Parametric Asset Allocation Thomas Dangl, Arne Westerkamp, Josef Zechner April 19, 2013 VERY PRELIMINARY AND INCOMPLETE! We study how a simple asset allocation policy in which the investment weights are determined only by two parameters regarding the short-term momentum and the value of each asset class aect the utility of investor with power utility U(r p ) = (1 + r p) 1 γ. (1) 1 γ We follow the idea of Brandt et al. (2008) and to determine a parameter vector ˆθ that optimizes the exposure of the investor to sentiment and value characteristics over the sample period ˆθ = max θ T 1 1 u T t=0 [ Nt ] ( w i,t + θ T ˆx i,t /N t )r i,t+1. (2) i=1 It can be seen that the weight on asset class i in period t is ( w i,t +θ T ˆx i,t /N t ), the scalar product of the estimated θ vector with the asset characteristics ˆx t, normalized by the number of asset classes N t. The optimization is performed in sample and out of sample, whereby the latter is characterized by an initial 10 year optimization only period and annual re-estimation thereafter. The returns rt+1 i are expressed in one basis currency, USD. They are derived from the original raw returns r i,ori t+1 expressed in the original currencies ORI in and hedged and an unhedged version for each asset class i r g,i,bas S t t+1 = r g,i,ori t+1, (3) S t+1 where S t is the exchange rate a time t expressed as units of ORI per unit of USD. Vienna University of Technology Vienna University of Economics and Business Vienna University of Economics and Business 1
As the return patterns of dierent asset classes dier widely in terms of means and volatilities, it is a challange to make the short-term momentum and value characteristics comparable accross asset classes. We follow the approach of Wang and Kochard (2011) and calculate for each asset class the Z-score, i.e., the quantile of its distribution of historical data up to month t 1 Z t = V t 1 mean(v 0,t 1 ). (4) Std(V 0,t 1 ) To ensure that the weights add up to one, there is an additional crosssectional normalization involved to attain to the standardized characteristics vector ˆx ˆx i,t = x i,t x t σ t. (5) In an extension, all or a subset of the asset classes, are subjected to a shortsale constraint. In that case, the weight on each asset class i is 1 Data w + i,t = max[0, w i,t ] Nt j=1 max[0, w i,t]. (6) Drawing on Asness et al. (2009), the main characteristics of each asset class are momentum (up to one year) and value. Unless otherwise noted the data are from GlobalFinancialData. Momentum Initially, this is just short-term momentum; in particular the cumulative return of the past 12 months, skipping the most recent month. [Alternatively, only the return of the months -12 to -7 as in Novy-Marx (2010), or past 3 months as in Barroso and Santa-Clara (2011)]. Value The negative of the cumulative return of the past ve years (excluding the most recent month). For currencies, the interest earned using 3-month LIBOR rates is accounted for when calculating the return. [Alternatively, we could follow Barroso and Santa-Clara (2011) and use the real currency appreciation.] The set of asset classes covers a broad spectrum of available to investors in global capital markets. Major equity markets Total returns of major aggregate equity market indices. The total return indices of Japan, Emerging Markets, the Eurozone, United Kingdom and the United States are used. Government bonds Total return indices for selected countries. 2
Corporate bonds Total returns on the BBB tranches of US and Europe Merrill Lynch indices as well as US-denominated sovereign emerging market Merrill Lynch index Currencies In contrast to the other asset classes, returns on currency strategies are excess returns and are calculated as the change in exchange rates, accounting for the interest rate dierential in LIBOR rates. Accordingly the weight in the benchmark is zero and currency strategies are generally excepted from short-sale constraints. Commodities Total return according to the US-denominated GSCI. Figures 1 to 3 show some examples of the return patterns and yields selected asset classes 2 Transaction costs Transaction costs are an important determinant of the strategy return and are considered in a number of ways. First, the return calculation given the portfolio weights can handle xed (per trade) and variable (proportional) fees. For the time being, these are the same for all assets but can vary over time. The eect of the xed costs on the portfolio return needs the specication of a portfolio volume. This can be a constant portfolio volume or alternatively, one can specify inows into and outows out of the fund. In the latter case the portfolio volume is also mechanically aected by the (net) portfolio returns. The calculation of transaction cost is detailed in Table 1 Proportional transaction costs can also be considered in the optimization. We follow Brandt et al. (2008) and implement a simple and a no-trade variant. The simple variant estimates the optimal θ given that the resulting weight changes and buy-and-hold returns result in proportional transaction fees. The no-trade specication builds on the idea that if the distance from the weights resulting from a candidate θ-vector in the optimization does not deviate suciently from the weights resulting from the previous weights, there will be no trade to the new optimum. In case of a suciently large deviation from the current weights, the new weights are either a direct outcome of the θ-parameter or a mixture a move towards the new weights up to the boundary of the notrade region. The empirical work of? introduces a mechanism to determine the optimal size and shape of the no-trade region. We follow Brandt et al. (2008) in specifying a quadratic region with an exogenous circumference as a rst approximation to the unkown optimum. For comparison, we also study the eect of a full move to the new optimum. 3 Results The base case uses the unhedged gross returns of the government bond and equity markets of US, UK, JP, and the Eurozone, the equity and sovereign US- 3
700 600 500 400 300 200 100 0 Figure 1: EuroStoxx Total Return 1950 1952 1953 1954 1955 1956 1957 1959 1961 1962 1963 1960 1964 1966 1967 1968 1969 1970 1971 1973 1974 1975 1976 1977 1978 1980 1981 1982 1983 1984 1985 1987 1988 1989 1990 1991 1992 1994 1995 1996 1997 1998 1999 2001 2002 2003 2004 2005 2006 2008 2009 2010 2011 2012 Source: GlobalFinancialData.com Total Return EuroStoxx (1991 = 100) 4
Figure 2: 10y US Treasury Bond Return 800 Total Return 10y US Treasury Bonds (1900=100) 700 600 500 400 300 200 100 0 1820 1824 1828 1832 1836 1840 1844 1848 1852 1856 1860 1864 1868 1872 1876 1880 1884 1888 1892 1896 1900 1904 1908 1912 1916 1920 1924 1928 1932 1936 1940 1944 1948 1952 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004 2008 2012 Source: GlobalFinancialData.com 5
Source: GlobalFinancialData.com Figure 3: Yield 10y Government Netherlands 6
Table 1: Fixed and Variable Transaction Costs in ow setting Calculation of xed (F) and variable (V) transaction costs depending on whehther there is a F low from or to the portfolio volume and/or whether optimization routine returned changed weights w. I denotes the indicator function, w bef the "bef"ore weights and w aft the weights "aft"er portfolio optimization (i.e., bef=aft if optimization routine is not called). T F is the time and assetspecic matrix of ticket fees and BAS is the time and asset-specic matrix of bid ask spreads. - Flow yes no Optim - yes xed (I[ w bef > 0] I[ w > 0]) TF I[ w > 0] TF variable w aft Flow+ w Vol bef 0.5BAS w Vol bef 0.5 BAS no xed I[ w bef > 0] TF None variable w aft Flow 0.5BAS None debt returns for the emerging markets, the total return indices of BBB-rated corporate bonds in the US and Europe as well as the USD-based Goldman Sachs commodity index. The transaction costs are based on a constant portfolio volume of EUR 10m, a bid-ask-spread of 100 bp until end of 2001 and 80 bp thereafter and a ticket fee per trade of EUR 100 until 1998 and of EUR 58 thereafter. The optimization is out of sample, with a training period of 25 years until February 1997. No negative weights are allowed. Two main congurations are without optimization of transaction costs (Figure 4) and with no-trade optimization of transaction costs (Figure 5). References Asness, C. S., Moskowitz, T. J., and Pedersen, L. H. (2009). Value and momentum everywhere. NBER Working Paper, 15205. Barroso, P. and Santa-Clara, P. (2011). Beyond the carry trade: Optimal currency portfolios. mimeo. Brandt, M. W., Santa-Clara, P., and Valkanov, R. (2008). Parametric Portfolio Policies: Exploiting Characteristics in the Cross Section of Equity Returns. Rev. Financ. Stud. Novy-Marx, R. (2010). Is momentum really momentum. Forthcoming in the Journal of Financial Economics. Wang, P. and Kochard, L. (2011). Using a Z-score Approach to Combine Value and Momentum in Tactical Asset Allocation. SSRN elibrary. 7
Figure 4: Out of Sample Performance without Optimization of Transaction Costs The black line ist the portfolio performance without consideration of transaction costs, the red line the performance with transaction costs deducted. The green and blue line represent the the benchmark without and with transaction costs. The turquois line represent an investment in the riskfree asset. 8
Figure 5: Out of Sample Performance with Optimization of Transaction Costs The black line ist the portfolio performance without consideration of transaction costs, the red line the performance with transaction costs deducted. The green and blue line represent the the benchmark without and with transaction costs. The turquois line represent an investment in the riskfree asset. 9