Original Article The value of stop-loss, stop- gain strategies in dynamic asset allocation Received (in revised form): 22nd July 2016

Austin Shelton is a Finance Ph.D student at Florida Atlantic University in Boca Raton, Florida. He holds a MS and BS in Finance from The University of Arizona. Prior to the Ph.D program, he worked as a Data Scientist at Dun & Bradstreet Credibility Corp. in Malibu, California and prior to earning his MS worked as a Brokerage Associate at the Vanguard Group in Scottsdale, Arizona.

Correspondence: Austin Shelton, Department of Finance, Florida Atlantic University, 777 Glades Rd, Boca Raton, FL 33431, USA E-mail: [email protected]

ABSTRACT Dynamic asset allocation strategies which utilize stop-loss and stop-gain rules may dramatically decrease risk and even increase -term return relative to other tradi- tional asset allocation strategies. I introduce a dynamic asset allocation strategy which shifts portfolio weights based on predefined stop-loss and stop-gain rules. The two-asset (S&P mutual fund and bond mutual fund) strategy tested from 1990 to 2012 produces an annual geometric return of 8.45% vs. 7.50% for the underlying S&P 500 Index fund with 50% less (9.41% vs. 18.76% for the S&P index fund). In addition, the strategy displays a positive and significant CAPM over the sample period. The strategy’s very strong results are robust to changes in the user-specified parameters, such as the level and number of stop placements. All findings indicate that portfolio stop-loss and stop-gain rule-based strategies comprise a promising dynamic asset allocation approach deserving of further research and development. Journal of Asset Management (2016). doi:10.1057/s41260-016-0010-y

Keywords: asset allocation; stop-loss; portfolio theory; risk management; dynamic asset allocation JEL Classification: G11; G11; G01; G17; G1

INTRODUCTION efficient frontier. Of course, the practical AND MOTIVATION problem is that asset returns are not easily, if Stop-losses and stop-gains have not even at all, predictable (Ferson et al, 2003). In traditionally been considered as part of an addition, Markowitz’ model and most asset allocation methodology. Instead, asset allocation models assume a static single- has been driven by period framework. mean–variance optimization since Harry Thus, I design and introduce an Markowitz revolutionized modern finance alternative method of asset allocation to be (Markowitz, 1952). Sophisticated used between major asset groups. My are rightfully focused on maximizing their alternative methodology is based on the return per-unit risk by investing along the placement of stop-losses and stop-gains set by

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management www.palgrave.com/journals Shelton

predefined rules. The implicit goal of my constant mix strategy to hedge against market method is to maximize return per-unit risk. collapses. That is, the sells as However, there is zero prediction of any they fall to cut his losses and buys them as return moment within my methodology. they rise. Perold and Sharpe show the rather Instead, I use stop-loss and stop-gain rules intuitive result that the type of market alone to shift asset allocations. Thus, no dictates which strategy is most effective and inferences regarding expected return are therefore no strategy is dominant across all made and no optimization of any sort is market states. For example, portfolio performed within my methodology. insurance strategies do better than constant Since my strategy is naturally dynamic in mix strategies in bear markets but will, of nature and therefore can be classified as a course, do this at the expense of tactical or dynamic asset allocation strategy, a underperforming constant mix strategies in brief review of the history of dynamic bull markets. asset allocation and risk management Herold et al (2007) show that a risk-based strategies, especially those involving stop- portfolio strategy that controls for portfolio losses and risk management rules, is quite shortfall risk enhances long-run performance. helpful. In their dynamic framework, investors have Dybvig (1988) shows that stop-losses tend the best risk-adjusted performance and lowest to be wealth destroying assuming that the portfolio shortfalls by allocating higher underlying asset can either go up or down proportions to stocks initially and then next period with equal probability and shifting to a more conservative strategy over magnitude. This is because when triggered a time. Although their approach does not stop-loss has a positive cost associated with it involve stop-losses, their intuition of due to the bid-ask spread and commissions. reducing portfolio shortfalls and reducing While sensible, this result relates to market portfolio risk is somewhat similar to my own. microstructure issues rather than using stop- Their approach has important real-world losses within a long-term dynamic implications as it is consistent with the asset allocation framework as done in this ensuing development and popularity of life- paper. cycle mutual funds which decrease an Perold and Sharpe (1988) authored the investor’s allocation to equities and increase defining work which started the discussion of his allocation to bonds as he nears retirement. dynamic asset allocation strategies. Perold and Clarke et al (2006) show the strong Sharpe identify four major types of dynamic performance of a minimum variance asset allocation strategies: (1) buy-and-hold, portfolio of US stocks over 456 months of (2) portfolio insurance, (3) constant mix, and historical data. A portfolio in which stocks (4) option-based portfolio insurance. Of are selected via an algorithm which course, buy-and-hold is essentially just a minimizes portfolio variance with no regard passive strategy in which an investor simply to expected return posts a Sharpe Ratio of buys a certain percentage of stocks, bonds, over .5 over the period versus .36 for the and/or other assets and holds them until he market portfolio. This suggests that even wishes to liquidate his holdings. In a constant when expected returns are not considered, mix strategy, an investor keeps his weightings minimizing the risk of one’s portfolio tends of all assets constant over time. Therefore, to add to risk-adjusted return. the investor sells stocks as they rise and buys In a recent work, Hocquard et al (2015) them as they fall in order to keep the introduce a dynamic asset allocation strategy proportion of equity holdings in his portfolio in which investors allocate funds between the constant. In portfolio insurance-type market and a risk-free asset to target a strategies, an investor does the opposite of a left-truncated Gaussian distribution with an

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management Dynamic asset allocation strategies exogenous volatility parameter. They show one’s bond portfolio, rather than just fixing the strategy has strong results in simulation investment in bonds (or a risk-free asset) for relative to the S&P 500 across varying arbitrary periods of time after equity stops are specifications of the targeted volatility triggered. Finally, stop-losses and stop-gains are parameter, especially in risk-adjusted terms. normalized throughout time based on lagged- While their results are impressive, they also return volatility within my model and my show that their strategy does not consistently model is completely generalizable to a portfolio outperform a very simplistic stop-loss based with any number of investment assets. strategy in risk-adjusted terms and suffers Most importantly though, I show that a from higher trading costs than such a strategy. stop-loss and stop-gain-based strategy is likely Most notably, Kaminski and Lo (2007) to actually be the dominant risk-adjusted develop a strategy in which one invests strategy, when the prior literature informs us entirely in equities until stop-losses are hit that no strategy is likely to dominate and and then shifts entirely to long-term therefore be a replacement for ‘‘buy-and- government bonds for set periods of time. hold’’ over the long run (Perold and Sharpe, They find their system has a significant 1988). As Perold and Sharpe argue, it is clear CAPM alpha and a historical tendency to that no particular dynamic asset allocation allocate to bonds preceding large equity strategy will dominate in all markets, be they market downturns, providing a logical basis ‘‘bull,’’ ‘‘bear,’’ or ‘‘flat.’’ However, I argue for asset allocation methodologies which that the performance of an asset allocation incorporate stop-losses. strategy in particular market states should not My marginal contribution to the relevant be one’s focus, but rather the expectation of literature is multiple. First, I implement a the strategy’s performance over the long run, dynamic asset allocation strategy that utilizes which naturally contains many potential a series of stop-losses and stop-gains rather market states. I show using Monte Carlo than just stop-losses alone. No major study I simulation that whether or not we introduce am aware of to date does this. This naturally jumps (‘crisis’ years in my methodology) into cuts the return distribution of the strategy markets, and even if we simulate markets as further from ‘left-truncated Gaussian’ to just less volatile than observed historically a stop- ‘truncated Gaussian.’ It also defeats the claim loss, stop-gain-based strategy produces a that the strategy is simply a synthetic form of much higher information ratio than a typical portfolio insurance. In my view, it may be ‘‘buy-and-hold’’ (S&P 500) or constant mix that capturing gains is just as important as (balanced 70–30 index) investment strategy. cutting losses, and therefore, stop-gains For any unfamiliar readers, the ‘information should be considered within an ratio’ is calculated as the asset’s mean return asset allocation paradigm in addition to stop- over volatility following Goodwin losses. (1998). Thus, my findings strongly suggest Next, the portfolio reallocations are that the risk-adjusted return of a dynamic designed to be modest within my model (a asset allocation strategy which constricts both movement from 40% stocks and 60% bonds gains and losses dominates the risk-adjusted to 46% stocks and 54% bonds would be a return of other commonly used typical change, instead of a 100% move from asset allocation strategies. stocks to bonds tested in Lo and Kaminski in Finally, I explain why my strategy has most other studies). As such, my strategy or a historically outperformed. I show that the variant of it has great potential to actually be time-varying correlation structure of stocks implemented by many portfolio managers. and bonds seems to be a major driving factor Another unique separating factor of my of the performance of my strategy. This model is that stops are incorporated within understanding of why the stop-loss, stop-gain

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management Shelton

asset allocation strategy outperforms is perhaps grade corporates to diverge during financial more important than the finding that it does. crises. Of course, the ‘flight to safety’ by In totality, I develop a new class of investors who demand safer and highly liquid dynamic asset allocation strategies which assets during volatile markets is what drives truncate one’s possible returns through the up the return of government and investment- placement of both stop-losses and stop-gains grade bonds and causes the ‘decoupling’ on all assets held. Although somewhat similar effect during large market downturns. Since, in intuition to portfolio insurance since one ‘decoupling’ is easily observable empirically, wishes to hedge against downturns in both one should expect a strategy which utilizes strategies, my strategy is distinct and unique stop-loss and stop-gain rules to shift with a different risk and return profile than allocations between an equity and a bond portfolio insurance. fund to take advantage of liquidity shocks and Intuitively, I was motivated to test the the ensuing ‘flight to safety’ observed during methodology for three reasons. First, since a financial crisis. That is, when stop-losses on stock returns are far from easily predictable, it equities are triggered during a significant is sensible to consider an asset allocation market downturn, the portfolio allocation to strategy which only targets the minimization bonds is naturally increased by such rules. At of risk rather than the accurate prediction of just this point in time, the correlation future return and minimization of risk. between stocks and bonds is often highly Second, the use of both stop-losses and stop- negative, and the expected return of bonds gains on one’s portfolio composed of stocks relative to stocks has increased. For this and bonds should naturally rebalance towards reason, we should observe that the asset with the higher expected return if asset allocation strategies between equities (1) assets have non-constant correlations and bonds which incorporate stop-loss and which diverge in financial crises, (2) equities stop-gain rules greatly outperform balanced have a positive serial correlation over the funds (‘‘constant mix’’) or straight investment term, and (3) equities exhibit long-term in equities (‘‘buy-and-hold’’) during financial . Of course, there is evidence crises’ and market downturns. that all these conditions may hold (Conrad Finally, stop-losses naturally provide value and Kaul, 1988; De Bondt and Thaler, 1987; in the presence of as one often is Fama and French, 1988; Gulko, 2002; ‘stopped-out’ before equities endure Jegadeesh, 1990; Jegadeesh and Titman, prolonged periods of significant downturn. 1993; Poterba and Summers, 1988). Since momentum of individual firm returns As far as the time-varying correlation and aggregate market returns at the monthly structure of major asset groups, it has been level are both well documented, it is only noted that during periods of large decline in natural to design an asset allocation strategy one major asset group (stocks) there is often a which may capture at least a portion of the ‘flight to safety’ in which investor funds flow ‘momentum alpha’ while simultaneously to the less risky major asset group (bonds) decreasing portfolio variance. Kaminski and (Gulko, 2002). During and after the 2008 Lo (2007) prove a very intuitive fact along financial crisis, this ‘decoupling’ effect these lines well worth proving formally: a originally noted by Gulko (2002) was widely portfolio composed of solely a risk-free asset commented on as the ‘flight to safety’ or and a risky asset that has a normally ‘flight to quality’ by the financial media distributed return and positive return serial (Noeth and Sengupta, 2010). correlation earns a positive excess return by Essentially, ‘decoupling’ is the tendency the use of stop-losses on the risky asset. for the correlation between the US stock Due to the non-constant correlation market and US treasury bonds and high- structure of major asset groups, the relatively

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management Dynamic asset allocation strategies low amount of predictability in individual determine the stop-loss and stop-gain level asset returns, momentum in equity returns on the at that point in time. over short horizons, and long-term mean Now, for clarity, let me give an example. reversion or reversal in equity returns, it is Let us assume we want to pick two worthwhile to reconsider dynamic investment assets to hold within our asset allocation strategies that incorporate portfolio: an S&P 500 mutual fund and an both stop-loss and stop-gain rules. In this investment-grade bond mutual fund work, I argue that such strategies may have a (N = 2). Let us also assume we choose I to place in investors’ portfolios and should be 4, J to be 4, and Z to be .5. That means further be explored as a new class of dynamic we set up I à J = 16 positions, each with a asset allocation strategy. unique stop-loss, stop-gain combination. The stop-loss and stop-gain levels on the positions in this example are shown in Table 1. ASSET ALLOCATION STOP- In total in this example, we have 16 LOSS, STOP-GAIN STRATEGY positions with stop-loss and stop-gain levels The asset allocation strategy I develop is of .5, 1, 1.5, and 2. Mathematically, the exact relatively simplistic. Within my strategy, an stop-levels and numbering of an investor’s investor holds a portfolio of N different positions is written below: investment assets, which are predetermined For i ¼ 1; ...; I & j ¼ 1; ...; J before the start of investment. For the sake of simplicity, and because many investors are ak ¼ i à Z & bk ¼ j à Z; primarily concerned with their portfolio d ¼ ðÞa ; b : allocations to stocks and bonds, I set N equal k k k to 2 throughout. In order to closely match The k positions are numbered so that the investments of many institutional k ¼ ðÞÃj À 1 J þ i. dk = ðak; bkÞ is a vector portfolio managers and retail investors alike, containing the unique stop-loss level, i, and the two investments I consider in this paper stop-gain level, j, of position k. are a large S&P 500 index mutual fund and a After an investor determines the assets he large investment-grade bond mutual fund. wishes to hold within his portfolio and I, J, Still, this strategy is easily extendible to any Table 1: Stop-loss and stop-gain levels when I = 4, number of investment assets. J = 4, and Z = .5 After an investor determines the assets he Position Stop-loss level Stop-gain level wishes to hold within his portfolio, he must determine the number of stop-losses (which I 1 0.5 0.5 2 1 0.5 will denote I), the number of stop-gains 3 1.5 0.5 (which I will denote J), and fixed spread 4 2 0.5 5 0.5 1 between both stop-losses and stop-gains 61 1 (which I will denote Z) that he wishes to 7 1.5 1 82 1 place on his portfolio. Since there are I à J 9 0.5 1.5 unique stop-loss, stop-gain combinations, the 10 1 1.5 investor holds I à J positions within his 11 1.5 1.5 12 2 1.5 portfolio. At any point in time, each position 13 0.5 2 is composed of only one of the N possible 14 1 2 15 1.5 2 investment assets. For each position, the 16 2 2 stop-loss and stop-gain level is multiplied by Note: For all positions, the stop-loss and stop-gain the 6-month lagged annualized return levels are multiples of the 6-month lagged (annualized) volatility of the asset currently held to volatility of the asset currently held.

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management Shelton

and Z, he may begin investment. On day 1, binary function simply signals whether a 1 he invests N of his capital in each of the stop-loss or stop-gain level was met or N assets he chooses. Thus, in our example, exceeded for all K portfolios at time t.Of with an S&P mutual fund and long-term course, t is initiallyÀÁ set to 1. investment-grade bond fund as our portfolio (2) 8k 2 K if fxk;t = 1, then a trade is assets, we originally invest 50% of our capital placed on the kth positionÈ at timeÈÉt +1 in the S&P mutual fund and 50% in the bond such that the pðxk;tþ1 ¼ 1; 2; 3; xk;t ; mutual fund. Since we have 16 positions we 1 ...; NgÞ ¼ NÀ1. That is, for each of the hold at any point in time within this portfolios in which a stop was triggered at specification of the strategy, we originally time t trade into a new asset (with equal, invest 8 positions in the S&P mutual fund random probability of choosing any new and 8 positions in the bond mutual fund. asset) at time t +1. After investment, it is straightforward to PK ÀÁ 1 follow the rules of the stop-loss, stop-gain (3) Let wk;tþ1 ¼ K if fxk;t  1. Equally strategy to determine the investor’s k¼1 asset allocation at any point in time. For each weight all K portfolios at time t + 1 if one of the I Ã J positions, on each day, if the or more stops were triggered at time t. holding period return of the asset currently (4) Repeat steps (1) through (3) for t = t +1. held is less than or equal to the stop-loss level The strategy has interesting implications set on the position or greater than or equal to in both the 2-asset and more complex N- the stop-gain level set on the position, then asset case. However, due to the ‘decoupling’ one trades from the asset currently held to effect observed between equities and bonds another of the (N - 1) available assets. If N is during market crises, I am most interested in greater than 2, than one simply picks another the performance of my strategy in a 2-asset of the N assets at random (with each of the case where the investment assets are an 1 remaining assets having probability of NÀ1 of equity index mutual fund and a bond index being drawn). In the 2-asset case, one simply mutual fund. In addition, since most trades to the other asset. In addition, when a investors allocate the majority of their stop-loss or stop-gain is triggered on any financial investments to diversified equity position, one reweights his entire portfolio so and bond funds, this becomes the most that all positions are equal-weighted. This natural place to start testing the strategy from process is repeated going forward at any point an applied portfolio management at which a stop-loss or stop-gain is hit. Thus, perspective. an investor can easily continue to follow the strategy for an indefinite amount of time. Mathematically, the rules of the strategy in DATA the N-asset case are written and explained As data for this study, I use historical daily below: Net Asset Values (NAVs) and yields  ÀÁ 1ifR  a or R  b for the dates from January 2, 1990 to Dec 31, (1) 8 k let fx ¼ k;t k k;t k ; k;t 0 o:w: 2012 for two large Vanguard mutual funds: where xk,t represents theÀÁ asset held as the VFINX, the Vanguard 500 Index Fund, and kth position at time t, fxk;t is a function VWESX, the Vanguard Long-Term which takes a value of ‘1’ if a stop was Investment-Grade bond fund. I gathered the triggered on the kth position at time t and data for both funds directly from Vanguard’s a value of ‘0’ otherwise, and Rk;t is the investor website. These data are also publicly holding period return of the asset being available from Yahoo! Finance as well as held as the kth position at time t. The other online sources. In addition, I use Ken

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management Dynamic asset allocation strategies

French’s data from his website for historical NAV. Thus, my strategy was actually Fama–French Carhart factors, as well as tradable over the dataset considered as it is historical estimation of the market return and guaranteed that one would be able to the risk-free rate over the period considered. exchange between the two mutual funds at Returns on the Vanguard funds were their reported NAVs. This same condition is calculated from daily NAVs and dividend not guaranteed when one uses the returns of yields assuming that all paid by the underlying index such as the S&P 500, either fund were paid at month end and individual stocks, or closed-end funds reinvested. Altogether, the data encompass including exchange traded funds (ETFs) as 5800 daily NAV observations for each of the one may not have been able to sell shares on mutual funds. Since stop-loss and stop-gain the open market at the daily closing price rules in my model are multiples of the rolling listed in a dataset such as CRSP. 6-month annualized standard deviation of In addition, as individual stocks and return of each of the underlying funds, only exchange traded funds would be the 127th (July 3, 1990) through 5800th continuously traded throughout the market (Dec 31, 2012) observations encompass the day, microstructure issues such as period of allowable testing for the from the bid-ask spread and commissions asset allocation strategy. could and to some extent would decrease the There were several reasons for choosing actual net return of the strategy through such two Vanguard index mutual funds as the two vehicles. However, such microstructure assets to represent a broad stock index and issues are completely avoided through the use broad bond index within my asset allocation of open-end fund returns. While when many strategy. First, Vanguard offers index funds at financial experts think stop-losses they think very low management fees and is considered microstructure, this paper is meant to the world’s premium provider of index encourage new methodologies in dynamic funds. It is obviously much less costly for a asset allocation which involve stop-loss and typical investor to buy the Vanguard 500 stop-gain rules. Index Fund than to buy the 500 companies Of course, the costs of holding a in the S&P 500 and rebalance the portfolio diversified portfolio, including incurring bid- on the investor’s own as a large index fund ask spreads and commissions to buy and sell can achieve large economies of scale. Second, positions and rebalance one’s portfolio as well both funds are very large, reputable, and as a very modest management fee, are already efficient funds commonly held by retail and encapsulated in the Vanguard NAVs. But, institutional investors alike. Finally, and most Vanguard, similar to other large mutual fund importantly, by using mutual funds it is companies has oftentimes allowed investors realistic instead of problematic to gather only to exchange between these no-load funds one potential stop ‘‘price’’ per day, the fund’s free of charge. So, I would argue returns (NAV). Thus, when a stop- reported already are net of trading costs as loss or stop-gain is triggered based on a fund’s they include the embedded bid-ask spread, closing NAV, an exchange occurs at the next commissions, and management fees from day’s (T + 1) NAV from VWESX to holding and exchanging the product. Taxes, VFINX or vice versa (for example, one of course, are not estimated, but I would note would receive the NAV calculated after that many retail investors may wish to today’s market close for a stop triggered by implement such strategies in nontaxable yesterday’s closing NAV). This is a perfectly accounts such as an IRA and therefore may realistic assumption without taking into find the strategy very appealing. account commissions, since mutual funds are The standard asset allocation stop-loss, only exchanged at one price per day, their stop-gain strategy I test is a specification in

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management Shelton

which I = 4, J = 4, and Z = .5. Asset 1 is was dominant from a return prospective as it the Vanguard 500 Index Fund (VFINX), and provided both superior risk-adjusted return asset 2 is the Vanguard Long-Term and higher raw return than VFINX, Index Fund (VWESX). VWESX, or any balanced index (‘‘constant Thus, since I Ã J = 16, 8 positions are mix’’ strategy) of the two. invested initially in VFINX and 8 into The baseline asset allocation strategy VWESX on July 2, 1990 in this baseline noticeably outperforms all other tested specification. Then, as stop-losses and stop- portfolios. It has a higher geometric return gains are triggered on any of the 16 positions than VFINX, VWESX, or the balanced held on days 127 (July 3, 1990) to 5800 index. The asset allocation strategy also has (December 31, 2012), those positions are 50% lower volatility than VFINX and a exchanged into VWESX from VFINX or 27.5% lower return volatility than the vice versa and all 16 positions are rebalanced. balanced index. In fact, the bond fund, VWESX, has only a marginally lower standard deviation of return than the asset allocation strategy (5.2% lower) which is RESULTS quite amazing. This is also suggestive of the The baseline specification above produces an fact that the strategy must have a higher annual geometric return of 8.45% over the historical allocation to bonds when 23-year period considered, with an correlations between stocks and bonds are annualized standard deviation of only 9.41% most negative. In terms of reward per-unit (all statistics reported use daily returns to risk as measured by the Information Ratio, estimate return volatility). In comparison, the base allocation strategy far outperforms VFINX, the Vanguard S&P 500 Index Fund, either of the underlying index funds (‘‘buy- produced a lower 7.50% annual geometric and-hold’’) or the balanced strategy return, with an annualized standard deviation (‘‘constant mix’’) with its Information Ratio of 18.76% over the period. The comparison of .90. Table 2 below details the annualized of returns to VFINX is important as VFINX geometric return, arithmetic return, standard represents the most basic ‘‘buy-and-hold’’ deviation, and Informatiom Ratio of the asset allocation strategy, that is, to hold an Asset Allocation stop-loss, stop-gain strategy, S&P 500 index fund as it is a proxy for the VFINX, VWESX, and a 70–30 balanced overall market. A 70–30 balanced portfolio of index of VFINX and VWESX from VFINX and VWESX produced a 7.16% 1990–2012. annual geometric return, with an annualized Perhaps the greatest potential benefit of standard deviation of 12.98% over the period. stop-loss and stop-gain-dependent This 70–30 balanced portfolio is important asset allocation strategies is also the most for comparison relative to the strategy as it obvious: their potential to mitigate portfolio represents a typical, basic ‘‘constant mix’’ variance and drawdown. While it is asset allocation strategy. Finally, VWESX reasonable to expect the asset allocation produced a 4.57% annual geometric return, strategy to match the return of a balanced with an annualized standard deviation of index or ‘‘constant mix’’ strategy with close 8.92%. Again, this can be viewed as another average allocations to the strategy itself, the very basic and conservative ‘‘buy-and-hold’’ real benefit of the strategy is to further investment strategy, in which one holds a decrease portfolio variance relative to an proxy for the aggregate long-term investment in a balanced index or straight investment-grade bond market instead of the equities, especially during highly volatile aggregate stock market. The stop-loss, stop- periods. In fact, the maximum drawdown or gain strategy with the baseline specification ‘peak-to-valley’ for the strategy from 1990 to

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management Dynamic asset allocation strategies

Table 2: Asset allocation stop-loss, stop-gain strategy: 22%, VWESX actually gained over 10%. I = 4, J = 4, Z = .5 And, in fact, when one separates the 23-year Annual geometric return 8.45% dataset into a 2002 and 2008 dataset which is Annual arithmetic return 9.01% Annualized standard deviation 9.41% representative of market crises, and a dataset Information ratio 0.9 including the other 21 years, one finds that VFINX, Vanguard 500 index fund Annual geometric return 7.50% the daily correlation between VFINX and Annual arithmetic return 9.06% VWESX was -.1 in the 21-year period Annualized standard deviation 18.76% Information ratio 0.4 excluding market crises, but roughly -.37 in Balanced 70–30 Portfolio (70% VFINX, 30% VWESX) the two years in which equity markets Annual geometric return 7.16% experienced a loss of over -20%, 2002 and Annual arithmetic return 7.71% Annualized standard deviation 12.98% 2008 (and this difference is highly statistically Information ratio 0.55 significant)! With a clear time-varying VWESX, Vanguard investment-grade bond fund Annual geometric return 4.57% correlation structure in place, at a time of Annul arithmetic return 4.87% crisis stop-losses on the asset declining swiftly Annualized standard deviation 8.92% Information ratio 0.51 in value (equities) will be triggered as the strategy naturally takes advantage of the Note: Information Ratios are simply defined as annual typical ‘flight to safety’ of investor funds into geometric return over annualized standard deviation of return (risk-free rates are not subtracted for Information the other asset (bonds). Thus, the strategy Ratio calculations, separating the calculation from the benefits from the ‘decoupling’ effect or Sharpe Ratio). nonconstant correlation structure between equities and bonds noted by Gulko (2002). 2012 was 30.58%, compared to 55.47% for In Table 3, the annual returns for the VFINX, the S&P 500 index fund. This strategy, VFINX, VWESX, and a 70–30 represents one of the largest benefits of the balanced index are listed. In Table 4,a strategy for institutional and retail investors CAPM alpha and for the strategy is who are, oftentimes, more concerned about estimated over the 23-year period. Results of not losing money than they are about a Fama-French Carhart regression are outperforming the stock market each and included in Table 5 below using the size every year. (SMB) and book-to-market (HML) factors In line with this, is the finding that the of Fama and French (1993) and the asset allocation stop-loss, stop-gain strategy momentum (UMD) factor of Carhart (1997). has experienced far better returns than the Notice, the strategy had some of its worst ‘‘constant mix’’ or ‘‘buy-and-hold’’ strategies raw returns in 2008 and 2002. However, as in significant market crises or ‘‘bear’’ markets. one may expect, the strategy outperformed Of course, my strategy mechanically VFINX by a wide in both years, over mitigates losses in large market declines, so 19% in 2008 and by over 15% in 2002. long as the correlation of the portfolio Also notice that the strategy has a investment assets diverges during these statistically significant CAPM alpha of declines. And, while it is common to hear approximately 2 bp (.02%) daily or 513 bp market participants speak of how assets (5.13%) annually. Again, this seems to be an converged toward correlations of 1 in 2008, impressive mark for a strategy that simply there is only very limited and misleading moves assets between stock and bond index truth to this statement (individual stocks did; funds based on predefined, fixed stop-loss however, broad asset groups did not). Notice, and stop-gain rules. In addition, the strategy VWESX only had a -2.46% return in 2008, has a similar, positive and significant Fama- when VFINX had a brutal -36.66% return. French Carhart alpha of roughly 2 bp a day. Similarly, in 2002 when VFINX fell over Thus, we can conclude that the strategy is

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management Shelton

Table 3: Annual Returns, Asset Allocation stop-loss, stop-gain strategy (I = 4, J = 4, Z = .5) compared to VFINX, VWESX, and a 70–30 balanced index of the two Year Strategy (%) VFINX Return (%) VWESX Return (%) 70–30 Index (%)

2012 13.81 15.21 7.33 12.85 2011 11.92 1.17 12.68 4.62 2010 18.46 12.81 7.19 11.12 2009 20.50 25.47 5.14 19.37 2008 -17.00 -36.66 -2.46 -26.40 2007 1.34 4.91 0.67 3.64 2006 7.62 14.67 -0.20 10.21 2005 8.09 4.34 2.30 3.73 2004 6.64 10.13 5.88 8.85 2003 25.42 28.23 3.32 20.76 2002 -6.65 -22.44 10.09 -12.68 2001 -6.40 -11.28 5.74 -6.17 2000 -0.14 -9.17 8.85 -3.76 1999 5.89 19.97 -9.11 11.24 1998 36.98 28.11 4.37 20.99 1997 9.66 32.70 10.93 26.17 1996 14.74 21.11 -2.90 13.90 1995 30.17 37.23 24.30 33.35 1994 -4.36 0.58 -7.55 -1.86 1993 13.08 9.57 8.77 9.33 1992 6.70 3.67 0.38 2.68 1991 19.51 27.26 10.10 22.11 1990 -8.72 -9.27 -0.72 -6.70

Note: The 1990 return is partial year (July 2–Dec 31).

Table 4: A CAPM regression of the daily returns of the Asset Allocation stop-loss, stop-gain strategy from 1990–2012 is provided below Coefficient Estimate t stat p value

Alpha .0002153*** 2.73 .0063 Beta -.0081599 -1.214 .2248 R2 .0002

* = significant at a p value of .10. ** = significant at a p value of .05. *** = significant at a p value of .01. Note: Ken French’s data available on his website is used in estimating Rm - Rf.

Table 5: A Fama–French Carhart regression of the daily returns of the Asset Allocation stop-loss, stop-gain strategy from 1990–2012 is provided below Coefficient Estimate t stat p value

Alpha .0002035*** 2.586 .0097 Beta -.005519 -.773 .4395 SMB .0007027*** 5.32 \.0001 HML .0004194*** 3.07 .0021 UMD -.0000125 -.13 .90 Adj. R2 .0057

* = significant at a p value of .10. ** = significant at a p value of .05. *** = significant at a p value of .01. Note: Ken French’s data, available on his website, is used in estimating Rm - Rf, SMB, HML, and UMD.

not simply profiting off of exposure to the simply allocates one’s portfolio between known Fama–French Carhart factors or equities and bonds, and not individual market risk. However, since the strategy stocks, it would make little sense if SMB,

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management Dynamic asset allocation strategies

HML, or even UMD explained away the perhaps even a 70–30 balanced fund, the strategy’s returns altogether as these factors positive effects of stop-loss and stop-gain explain the cross section of stock returns, rules go beyond a reduction in portfolio rather than aggregate stock or bond market volatility. For instance, even if the strategy returns. A ‘’ factor would do only matched the return of equities or even a more to explain the returns of my strategy; balanced fund, one could easily argue that it however, there is ironically no prediction of is an improvement on either since it has market returns within my model (that is the lower tail-risk as measured by maximum point)! The point is that intelligent (or even drawdown and other basic statistical measures fairly unintelligent, simplistic, and (it also has less negative skewness). The tail- mechanical…) risk-controls utilizing stop- risk benefit of the strategy is well portrayed in losses and stop-gains seem to ‘time’ the a simple graphical setting. Therefore, I market rather well. And, this again can only display the monthly return distribution of the be due to a nonconstant correlation strategy and VFINX and over the sample structure between the assets held in one’s period with histograms in Figures 1 and 2 portfolio. While Kaminski and Lo (2007)do and encourage one to pay particular attention not focus on this intuition, it is in line with to both the negative return outliers and scale their finding of a stop-loss strategy in each figure. (seemingly oddly) avoiding market Finally, it is very important to note that downturns. As I will show, the appropriate the much lower variance inherent in the explanation for at least a portion of the strategy leads to higher long-term returns and strategy’s alpha is the nonconstant higher annual geometric returns than equities correlation structure between equities and even if the average daily or monthly returns bonds, and in particular the divergent of the strategy are only approximately the correlations typically witnessed in market same, or even slightly lower than equities. crises. Mathematically, this is just an application of While it is not surprising that the Jensen’s Inequality and the resulting gap asset allocation strategy reduces variance between geometric and arithmetic returns alone relative to an investment in equities or (Jensen, 1906).

Figure 1: Histogram of Asset Allocation stop-loss, stop-gain strategy monthly returns with normal curve.

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management Shelton

The five, boxed monthly returns are all below-11%, and below the minimum monthly return of the strategy realized throughout the entire dataset.

Figure 2: Histogram of VFINX monthly returns with normal curve.

Reasonable, long-term investors also boosts long-term real returns quite interested in the optimal asset allocation of dramatically to an investor relative to their portfolio will be concerned with long- investment in equities (‘‘buy-and-hold’’) or term geometric rather than short-term even balanced funds (‘‘constant mix’’). In arithmetic returns as long-term geometric addition, this benefit is realized most when returns are an actual measure of the one demands it most, in volatile markets. compounding of one’s money or portfolio Thus, ‘smoothing’ of returns within the value over time. Daily returns should be all strategy is another benefit to investors but unimportant to a prudent, rational long- rightfully concerned with the long-term term investor utilizing zero leverage within a return, volatility, and maximum drawdown conservative strategy. Since, mechanically, of their portfolios. my strategy ‘smoothes’ returns, the gap between the strategy’s average geometric return and its average arithmetic return over ROBUSTNESS CHECK any time period is much smaller than the gap While the risk-adjusted returns of my between the average geometric return and strategy are enticing, there is certainly an average arithmetic return of a more volatile absolute need to show the robustness of the strategy. More so, this gap between actual or results I present. In particular, I perform tests geometric return and a simple average of to alleviate any concern that the encouraging return grows the longer the time period results from my strategy are not just a considered (more time for returns to byproduct of lucky (or even purposeful) compound). Of course, the higher the selection of the exogenous inputs. The variance of an asset, the higher the difference baseline strategy considered has 4 stop-loss between its geometric and arithmetic returns levels (I = 4), 4 stop-gain levels (J = 4), and across any time period and the longer an fixed-spreads between stop-loss and stop- investor’s investment horizon the larger the gain levels of ‘ of the 6-month lagged economic impact to the investor. Thus, the annualized standard deviation of return of the ‘smoothing’ of returns inherent in this long- asset held in each position (Z = .5). Thus, term strategy not only mitigates risk, but it the strategy is composed of 16 positions with

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management Dynamic asset allocation strategies

stop-loss and stop-gain levels of d1 = (.5, bonds, and therefore no diversification .5), d2 = (.5, 1),…, d16 = (2, 2). Since a benefit could be derived. For that reason, an two standard-deviation return outlier (in I = 1 and J = 1 specification is considered to terms of annualized volatility of the 6-month clearly be suboptimal, as it does not clearly lagged return) is obviously relatively rare and add to the expected return of the strategy yet a ‘ standard deviation return occurs clearly adds to expected volatility relative to relatively frequently, the baseline all other specifications. asset allocation stop-loss, stop-gain strategy Within the robustness checks Z is was purposely specified to neither have such specified at four different levels: Z = .25, .5, tight stops placed on positions that .75, and 1. Thus, 16 total specifications are reallocations are constantly triggered nor to performed (15 of which are true robustness have such wide stops on positions that checks, and one of which is the baseline, reallocations are never triggered. I = 4, J = 4, and Z = .5). The results of all Thus, my robustness check is simply a are shown in Table 6. number of reasonable respecifications of I, J, Notice, the results of the robustness check and Z, with the investment assets unchanged strongly confirm the strength of the so that asset 1 is still VFINX and asset 2 asset allocation stop-loss, stop-gain strategy VWESX. The goal is to verify that similar and reject the reasonable concern that I, J, strong return and mitigated risk and Z were simply ‘hand-picked’ or characteristics are exhibited by the strategy calibrated in order to produce favorable or across different reasonable combinations of even optimal results. the exogenous inputs. Since an investor in All 16 specifications have significantly less my stop-loss, stop-gain strategy must specify variance than both the balanced fund and the the number of stop-loss and stop-gain levels S&P index fund. Thus, it seems the stop-loss, as well as the spread between levels prior to stop-gain strategy decreases risk relative to investment, it is extremely prudent to show both ‘‘buy-and-hold’’ and ‘‘constant mix’’ that the strategy’s strong long-term results are investing. In addition, 14 out of the 16 not overly sensitive to reasonable (87.5%) of specifications tested beat the raw respecifications of I, J, and Z. return of the balanced index and 12 out of 16 Therefore, for the robustness check, I and (75%) beat the raw return of VFINX (S&P J are respecified at (I, J) = {(2, 2), (3, 3), (4, index fund) over the period from July 2, 4), (5, 5)}. Notice that only specifications in 1990 to Dec 31, 2012. According to US which I and J are equal are considered. News, from 2008 to 2012 only 27% of equity Specifications of I and J in which I \ J so that mutual funds beat their relevant S&P index losses are cut quickly and gains are left on the benchmark (Silverblatt, 2012). And, this is table are not. Similarly, specifications in without even considering the added risk which J \ I are also not considered. While many funds may have taken on in an attempt the question of optimal stop-loss and stop- to do so. In the robustness check, all gain placement on asset groups and whether portfolios have an annualized return standard losses should be cut quickly and ‘winners left deviation between 8% and 12%. All have to run’ is certainly an interesting one which Information Ratios which are greater than demands more work and may be that of VFINX, VWESX, or a balanced testable from my model, it is also second- index, indicating that the strategy has order question beyond the scope and purpose historically completely dominated ‘‘buy-and- of this paper. Also, notice that (I, J) = (1, 1) hold’’ and ‘‘constant mix’’ strategies in terms is not considered. In an I = 1 and J = 1 of risk-adjusted return. specification, there is only one position held In short, the asset allocation stop-loss, at any point in time, here solely equities or stop-gain strategy I create is quite robust to

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management Shelton

Table 6: Overall statistics for the 15 respecifications of the Asset Allocation stop-loss, stop-gain strategy as well as the baseline (I = 4, J = 4, Z = .5) I J Z Annualized Annualized P(F\=f) Return [70–30? Return [ S&P? Information geometric return (%) r (%) ratio

2 2 0.25 6.19 10.28 \.0001 No No 0.6 3 3 0.25 7.04 10.85 \.0001 No No 0.65 4 4 0.25 7.44 10.98 \.0001 Yes No 0.68 5 5 0.25 8.77 11.20 \.0001 Yes Yes 0.78 2 2 0.5 7.35 10.41 \.0001 Yes No 0.71 3 3 0.5 7.97 10.52 \.0001 Yes Yes 0.76 4 4 0.5 8.45 9.41 \.0001 Yes Yes 0.9 5 5 0.5 9.56 9.85 \.0001 Yes Yes 0.97 2 2 0.75 8.06 9.65 \.0001 Yes Yes 0.84 3 3 0.75 8.68 8.96 \.0001 Yes Yes 0.97 4 4 0.75 8.48 9.44 \.0001 Yes Yes 0.9 5 5 0.75 8.64 10.67 \.0001 Yes Yes 0.81 2 2 1 9.37 8.81 \.0001 Yes Yes 1.06 3 3 1 8.63 9.13 \.0001 Yes Yes 0.94 4 4 1 8.16 9.82 \.0001 Yes Yes 0.83 5 5 1 8.09 11.62 \.0001 Yes Yes 0.7 Average 8.18 10.10 87.5% ‘‘Yes’’ 75% ‘‘Yes’’ 0.82

Note: The P(F \=f) gives the probability that the variance of the daily returns of the respecification is greater than that of the 70–30 balanced index. The ‘‘Return [70–30’’ and ‘‘Return [ S&P’’ columns display ‘‘Yes’’ if the specifi- cation’s raw geometric return beats that of the 70–30 balanced index or VFINX, respectively. All results are for the period of 1990–2012.

reasonable changes in the exogenous gain dynamic asset allocation strategy’s parameters picked, indicating that it, or a performance within differing market states variant of it, likely has valuable use as an relative to ‘buy-and-hold’ investment asset allocation tool in practice. The strategy strategies. maintains its very low volatility and superior Within the MC simulations, my primary risk-adjusted return characteristics given assumption is that there are two possible reasonable respecifications of the necessary states for the stock market: a ‘normal’ market exogenous parameters. or a ‘crisis’ market. Within ‘normal’ markets, It is surprising to many that such a the daily returns of bonds have a constant simplistic strategy, focused only on risk- correlation of pn to the daily returns of control rules, can outperform other equities. Within ‘crisis’ markets, in which an asset allocation frameworks. In order to help economic shock occurs, the daily return justify my argument that a significant portion correlation of bonds to equities is pc. Thus, of the strategy’s alpha is due to the my Monte Carlo simulations are able to nonconstant correlation structure between model the nonconstant correlation structure stocks and bonds, I create a Monte Carlo between stocks and bonds and the model and run the strategy on thousands of ‘decoupling’ effect between major asset years of simulated equity and bond returns groups noted by Gulko (2002). across different market scenarios. As noted above, at any point in time within each MC simulation the state of the stock market is either ‘normal’ or ‘crisis.’ Furthermore, the market may only transition MONTE CARLO SIMULATION states on an annual basis (every 252 trading I create and run MC simulations to model days). The probability of a stock market the daily returns of stocks and bonds over ‘crisis’ in any given year is designated k. 20-year periods and test the stop-loss, stop- Therefore, the probability of a ‘normal’ stock

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management Dynamic asset allocation strategies market in any given year is 1 - k. More written as formally, (1) The state of the stock market, S, in any Xq Xp a a 2 : year is a binomial random variable with vt ¼ 0 þ i 2tÀ1 þ bivtÀ1 i¼1 i¼1 probability of a ‘crisis’ k: S  BðÞ1; k .If S = 1, the market is a ‘crisis’ market (c) By setting p = 1 and q = 1, only last during the given year, and if S = 0, the period’s squared error-term and variance market is a ‘normal’ market during the remain so that the GARCH(1, 1) model given year. is condensed to the following autore- In ‘normal’ years, the mean daily return of gressive stochastic process: the stock market is assumed to be the his- 2 torical mean daily return of the S&P 500 vt ¼ a0 þ a1 2tÀ1 þb1vtÀ1: from 1990 to 2012, excluding 2002 and 2008 as these were significant stock market Using maximum likelihood estimation downturns of over -20% annually for the (MLE),a^0, a^1, and b^1 are estimated using S&P 500 Index and are considered ‘crisis’ 1990–2012 S&P 500 Index daily return data years. The stock market’s return variance in excluding 2002 and 2008. Then, in each ‘normal’ years follows a GARCH (1, 1) ‘normal’ year a random stock market process with the parameters a0, a1,andb1 stochastic volatility path over 252 trading calibrated to historical 1990–2012 S&P 500 days (one year) is simulated in which stock Index daily return data with 2002 and 2008 market volatility follows a GARCH(1, 1) discarded. Following GARCH (Generalized process with parameters equal to a^0, a^1, and Autoregressive Conditional Heteroscedas- b^ . ticity), stock market volatility at time t fol- 1 In ‘crisis’ years, annualized stock market lows a stochastic process defined by returns are assumed to follow a normal Bollerslev and Mikkelsen (1996) and well- distribution. Average ‘crisis’ stock market described by Hansen and Lunde (2005). return and stock market volatility are both Specifically, the GARCH(p, q) model states exogenously set. I explain the reasoning for that variance at time t, here denoted v ,isa t the ‘crisis’ stock market average annualized function of the level of variance over the last return and annualized return volatility p periods, and volatility shocks or model parameters I pick within the scenarios I error-terms over the past q periods. More define below. Formally, within the MC formally, the GARCH(p, q) model states the simulations the stock market return dynamics following: are as follows: (a) 2 is the error-term of the stochastic t (2) If S = 1, then the stock market is in a variance process at time t: 2 is normally t ‘crisis’ market. Within a ‘crisis’ market, distributed with variance, vt. Therefore, pffiffiffiffi daily stock market returns are normally 2t ¼ ut vt distributed and IID. Average annual return and annualized volatility of the where u is a random draw from a stan- t stock market in a ‘crisis’ market are both dard normal distribution so that exogenously set. Stock market returns are u  NðÞ0; 1 . t normally distributed in the ‘crisis’ market (b) Variance at time t is a function of the last with an annualized mean return l and q period’s squared error-terms, and the M~ annualized variance r2 . Therefore, the last p period’s realized variances. There- M~ fore, the GARCH(p, q) model can be annualized crisis return of the stock

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management Shelton

market (M) in year i is denoted as follows In scenario (a), the correlation in ‘crisis’ (the tilde denotes a ‘crisis’ market): markets, qc, is set to -.4, as -.37 was the realized correlation between the S&P 500 ÀÁ and VWESX in 2002 and 2008. In the other r  N l ; r2 : i;M~ M~ M~ scenarios, (b) and (c), the correlation in ‘crisis’ markets is set to -.1, representing no (3) If S = 0, then the state of the stock ‘divergence’ at all or a static correlation market is ‘normal.’ Daily stock market structure between the stock and bond return volatility is determined by a ran- markets. Therefore, more formally, dom stochastic volatility path following a (4) If S = 0, the correlation between stocks GARCH(1, 1) model over 252 trading and bonds over the year is the ‘normal’ days (one trading year) with parameters correlation, qn, set to -.1, the approximate equal to a^0, a^1, and b^1. The mean daily realized correlation of the S&P 500 to return of the stock market is calibrated to VWESX over 1990–2012 excluding 2002 the mean daily return of the S&P 500 and 2008. If S = 1, the correlation Index over 1990–2012 excluding 2002 between stocks and bonds is the ‘crisis’ and 2008. Therefore, the daily return of correlation, qc, which is exogenously set by the stock market on any day t : t 2 the user, and takes on values of -.4 or -.1 fg1; ...; 252 within a ‘normal’ year i is within scenarios (a), (b), and (c) tested. For simplicity, bond returns are assumed to be ri;t;M  NðÞlM ; vt ; normally distributed in both ‘normal’ and ‘crisis’ markets. In ‘normal’ markets, average where l is the average daily return of M annualized bond returns and annualized bond the S&P 500 over 1990–2012 excluding volatility are calibrated to the realized values ‘crisis’ periods of 2002 and 2008 and v is t for VWESX over the ‘normal’ sample period the stochastic variance of the stock mar- of 1990–2012 excluding 2002 and 2008. ket on day t. v v are determined from 1... 252 Formally, a random stochastic volatility path fol- lowing a GARCH(1, 1) model with (5) If S = 0, bonds are assumed to follow a parameters a^0, a^1, and b^1 calibrated to normal distribution such that the return daily S&P 500 Index data from 1990 to of bonds (B) in year i is 2012 excluding 2002 and 2008. ÀÁ 2 As stated initially, within ‘normal’ markets ri;B ¼ N lB; rB ; daily bond returns have a correlation of q to n where l is the average annualized return daily stock market returns. Within ‘crisis’ B of VWESX over 1990–2012 excluding markets, this correlation diverges to qc. The 2 2002 and 2008 and rB is the annualized ‘normal’ correlation in all simulations, qn,is estimated based on the correlation of the variance of VWESX over the same S&P 500 Index and VWESX from 1990 to period. 2012 excluding 2002 and 2008. The corre- (6) If S = 1, bonds are assumed to follow a lation between the S&P 500 and VWESX normal distribution such that the return over the ‘normal’ portion of this sample of bonds (B) in year i is period is roughly -.1. The ‘crisis’ correlation ÀÁ 2 between stocks and bonds, qc, varies between ri;B~ ¼ N lB~; rB~ ; scenarios in order to compare the effect of 2 time-varying correlations on the strategy’s where lB~ and rB~ are exogenously set and performance. lB~ varies between scenarios as the

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management Dynamic asset allocation strategies

profitability of the stop-loss, stop-gain the results of the Monte Carlo simulation for asset allocation strategy is heavily depen- scenarios (a), (b), and (c) are displayed. dent on mean ‘crisis’ bond returns, lB~ (as In the first scenario with results reported well as qc). in Table 8, I estimate k to be .05, which is a The Monte Carlo simulation defined above conservative estimate of the probability of a is run for 1000 simulations of 20 years of ‘crisis’ market in line with 1990–2012 daily data per scenario. Three different sce- historical data. 2002 and 2008 are easily narios are tested: a ‘Decoupling’ scenario identified as ‘crisis’ years within this period as which most closely mirrors the realized data in each year the S&P 500 declined by over over 1990–2012 in which significant diver- -20%. In addition, there is a reasonable gence in US stock and bond market corre- argument to be made that 1999 should lations and realized returns occurred during similarly be defined as a ‘crisis’ year. financial crises (scenario (a)), a ‘Mild decou- However, even with only 2002 and 2008 pling’ scenario (scenario (b)), and a ‘No identified as crises, 8.7% of years (2/23) are Decoupling’ scenario (scenario (c)). The ‘crises’ in the sample period of 1990–2012. scenario parameters are discussed in further Of course, with so few crisis years and a detail below. relatively small dataset, it is entirely By using realistic, yet conservative inputs impossible to accurately estimate k with any for the simulations in scenarios (a), (b), and reasonable precision. Still, 5% or 1 in (c), we are better able to determine if the 20 years seems to be a reasonable, if not even strategy is likely to continue to attain returns conservative, starting point for the in line with its strong historical performance probability of a ‘crisis’ market. In addition, in over the 1990–2012 sample period. In scenario (a), I set the crisis correlation, qc,to particular, these Monte Carlo simulations can -.4 which is closely in line with the help validate that the period did not just correlation of equities to bonds in the ‘crisis’ happen to be a particularly fortunate time- years of 2002 and 2008 (-.37). I set the series for the strategy. In addition, by normal correlation, qn,to-.1 since this is changing the inputs, particularly the approximately the daily correlation of probability of a ‘crisis’ market occurring, k, equities to bonds in the 1990–2012 period or the correlation between stocks and bonds excluding 2002 and 2008. The mean during crises, qc, we may better understand annualized stock market ‘crisis’ return, lM~ ,is how the effect of broad asset ‘decoupling’ in set to -20% with a 15% standard times of financial crisis affects the profitability deviation,rM~ . The mean annualized bond of stop-loss and stop-gain rule-based dynamic market ‘crisis’ return, lB~, is set to 0% with a asset allocation strategies. In Table 7 10% annualized standard deviation, rB~. Both definitions of the MC simulation estimators parameters are sensible based on the are given and in Tables 8, 9, and 10 below 1990–2012 sample period but in the same

Table 7: MC simulations Within the simulations below the estimators denote the following l^M = Estimated annualized geometric return of the Stock Market l^B = Estimated annualized geometric return of the Bond Market l^SL = Estimated annualized geometric return of the baseline Asset Allocation stop-loss, stop-gain strategy r^M = Estimated annualized volatility of the Stock Market r^B = Estimated annualized volatility of the Bond Market r^SL = Estimated annualized volatility of the baseline Asset Allocation stop-loss, stop-gain strategy c IRM = Information Ratio of the Stock Market c IRB = Information Ratio of the Bond Market c IRSL = Information Ratio of the baseline Asset Allocation stop-loss, stop-gain strategy

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management Shelton

Table 8: MC simulation results of the ‘Decoupling’ Scenario (scenario (a)). The table contains the results from 1000 simulations of 20 years of equity and bond data, and the performance of the Asset Allocation stop-loss, stop- gain strategy on this simulated data Estimator l^ (%) r^ (%) c l^ (%) r^ (%) c l^ (%) r^ (%) c M M IRM B B IRB SL SL IRSL

MC estimate 5.94 18.29 0.33 4.37 8.98 0.49 6.45 11.19 0.58 SE 0.14 0.05 0.01 0.07 \0.01 0.02 0.09% 0.04% 0.01

Note: Results are reported for equities, bonds, and the baseline Asset Allocation stop-loss, stop-gain algorithm. The normal correlation is set to -.1 and crisis correlation is set to -.4. The mean returns and volatilities of stocks and bonds in normal times are calibrated from the 1990–2012 data. The annual crisis probability is 5%. The mean crisis annualized equity return is -20% with a 15% standard deviation. The mean crisis bond return is 0% with a 10% standard deviation.

Table 9: MC simulation results of the ‘Mild Decoupling’ Scenario (scenario (b)). The table contains the results from 1000 simulations of 20 years of equity and bond data, and the performance of the Asset Allocation stop-loss, stop- gain strategy on this simulated data Estimator l^ (%) r^ (%) c l^ (%) r^ (%) c l^ (%) r^ (%) c M M IRM B B IRB SL SL IRSL

MC estimate 6.09 18.32 0.34 3.79 8.98 0.42 6.11 11.22 0.55 SE 0.14 0.05 0.01 0.07 \0.01 0.01 0.09 0.04 0.03

Note: Results are reported for equities, bonds, and the baseline Asset Allocation stop-loss, stop-gain algorithm. Both the normal and crisis equity correlation is set to -.1. The mean returns and volatilities of stocks and bonds in normal times are calibrated from the 1990–2012 data. The annual crisis probability is 5%. The mean crisis annu- alized equity return is -20% with a 15% standard deviation. The mean crisis bond return is -10% with a 10% standard deviation.

Table 10: MC simulation results of the ‘No Decoupling’ Scenario (scenario (c)). The table contains the results from 1000 simulations of 20 years of equity and bond data, and the performance of the Asset Allocation stop-loss, stop- gain strategy on this simulated data Estimator l^ (%) r^ (%) c l^ (%) r^ (%) c l^ (%) r^ (%) c M M IRM B B IRB SL SL IRSL

MC Estimate 7.55 18.23 0.42 4.62 8.92 0.52 7.38 11.18 0.67 SE 0.14 0.05 0.01 0.07 \0.01 0.02 0.09 0.04 0.01

Note: Results are reported for equities, bonds, and the baseline Asset Allocation stop-loss, stop-gain algorithm. The normal correlation is set to -.1 and the mean returns and volatilities of stocks and bonds in normal times are calibrated from the 1990–2012 data. The annual crisis probability is 0%. As such, no other crisis parameters need to be defined.

manner as k are naturally imprecise as we effect is simulated. Instead of an expected have few ‘crisis’ data points to consider. annualized bond return of 0% during crises, I Thus, in scenario (a), the significantly higher conservatively estimate the mean annualized expected return of bonds versus equities bond return during crises, lB~, to be a less during crises along with the divergent favorable -10% (‘less favorable’ to the stop- correlation structure of bonds to equities loss, stop-gain asset allocation strategy return during financial crises is roughly in line with relative to that of a ‘‘buy-and-hold’’ strategy). what was actually observed over the In addition, within scenario (b) the 1990–2012 period. The scenario fully, yet correlation of equities to bonds during crises, conservatively models ‘decoupling’ in line qc, stays fixed at -.1 instead of diverging to with historical data. Therefore, scenario (a) is -.4, eliminating the diverging correlations referred to as the ‘Decoupling’ scenario. component of the ‘decoupling’ effect. All In the second scenario with results other parameters in scenario (b) are identical reported in Table 9, a ‘Mild Decoupling’ to those set in scenario (a) above. Thus,

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management Dynamic asset allocation strategies within scenario (b) there is a much milder correlations, we see a 6.11% geometric return simulated ‘flight to quality’ from equities to to the strategy vs. 6.09% to equities. Finally, bonds during crisis markets and less safe in scenario (c), the ‘No Decoupling’ scenario harbor within fixed income assets than in which market crises never occur, we see a within scenario (a). Therefore, scenario 7.38% return to the strategy vs. 7.55% for (b) very mildly and to a limited extent equities. These results further reiterate the simulates the effect of ‘decoupling’ during notion that the existence of nonconstant asset financial crises and is referred to as the ‘Mild correlations and a flight to safety from Decoupling’ scenario. equities to bonds during market crises In the third scenario with results reported benefits the return of the stop-loss, stop-gain in Table 10, I eliminate the ‘decoupling’ dynamic asset allocation strategy relative to a effect altogether by setting the probability of a ‘buy-and-hold’ investment strategy. market ‘crisis,’ k,to0%.Thus,inscenario Of course, no scenario or MC simulation (c) the volatility path of equities always follows of any sort matches the return dynamics of a GARCH(1, 1) process with parameters the equity and bond markets perfectly and calibrated to the 1990–2012 data, excluding one may rationally argue that the actual equity the ‘crisis’ years of 2002 and 2008. In addition, and bond market dynamics are somewhere the correlation of bonds to equities stays fixed between scenario (a) and scenario (c). While at the ‘normal’ correlation, qn,of-1. Of I was purposely conservative in specifying the course, there is no need for any specification parameters within all scenarios, I fully of the mean stock or bond market ‘crisis’ understand and accept this rationale. That return or variance parameters within scenario said, I point out that the information ratio of (c) as market crises simply do not exist within the stop-loss, stop-gain strategy dominates the scenario. All ‘normal’ parameters for the that of both equities and bonds in all three return and variance of stocks and bonds within scenarios by a large and statistically significant scenario (c) are the same as those used in margin. This finding that the risk-adjusted scenarios (a) and (b) above. This scenario of return of a stop-loss, stop-gain rule-based ‘No Decoupling’ is a very purposeful dynamic asset allocation strategy outperforms oversimplification. Market crises are that of equities or bonds even in completely completely avoided in order to test the normal markets devoid of jumps, large downturns, performance of the stop-loss, stop-gain or time-varying correlations is a surprising dynamic asset allocation strategy in ‘normal’ benefit of such a strategy. The likely markets alone. explanation is that there is still enough equity Reasonably, one should expect the market volatility through the use of the estimated stop-loss, stop-gain strategy GARCH(1,1) model within normal markets geometric return and information ratio for an investor to benefit by decreasing his relative to those of a ‘‘buy-and-hold’’ strategy allocation to equities when the lagged return to be best in the first scenario (‘Decoupling’), to equities is quite volatile. In other words, then the second (‘Mild Decoupling’), and implicit downside volatility protection is still then the third (‘No Decoupling’). The results a benefit of such a strategy due to the bear this out exactly. stochastic nature of volatility in which large We observe a 6.45% geometric return to volatility spikes tend to be preceded by the strategy versus 5.94% to equities in smaller spikes. This finding only helps scenario (a), the ‘Decoupling’ scenario, strengthen the robustness of the result that which incorporates decoupling and asset allocation strategies which utilize stop- nonconstant asset correlations. In scenario losses and stop-gains need more investigation (b), the ‘Mild Decoupling’ scenario, which as an alternative and viable form of dynamic assumes mild decoupling but constant asset asset allocation.

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management Shelton

CONCLUSION Carhart, M.M. (1997) On persistence in mutual fund perfor- mance. The Journal of Finance 52(1): 57–82. In totality, we see that asset allocation Clarke, R., De Silva, H. and Thorley, S. (2006) Minimum- strategies which utilize stop-loss and stop- variance portfolios in the US equity market. The Journal of gain rules in order to mitigate risk may have Portfolio Management 33(1): 10–24. Conrad, J. and Kaul, G. (1988) Time-variation in expected significant potential to enhance portfolio returns. Journal of business 61: 409–425. risk-adjusted return and limit portfolio De Bondt, W.F. and Thaler, R.H. (1987) Further evidence on drawdown. By allocating assets in one’s investor overreaction and stock market seasonality. The Journal of Finance 42(3): 557–581. portfolio through the use of stop-loss and Dybvig, P. (1988) Inefficient dynamic portfolio strategies or stop-gain rule-based algorithms, one may be how to throw away a million dollars in the stock able to reduce portfolio variance, while not market. Review of Financial studies 1(1): 67–88. Fama, E.F. and French, K.R. (1988) Permanent and tempo- sacrificing long-term return. rary components of stock prices. The Journal of Political Given that the results I observe from Economy 96: 246–273. calculating historical returns, estimating Fama, E.F. and French, K.R. (1993) Common risk factors in factor model regressions, conducting the returns on stocks and bonds. Journal of Financial Economics 33(1): 3–56. robustness checks, and developing and Ferson, W., Sarkissian, S. and Simin, T. (2003) Is stock return running MC simulations for varying market predictability spurious. Journal of Investment Manage- scenarios are all broadly consistent; I believe ment 1(3): 1–10. Goodwin, T.H. (1998) The information ratio. Financial that asset allocation strategies which Analysts Journal 54(4): 34–43. incorporate stop-loss and stop-gain rules Gulko, L. (2002) Decoupling. The Journal of Portfolio Manage- likely offer a valuable alternative approach to ment 28(3): 59–66. Hansen, P.R., and Lunde, A. (2005) A forecast comparison dynamic asset allocation. Mutual fund of volatility models: does anything beat a GARCH (1, 1)? companies and ETF providers may be wise to Journal of Applied Econometrics 20(7): 873–889. consider the development of asset allocation Herold, U., Maurer, R., Stamos, M. and Vo, H.T. (2007) funds which allocate investors’ assets, at least Total return strategies for multi-asset portfolios. Journal of Portfolio Management 33(2): 60. in part, based on stop-loss and stop-gain or Hocquard, A., Papageorgiou, N. and Remillard, B. other risk minimization rules. More research (2015) The payoff distribution model: an application to is needed to investigate why dynamic dynamic portfolio insurance. Quantitative Finance 15(2): 299–312. asset allocation stop-loss, stop-gain strategies Jensen, J.L.W.V (1906) Sur les fonctions convexes et les tend to outperform as these strategies show ine´galite´s entre les valeurs moyennes. Acta strong long-term risk-adjusted returns, even Mathematica 30(1):175–193. Jegadeesh, N. (1990) Evidence of predictable behavior in historically nonvolatile markets. of security returns. The Journal of Finance 45(3): 881–898. Jegadeesh, N., and Titman, S. (1993) Returns to buying winners and selling losers: Implications for stock market ACKNOWLEDGMENTS efficiency. The Journal of Finance 48(1): 65–91. I would like to thank Dr. Luis Garcia-Feijoo Kaminski, K., and Lo, A.W. (2007) When Do Stop-Loss (Florida Atlantic University), Dr. Scott Rules Stop Losses? European Finance Association General Assembly; 24 August, Ljubljana, Slovenia. Cederburg (University of Arizona), Dr. Gary Markowitz, H. (1952) Portfolio selection. The Journal of Sanger (Louisiana State University), and Dr. Finance 7(1): 77–91. Judson Russell (University of North Carolina Noeth, B.J. and Sengupta, R. (2010) Flight to safety and US Treasury securities. The Regional Economist 18(3): Charlotte) for their thoughtful advice and 18–19. feedback on this project. Perold, A.F. and Sharpe, W.F. (1988) Dynamic strategies for asset allocation. Financial Analysts Journal 44(1): 16–27. Poterba, J. M., and Summers, L.H. (1988) Mean reversion in REFERENCES stock prices: Evidence and implications. Journal of Financial Economics 22(1): 127–59. Bollerslev, T. and Ole Mikkelsen. H.O. (1996) Modeling and Silverblatt, R. (2012) Index Funds Still Beat Most Managers. pricing long memory in stock market volatility. Journal of US NEWS, 12 October. Econometrics 73(1): 151–184.

Ó 2016 Macmillan Publishers Ltd. 1470-8272 Journal of Asset Management