International Portfolio Diversification Using Parametric Portfolio Policies

International Portfolio Diversification Using Parametric Portfolio Policies

Chao Wang

Research Master in Economics and Business

Faculty of Economics and Business

Rijksuniversiteit Groningen Groningen, the Netherlands

September 1, 2013

A thesis submitted to Rijksuniversiteit Groningen

in partial fulfillment of the requirements of the degree of Research Master in Economics and Business

Copyright c ⃝ Chao Wang, 2013

Abstract

Portfolio theory is the quantitative analysis of how investors can diversify their portfolio in order to minimize risk and maximize returns. The traditional mean-variance analysis has a fatal disadvantage that it is not only infeasible to implement for a large number of assets but also yields noisy and unstable results. This thesis is about international portfolio diversification using parametric portfolio policies, from the perspective of a U.S. investor. Following the novel approach of optimizing portfolios with large numbers of assets, proposed by Brandt et al.

(2009), we study international asset allocation in the universe consisting of 53 country equity indices. In contrast to the traditional mean-variance method, we model directly the portfolio weight in each asset as a function of the assets characteristics: size, value, and momentum anomalies. First, using zero-cost long-short portfolios based on these anomalies (Asness et al.

2009), we explore the reason why these characteristics are selected. Then the performance of the optimal parametric portfolio policies with market and equal-weighted benchmark are evaluated, using historical financial data from 1973 to 2013. We also compare their performance with the joint equal-weighted characteristic-based long-short portfolio strategy. Our results show that parametric portfolio policies are simple to implement and produces robust returns both in and out of sample. At the same time, the joint equal-weighted characteristic-based strategy is also easy to implement, and generates excellent returns as well ¹ .

1

I want to thank my supervisor Prof.Paul Bekker for his help with this thesis.

1 Introduction

Portfolio diversification, also known as asset allocation, is very relevant because the knowledge of how to invest is necessary for both institutional and individual investors. Investors must decide how much to invest in the financial markets and how to allocate that amount among many available financial assets. We study international portfolio diversification at the country equity indices level, with 53 indices ² in the investable universe. The motivation of this research is that in the progress of the global financial integration of recent decades, more and more investors take the advantage of investing abroad, since it is generally believed that the gains from international diversification are large. For example, Harvey (1995) shows that from a U.S. perspective, large benefits can be obtained from investing in emerging markets, because adding emerging market assets to the portfolio problem significantly enhances the investment opportunity set. The characteristic-based approach proposed by Brandt, Santa-Clara, and Valkanov (2009) makes it possible to allocate wealth among a large amount of assets. Using this strategy, they studied the universe of all listed stocks in the United States from January 1964 to December 2002.

Another interesting portfolio strategy is the joint equal-weighted characteristic-based long-short portfolio, which combine different single-characteristic long-short portfolios equally. Using these portfolios, this thesis investigate international asset allocation in 53 country equity indices.

The selection of the characteristics to include in the portfolio policy specification is an important stepping stone. For Stocks, the characteristics, such as the firms lagged return, market capi- talization, price-to-earning ratio are related to the stocks expected return and variance (Fama and French 1996, Chan, Karceski, and Lakonishok 1998). However, these characteristics, are also known as anomalies from the empirical literature long before these works, because they challenge the market efficiency hypothesis. They are named as small-firm effect, value versus growth effect, and momentum and reversal. For the behavioral explanations for these anomalies, please refer to Ackert and Deaves (2010).

2

The country equity indices are: Argentina, Australia, Austria, Belgium, Brazil, Bulgaria, Canada, Chile,

China A, China H, Colombia, Cyprus, Czech Rep., Denmark, Finland, France, Germany, Greece, Hungary, Hong

Kong, India, Indonesia, Ireland, Israel, Italy, Japan, Korea, Luxembourg, Malaysia, Mexico, Netherland, New

Zealand, Norway, Pakistan, Philippine, Peru, Poland, Portugal, Romania, Russia, Singapore, Slovenia, South

Africa, Spain, Sri Lanka, Sweden, Switzerland, Taiwan, Thailand, Turkey, U.K., U.S., Venezuela.

The small-firm effect is the tendency that firms with small market capitalization would earn excess returns after accounting for market risk. In the U.S. markets, a zero-cost long-short size portfolio, which long the smallest firms and short the largest firms, was able to earn 1.52% per month during 1931-1975 (Banz 1981). But for U.S. markets, the small-firm effect is gradually disappearing in the last 20 years or so. One possibility is that arbitrageurs and investors have systematically exploited the profitable opportunities after they are revealed by published research (Easterday 2007). But as we will show later, small size effect still exists in the global country indices.

Value investing is the tendency to overweight value stocks relative to growth stocks, because value stocks tends to outperform growth stocks. Value stocks are the stocks with prices that are low relative to accounting measures, such as earnings, cash flows, and book value. This implies that they are undervalued. On the other hand, growth stocks, those glamour stocks that draw lots of public attention, are stocks with prices that are high relative to earnings, cash flows and book value. They tend to be overpriced, partially because the market anticipates high future growth. A study by Sanjoy Basu (1977) focuses on price-to-earnings (P/E) ratios. Sampling an average of 500 stocks per year over 1956-1969, he groups them into quintiles on the basis of P/E ratios. The results show that returns and P/E ratio are negatively correlated. The quintile with the highest P/E ratio has lower returns (9.34% per year) than does the lowest quintile (16.30% per year), after controlling for the risk, in the sense of CAPM beta risk. Researches in term of other related characteristics, such as book-to-market price ratio (B/P), cash flow- to-price ratio (CF/P) also share the similar results (Lakonishok 1994, Fama and French 1998).

Intuitively, these ratios compare fundamentals (earnings, book values, and cash flows) per share with market values per share. Holding these accounting fundamentals constant, a lower P/E ratio, or a higher B/P (CF/P) ratio, implies a lower stock price, and, according to empirical evidence, leads to higher future returns.

Momentum and reversal also challenge the market efficiency hypothesis, because it stipulates

that returns should not be predictable by lagged returns (Ackert and Deaves 2010). Momentum

exists if returns are positively correlated with past returns, while reversal exists when return-

s are negatively correlated with past returns. Well-documented empirical evidences indicate

that there are reversal for short-term intervals (one-month) (Jegadeesh 1990), momentum for

medium-term intervals (3-12 months) (Jegadeesh and Titman 1993), and reversal for long-term intervals (about 3-5 years) (De Bondt and Thaler 1985). The approaches of these works are similar, they form portfolios of winner and loser portfolios based on past stock market perfor- mance over short, medium or long term intervals, respectively. For instance, based on five-year formation periods, De Bondt and Thaler (1985) document that the difference between winners and losers is stark, because past losers substantially outperform past winners. Intermediate- term (3-12 month) momentum is documented by Jegadeesh and Titman (1993). By buying winners and selling losers, they form a long-short zero-cost portfolio on the basis of returns over the previous six months earned an average excess return of 0.95% per month over the next six months. Moreover, momentum and other anomalies have been found to be robust internationally, which alleviate the charge of data snooping. For example, in an examination of 12 European countries, a zero-cost long-short momentum strategy generate return of 1% per month (Rouwenhorst 1998).

We have discussed the characteristics of stocks, but for country indices, the list of anomalies may have some minor differences. In the next section, the anomalies in the set of 53 country equity indices will be examined, using zero-cost long-short portfolios. As we will show later, momentum (mid-term), value and growth effect, and size effect are still robust. However, instead of having a short-term reversal, there is a short-term momentum for country indices. Moreover, the long-term reversal may still exists, but since it is not significant from zero, we will not include it as one of the characteristics in the parametric portfolio policies.

One may have some concerns with the role that foreign exchange rate will play in our research, as we are studying international portfolio diversification. But it is much easier to solve than we first expect. For each nominal amount invested in a foreign asset the same amount of foreign currency must be purchased, in other words, at a given time t, in order to purchase a foreign asset, the price that we need to pay in term of the domestic currency (in our case, U.S. dollars) equals to the price in foreign currency times the exchange rate:

S D,t = S F,t X t , (1)

where S _D,t and S _F,t are the prices in domestic and foreign currencies, respectively, and X _t is the

exchange rate. To facilitate our research, we convert the prices of foreign assets in their own

currency (S _F,t ) to the prices in the domestic currency (S _D,t ), using the foreign exchange rates

at time t. The relationship above suggests, using the price S _D,t , which is in domestic currency, we don’t need to worry about the exchange rate issue when studying international portfolio diversification, because the return in domestic currency r _D,t has already taken the change of the exchange rate X _t into account.

Recently, Asness, Moskowitz and Pedersen (2009) use equal-weighted characteristic-based long- short portfolio strategies to study value and momentum jointly. Their analysis across different asset classes and countries shows that using a portfolio strategy based on both of these char- acteristics strongly outperforms each of the individual characteristic strategies. In a similar framework, Hjalmarsson (2009) study U.S. stocks with long-short portfolio strategies formed on seven different stock characteristics representing various measures of past returns, value, and size. Although the analysis of using simple long-short portfolio strategies sheds some light on international portfolio diversification, it is far from complete. On the other hand, using Markowitz’s mean-variance framework, DeRoon, Nijman, and Werker (2001) analyze interna- tional diversification benefits for a domestic investor with mean-variance utility, from a U.S.

investor’s perspective. Following the same regression framework, Driessen and Laeven (2007) estimate the regional and global diversification benefits when allowing the investor to invest in equity indices for Europe, the US, and Far East. But the Markowitz’s mean-variance frame- work has a fatal shortcoming, as we will discuss later. As far as we know, no one has apply parametric portfolio policies advanced by Brandt et al. (2009) to study international portfo- lio diversification, with a set of county equity indices. Therefore, this thesis on international portfolio diversification will fill this gap. It gives robust and easy to implement results, which should be useful and valuable in practice as well.

2 Methodology

The traditional approach to optimize portfolio allocation is the mean-variance analysis by

Markowitz (1952), which requires modeling the expected returns, variances, and covariances

of all assets. For a problem with N stocks, the traditional Markowitz approach requires model-

ing N first and (N ² + N )/2 second moments of returns. With 500 assets, the return modeling

step involves more than 125,000 parameters. Even for the portfolio with only 50 assets, the

modelling step needs to estimate more than 1250 parameters. This is a formidable econometric problem given the large number of moments involved and the need to ensure the positive def- initeness of the covariance matrix, moreover, the results of the procedure are also notoriously noisy and unstable (e.g., Michaud 1989). Although a number of fixes for the Markowitz ap- proach have been proposed, such as shrinkage of the estimates (e.g., Jagannathan and Ma 2002, Ledoit and Wolf 2003, 2004), formal portfolio optimization based on mean-variance analysis is seldom applied by asset managers or investors in practice, because it requires substantial resources.

Brandt et al. (2009) proposes a new approach to portfolio optimization based on stocks’ char- acteristics. They parameterize the portfolio weight of each stock as a function of the firms characteristics and estimate the coefficients of the portfolio policy by maximizing the expected utility the investor would have obtained by implementing the policy over the historical sample period. By this means they directly draw inferences about the optimal portfolio weights from the data, and skip the intermediate auxiliary step of modeling returns. Their research focuses on all the U.S. stocks in the CRSP-Compustat data set from 1974 to 2002. The results show that their approach substantially improves the efficiency of the portfolio, and it is likely to perform almost as well out of sample as the in-sample analysis suggests.

The basic idea of Brandt et al. (2009) is as follows, suppose at date t, there is a large number (N _t ) of stocks in the investable universe. Asset i has return of r _i,t+1 from date t to t + 1 and is associated with a vector of characteristics x _i,t observed at date t. For example, the characteristics could be the market capitalization of the stock, the price-earning ratio of the stock, momentum (medium term interval), short-term and long-term reversals of the stock.

Once the characteristics of the assets have been selected, the investor’s problem is to choose the portfolio weights w i,t to maximize the conditional expected utility of the portfolio’s return r _p,t+1 :

max

{w

}

E _t [u (r _p,t+1 )] = E _t [

u ( _N

∑

i=1

w _i,t r _i,t+1 )]

, (2)

where

r _p,t+1 =

N

∑

i=1

w _i,t r _i,t+1 . (3)

The optimal portfolio weights can be parameterized as a function of the stocks’ characteristics.

In a simple linear specification, we have

w _i,t = w _i,t + 1

N _t θ ^⊤ ex i,t , (4)

where w _i,t is the weight of asset i at date t in a benchmark portfolio, θ is a vector of coefficients to be estimated, and ex i,t are the characteristics of stock i, standardized cross-sectionally to have zero mean and unit standard deviation across all stocks at date t. In order to standardize the characteristics, we run a cross-sectional regression each period of each given characteristic on others, and take the normalized residuals (zero mean and unit variance) of that regression as inputs to the portfolio policy (4). These residuals are the component of the characteristic that are orthogonal to other characteristics, since the regression removes all commonality in the regressand due to the regressors. The second term in (4) represents the deviations of the optimal portfolio from the benchmark. The term 1/N _t is a normalization that allows the portfolio weight function to be applied to an arbitrary number of stocks. Since the number of assets may be varying over time, without this normalization, changing the number of assets without otherwise changing the cross-sectional distribution of the characteristics results in totally different asset allocations, even though the investment opportunities are fundamentally unchanged. The most important aspect of our parameterization is that the coefficients θ are constant across assets and through time.

The corresponding sample analog of the optimization problem (2) is

max θ

1 T

T −1

∑

t=0

u ( _N

∑

i=1

(

w _i,t + 1 N _t θ ^⊤ ex i,t

) r _i,t+1

)

. (5)

The estimator b θ should satisfy the first-order condition:

1 T

T −1

∑

t=0

u ^′ (r _p,t+1 ) ( 1

N _t ex ^⊤ t r _t+1 )

= 1 T

T −1

∑

t=0

u ^′ ( _N

∑

i=1

(

w _i,t + 1 N _t θ ^⊤ ex i,t

) r _i,t+1

) ( 1

N _t ex ^⊤ t r _t+1 )

= 0. (6)

Parameters b θ and the covariance matrix of coefficients b Σ _θ in this equation can be estimated using Generalized Method of Moments (GMM) (Hansen (1982)). We define

h(r _t+1 , ex t ; θ) ≡ u ^′ (r _p,t+1 ) ( 1

N _t ex ^⊤ t r _t+1 )

, (7)

m(r _t+1 , ex t ; θ) ≡ 1 T

T −1

∑

t=0

u ^′ (r _p,t+1 ) ( 1

N _t ex ^⊤ _t r _t+1 )

, (8)

and

G(r _t+1 , ex t ; θ) = ∂m(r _t+1 , ex t ; θ)

∂θ = 1

T

T −1

∑

t=0

u ^′′ (r _p,t+1 ) ( 1

N _t ex ^⊤ _t r _t+1 ) ( 1

N _t ex ^⊤ _t r _t+1 ) _⊤

. (9)

Thus vector m(r _t+1 , ex t ; θ) has the property:

m(r _t+1 , ex t ; θ) ∼ N (0, V ) , (10) where V is estimated by

V = b 1 T

T −1

∑

t=0

h(r _t+1 , ex t ; θ)h(r _t+1 , ex t ; θ) ^⊤ . (11) With some starting value θ (0) , the coefficients θ can be estimated by computing the following equation iteratively:

bθ _(j+1) = b θ _(j) − {

G ^⊤ (r _t+1 , ex t ; θ) b V ⁻¹ G(r _t+1 , ex t ; θ) } ₋₁

G ^⊤ (r _t+1 , ex t ; θ) b V ⁻¹ m(r _t+1 , ex t ; θ) . (12) Generally, it converges quickly to b θ _{GM M} after a sufficient number of iterations. The asymptotic distribution of this estimator is

√ T (b θ GM M − θ) ∼ N (

0, plim

n →∞

{

G ^⊤ (r t+1 , ex t ; b θ GM M ) b V ⁻¹ G(r t+1 , ex t ; b θ GM M ) } ₋₁ )

. (13)

Hence its asymptotic covariance matrix is bΣ _θ = 1

T {

G ^⊤ (r _t+1 , ex t ; b θ _{GM M} ) b V ⁻¹ G(r _t+1 , ex t ; b θ _{GM M} ) } ₋₁

. (14)

Having estimated the parameters θ, we can use it in the simple linear specification of portfolio weights (4).

There are several benefits from focusing directly on the portfolio weights. First, it avoids

the Achilles heel of the traditional econometric approach: the return modeling step. Second,

parameterizing the portfolio policy leads to a tremendous reduction in dimensionality, and

therefore limits the room for model misspecification and estimation error. The coefficients will

only deviate from zero if the respective characteristics offer an interesting combination of return

and risk consistently across stocks and through time. In contrast to the Markowitz approach,

the approach of Brandt et al. (2009) only involves estimating coefficient θ regardless of the

investors preferences and the joint distribution of asset returns. From a practical perspective,

this approach is simple to implement and produces robust results in and out of sample.

3 Characteristics Selection

We want to apply the parametric portfolio policies to study international portfolio diversification in the investable university consisting of 53 country equity indexes. The monthly data from July 1973 to July 2013 is obtained using Datastream. The first step of our analysis is to select a finite set of characteristics that should be included in the portfolio policy specification (4).

The are a number of characteristics that can be chosen. Five different characteristics are ex- amined in this section using zero-cost long-short portfolios. The first three represent momen- tum/reversal in returns: short-term momentum (ST-Mom), defined as the prior month’s (t − 1) return; medium-term momentum (Mom), defined as the returns from month t − 12 to t − 1; and long-term reversals (LT-Rev), defined as the returns from month t − 36 to t − 13. The last two are value ratio: price-earning ratio (P/E), and the size effect, measured by market capitaliza- tion (MV). We need to point out that the empirical evidence shows that country indices have short-term momentum, as opposed to short-term reversals for stocks. The disappearance of the short-term reversal for country indices is reasonable, because each country index aggregates all the available stocks.

Each period (every month), we sort and group the country indices into quantiles according to different characteristics at the beginning of that period. The returns on a zero-cost long- short portfolio for each characteristic are constructed by calculating the difference between the returns on the top decile portfolio and the bottom decile portfolio. The long-short portfolios are scaled to one dollar long and one dollar short. In cases of ST-Mom and Mom, the returns are expected to increase in the characteristic, hence the returns of the high-minus-low portfolios are constructed (by longing the top decile portfolio, and shorting the bottom decile portfolio);

on the other hand, for LT-Rev, P/E, and MV, the returns are expected to decrease in the characteristic, thus the returns on the low-minus-high portfolios are constructed.

The first part of table 1 reports summary statistics for the (annualized) returns of all the

zero-cost single-characteristic portfolios. The second row of this table indicates that whether

it is the return of the top decile portfolio minus that of the bottom decile portfolio (high-

minus-low), or the other way around (low-minus-high). Long-short LR-Rev portfolio has the

LT-Rev Mom ST-Mom P/E MV Joint low-high high-low high-low low-high low-high

¯

r 0.0724 0.1033 0.1233 0.1547 0.1153 0.1022

Std. Err. (0.0558) (0.0486) (0.0504) (0.0459) (0.0546) (0.0226)

σ(r) 0.3393 0.3036 0.3145 0.2868 0.3411 0.1409

Info. Ratio 0.2135 0.3402 0.3919 0.5394 0.3380 0.7252 Correlations of the single-characteristic portfolios

LT-Rev 1.0000

Mom -0.0010 1.0000

ST-Mom -0.0509 0.0911 1.0000

P/E 0.0046 -0.2861 -0.1970 1.0000

MV -0.0069 0.0100 0.1227 0.3448 1.0000

Table 1: Single-characteristics zero-cost long-short portfolios

lowest mean return 7.24%. For other characteristics: Mom, ST-Mom, P/E and MV, the mean returns are 10.33%, 12.33%, 15.47% and 11.53%, respectively. The mean returns of zero-cost long-short portfolios based on Mom, ST-Mom, P/E and MV generate average returns that are statistically different from zero, because they are more than two standard deviations away from zero. The only exception is the LR-Rev based portfolio, for which the null hypothesis of zero average returns cannot be rejected at standard significance levels. The following row of table 1 presents a performance measure of the risk-adjusted return: information ratio (IR).

It is defined as expected excess return from the long-short portfolio divided by its standard deviation. Long-short P/E portfolio has the greatest information ratio (0.5394), which means that its risk-adjusted return is the highest among all the long-short portfolios, and long-short LR-Rev portfolio has the smallest IR (0.2135), others lie midway between them. Since the mean return of LR-Rev portfolio is not significant different from zero, as well as its zero-cost long-short portfolio has the lowest information ratio, we will not include LR-Rev in the characteristics set for the parametric portfolio policies.

The second part of table 1 shows the correlations of the monthly returns on the single-characteristic

zero-cost long-short portfolios. Notice that there are negative correlation between value (P/E)

and momentum, value and ST-Mom portfolios. These two pair of strategies are negatively corre-

lated with each other, and each of them consistently produces positive mean return. Moreover,

none of these anomalies are highly correlated. The highest one, 0.3448, is correlation coefficient

between MV and P/E portfolios. Therefore it is beneficial to exploit these these anomalies jointly. The last column labeled “joint” is the equal-weighted zero-cost long-short portfolio across four characteristics (Mom, ST-Mom, PE and MV), which strongly outperforms all the single-characteristic long-short portfolios, measured by the information ratio.

It’s worth emphasizing that initially the investor does not need to pay anything for the zero-cost long-short portfolios, but the zero-cost (Mom, ST-Mom, P/E and MV) portfolios will generate positive returns, which are significantly different from zero. And yes, this challenges the effi- cient market hypothesis. Investors can take advantage of these anomalies by investing in them.

One thought is that we simply exploit these characteristics jointly in the long-short portfo- lio. As shown before, joint equal-weighted characteristic-based long-short portfolio outperforms single-characteristic long-short portfolio, because there are great diversification benefits from combining long-short strategies based on several different characteristics. Another though is using these characteristics in the parametric portfolio policies, because this method will deliver the optimal portfolio weights (4) by solving the optimization problem (2).

4 Empirical Application

In order to illustrate the effectiveness, and compare the performance of different strategies, we apply parametric portfolio policies (PPP) and joint equal-weighted characteristic-based long- short portfolios (joint LS) to study international portfolio management in 53 country equity indices. The data on country indices comes from Datastream, from July 1973 to July 2013.

Please refer to the first section for the list of the indices. In the optimization problem (2), constant relative risk aversion (CRRA) utility function is used, with the relative risk aversion γ = 5.

The best way of testing the robustness of parametric portfolio policies approach is through

out-of-sample experiment. We use the first ten years of data, from July 1973 until July 1983,

to estimate the coefficients for the initial out-of-sample portfolio policy, then apply these pa-

rameters to form portfolio every month in the following year. We repeat this procedure every

subsequent year by re-estimating the coefficients with the enlarged sample and applying it to

PPP, MW benchmark PPP, EW benchmark MW EW Joint LS In sample Out of sample In sample Out of sample

θ

- - - 1.3568 1.1140 1.3557 1.1915

Std. Err. - - - (0.5671) (0.6173) (0.5746) (0.6240)

θ

- - - 0.9710 1.1816 1.0157 1.1292

Std. Err. - - - (0.6989) (0.8322) (0.7269) (0.8384)

θ

- - - -3.8009 -2.1192 -2.8379 -1.4150

Std. Err. - - - (0.9242) (0.9654) (0.9272) (0.9753)

θ

- - - -0.5333 0.3340 1.6177 2.2313

Std. Err. - - - (1.3763) (1.3684) (1.3744) (1.3701)

|ω

| 0.0189 0.0189 0.0327 0.0557 0.0475 0.0535 0.0446

max ω

0.4369 0.0330 0.3437 0.2994 0.4321 0.2467 0.3223

min ω

0.0000 0.0330 -0.0605 -0.2547 -0.1745 -0.2547 -0.1668

∑ ω

I(ω

< 0) 0.0000 0.0000 -0.3551 -0.9762 -0.7581 -0.9180 -0.6815

∑ |ω

− ω

| 0.0279 0.0047 0.8244 1.4815 1.4698 1.4650 1.4638

¯

r 0.1249 0.1710 0.2535 0.3521 0.2530 0.3676 0.2725

Std. Err. (0.0291) (0.0350) (0.0401) (0.0647) (0.0517) (0.0660) (0.0508)

σ(r) 0.1820 0.2188 0.2505 0.3545 0.2831 0.3613 0.2784

Sharpe Ratio 0.4053 0.5436 0.8737 0.8849 0.7499 0.9142 0.8353

Info. Ratio - 0.3932 0.7252 0.7691 0.6727 0.7954 0.7735

α - 0.0567 0.1349 0.2363 0.1280 0.2538 0.1507

β - 1.0194 0.9871 0.9511 1.0683 0.9260 1.0283

σ(ϵ) - 0.1207 0.1561 0.3021 0.1979 0.3115 0.1976

Cumulative returns 19.1 75.8 487.6 2940.1 392.9 4111.3 643.4

Table 2: Summary of different portfolio policies

from the new portfolio policies in the following year. The out-of-sample results of the optimal parametric portfolio policy can be obtained, and compared with the in-sample results. It means that now the data sample goes from July 1983 to July 2013, since we lose the first ten years of data to estimate the initial portfolio for the out-of-sample experiments. Moreover, because our research is from a U.S. investor’s perspective, we use the three-month Treasury bill rate as the risk-free rate. It is available freely from the website of the Federal Reserve Bank of St. Louis.

For ease of interpretation, all measures are annualized.

Table 2 presents the results of different portfolio startegies for the international portfolio di-

versification in the investable universe consisting of 53 country indices. The table is divided

into three sections describing separately the (i) parameter estimates and standard errors, (ii) distribution of the portfolio weights, (iii) performance of the portfolio returns. The columns labeled “MW”, “EW”, “Joint LS”, “PPP, MW benchmark”, and “PPP, EW benchmark” dis- play statistics of the market-weighted portfolio, equal-weighted portfolio, joint equal-weighted characteristic-based portfolio, the optimal parametric portfolio policies with market-weighted and equal-weighted portfolio as benchmark, respectively. Each optimal parametric portfolio policy are further divided into two columns: “in sample” and “out of sample”. Another thing to notice is that, different from the zero-cost long-short portfolios in the last section, in this section, the joint equal-weighted characteristic-based long-short portfolio is the combination of two parts: first part is the market portfolio that worth one dollar; the other part is the zero-cost equal-weighted long-short portfolio, which are scaled to one dollar long and one dollar short, based on four characteristics (Mom, ST-Mom, PE and MV).

The first few rows of table 2 present the estimated coefficients of the portfolio policies along with their standard errors. The coefficients determine the deviations of the optimal portfolio from the benchmark. The over- or underweighting of each asset, relative to the benchmark portfolio, depends on momentum, short-term momentum, price-earning ratio and market capi- talization, using the policy function in equation (4). In the fourth column, the deviations of the optimal weights from the market weights (benchmark) increase with momentum and short-term momentum, and decrease with P/E ratio (value) and market capitalization (size). The signs of the estimates are consistent with the discussion on momentum, value and size effect in the pre- vious section. The sixth column describes the coefficients for the parametric portfolio policies with equal-weighted benchmark. Comparing with the fourth column, there is a minor differ- ence: the coefficient of the characteristic MV θ _{M V} is positive. This is due to the fact that the benchmark, equal-weighted portfolio, considerably underweights the indices with large market value, relative to the market portfolio. Note that not all coefficients are significant, but this is not a big concern, because the coefficients are only needed for the portfolio policy specification (4). The most convincing way to establish the robustness of a portfolio strategy is through an out-of-sample experiment.

The next section of table 2 show statistics of the portfolio weights averaged across time. These

statistics include the average absolute portfolio weight, the average minimum and maximum

portfolio weights, the average sum of negative weights in the portfolio, and the turnover in the portfolio. The average absolute weight of the parametric portfolio is nearly three times that of the market and equal-weighted portfolios (1.89% versus 5.57% and 5.35% for parametric portfolio with market and equal-weighted benchmark, respectively), while the average absolute weight of the joint characteristic-based portfolio is 3.27%. Comparing with market and equal- weighted portfolios, the active portfolio takes larger positions, which mean the asset allocations are more aggressive. But the positions of these portfolios are not extreme. The time averaged maximum weights of the market portfolio are 43.69%, which is due to the fact that U.S. market takes up a large portion of the whole investable universe. The maximum weight of equal- weighted portfolio is 3.30%, which means it significantly underweights indices with large market value. For other portfolios, the maximum weights are slightly smaller than that of the market portfolio. They range from 24.67% to 34.37%. Moreover, the minimum weights are 0.00%, -6.05% and -25.47% for the market, joint characteristic-based, and the parametric portfolios, respectively. The average sum of negative weights in the two parametric portfolios are -97.62%

and -91.80%, which means that the sum of long positions are 197.62% and 191.80%. For joint characteristic-based portfolio, the average sum of negative weights is -35.51%. These results make sure that these portfolio policies do not require unreasonably large trading activity.

For a given policy such as (3), the turnover of each period is defined as the sum of all the absolute changes in portfolio weights from one period to the next:

T t =

N

∑

i=1

|ω i,t − ω i,t −1 | . (15)

The time averaged turnovers of the joint characteristic-based and the optimal parametric port- folios are around 82.44% and 147% per year, respectively, in contrast to an average turnover of 2.79% per year for the market portfolio, and 0.47% per year for the equal-weighted portfolio.

As the transaction cost is approximately at the magnitude of 0.1% of the turnover, the returns are unlikely to be affected much by trading costs.

The last section of table 2 reports the performance of all these portfolios. The mean returns

of the market, equal-weighted, joint characteristic-based, optimal parametric portfolios with

market and equal-weighted benchmark are 12.49%, 17.10%, 25.35%, 35.21%, and 36.76%, re-

spectively, and the volatility of their returns are 18.20%, 21.88%, 25.05%, 35.45%, and 36.13%

respectively. The parametric portfolio with equal-weighted benchmark has the highest return and volatility, while the market portfolio has the lowest return and volatility. Notice that there is a trend that the return increase with the volatility. In order to evaluate their performance, we use Sharpe ratio and information ratio. The definition of Sharpe ratio is

SR = E[r _p − r f ]

√ Var(r _p − r f ) , (16)

where r _p is the return of the portfolio, and r _f is the risk-free rate. It measures the excess return (with respect to the risk-free rate) per unit of risk in a portfolio. The information ratio is

IR = E[r _p − r m ]

√ Var(r _p − r m ) , (17)

where r _m is the return of the market portfolio. It is a measure of the risk-adjusted return of a portfolio, with respect to a selected benchmark, which is the market portfolio in this definition.

In terms of Sharpe ratio, the parametric portfolios with equal-weighted benchmark has highest Sharpe ratio (0.9142). It is more than twice the Sharpe ratio of the market portfolio (0.4053), which is the lowest. The Sharpe ratios of the parametric portfolio with market benchmark and the joint characteristic-based portfolio are slighted lower (0.8849 and 0.8737), but still much higher than the second lowest one (0.5436), which is the Sharpe ratio of the equal-weighted portfolio. The information ratio of these portfolios tell us a similar story: the parametric portfolios with equal-weighted benchmark has the highest information ratio (0.7954), which is higher than other two active investment strategies, while the equal-weighted portfolio has the lowest one (0.3932).

The CAPM alpha, beta and residual risk of a portfolio are obtained by regressing the excess return of the portfolio on the excess return of the market:

r _p,t − r f = α + β(r _m,t − r f ) + ϵ _t . (18)

In an efficient market, the expected value of the alpha is zero. Therefore the alpha coefficient

indicates how a portfolio has performed after accounting for the risk it involved. When alpha

is greater than zero, it means the portfolio has a return in excess of the reward for the assumed

risk. The beta coefficient represents risk. Generally speaking, higher-beta portfolios tend to be

more volatile and therefore riskier, but provide the potential for higher returns, while portfolios

with lower beta are less volatile. Notice that in table 2, all the active portfolios and the equal- weighted portfolio produce positive alpha. Moreover, the beta coefficients for these three active portfolios are all smaller than one. The parametric portfolio with equal-weighted has the biggest alpha, 0.2538, and the smallest beta, 0.9260, which means its performance is the best. The last row of table 2 reports the cumulative returns to different portfolio strategies. Returns of the in-sample optimal parametric portfolios are two orders of magnitude greater than that of the passive portfolios (market and equal-weighted portfolios).

To sum up, the performance of the optimal parametric portfolio with equal-weighted benchmark is the best, which outperforms the parametric portfolio with market benchmark. This may due to the reason that equal-weighted portfolio has better performance than the market portfolio (DeMiguel, Garlappi, and Uppal 2007). The joint equal-weighted characteristic-based portfolio also delivers excellent returns. These three active investment strategies are much more efficient than the market and equal-weighted portfolio. Last but not least, it should not be very surprising that the optimal parametric portfolio outperforms the market and equal-weighted portfolios because we are optimizing in sample and have chosen characteristics that are associated with returns.

Now we try to establish the robustness of these portfolio strategies through out-of-sample ex-

periments. As we have stated before, we use the first ten years of data to estimate the initial

coefficients, which are used to form the first year’s portfolios. Every subsequent year, we enlarge

the sample, reestimate the coefficients, and apply them to form portfolios in every month of the

following year. The out-of-sample results are presented in the fifth and seventh column of table

2. The coefficients for the characteristics are roughly similar to the in-sample estimates of the

parametric portfolio with equal-weighted benchmark. The distribution of the portfolio weights

of both in- and out-of-sample portfolios are analogous. Moreover, the time averaged turnovers

are approximately the same for both in- and out-of-sample. For the properties of returns, both

mean returns and volatilities of the out-of-sample portfolios are lower than that of the in-sample

portfolios. The Sharpe ratios and information ratios for the out-of-sample parametric portfolios

are also lower than the corresponding in-sample portfolios, but the differences are not too big,

which means that there is no large deterioration in the performance of returns. Another thing

to point out is that in terms of different measures of performance, equal-weighted characteristic-

July 1983 0 July 1988 July 1993 July 1998 July 2003 July 2008 July 2013 10

20 30 40 50 60

70 market−weighted equal−weighted US

Figure 1: Cumulative returns of the market, equal-weighted portfolios, and U.S. index based portfolio and out-of-sample parametric portfolio with equal-weighted benchmark are com- parable, although for the in-sample performance, the latter outperforms the former.

The last row of table 2 also reports the cumulative returns of joint characteristic-based portfolio and out-of-sample optimal parametric portfolio policies. They are one order of magnitude smaller than the in-sample cumulative returns, but still one order of magnitude greater than that of the market and equal-weighted portfolios. Figure 1 plots the cumulative returns of the market, equal-weighted portfolios, and U.S. index. The curves for the market portfolio and U.S. index highly coincide, and that for equal-weighted portfolio is considerably higher than the other two. Figure 2 reports the cumulative returns of in- and out-of-sample active portfolios, in both Cartesian and semi-log coordinates. In the legend of the figure, “PPP”, “MW”, “EW”, and “OS” represent “parametric portfolio policy”, “market-weighted”, “equal-weighted”, and

“out-of-sample”, respectively. The curves for in-sample parametric portfolios rise really fast,

but are also more volatile. The joint characteristic-based long-short portfolio and the out-of-

sample parametric portfolios deliver similar cumulative returns. One thing to notice is that

since an investment in different portfolios would assume varied amount of risk. Moreover, it

is well known that high level of potential return is associated with high level of risk. As the

volatilities or risks of these portfolios are very different, we should not read too much into the

July 1983 0 July 1988 July 1993 July 1998 July 2003 July 2008 July 2013 500

1000 1500 2000 2500 3000 3500 4000

Joint LS PPP MW PPP MW OS PPP EW PPP EW OS

July 1983 July 1988 July 1993 July 1998 July 2003 July 2008 July 2013 10

10

10

10

Joint LS PPP MW PPP MW OS PPP EW PPP EW OS

Figure 2: Cumulative returns of the active strategies, in- and out-of-sample

cumulative returns in figures 1 and 2, which are only for illustrative purpose.

5 Conclusion and Possible Extensions

We document the positive returns of the zero-cost long-short portfolio based on different charac- teristics: market capitalization, price-to-earning ratio, momentum and short-term momentum.

We use the joint equal-weighted characteristic-based portfolio, and the parametric portfolio policies to study international portfolio diversification in the investable universe consisting of 53 country equity indices. The parametric portfolio policies proposed by Brandt et al. (2009) models the portfolio’s weights in each stock as a function of the asset’s characteristics. The coef- ficients of this function are found by optimizing the investor’s expected utility of the portfolio’s return over a given sample period.

The conclusion is that the parametric portfolio policies substantially improves the efficiency of the portfolio without a significant loss in terms of out-of-sample robustness, since it performs almost as well out of sample as the in-sample analysis suggests. For the parametric portfolio poli- cies, the equal-weighted portfolio is a better benchmark comparing with the market benchmark.

Moreover, the simple strategy of joint equal-weighted characteristic-based long-short portfolio can deliver excellent returns, whose performance is comparable with that of the out-of-sample parametric portfolio strategies.

For further researches, we can consider some extensions, such as short-sale constraints, time- varying coefficients, and transaction costs. Short-sale constraints are very common in practice.

In order to impose the no-short-sale constraint, we need to set the negative portfolio weights

to zero. But in doing so, the portfolio weights no longer sum to one. Hence we need to

renormalize the portfolio weights to insure their sum is one. With respect to including time-

varying coefficients, the reason is that, although the coefficients in equation (4) are assumed to

be time invariant, in fact, macroeconomic variables related to the business cycle have certain

level of influence on asset returns. Trying to accommodate such kind of time variation in the

coefficients of the portfolio policy will refine the analysis. Finally, another extension is to try

to incorporate transaction costs, for example, by using approaches as described by Davis and

Norman (1990), Shreve and Soner (1994), Akian, Menaldi and Sulem (1996).

Reference

Ackert, L., Deaves, R. 2010. Behavioral Finance, Psychology, Decision-Making and Markets.

Mason, Ohio: South-Western Cengage Learning.

Akian, M., J. Luis Menaldi, and A. Sulem. 1996. On an Investment-Consumption Model with Transactions Costs. SIAM Journal of Control and Optimization 34:329-64.

Asness, C.S., T.J. Moskowitz, and L. Pedersen. 2009. Value and Momentum Everywhere.

Working Paper, Stern School of Business, New York University.

Banz, R. W., 1981. The relationship between return and market value of common stocks, Journal of Financial Economics 9:3-18.

Basu, S., 1977, Investment performance of common stocks in relation to their price-earnings ratios: A test of the efficient market hypothesis, Journal of Finance 32, 663-682.

Brandt, Michael W., Pedro Santa-Clara, and Rossen Valkanov. 2009. Parametric Portfolio Policies: Exploiting Characteristics in the Cross Section of Equity Returns. Review of Financial Studies 22:3411-3447.

Chan, K. C., J. Karceski, and J. Lakonishok. 1998. The Risk and Return from Factors. Journal of Financial and Quantitative Analysis 33:159C88.

Davis, M. H. A., and A. R. Norman. 1990. Portfolio Selection with Transaction Costs. Mathe- matics of Operations Research 15:676-713.

De Bondt, W. F. M., and R. Thaler. 1985. Does the stock market overreact? Journal of Finance 40, 793-807.

DeMiguel, V., L. Garlappi, and R. Uppal. 2009. Optimal Versus Naive Diversification: How

Inefficient is the 1/N Portfolio Strategy? Review of Financial Studies 22, 1915-1953.

De Roon, Frans A., Theo E. Nijman, and Bas J.M. Werker. 2001. Testing for Mean- Variance Spanning with Short Sales Constraints and Transaction Costs: The Case of Emerging Markets.

Journal of Finance 56(2):721-742.

Driessen, J. and L. Laeven. 2007. International Portfolio Diversification Benefits: Cross- Country Evidence. Journal of Banking and Finance, 31(6):1693-1712.

Easterday, K. E., P. K. Sen, and J. A. Stephan. 2007. The small firm/January effect: Is it disappearing in U.S. markets because of investor learning? Working paper.

Fama, E. F., and K. R. French. 1996. Multifactor Explanations of Asset Pricing Anomalies.

Journal of Finance 51:55-84.

Fama, E. F., and K. R. French. 1998. Value vs. growth: The international evidence, Journal of Finance 53, 1975-99.

Hansen, L. P. 1982. Large Sample Properties of Generalized Method of Moments Estimators.

Econometrica 50:1029-53.

Harvey, Campbell R. 1995. Predictable Risk and Returns in Emerging Markets. Review of Financial Studies 8(3):773-816.

Hjalmarsson, E. 2011. Portfolio Diversification Across Characteristics. Journal of Investing 20(4):84-88

Jagannathan, R., and T. Ma. 2002. Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps. Journal of Finance 58:1651-84.

Jegadeesh, N., 1990. Evidence of predictable behavior of security returns, Journal of Finance 45, 881-898.

Jegadeesh, N., and S. Titman. 1993. Returns to buying winners and selling losers: Implications

for stock market efficiency. Journal of Finance 48, 65-91.

Lakonishok, J., A. Shleifer, and R. Vishny. 1994. Contrarian investment, extrapolation and risk. Journal of Finance 49, 1541-78.

Ledoit, O., and M. Wolf. 2003. Improved Estimation of the Covariance Matrix of Returns with an Application to Portfolio Selection. Journal of Empirical Finance 10:603-21.

Ledoit, O., and M.Wolf. 2004. Honey, I Shrunk the Sample Covariance Matrix. Journal of Portfolio Management 30:110-19.

Markowitz, H. M. 1952. Portfolio Delection. Journal of Finance 7:77-91.

Michaud, R. O. 1989. The Markowitz Optimization Enigma: Is Optimized Optimal? Financial Analyst Journal 45:31-42.

Rouwenhorst, K. G., 1998. International momentum strategies. Journal of Finance 53, 267-284.

Shreve, S. E., and H. M. Soner. 1994. Optimal Investment and Consumption with Transactions

Costs. Annals of Applied Probability 4:909-62.