contrasts the consensus philosophy “buy low and sell high”. When there is momentum in stock returns, prices will further increase when they did in the period before and fall further when they decreased in the period before. According to Jegadeesh and Titman (1993) there is evidence for momentum in returns. The strategy that an investor should follow is to buy when prices increase and sell, or even go short, when prices decrease.
Momentum is explained by the idea of irrational investors (DeBondt and Thaler, 1985). Investors tend to overreact on news. Bad news is interpreted more badly than should and good news is interpreted too well (Bareberis et. al, 1998). Griffin and Tversky (1992) reason that people are too much convinced by the evidence they are presented with in making forecasts and pay too little attention to the statistical. Though Crombez (2001) finds momentum even when investors are rational and markets are efficient. His argument is that information markets have imperfections. Whether momentum exists in every stock market around the world is not clear. Cultural differences can affect the psychology behind investing and the level of momentum in stock markets (Chui et al., 2010).
9
The minimum variance portfolio (MVP) is the portfolio with the lowest volatility possible that lies on the efficient frontier, see Figure I. Another advantage of the MVP is that the MVP is the only efficient stock portfolio whose weights do not depend on the expected returns. As already mentioned, estimating future returns is always associated with errors. A portfolio that only needs a prediction on volatilities and correlations can decrease the level of estimation bias in the weightings of the stocks.
Constant mix
The constant mix can be compared with the EWP. The EWP includes stocks at the same weight, where the constant mix keeps the same proportion of the investment in stocks and bonds (Merton, 1969 and Perold and Sharpe, 1988). Due to rebalancing, the constant mix has the same risk profile every period. When the stock market increases in value part of the position in the stock market will be sold and more of the risk-free asset will be bought and vice versa. A constant proportion to risky assets ensures that investors buy when prices are low and sell when they rise in the same way as the EWP does.
Constant Proportion Portfolio Insurance
The constant proportion portfolio insurance (CPPI) was first introduced by Perold (1986) and Black and Jones (1987). The strength of the CPPI is the limitation of downside risk. When the portfolio hits the floor, the entire portfolio will be invested in the risk-free asset. Hence, in theory, the portfolio will not decline anymore and will not have a final value below the value of the floor. Therefore, the CPPI is also called an insurance portfolio. Only when the stock price falls with a large amount before rebalancing, the value of the portfolio can be lower than the floor.
10
residual amount is invested in a risk-free asset. The floor is predetermined and does not change over time. Some research argues that the fixed floor is a disadvantage and advocates for a time-varying floor (Hamidi et al, 2009).
A problem with the CPPI is that there is no theoretical background for choosing the floor and the multiplier, which makes the analysis somewhat subjective. Merton (1971) used utility functions to calculate the strategy that maximizes the utility of the investor. Without the assumptions that markets are frictionless and that there are no borrowing constraints, this calculation is extremely difficult due to the path dependency of the strategy.
Time Invariant Portfolio Protection
The time invariant portfolio protection (TIPP) is not very different from the CPPI and is known as a portfolio insurance strategy as well (Estep and Kritzman, 1988). TIPP also uses a floor as insurance in such a way that the value of the portfolio will not fall under the floor and the allocation to the risky asset is calculated in the same way. Only, the floor is adjusted every rebalancing period. It is a proportion of the highest value the portfolio has reached during the investment period. Part of the gains of the returns is insured as well. The floor of a TIPP portfolio is therefore the same or higher than the floor of a CPPI portfolio. Because of the extra insurance, the TIPP is more conservative strategy with respect to the CPPI. Dichtl and Drobetz (2011) find evidence that the TIPP not only achieves floor protection, but protection of capital gains as well. On the other hand, this mechanism leads to opportunity costs due to a reduced investment in an upward moving market.
Path dependency
11
strategy. Path dependent strategies are not favourable for investors. The fluctuations a stock price made to reach the final value might lower the return of the portfolio while the markets overall increased. The CPPI and TIPP are clear examples of path dependent strategies. When prices first decrease, the amount invested in stocks decreases. When the stock prices increase in the second period the return is less than it would be without rebalancing. The other strategies are path dependent because of the rebalancing scheme as well.
Tactical asset allocation versus stock picking strategies
Additionally, another difference between the strategies is that some of them focus on stock picking and others on tactical asset allocation. A stock-picking strategy sets rules about which stocks should be taken and what the weights of the stocks should be. The buy-and-hold, EWP, momentum and MVP are examples of portfolio strategies that provide a method to select stocks and their weights. Tactical asset allocation gives constraints for distributing the portfolio into different asset classes. The constant mix, CPPI and TIPP are strategies that consist of a combination of the risk-free rate and the market portfolio. The portfolios of tactical asset allocation strategies should reach a higher return per unit of risk than strategies that do not lie on the CML.
Mean reverting, momentum and payoff structures
12
Momentum and mean reverting might be in contradiction, although momentum is referred to as a short term (Jegadeesh and Titman, 1993) and mean reverting a long term process. Thus, they might not be contradictory, but apply for a different investment period. The EWP and constant mix are examples of strategies which are made to gain from mean reverting. The momentum strategy completely focusses on momentum in stock returns.
Figure II.
Figure II presents the payoff paths of different investment strategies with respect to the value of the stock market.
13
the portfolio. In a bear market or when a market crash occurs, insurance strategies do really well relative to other strategies, because they have a limited downside risk. They are robust to market crashes (Bird et al, 1990). The buy-and-hold gives a straight line with respect to the outcome of the market.
Implications of the strategies
Based on the discussion of active and passive investing, a strategy that has a high turnover and involves a lot of effort has to achieve higher returns. The buy-and-hold is the easiest investment strategy to implement. At the start of the investment period the stocks are bought and the portfolio does not need to be rebalanced. The transaction costs are therefore very low. Every other strategy should have a lower volatility, better downside risk protection or higher return than the buy-and-hold. The EWP invests in the same stocks as the buy-and-hold and has no explicit downside protection. The implementation of the EWP is quite easy, but it has a relatively high turnover. The transaction costs and the effort of the investor to rebalance the portfolio should be compensated by extra returns. Otherwise, the buy-and-hold is a better strategy.
The EWP does not gain from momentum since the gain of a stock will be sold when the portfolio is rebalanced. It does gain from mean reverting. In flat markets, with mean reverting stock returns, the EWP should earn returns above average. The ‘winners’ are sold and the ‘losers’ are bought. The buy-and-hold strategy gains from momentum. An upward move of the stocks will be reflected completely in the return of the portfolio. In bull markets the buy-and-hold normally makes a higher return.
Jobson and Korkie (1980) mention that the EWP outperforms mean-variance portfolios due to smaller estimation errors. Duchin and Levy (2009) find that the EWP is a sufficient investment strategy for small investors. Though, the mean-variance portfolios dominate the EWP for larger investments. DeMiguel, Garlappi and Uppal (2009) find that EWP have higher Sharpe ratios than mean-variance portfolios.
14
has to time the purchase and sale of the stock very well and when the bubble bursts, the investor is too late and loses a large part of his investment. This risk and the lack of downside protection have to be compensated by a high average return. Momentum works the best in bull markets, when people are overoptimistic and prices are rising. In a bear market there are only a few stocks which increase. Earning returns with shorting stocks that decreased the most in the prior period has the same practical problems as short selling in general. Jegadeesh and Titman (1993) find evidence that momentum strategies have higher returns than the market portfolio. That should result in a positive and significant Jensen’s alpha in this research. Interestingly, MacDonald and Power (1993) find that stock returns in the UK are characterized by a random walk process, which suggests no momentum in stock returns. Then, the EWP might be a better strategy than the momentum strategy.
The average return is not the main objective of the MVP. The MVP tries to lower volatility. In theory the MVP should therefore result in a low standard deviation, a high Sharpe ratio and high downside risk protection. The average return is probably relatively low. Besides, it takes some mathematical knowledge to implement and rebalance the MVP every period and turnover is high as well. Therefore the costs of running the MVP are higher than the other strategies.
15
measures should be determined to incentivize investors for choosing stocks with the highest Sharpe ratio. Hence, when high volatility stocks are overvalued investors should choose less volatile stocks.
The constant mix, CPPI and TIPP include a tracker of the stock market and the risk-free rate. Because of that, the portfolios have low transaction costs and rebalancing does not take a lot effort. If returns are mean reverting and the volatility in a market is high the constant mix should earn higher returns than the other portfolios (Demptser et al, 2009). In a bull market stocks will be sold to invest in the risk-free rate. The constant mix does not gain from momentum in the market since the return of the stock is sold when the price increased. Strategies that move proportional to the market, like the buy-and-hold, will gain from momentum, but will remain at the same value in a sideward moving market. Therefore the constant mix should have a higher return than the buy-and-hold in a flat market, but less in a bull or bear market. The downside protection of the constant mix is poor, since more stocks will be bought in a bear market.
The inclusion of the insurance mechanism in the CPPI and TIPP must result in better downside risk protection but lower average return and lower upside potential than the other strategies. The MVP also has the main objective of downside risk protection and low volatility although some academics argue that the returns are very high as well (Jegadeesh and Titman, 1993). Annaert et al. (2009) also find that the CPPI outperforms the buy-and-hold in terms of downside risk and Sharpe ratio.
16
Although the TIPP gains from the extra insurance, the portfolio will also suffer when all the assets are transferred to the riskless asset at the point the floor is reached (Choie and Seff, 1989). In that way the TIPP, more often than the CPPI, ends up with all the assets in the risk-free asset and there is no upward potential towards a market increase left for the portfolio. The TIPP not specifically gains from momentum. When the stock prices increase, part of the investment in stocks is sold and invested in the risk-free rate.
17 Data
The stocks of the S&P 500, a tracker of the S&P 500 and the three month US Treasury Bills (after: T-Bills) are used to construct the portfolios. All the stocks of the S&P 500 will be used in the buy-and-hold portfolio and the EWP. The MVP and momentum strategy select fifty of the five hundred stocks of the S&P 500. The constant mix, CPPI and TIPP portfolio consist of different fractions of the tracker of the S&P 500 and T-bills.
On the one hand, adding stocks from other parts of the world than the USA to the portfolio can decline the risk within the portfolio due to extra diversification. On the other hand, including stocks traded in other currencies than the dollar will insert currency risk in the portfolios. To avoid an analysis that is biased because of currency risk an analysis of the USA will be made. Besides that, a better comparison with other research in the field of portfolio theory can be made since most of the research is done on the American stock markets. Furthermore, for the US market there are plenty of data available, while for other parts of the world there are less data available. In any case, the tested portfolios will consist of at least fifty stocks to diversify away the idiosyncratic risk within the US stocks.
The stocks in the S&P 500 differ over time. Some stocks are removed or default and other stocks appear later in the S&P. To get a proper view on the performance of the different strategies the portfolio will be adjusted every rebalancing period for the disappearing stocks and the newcomers. The available list of constituents of the S&P 500 in DataStream starts in 1989. Therefore this research will use the data from 1989 till 2013 in the analysis.
The buy-and-hold strategy in our research includes the same stocks as the S&P 500 at the beginning, but does not change the stocks over time. Besides that, the buy-and-hold is equally weighted at the beginning instead of the value weighted S&P 500.
18
19 Constructing the portfolios
In this Section the estimation, investment and rebalancing period are explained first. Thereafter the transaction costs and the methods of constructing the portfolios are considered. In the calculations geometric returns are used, see equation (1). A positive sum of returns over time means that at least the initial amount is paid back.
ln
(1)
Where is the geometric return, the price at 1 and the price at 0.
Estimation and investment period
The estimation period is the period before the investment period used to estimate the weights in the portfolio. The MVP is based on realized returns of the past five years. Clarke et al (2006) use an estimation period of five years as well. The momentum strategy only needs the returns half a year prior to the investment to estimate which stocks are chosen in the portfolio, see Jegadeesh and Titman (1993). The other strategies do not need historical data for the estimation of weights. The estimation period is rolling form the earliest point of available data and will have the same length all the time.
The investment period in this research will start directly after the estimation period and will last for five years. Private equity often uses five years as an investment period. Fama and MacBeth (1973) and Hagen and Baker (1991) use a testing period of five years as well. Every month a new portfolio starts except from the MVP. They start every quarter.
20
point made by Samuelson (1969). Time variation in expected returns induces mean reversion in returns, slowing the growth of conditional variances of long-horizon returns. Therefore, stocks appear less risky at long horizons, and hence more attractive to the investor (Bareberis, 2000).
Rebalancing period
The rebalancing period is the time after which the portfolio will be updated in line with the strategy. Merton’s (1969) main point is that portfolios should be rebalanced continuously. The less frequently investors revise their portfolios the more likely their portfolio will drift relative to their target weight and desired level of risk aversion (Jones and Stine, 2005). Thus, rebalancing decreases the volatility in the outcomes of the portfolios of one strategy. In practice it is impossible to manage continuously rebalancing. Besides that, the transaction costs increase to unacceptable amounts. According to Tsai (2001) portfolios should be rebalanced periodically, because neglecting rebalancing results in lower Sharpe ratios. Rewards are smaller relative to the additional risk taken without rebalancing.
The momentum strategy portfolios are rebalanced on a half yearly basis. The stocks are sold after half a year. Thereafter, new stocks are bought. The MVP is rebalanced on a quarterly basis, like Haugen and Baker (1991) did. The buy-and-hold portfolio buy-and-holds the same stocks the entire investment period and is not rebalanced. The other strategies, EWP, constant mix, CPPI and TIPP are rebalanced monthly.Although rebalancing every day would provide the best protection, Annaert, Van Osselaer and Verstraete (2009) find that a monthly rebalancing of the CPPI compensates lower protection with higher average excess return.
Transaction costs
21
Nevertheless, the transaction costs are only a little fraction of the total investment. Excluding transaction costs won’t be the cause of a huge difference between the strategies. Therefore the transaction costs are not taken into account in the calculations of the portfolio returns. In the discussion of the results expressly attention will be paid to the transaction costs of the different strategies.
Buy-and-hold
The buy-and-hold portfolio contains all the stocks in the S&P 500 and holds them for the rest of the investment period. The portfolio is not rebalanced during the investment period. Even when the stocks are removed from the S&P 500 they will stay in the portfolio. The portfolio starts with equally weightings of the stocks.
Equally weighted portfolio
The EWP holds an equal proportion of all the stocks of the S&P 500 in the portfolio. Due to rebalancing the portfolios are changed every month in such a way that the stocks have equal weightings again. When a stock is excluded from the S&P the stock will be excluded from the portfolio as well. The new stock in the S&P 500 will take the place of this stock. The EWP starts with the same portfolio as the buy-and-hold.
Momentum strategy
The momentum strategy buys stocks based on their past six months return and sells them after six months. Jegadeesh and Titman (1993) advocate this method as a good indicator of momentum strategies. The fifty best performing stocks, in terms of total return, of the S&P 500 over the past six months will be selected. Summing up the constraints it means that the stock will be selected when the stock:
1. Is present in the S&P 500 at the date the portfolio starts;
2. Belongs to the fifty best performing stocks over the past six months.
22
A normal third constraint would be that the stock had a positive return over the past six months. A declining stock does not signal positive momentum. Though, this constraint would cause that several periods have less than ten stocks in the portfolio and the idiosyncratic risk is not diversified. The total risk of the portfolio will increase for the same amount of expected return. Leaving this constraint the portfolio will contain fifty stocks every period.
Our research does not contain short selling in the momentum strategy. Hong et al. (2000) find evidence that momentum strategies primarily gain from the short positions in the portfolios rather than from the long positions. Though, Israel and Moskowitz (2012) find that the short and long positions both earn half of the profits. They advocate that the result of Hong, Lim and Stein (2000) are sample specific.
Minimum variance portfolio
The MVP only contains stocks of the S&P 500. Next to that, previous stock returns are necessary for the estimation of the weights of the stock returns. Each stock must have sixty months, five years, of stock returns available. Furthermore, constructing a MVP with all the stocks in the S&P 500 is not possible. The covariances between the stocks do not allow making a minimum variance portfolio with so many stocks. Therefore a selection of the stocks is made. The fifty largest companies of the S&P 500 with respect to market capitalization are selected. This measure is a common phenomenon for limiting the MVP (Haugen and Baker, 1991, and Clarke et al, 2006). Finally, the minimum variance portfolio selection criteria are:
1. The company is present in the S&P 500;
2. The stock of the company has data available for at least five years previous to the portfolio date;
23
variance and covariance matrix of all the stocks in the portfolio. One should minimize the portfolio variance with the matrix notation:
, ,
(2) Where , … , !"′ is a vector of portfolio weights and $ a vector of ones.
Kempf and Memmel (2008) suggested another estimation method for the portfolio weights of the MVP. They show how to calculate the weights of the MVP with the regression:
,! %& ', (,! , ) & … & ',! (,! ,! ) & * 1, … , + (3) Where N is the amount of stocks, t a certain point in time, ,! the return of a stock in the portfolio. The regression coefficients ',, are the weights of the stocks in the portfolio. The weights of some stocks can be negative, which means that the stock is shorted. Finally the sum of all ',,′- should be one and hence equal to hundred percent. Therefore, the weight of the stock on the left side of the equation can be calculated by one minus the sum of the other weights. An assumption underlying this technique is that the error term does not correlate with the return differences used in the regression, ,! ,,.
Because the method of Kempf and Memmel is much easier to implement than the traditional approach, this method will be used to estimate the portfolio weights. Shrinkage methods as advocated by several researchers are omitted (e.g. Jorion, 1986, Ledoit and Wolf, 2003, and DeMiguel et al., 2009). Because returns of the past five years are used, the MVP is the portfolio with the minimum volatility over the last sixty months. The assumption is that the historical covariance matrix will be an indicator for the future covariance.
24
in order to come up with the portfolio. Stocks that pass out of the S&P 500 are replaced by the new stocks that enter the S&P 500.
There are no limitations to short positions in the portfolio. Green and Hollifeld (1992) find evidence that without making a constraint on short positions the systemic and idiosyncratic risk is reduced. Though, Jagannathan and Ma (2003) advocate that it does not matter whether one constructs the portfolio with or without the constraint.
The liquidity of stocks in the S&P 500 is high. High liquidity of stocks increases the possibilities of shorting the stocks. When the MVP will be used with other stocks than the S&P 500 one should take the liquidity of these stocks and the possibility of going short into account.
Constant mix
The constant mix consists for seventy per cent of the tracker of the S&P 500 and thirty per cent of the risk-free asset. Every rebalancing period the portfolio will be reconstructed in such a way that the portfolio equals these proportions.
CPPI
The allocation to risky assets in the CPPI is calculated with:
. / + 0" (4)
25
is twenty1 percent allocated to stocks. Suppose that the value of the portfolio would
drop to 95 due to a decrease of the value of the stock market. The value of the stocks will drop to fifteen and the value of the bonds will remain eighty, hence the portfolio value will become 95. In order to hold the same exposure to stocks the portfolio is rebalanced as follows. The amount in stocks becomes ten and the amount in bonds becomes 85.2 Thus, when the stock market drops, the allocation to stocks in
absolute terms declines and the allocation to bonds rises and vice versa.
Every month a new portfolio will start that lasts for five years. The portfolio will be rebalanced every month. Due to the difference in allocation to stocks over time, every new started portfolio has a different return during the same month. The floor will be the underlying value that the portfolio at least should return after the investment period. The only exception is when the stock market drops by more than 1//. With a multiplier of two, the market can drop up to fifty percent before rebalancing without breaching the floor. Short positions in the risk-free rate are allowed. When the difference between the portfolio value and the floor is higher than 1// times the portfolio value, more than hundred per cent is invested in the market portfolio. The amount that exceeds the hundred per cent will be shorted in the risk-free rate.
TIPP
The implementation of the TIPP is quite similar to the CPPI. The allocation to the risky asset is calculated with the same formula as CPPI. The difference is the calculation of the floor. The floor of the TIPP is calculated with:
0 /23 + ∙ 5, 0 " (5)
Where + is the total value of the portfolio, 5 the insured proportion at the beginning and 0 the floor of the previous period. The floor in the CPPI doesn’t change, where the floor of the TIPP increases over time. When the floor value at point t is
1
2(100-90) = 20
2
26
lower than the floor of the previous period, at point t-1, the floor of the previous period will be used to calculate the allocation to stocks and bonds. The value of the portfolio is lower than the floor when the stock prices decline with more than 1/m.
Similar to the CPPI, the TIPP will be rebalanced every month and every month a new portfolio will start for a period of five years. In the TIPP portfolio the monthly returns of every portfolio are often very similar because the allocation to stocks and bonds remains equal when the value of the stocks did not change or increased during the previous month. Only when the value of stocks decreased the allocation to bonds will be higher than the pre-announced percentage.
27 Performance measurement
In the comparison of the different strategies the average return of the strategy only summarizes part of the picture. There are several performance measures which add information. They show other advantages and disadvantages of the strategies. The considered performance measures are the Sharpe ratio, the Treynor ratio, Jensen’s alpha, the information ratio, the Sortino ratio, the upward potential ratio and the VaR. The four factor model of Cahart (1997) will be analysed as well.
Sharpe ratio
In theory, academics state that the higher the expected return of a portfolio is, the higher the risk of the portfolio should be. Strategies that imply high risk should, therefore, have a higher return. A measure to deal with this problem is the Sharpe ratio. The Sharpe ratio uses the CML as a benchmark to evaluate the performance of the investor (Sortino and van der Meer, 1991). A higher Sharpe ratio means a higher return per unit of risk. Equation 6a shows the return per unit of risk. Equation 6b compares the return that exceeds the risk-free return with the standard deviation.
67 .
(6a)
67 . .8
(6b)
28 Treynor ratio
Beta can be interpreted as the risk of an asset in the portfolio relative to the movements of the market portfolio (Fama and MacBeth, 1973). The systemic risk, measured by Beta, is the residual risk in the portfolio once the idiosyncratic risk is assumed to be diversified. The security market line (SML) relates return to Beta and the Treynor ratio is derived from the SML (Treynor, 1965). This ratio represents the slope of the line from the actual portfolio. So it divides the portfolio return higher than the risk-free return by Beta. The equation is:
+ .' .8
(7)
Where ' is the Beta of the portfolio.
Jensen’s alpha
Jensen’s alpha is derived from the CAPM and the SML (Jensen, 1968). It measures the performance of the portfolio with a comparison between the return that exceeds the risk-free return and the return from the market. If a strategy adds value the portfolio should be at a higher level than the SML. In short, Jensen’s alpha measures the return a portfolio received upon the return the portfolio earned due to market movements and the risk-free rate. The regression is:
., .8, %, '(., .8,) & * (8)
Where %, is the alpha of the portfolio, ., the return of the market portfolio and * the idiosyncratic risk.
Information ratio
29
resembles a T-test that the portfolio returns do not differ from the returns of the benchmark. The difference between the portfolio return and the return of the benchmark is divided by the tracking error. The tracking error measures the divergence of the portfolio with the benchmark or the risk relative to the benchmark. The information ratio can be interpreted as the abnormal return of the manager's portfolio divided by the amount of risk that the manager takes relative to the benchmark. The IR is modelled by:
9. . .
: (9)
Where : is the tracking error and can be calculated with:
: ;1+ <(., .,) =
>
(10)
Sortino ratio
A problem with the Sharpe ratio, Jensen’s alpha and the Treynor ratio is that they assume normally distributed returns. The Sortino ratio doesn’t depend on this assumption and measures the downside risk of the portfolio. A higher Sortino ratio means a better protection against a downside turn of the portfolio. It uses an ex-ante determined reference point as minimal acceptable return to compare with the return of the portfolio.
6? . .B @A
(11)
30 B ;+ < .1 , .@A" = > ∀ .,D .@A (12)
Upside potential ratio
The upside potential ratio measures the upside potential per unit of downside risk of the portfolio (Sortino, van der Meer and Plantinga, 1999). The upside potential is measured by the returns that exceed the .@A and the downside risk is measured by the returns that are equal or below the .@A. The .@A should therefore represent the goal of the portfolio. The UPR is calculated by:
E. ∑ $ G + ., .@A" = > ∑ $ + ., .@A" = > (13)
Where $G is one for the returns above the .@A and zero otherwise. $ is zero for the returns above the .@A and one otherwise.
Value at risk
Value at risk (VaR) is a measure that measures the amount of the portfolio that is at risk (Jorion, 2001). The VaR represents the minimal return for a certain confidence level. Consider a portfolio with a VaR of two per cent. With 95 per cent confidence, the return will then be two per cent or higher. VaR assumes that returns are normally distributed. Though stock returns are slightly skewed and leptokurtic (Brown and Warner, 1985). Insurance portfolios, which use a floor, are skewed due to the insurance mechanism.
Four factor model of Cahart
31
based on the three factor model of Fama and French (1993). The three factor model adds two factors to the CAPM, 6HI and JHK. 6HI is the difference in returns between diversified portfolios with small and large stocks and JHK is the difference in returns between diversified portfolios with high and low book to market ratios.
Banz (1981) find that smaller firms with respect to market capitalization earn higher returns. This phenomenon is also known as the size premium. The economic reasoning is that smaller firms are riskier, but on average earn higher returns to compensate for the risk.
Value stocks have high book to market ratios and growth stocks have low book to market ratios. There is evidence that value stocks earn more return than growth stocks (Fama and French, 1992). This extra return is called the value premium. Value stocks have high book to market ratios and growth stocks have low book to market ratios. Value investing in general means buying stocks that are undervalued based on fundamentals like the book to market ratio (Graham, 1949). Cahart (1997) adds a momentum factor to the three factor model of Fama and French based on the evidence of Jegadeesh and Titman (1993) that stocks gain from momentum. The difference in returns in month t between diversified portfolios with the best and worst performing stocks of the year before determine the momentum factor. The Beta in the four factor model of Cahart has another value than in the CAPM. The 6HI and JHK add information to the returns but also correlate with Beta and cause that Beta values differ (Fama and French, 1993). The regression of the four factor model is:
, 8,& '(., .8,) & -6HI & LJHK& /H?/& *, (14) Where 6HI represents the size factor, JHK the book to market factor and H?/ the momentum factor.. The average monthly returns of the portfolios are used for the regression.
32
caps should have a negative sign on the SMB factor. A positive and significant HML factor shows that the strategy joins the concept of value investing. The momentum factor signals gains from momentum when the sign is positive and significant.
33 Results
Performance measurement
The results are presented in Table I. The buy-and-hold earns a high return of 46 per cent on average. Only three per cent of the portfolios lost part of the initial investment. The VaR, with -3.77%, is not relatively low and the Treynor ratio is the highest of all portfolios.
The theory suggests that the EWP earns a higher return than the buy-and-hold. Almost the entire investigated period the EWP’s had a higher profit than the buy-and-hold portfolios, see Graph Ia. In the results is shown that the return is higher and volatility approximately the same. That results in a higher Sharpe ratio, see Figure III. The disadvantage of the EWP is that it has a lower minimum than the buy-and-hold and a lower upside potential. What strikes is the highest positive Jensen´s alpha of all strategies for the EWP. The VaR is relatively high and the amount of portfolios beneath zero return is only three per cent.
The momentum strategy has an average return of 35 per cent, which is lower than the buy-and-hold and the EWP. Besides, the volatility is the highest of all and therefore the Sharpe ratio is the lowest. Jensen’s alpha is positive, but not significant and almost 25 per cent of the momentum portfolios ended with a loss. The momentum strategy is more volatile and has a lower average return than the S&P 500 as shown in Figure III. The VaR is very low which means that the probability that a momentum portfolio will not reach a positive return is relatively high. On the other hand, the highest portfolio value measured during the entire research period used the momentum strategy.
34 Table I.
Table I presents the characteristics of the portfolios over an investment period of five years. The numbers are not annualized. Table I shows the average portfolio return, standard deviation of the portfolio returns, maximum portfolio return, minimum portfolio return, the percentage of portfolio with a return less than zero, the Sharpe ratio without the free rate, the Sharpe ratio including the risk-free rate, Jensen’s alpha, the Treynor ratio, the Sortino ratio, the information ratio, the upside potential ratio (UPR), the value at risk (VaR), a downside risk measure (δ) used for the calculations of other ratios, Beta (β), the tracking error (:), and the minimal accepted return (.@A) and the confidence level used for the calculation of the VaR. These characteristics are presented for the buy-and-hold (BH), equally weighted portfolio (EWP), momentum strategy (Mom), minimum variance portfolio (MVP), constant mix (CM), constant proportion portfolio insurance (CPPI) and the time invariant portfolio protection (TIPP).
BH EWP Mom MVP CM CPPI TIPP
Average return 0.46 0.51 0.35 0.07 0.36 0.30 0.23 Standard deviation 0.30 0.30 0.45 0.03 0.30 0.28 0.13 Maximum 0.96 1.03 1.13 0.12 0.96 0.95 0.45 Minimum -0.20 -0.46 -0.64 0.03 -0.17 -0.03 0.00 Portfolio values < 0 3.2% 3.2% 24.7% 0.0% 10.4% 2.1% 0.4% Sharpe ratio 1.52 1.69 0.78 2.63 1.18 1.07 1.72 Sharpe ratio (-rf) 1.30 1.12 0.40 -3.68 0.61 0.45 0.42 Treynor ratio 0.47 0.51 0.22 -5.81 0.27 0.19 0.19 Jensen's alpha 0.16*** 0.20*** 0.03 -0.08*** 0.02*** -0.02*** 0.00 Information ratio 0.20 0.41 -0.18 -0.91 -0.41 -0.57 -0.53 Sortino ratio 1.89 1.83 0.15 -0.99 0.34 -0.03 -0.52
Upside potential ratio 38.47 19.79 2.05 0.00 5.89 3.97 1.34
VaR -3.8% 1.3% -38.8% 2.7% -14.0% -16.0% 1.0% δ 0.08 0.11 0.33 0.23 0.17 0.17 0.14 β 0.62 0.66 0.82 0.02 0.68 0.64 0.30 : 0.22 0.24 0.33 0.11 0.13 0.20 0.35 RNOP 30% 30% 30% 30% 30% 30% 30% Confidence level 95% 95% 95% 95% 95% 95% 95%
*** Significant at a one per cent significance level.
35
only a few portfolios with a lower final value than the initial investment for the CPPI and the TIPP. Besides that, the lowest return measured is close to zero. The Sortino ratio is beneath zero, because the average return of the portfolio was lower than the .@A. VaR does not measure what we want to know for the TIPP and CPPI. Since these strategies have a floor the distribution of the returns differs from the normal distribution. Therefore it is hard to give an economic explanation of the numbers of the VaR. The CPPI has a higher average return, a higher maximum portfolio return, a higher upside potential, but a much higher risk level as well. But, volatility more than doubled with respect to the TIPP. The TIPP shows a relatively high return in bear markets, see Graph Ic. Furthermore, the TIPP has an even higher Sharpe ratio than the buy-and-hold and the EWP, see Table I.
Figure III.
Figure III shows the return per unit of volatility, which is the Sharpe ratio. The Sharpe ratios are presented for the buy-and-hold (BH), equally weighted portfolio (EWP), momentum strategy (Mom), minimum variance portfolio (MVP), constant mix (CM), constant proportion portfolio insurance (CPPI) and the time invariant portfolio protection (TIPP).
Graph I.
Graph I shows the final logarithmic return (Ln(r)) of the investigated portfolios. All the portfolios are investments for five years and the returns are not annualized. Rf represents portfolios that are invested in the risk-free rate, S&P the returns of portfolios that invest in a tracker of the S&P 500. In Graph Ia, BH shows the returns of the buy-and-hold and EWP the returns of an equally weighted portfolio. BH is not rebalanced and the EWP is rebalanced every month. In Graph Ib, Mom represents the returns of portfolios that invested in the fifty best stocks of the past six months, according to a momentum strategy. MVP shows the returns of the minimum variance portfolio. The momentum strategy is
36
rebalanced every six months and the MVP quarterly. In Graph Ic, CM shows the return of constant mix portfolios, CPPI the returns of portfolios adapting the constraints of the constant proportion portfolio insurance and TIPP the returns of the time invariant portfolio protection (TIPP). The CM, CPPI and TIPP portfolios are rebalanced every month.
37 Four factor model of Cahart
The results of the regressions of the four factor model of Cahart are shown in Table II.All the factors have significant influence on the stock returns of the buy-and-hold strategy. The buy-buy-and-hold returns are positively related to the market, is a small stock strategy and behaves as a value investor. The strategy invests in stocks with a high book to market ratio. The momentum factor is negative which means that the buy-and-hold suffers from momentum. The EWP shows almost the same results as the and-hold. The outcomes are only slightly more extreme. The buy-and-hold and the EWP consist of equally weightings for all the stocks at the beginning. The market portfolio is value weighted and has a higher emphasis on large companies. Therefore it is not surprising that the buy-and-hold and the EWP have a positive and significant coefficient on the SMB factor. The EWP suffers from momentum because an increasing stock is sold partly when the portfolio is rebalanced. The momentum of that stock will only partly be reflected in the portfolio return.
Table II.
Table II presents the results of the four factor model of Cahart (1997). Rf represents the risk-free rate, Mkt the market risk premium, SMB the small minus big factor to measure for size, HML the high minus low factor for the book to market ratio and momf the momentum factor. The results are presented for the buy-and-hold (BH), equally weighted portfolio (EWP), momentum strategy (Mom), minimum variance portfolio (MVP), constant mix (CM), Constant Proportion Portfolio Insurance (CPPI) and the Time Invariant Portfolio Protection (TIPP).
BH EWP Momp MVP CM CPPI TIPP
Rf 0.60** 0.67** 2.59*** 0.37*** 0.34** 0.25 0.33***
Mkt 0.17** 0.19** 0.40* 0.00 0.57*** 0.60*** 0.08***
SMB 0.44*** 0.49*** 0.03 -0.01 0.01 -0.03 0.03**
HML 0.37*** 0.40*** 1.15*** -0.02 0.07 0.00 0.04**
Momf -0.15*** -0.16** 0.67*** 0.00 -0.05* -0.06* 0.00 * Significant at a ten per cent significance level.
** Significant at a five per cent significance level. *** Significant at a one per cent significance level.
38
and significant for the HML factor. Therefore the momentum strategy can also be classified as a value investing strategy. The SMB coefficient is not significant. Our result is in line with Israel and Moskowitz (2012), who find evidence that momentum does not increase by firm size or decrease.
39 Discussion
Which investment objective and which strategy should be linked?
The portfolios that, according to the theory, have a high average return as their main objective are the buy-and-hold strategy, the EWP, the momentum strategy and the constant mix. When we discuss the results of the buy-and-hold portfolios we see that they validate the theory. The buy-and-hold, a passive investment strategy with low transaction costs, had an average return of 46 per cent on five year investments. The EWP, a more active investment strategy, has a higher return and slightly better downside protection, but needs more effort of the investor and incurs high transaction costs. The buy-and-hold does not need any effort during the investment period and only has transaction costs at the start and at the end of the investment. The trade-off between these strategies lies in the amount of effort and the transaction costs. An optimal rebalanced EWP has transaction costs that can exceed the extra returns of the EWP over the buy-and-hold. Less rebalancing takes less effort and might be cheaper, but can decrease the return of the portfolio.
The results of the momentum strategy do not show the results indicated by the theory. They should present a higher return than the other strategies, but the average return was relatively low, 35 per cent. Volatility was high and downside protection bad. By performing worse than the market, our result is not in accordance with the research of Jegadeesh and Titman (1993) who find evidence that momentum strategies can beat the market portfolio. It might be the case that using other implementation rules of the momentum strategy gives results more in line with their research. Next to the long positions in stocks, they short sell the worst stocks of the previous period which is not done in this research. Finally, putting effort in making the momentum portfolio does not give a higher return than the buy-and-hold. Therefore, concluding in line with this research, the momentum strategy is not recommended to investors.
40
an investor has the main objective to achieve a high average return, the buy-and-hold and the EWP are the most appropriate strategies.
The strategies that try to limit downside risk and lower volatility are the MVP, CPPI and TIPP. The MVP shows the minimum amount of risk in a portfolio with only stocks. Although the downside risk protection is really good and volatility was really low, one must keep in mind that investing in the risk-free rate would have reached a higher return in almost the entire research period, see Graph Ib. Only from 2010 till 2013, the MVP has better results than the risk-free rate. In years when the interest rate is really low, the MVP might be a better investment opportunity than the risk-free rate. In the financial crisis and the debt crisis the interest rates are almost zero and the MVP had a higher return than a risk-free rate investment. Keeping in mind the difficulties of calculating the weights of the portfolio, the MVP is not a strategy that should be used by private investors. The risk-free rate is a better investment for them. Institutional investors have the knowledge to implement the MVP. They might use the MVP when the risk-free rate is very low, because of the stable returns it delivers.
In contrast with this research, Haugen and Baker (1991) and Clarke, et al. (2006) find higher returns for the MVP than a pre-specified benchmark. Different methodology might explain the difference in the results, but the gap might be too large to subscribe to a different methodology. Clarke, et al. (2006) argue that the MVP has a value and small size bias. Our research finds no evidence that confirms this statement. The similarity with this research is that we both find evidence that the MVP has low volatility.
41
choose for the TIPP. When an investor chooses an insurance strategy and increases downside protection, the upward potential is probably not a main objective. Then, the TIPP serves the main objectives of downside risk protection better, lowers volatility further and adheres more to his investment philosophy than the CPPI. Because the MVP has a lower average return than a risk-free investment, we recommend the TIPP for downside protection of portfolios.
Which cyclical phase and which strategy should be linked?
Bull, bear or flat markets require other strategies to serve the objectives of the investor. When investors have an idea of the direction of the market, they want to invest according to their thoughts.
In a bull market, the upside potential and a high average return of the strategy is important, since a downturn of the market does not have to be covered. As already mentioned, the buy-and-hold and the EWP show the highest levels of these characteristics.
A bear market has declining stock returns and the portfolios need downside protection. As we already saw, the risk-free rate, MVP or TIPP are the best strategies when the investor requires downside protection.
The arising question in the results of the MVP is how it is possible that a portfolio, only including stocks, always returns the initial investment plus three per cent and has a very low volatility in returns over an investigated period of more than twenty years. The answer might be that in good times the long positions will have a higher return than the short positions and that in bad times the short positions compensate for the losses of the long positions. When this research would be done with the constraint that short selling is not allowed, the result should contain less downside protection. Then, the MVP should have some negative returns in bear markets and higher returns in bull markets. Because of the downside protection and limited upside potential, the MVP is a better strategy in bear markets than bull markets.
42
and average return are not important to the investor, the risk-free rate is the best investment. Although in times with a really low interest rate, the MVP is more appropriate.
In flat markets theory subscribes two investment strategies, the EWP and the constant mix. Our research does not give explicit results for the strategies in flat markets. But, according to our results, the EWP has an overall better performance than the constant mix. Therefore we recommend the EWP and not the constant mix in flat markets.
An explanation of the bad performance of the constant mix can be that the investment period is too long to gain from the constant mix. Theory suggests that the constant mix will gain from volatile markets moving around their mean, a flat market. In a bear market the strategy does really badly because more stocks are bought every rebalancing period, but the stock prices decline further. In a bull market the upward moving stocks are sold partly and not the entire upstream is reflected in the portfolio. Over an investment period of five years the market probably will be in a bear or bull market instead of a flat market. A shorter investment period might increase the probability of a flat market and a better average return of the constant mix.
43 Conclusion
In this research we test different portfolios strategies out-of-sample for a five year investment. The strategies are the buy-and-hold, equally weighted portfolio (EWP), momentum strategy, minimum variance portfolio (MVP), constant mix, CPPI and TIPP are presented. All the portfolios are based on the S&P 500 and the research lasts from 1989 till 2013.
The EWP has the highest average return, fifty per cent on a five year investment, and a higher Sharpe ratio, 1.69, than the buy-and-hold, 1.52. The MVP has the best downside protection. Although, the average return lower than the risk-free rate. Therefore the risk-risk-free rate or the TIPP might be better investment strategies when investors want to reduce their downside risk, but care about sufficient returns. The amount of TIPP’s with less than zero return is 0.41 per cent. Besides that, the volatility is very low, thirteen per cent, and the Sharpe ratio is the highest of all the strategies, except from the MVP. The momentum strategy has the highest maximum portfolio return, but the average return is lower than the EWP and the buy-and-hold and the Sharpe ratio is even lower than the S&P 500, 0.78. Next to the momentum strategy, the constant mix also showed a very poor performance. The average return and the Sharpe ratio are lower than the buy-and-hold and the equally weighted portfolio as well. Therefore when an investor wants to have high upside potential it is better to invest following the EWP.
From the regression of Cahart’s four factor model we can assume that the buy-and-hold and the EWP are positively related to the market, have a focus on small firms, are value investors and react negatively on momentum. The momentum strategy tends to be value investing and the MVP is not related to any of the factors of Cahart’s model. The constant mix and CPPI are positively related to the market and slightly negative to momentum. The TIPP is significantly related to all the factors, except from the momentum factor.
44
implementing these strategies should be relatively low, although the buy-and-hold strategy should have even lower fees. The MVP and momentum are harder to implement and show results worse than the EWP and TIPP. The higher fee for the fund manager will not be compensated by extra return.
Further research
45 References
Annaert, J., Van Osselaer, S. Verstraete, B., 2009. Performance evaluation of portfolio insurance strategies using stochastic dominance criteria. Journal of Banking and Finance 33(2), 272-280.
Arnott, R., Hsu, J., Moore, P., 2005. Fundamental indexation. Financial Analyst Journal 61(2), 83-99.
Asness, C., Moskowitz, T., Pedersen, L., 2013. Value and momentum everywhere. Unpublished working paper, Journal of Finance.
Baker, M., Bradley, B., Wurgler, J., 2011. Benchmarks as limits to arbitrage: understanding the low-volatility anomaly. Financial Analysts Journal 67 (1), 40-54.
Banz, R., 1981. The relationship between return and market value of common stocks. Journal of Financial Economics 9, 3-18.
Barberis, N., Shleifer, A., Vishny R., 1998. A model of investor sentiment. Journal of Financial Economics 49(3), 307-343.
Barberis, N., 2000. Investing tor the long run when returns are predictable. Journal of Finance 55(1), 225-264.
Beber A., Pagano, M., 2009. Short-selling bans around the world: evidence from the 2007-09 crisis. Unpublished working paper, Journal of Finance, University of Naples, Italy.
Bird, R., Cunningham, R., Dennis, D., Tippett, M., 1990. Portfolio insurance: a simulation under different market conditions. Insurance: Mathematics and Economics 9, 1–19.
Black, F., Jones, R., 1987. Simplifying portfolio insurance. Journal of Portfolio Management 14(1), 48-51.
Black, F., Perold, A., 1992. Theory of constant proportion portfolio insurance. Journal of Economic Dynamics and Control 16, 403-426.
Blitz, D., Van Vliet, P., 2007. The volatility effect: lower risk without lower return. Journal of Portfolio Management 7, 102-113.
46
Brown, S., 1976. Optimal portfolio choice under uncertainty: a Bayesian approach. Unpublished dissertation, University of Chicago.
Cahart, M.M., 1997. On Persistence in mutual fund performance. Journal of Finance 52(1), 57-82.
Campbell, J.Y., 2000. Asset pricing at the millennium. The Journal of Finance 55(4), 1515-1567.
Chui, A., Titman, S., Wei, K.C.J., 2010. Individualism and momentum around the world. Journal of Finance 65, 361-392.
Clarke, R., De Silva, H., Thorley, S., 2006. Minimum-variance portfolios in the U.S. equity market. Journal of Portfolio Management, 33, 10-24.
Cox, J.C., Leland, H.E., 2000. On dynamic investment strategies. Journal of Economic Dynamics & Control 24, 1859-1880.
Crombez, J., 2001. Momentum, rational agents and efficient markets. The Journal of Psychology and Financial Markets 2(4), 190-200.
DeBondt, W., Thaler, R., 1995. Financial decision-making in markets and firms: a behavioral perspective. In: Jarrow, R. et al. (Ed.), Handbooks in OR and MS 9, Elsevier, Amsterdam, 385-410.
DeBondt, W., Thaler, R., 1985. Does the stock market overreact? Journal of Finance 40, 793-805.
DeMiguel, V., Garlappi, L., Uppal, R., 2009. Optimal versus naive diversification: how inefficient is the 1/N portfolio strategy? Review of Financial Studies 22, 1915-1953.
Dempster, M., Evstignev, I., Schenk-Hoppe, K., 2007. Volatility-induced financial growth. Quantitative Finance 7(2), 151-160.
Dichtl, H., Drobetz, W., 2011. Portfolio insurance and prospect theory investors: popularity and optimal design of capital protected financial products. Journal of Banking and Finance 35, 1683-1697.
Duchin, R., Levy, H., 2009. Markowitz versus the talmudic portfolio diversification strategies. Journal of Portfolio Management, 71-74.
Eun, C., Resnick, B., 1988. Exchange rate uncertainty, forward contracts, and inter-national portfolio selection. Journal of Finance, 43, 197-215.
47
Fama, E., 1970. Efficient capital markets: a review of theory and empirical work. The Journal of Finance 25, 383-417.
Fama, E., French, K., 1992. The cross-section of expected stock returns. Journal of Finance 47(2), 427-465.
Fama, E., French, K., 1993. Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33, 3-56.
Fama, E., French, K., 2012. Size, value, and momentum in international stock returns. Journal of Financial Economics 105, 457-472.
Fama, E., MacBeth, J., 1973. Risk return and equilibrium empirical tests. The Journal of Political Economy 81(3), 607-636.
Frankfurter, G., Phillips, H., Seagle, J., 1971. Portfolio selection: the effects of uncertain means, variances and covariances. Journal of Financial and Quantitative Analysis, 6, 1251-1262.
Goodwin, T., 1998. The information ratio. Financial Analysts Journal 54(4), 34-43.
Graham, B., 1949. The intelligent investor. Harper & Brothers, New York.
Green, R., Hollifeld, B., 1992. When will mean-variance efficient portfolios be well diversified? The Journal of Finance 47(5), 1785-1809.
Grossman S., Stiglitz, J., 1980. On the impossibility of informationally efficient markets. The American Economic Review 70(3), 393-408.
Hamidi, B., Maillet, B., Prigent, J., 2009. A dynamic autoregressive expectile for
time-invariant portfolio protection strategies. Unpublished working paper, University of Cergy.
Haugen, R., Baker, N., 1991. The efficient market inefficiency of capitalization-weighted stock portfolios. Journal of Portfolio Management 17(3), 35-40.
Hong, H., Lim, T., Stein J., 2000. Bad news travels slowly: Size, analyst coverage, and the profitability of momentum strategies. Journal of Finance 55, 265-295.
Israel, R., Moskowitz, T., 2013. The role of shorting, firm size, and time on market anomalies. Unpublished working paper. Journal of Financial Economics.
