2.1 MSP model formulation

Using this notation for decision trees, a multiperiod stochastic program can be formulated recursively by considering the nodal decision problems. A linear stochastic programming model is presented for clarity; extensions to a general nonlinear framework is quite straightforward, see, e.g., formulations in Edirisinghe and Ziemba (1992) and Frauendorfer (1996). The decision process up to (tree) node has observed the realization sequence indexed as , and corresponds to the decision vector sequence . The nodal value function for a period t, , is the optimal profits to be realized after observing the event indexed by and by choosing the decision vector to optimally trade off current profits and the expected value of optimal profits at the child nodes, i.e., for each of . This nodal decision problem is formulated as follows, for .

(1)      

The nodal value function, , for a node at terminal period T, is

(2)      

Array dimensions are not explicitly listed for reasons of brevity and all arrays are assumed to have shapes and sizes compatible with the depicted operation. Then, the multistage stochastic program can be equivalently represented as the following compact recursive value function formulation.

(3)      

3.1 Transactions and slippage costs

Usually, fund managers face transactions costs in executing the trade vector, , which leads to reducing the portfolio net return. Placing a trade with a broker for execution entails a direct cost per share traded, as well as a fixed cost independent of the trade size. For instance, a $10 fixed cost and a 2-cent per-share cost may be levied. In addition, there is also a significant cost due to the size of trading volume , as well as the broker’s ability to place the trading volume on the market. If a significant volume of shares is traded (relative to the market daily traded volume in the security), then the trade execution price may be adversely affected. A large buy order usually lead to trade execution at a price higher than intended and a large sell order leads to an average execution price that is lower than desired. This dilution of the profits of the trade is termed the market impact loss, or slippage. This slippage loss generally depends on the price at which the trade is desired, trade size relative to the market daily volume in the security, and other company specifics such as market capitalization, and the beta of the security. See Keim and Madhavan (1996, 1997, 1998), Loeb (1983), and Torre and Ferrari (1999), for instance, for a discussion on market impact costs.

In our model, a quadratic simplification for the market impact cost is utilized. That is, the slippage costs are proportional to the square of the volume traded in the market, and the constant of proportionality depends directly on the intended execution price, and inversely on the (daily) share volume in that security. For trading shares of security j (with the intended trade price when the expected total daily volume is shares), thus, leads to a market impact cost of

where is a constant calibrated to the market data. The direct cost of trading is represented by where is the per-share transactions cost. The fixed costs of trading are ignored here, and then, the total transactions and slippage loss function is

     (4)

Therefore, the total loss due to portfolio rebalancing at time t, given the historical scenario of security price evolution, is

     (5)

3.2 Budget and portfolio wealth

Portfolio rebalancing from period-to-period is assumed to carry forward any cash that is accumulated (i.e., the riskless security) and the total cost of trading at the beginning of period t is thus limited by the cash carried forward from period , denoted by the nonnegative variable , and any exogenous funds available at the beginning of period t, denoted by . Then, the following budget constraint must hold.

     (6)

The transposition of the price vector is suppressed for clarity of exposition in the above expression, as well as throughout the remainder of the chapter. When applying the budget constraint (6) for trading at the root node , by convention, , the initial security positions, and , the initial cash position. Note that “cash” carried forward yields a deterministic monetary return from period to period given by the rate (⩾1), which is already adjusted to the length of the period t. Thus, is the excess cash carried forward and available for portfolio rebalancing at node of period . In the event that trading is required to be self-financing, one must set . Alternatively, if a certain dollar amount, say , must be taken off the stock market (to satisfy a certain known liability at the current time period), then, one must set .

Given portfolio rebalancing at the current decision node to obtain revised positions , the portfolio gain under (future) return scenario vector , is

     (7)

where represents the diagonal matrix in which the jth diagonal element is . Observe that the uncertainty of the return , conditional upon the history being followed, endows the portfolio gain variable with uncertainty. Consequently, the projected portfolio wealth, denoted by , under return scenario satisfies

     (8)

where denotes the portfolio wealth (consisting of cash and risky securities) at the end of period , evaluated before trading at the current decision node using the state price vector . When applying (8) for , the initial portfolio wealth prior to trading at the root node is . It must be noted that, at some period t under some return scenario , it is possible that the portfolio projected wealth variable , which thus indicates the termination of the portfolio. Consequently,

     (9)

is specified as a required feasibility condition under all realizations of the scenario tree.

3.3 Limiting positions

The nonnegative trade-size in security j is . Since the investment objective is to maximize the total portfolio (expected) return, less the convex quadratic slippage loss in (5), the following linear constraints can be used to replace the nonlinear (absolute-value) expressions for the trade-size vector:

     (10)

Positions , after portfolio revision, must satisfy certain minimum/maximum limits due to policy requirements. This often arises from requiring that no more than (or no less than for the purposes of short positions) a certain dollar allocation be made in a particular security. In share volume terms, these maximum and minimum position limits are and , and thus the resulting constraints are

     (11)

Alternatively, long and short positions in the portfolio may be controlled in aggregate by using the long leverage and short leverage restrictions. That is, a prespecified nonnegative $ limit may be applied for the total long position and for the total short position, when trading for period t, which yields

A linear representation of the above is obtained by defining pairs of nonnegative (long/short) variables and , for security j, such that

     (12)

Under the variable description above, the following pair of linear constraints can be used to represent the leverage constraints.

     (13)

Furthermore, the total short position may be controlled relative to the total long position, and vice versa. The primary reason for such joint control is to limit portfolio risk exposure and it is often dependent on fund manager’s investment style or policy. Consequently, we identify two major forms of portfolio risk control; those arising from direct position control due to policy and those that control investments due to portfolio risk measurement. We consider several forms of direct position control first; in Section 4, several risk metrics for portfolio risk control are discussed.

3.4 Excessive shortsale risk

While position limits or leverage constraints themselves are forms of policy-oriented risk controls, the strategies that belong to this category control relative positions of long and short exposure. The first of this type considered here is excessive shortsale risk, or ESR. In this type of control, the total short position is controlled not to exceed a certain fraction of the total long position, within a pre-identified group of securities. For instance, in order to limit the short-selling risk within a volatile sector, such as the Internet stocks, the managerial policy may be that no more than 30% of the long position in Internet stocks may be tied in short positions within the same sector. Such a policy may also be interpreted as some sort of margin requirement within a certain group of securities.

In a more generalized setting, such margin requirements may apply to different groups of stocks differently. Suppose there are K groups of stocks whose index set is denoted by , for , such that . Given the value of the allowable shortsale fraction for group of stocks as , these constraints are

     (14)

Although the allowable fractions may be specified such that they are dependent on the time period or the historical scenario being followed, such dependencies are ignored here for the ease of exposition. Noting that (14) is nonlinear, an alternative linear formulation is available under the condition that .

Proposition 3.1 If for , then the nonlinear margin constraints in (14) have the following equivalent linear representation

     (15)

Proof Let () be a feasible solution in (14). Then, construct the nonnegative pair of vectors such that, for , and . Observe that the pair satisfies the inequality in (15); furthermore, it satisfies the equality in (12) since

hence, a feasible solution to the linear representation of the shortsale constraints.

Conversely, it will be shown that given a pair that is feasible in (15), one obtains a feasible solution in (14) using the construction:

Defining, , it follows that and

     (16)

Using this “hat” solution and substituting from (16) in (14):

 is feasible in (15)

This completes the proof.

The condition that implies that the total short position in the group cannot exceed the total long position. Thus, the use of ESR constraints allows the fund manager to keep the short investment within the total long investment with respect to a given group of securities. However, often a portfolio is required to be neutral in its long and short investments so as to immune the portfolio from any particular directional risk. Two types of portfolio neutral strategies will be discussed in the next section: beta neutrality and dollar neutrality.

3.5 Degree of portfolio neutrality

Portfolio neutrality is provided by hedging strategies that balance investments among carefully chosen long and short positions. Fund managers use such strategies to balance the portfolio so as to buffer it from severe market swings, for instance, see Nicholas (2000) and Jacobs and Levy (2004). Alternatively, a prescribed level of imbalance or nonneutrality may be specified in order for the portfolio to maintain a given bias with respect to market swings. An important metric of portfolio bias relative to the broader market is the portfolio beta. A balanced investment such that portfolio beta is zero is considered a perfectly beta neutral portfolio and such a strategy is uncorrelated with the market return. Beta is the measurement of a stock’s volatility relative to the market. A stock with a beta of 1 moves historically in sync with the market, while a stock with a higher beta tends to be more volatile than the market and a stock with a lower beta can be expected to rise and fall more slowly than the market. Many practitioners of beta neutral long/short trading balance their positions in the same sector or industry. By being sector neutral, the risk of market swings affecting some industries or sectors differently than others may be avoided.

The degree of market-neutrality of the portfolio measures the level of correlation of performance of the portfolio with an underlying broad-market index. Typically, the S&P 500 index may be used as the market barometer; alternatively, if the portfolio is constructed out of the S&P 100 stocks, the S&P 100 index may serve as the underlying market performance metric. Let be the beta of the security j given the sequence of historical prices observed, as indicated by scenario . Then, is the (estimated conditional) covariance of the rates of return between the security j and the chosen market barometer (index), scaled by the variance of the market rate of return. Since is the random variable representing the rate of return of security j, by denoting the market index rate of return by the random variable , it follows that

     (17)

At a portfolio rebalancing decision node , the resulting portfolio beta can be obtained by

Proposition 3.2 At some trading node , having observed the price vector at the beginning of period t on the portfolio positions , let the portfolio value before trading be . Suppose the portfolio is rebalanced at the current node to establish the new security positions . Then, the resulting portfolio beta, , is

     (18)

Proof Since the projected portfolio value at period t is given by (8), the rate of return of portfolio for period t is the random variable and thus, . Referring to (7), the conditional covariance given and , is

since the cash rate is assumed fixed (and thus it has zero correlation with the market). The result in the proposition then follows.

To control the portfolio beta at a level , the constraints must be imposed at each trade rebalancing decision node . By the nonnegativity in (9), the required portfolio beta constraints are

     (19)

Portfolio beta is set at with a tolerance level of . With , rebalancing strives for a portfolio beta of . In particular, with , the model attempts for a perfectly beta-neutral dynamic investment in the portfolio.

In addition to the (degree of) market neutrality afforded by the above beta constraints, fund managers often seek additional neutrality through dollar neutrality. In order to create a portfolio with dollar neutrality, equal amounts of long and short investments are made, or alternatively, a specific dollar imbalance may be prescribed as policy. Given a targeted maximum dollar imbalance of , the “degree of dollar neutrality” constraint is

Under the variable description in (12), the equivalent linear constraints are:

     (20)

If portfolio risk control is solely via market neutral long/short trading as an act of portfolio balancing, then it would generally involve a large amount of buying and selling. Such a strategy naturally leads to additional risks as the fund manager now needs to have the ability to execute trades efficiently as well as to keep brokerage costs from severely affecting trading profits. Furthermore, fund managers must also trade in very liquid stocks, indicating a high level of daily volume, in order to ensure they can get quickly in and out of positions, as happens in frequent trading. Therefore, portfolio risk management should not be viewed solely through such policy or balancing strategies. Furthermore, in order to keep trading costs relatively low, relative to short term rates of returns of securities, a multiperiod trading view must be taken when portfolio rebalancing and risk controls must be applied within such a setting.

4.1 Static risk control (SRC)

Two types of risk metrics are considered in the context of SRC, where risk expressions can be obtained in closed-form based on the forecasted parameters of uncertainty at a given decision node . Such SRC may be applied at all rebalancing nodes in the decision (scenario) tree. The first is a variant of portfolio variance, and the second is what is termed the portfolio catastrophic risk.

The use of portfolio variance as a risk expression and using it within a static mean variance trade-off optimization model date back to Markowitz (1959). Its multistage extensions, where mean-variance trade-off is specified at each of the decision nodes in a scenario tree, too are considered in the literature, see, e.g., Steinbach (2001) and the many references therein, and also Gulpinar, Rustem and Settergren (2003). The mean-variance trade-off is a quadratic programming model, and thus, the application of quadratic programming on a multistage stochastic programming model is computationally tedious.

In our development here, a generalized approach to portfolio (quadratic) tracking penalty is considered and portfolio variance is obtained as a special case. Consider a target return that portfolio wishes to track. Such targets can be either deterministic or stochastic, and also, they can be exogenously or endogenously specified, see Geyer et al. (2003). For instance, tracking the portfolio mean is a target specified endogenously, but deterministic. An example of a stochastic (and exogenous) target is the return on a broad market index, which may be correlated with the portfolio securities. Index target is considered exogenous because it is assumed that portfolio rebalancing does not affect the broader market index to any appreciable extent. Benchmark tracking for portfolio management has been well-studied in the literature, see, for example, Dembo and King (1992), Frino and Gallagher (2001), Jansen and van Dijk (2002), Roll (1992), and El-Hassan and Kofman (2003). However, the approach taken here is slightly different, as discussed below.

4.1.1 Tracking risk control

Consider a benchmark (stochastic) target, such as the return on a broad market index, say the S&P 500 index. Let the conditional rate of return (RoR) on the benchmark index, at time t at the rebalancing decision node , be , which is a univariate random variable. In a pure investment in the benchmark index, the total investment in the portfolio, given by , yields the target total $ return as

     (21)

is a univariate random variable and it serves as the target return for the portfolio for period t. The conditional expectation of the quadratic tracking penalty for portfolio return on risky investments not following the above target return is

     (22)

Denoting the portfolio standard deviation by , its variance is

     (23)

where is the (conditional) variance–covariance matrix of the random vector .

Proposition 4.1 An upper bounding risk metric on the tracking penalty in (22) is

     (24)

where is the index rate of return and is the index standard deviation.

Proof Note that

Using the expression

and upon algebraic manipulation, it follows that

due to the nonnegativity of the second term in the above equality expression.

Observe that the risk metric can be interpreted as the sum of the three components,

beta not being 1.0 

In particular, when the target being tracked is simply the portfolio mean itself, then the above risk metric is

Corollary 4.2 If the target is specified as , then the risk metric becomes , i.e., the usual portfolio variance.

The risk metric in (24) is of static-type as it is computable under the forecasted parameters, and also it is convex quadratic in the portfolio positions . The use of such a risk metric in the portfolio model is to seek a trade-off between the risk metric and the portfolio expected net return after rebalancing. This objective, at the current rebalancing node, is

     (25)

where the function is defined in (7), and is herein termed a (static) tracking risk aversion (TRA) coefficient, which is nonnegative.

4.1.2 Catastrophic risk control

A second form of significant portfolio risk is next considered, largely motivated by practitioners. This form of risk is concerned with the direction of (the future) price of a security being opposite to the sign of the established position in the portfolio. That is, securities in a long portfolio fall in price while the securities in a short portfolio rise in price. Such risk is often the result of error in forecasting the direction of stock price movement. This would entail observing a drop in price for long securities and an increase in price for shorted securities. Generally, there is no formal mechanism to safeguard against the risk posed by such a catastrophic event. Controlling the portfolio variance, or more generally the tracking risk metric discussed above, does not necessarily counter the effects of catastrophic risk, abbreviated herein as Cat risk. The effects of Cat risk control in a portfolio is demonstrated under the section on model application.

We define Cat risk as the anticipated total dollar wealth loss in the event stock returns move against the portfolio positions by one standard deviation, denoted by , where the positions are established at event node for period t. That is,

     (26)

which is thus a form of static risk control. While the Cat risk expression is free of correlations among securities, it can be shown that it provides an upper bound on the portfolio standard deviation.

Proposition 4.3 The portfolio standard deviation is a lower bound on the catastrophic risk, i.e., where is given by (23).

Proof

Defining the random variable , variance of the portfolio is . Denoting the standard deviation of by and the correlation between and by ,

Noting that , and since the prices are nonnegative, we have . Therefore, it follows that the portfolio variance is bounded from above by .

Since portfolio standard deviation is only a lower bound on Cat risk, controlling portfolio variance, as in Markowitz-style, is not guaranteed to provide adequate protection against catastrophic risk. In contrast, controlling Cat risk will certainly control the portfolio variance directly, and thus, Cat risk metric plays a dual role in portfolio rebalancing. The two risk metrics, and , however, have distinct characteristics in shaping portfolio positions, as will be demonstrated numerically later. Geometrically, Cat risk (as a function of portfolio positions) bounds the portfolio standard deviation by a polyhedral convex cone with apex at the origin.

While it is possible to incorporate in to the objective function with a risk aversion coefficient, similar to (25), it is easier to specify the maximum allowable $ loss in the portfolio for a catastrophic move (of, say, one standard deviation) as a constraint. The Cat risk constraint is then

     (27)

where $ is the pre-specified dollar loss. Since is nonlinear in its arguments, a linear representation is obtained by appealing to the pairs of nonnegative (long/short) variables given in (12). Then, the following linear constraint replaces the Cat risk constraint in (27),

     (28)

Since it is possible to easily argue that (27) and (28) provide equivalent expressions for risk control, details are skipped here. Note that (28) limits positions (long or short) in securities with high volatility (as measured by ), and thus provides a form of indirect position control.

4.2 Dynamic risk control (DRC)

As defined earlier, in DRC, risk metric computation requires the availability of a scenario sample (conditional upon the history ), in addition to the forecasted parameters, at a rebalancing decision node . In the tracking risk metric, portfolio wealth variation around a target is penalized in either direction. In the case when the target is the portfolio mean, the result is the Markowitz mean-variance trade-off. Mean-variance optimal portfolios are shown to be (stochastically) dominated by carefully constructed portfolios. Such is the case when one penalizes only the variation of portfolio value on the downside, i.e., the risk metrics based on downside deviation.

The relative merits of using Markowitz mean-variance type models and those that trade off mean with downside semi-deviation are examined in Ogryczak and Ruszczynski (1999). The semi-deviation risk trade-off approach yields superior portfolios that are efficient with respect to the standard stochastic dominance rules, see Whitmore and Findlay (1978). When quadratic penalty is applied on the downside deviations, with target defined at the portfolio mean, it is called the downside semi-variance risk metric. The concept of semi-variance was described by Markowitz (1959, Chapter IX). Semi-variance fails to satisfy the positive homogeneity property required for a coherent risk measure, see Artzner et al. (1999) and Rockafellar, Uryasev and Zabarankin (2002).

4.2.2 Drawdown risk control (DDR)

A second type of dynamic risk control of paramount importance to fund managers is the so-called portfolio drawdown control. Investors and fund managers do not wish to see the value of the portfolio decline considerably over time. Such drastic declines in portfolio value may lead to perceptions that the fund is too risky; it may even lead to losing important client accounts from the fund. Portfolio drawdown is defined as the relative equity loss from the highest peak to the lowest valley of a portfolio value decline within a given window of observation. Mathematically, given the current rebalancing node and the portfolio wealth trajectory , for where , the maximum portfolio value in the time window is

     (36)

The portfolio, under the revised portfolio positions , is said to have -period drawdown under return vector if holds, and in this case, the relative drawdown is measured by

For instance, suppose a fund manager has logged the portfolio wealth at the beginning of a year at $5 million, which reaches a peak in June to $8 million, and then loses its value to $6 million by the end of the year. Thus, for the period of one year, the fund had a relative drawdown of , or 25%, while the fund has an annual RoR of , or 20%. Portfolio performance, as measured by the reward-to-drawdown (RTD) ratio, is , and in this example, it is indicative of poor portfolio performance. Successful money managers strive for ratios of 2 or better. Another important issue in portfolio drawdown is the time it takes to recover from the (drawdown) losses, termed as the drawdown recovery time, relative to the drawdown duration time. Indeed, shorter recovery times are preferable in money management. Aiming for the highest portfolio return, usually, does not translate to a high RTD ratio. While individuals may have the intention for fast portfolio growth with a “stomach” for large and repeated drawdowns, a low RTD ratio is not a trait of a successful money manager.

Regulatory control in portfolio drawdown makes fund manager’s task quite challenging. For instance, according to Chekhlov, Uryasev and Zabarankin (2003), “it is highly uncommon, for a Commodity Trading Advisor (CTA) to still hold a client whose account was in a drawdown, even of small size, for longer than 2 years. By the same token, it is unlikely that a particular client will tolerate a 50% drawdown in an account with an average- or small-risk CTA. Similarly, in an investment bank setup, a proprietary system trader will be expected to make money in 1 year at the longest, i.e., he cannot be in a drawdown for longer than a year. Also, he/she may be shut down if a certain maximal drawdown condition will be breached, which, normally, is around 20% of his backing equity. Additionally, he will be given a warning drawdown level at which he will be reviewed for letting him keep running the system (around 15%). Obviously, these issues make managed accounts practitioners very concerned about both the magnitude and duration of their clients’ accounts drawdowns.”

Despite this apparent significance of drawdown, theoretical development on a drawdown risk metric is relatively sparse. Grossman and Zhou (1993) consider an exact analytical solution of a maximal drawdown problem for a one-dimensional case under lognormality assumptions. Subsequently, Cvitanic and Karatzas (1995) generalized this work for multiple dimensions. In contrast, Chekhlov, Uryasev and Zabarankin (2003) developed a metric, termed the conditional drawdown-at-risk (CDaR), and applied it along sample paths of portfolio returns with no assumptions on underlying distributions. Nevertheless, all these deal with either a static model that is rolled over in time, or holding constant portfolio weights over the duration of the sample paths.

Since the portfolio drawdowns in multiple periods in the future are affected by the current rebalancing decisions (in the presence of trading costs), applying drawdown risk control within a multistage framework is expected to improve portfolio reward-to-drawdown (RTD) ratio considerably. This is highlighted in the numerical experiments reported in Section 6. To develop our drawdown risk (DDR) metric, it is assumed, without loss of generality, that the drawdown window begins from the root node at time , i.e., holds in (36) for maximum portfolio value. Define the drawdown random variable at node by , i.e.,

     (37)

Our drawdown risk metric is determined with respect to a prescribed level of acceptable relative drawdown in the portfolio, herein denoted by the (nonnegative) fraction π. Thus, the DDR metric, denoted by , takes on value zero if , and it is positive otherwise. Therefore, defining the ($) violations from the acceptable level of drawdown by ,

     (38)

The drawdown (quadratic) risk metric is then

     (39)

Therefore, the nodal trade optimization problem with drawdown risk control at node has the objective

     (40)

where the function is defined in (7), and is herein termed a (dynamic) drawdown risk aversion (DRA) coefficient, which is nonnegative. The constraints that must be imposed to determine the drawdown risk correctly are

     (41)

where must satisfy the following linear constraints along the scenario path up to node

     (42)

The drawdown constraints in (41)-(42) are free of the drawdown variables in (37). Moreover, the constraints (41) must be imposed along each scenario outcome descendant from node , i.e., for all , , which thus makes the computation of the risk metric directly dependent on the generated scenario tree of security returns.

5.1 Key issues in model rollover

When the model in (43) is applied for portfolio rebalancing at some point in time, say , then the investor must first choose a suitable number of trading model periods T as well as respective period lengths (say, a day) for each of the T periods. Then, the model is initiated with root node at time . For ease of exposition, suppose the money manager is interested in (actual) rebalancing of the portfolio at pre-specified time epochs (). To assure optimized rebalancing at each of these time epochs, the model in (43) is applied at each time , , with its own multistage scenario tree. We will assume that each of these trees will have T periods, see Figure 4. Let the sequence of optimized rebalanced positions determined for each trading time , using the model in (43), is herein denoted by , , which are established in the market for the length of time , referred to as the ith trading segment. Thus, it is only appropriate that the model in (43), applied at trading time , has specified a first period of length . Period lengths are indirectly incorporated into the model via the required forecasted parameters such as the mean vector , variance–covariance matrix , security beta vector , and the market index parameters and . In addition, a scenario generator is required that is adapted to the period lengths in the model.

Fig. 4 Rolling horizon implementation of the model.

There are several pertinent issues that must be addressed in a successful implementation. The treatment here is cursory, as the focus in this chapter is model development and its performance analyses. First, while the first period length is set to (at trading time ), the remaining period lengths (of the periods) need not necessarily correspond to the actual trading epochs. One reason for this is that the computational solution of the model (43) is realistic only for a modest number of periods, for instance, T being not more than 3 or 4. But in order to capture the price dynamics in the future, periods of increasing length may be specified. For infinite horizon problems modeled as finite stage models, Grinold (1986) describes constructing an end effects period to capture the dynamics of the future after a finite number of model periods. For instance, see Cariño, Myers and Ziemba (1998) for an application in financial planning that incorporate end effects periods. In the implementation and the computational results reported in the next section, we pick the period lengths quite arbitrarily, and these lengths remain fixed at all subsequent model rollovers.

The second important issue is the estimation of parameters (of random vectors) for the chosen periods, conditional upon the observed history of actual price series. Suppose the actual price (vector) is denoted by at time τ. Thus, when forecasting the parameters for trading segment 1 at time , one must use historical prices (only) from the realized history , which will be applied at node of the model. Thereafter, for all forecasts at an arbitrary node of the model, only the actual observations must be used; however, they may be adapted to the particular (generated) scenario being followed in the scenario tree.

For the implementation in Section 6, second moment forecasts (variances and correlations) remain fixed at all nodes and set equal to those at the root node . However, the first moment forecasts computed at the root node are progressively adapted to the scenario outcomes being followed up to a given node . This adaptation procedure or the computation of the root node forecasts themselves are highly specialized and are omitted from further discussion in this chapter. The subject of parameter estimation in the context of financial market data has received considerable attention and there is a wealth of literature on various methods. The interested reader is referred to, for instance, Andreou and Ghysels (2002), Bai, Russell and Tiao (2001), Cohen et al. (1983), Foster and Nelson (1996), Ledoit and Wolf (2003), Merton (1980), and Scholes and Williams (1977).

Between two model rollovers, say from to , the optimized portfolio must be evaluated based on the actual price series, herein termed portfolio (out-of-sample) simulation and it represents the actual performance of the portfolio under model-specified trade-sizes. Therefore, the actual (initial) portfolio wealth for a model-run beginning at the trading epoch at , denoted by , is given by , where is the discount rate adjusted to the trading segment length . is the wealth at the inception of the portfolio at the beginning of the first trading epoch . Then, the portfolio value series, under the model-based continuous rollover rebalancing, is the wealth sequence , which is evaluated for performance.

5.2 Portfolio performance metrics

The rebalancing model in (43) is evaluated by computing performance metrics for the (simulated) wealth series of the managed portfolio. These performance metrics are

• maxDD (portfolio maximum drawdown): for trading segment is , scaled by , where .

The last two drawdown metrics have not been considered in the literature, to the best of the author’s knowledge. However, they provide key metrics of performance evaluation that are critical for successful money management.

6.1 Single stage models

The first set of experiments is concerned with analysing the one period model version with varied risk metrics to highlight the relative effects of risk controls. With only SRC applied on one period models, scenarios of uncertainty are not required. These experiments compare SRC risk metrics: TRARISK in (24), CATRISK in (26), and the (variance) risk metric , MARKOW. Pure SRC control means only one SRC metric is applied at a time. Table 3 provides a representative summary of portfolio performance for this case, where each column corresponds to a particular value of the objective risk aversion coefficient. See Figures 6 and 7 for a comparison of main portfolio performance measures across pure SRC metrics.

Table 3 Performance for one period model rollover with Pure SRC

Fig. 6 Pure SRC risk (1 day horizon): Sharpe vs. Std Deviation.

Fig. 7 Pure SRC risk (1 day horizon): Reward-to-Drawdown vs. max Drawdown.

Pure (Markowitz) variance risk metric performs extremely poorly at all attempted coefficients of risk aversion, and none of the performance metrics appear to be within acceptable ranges for fund management. The strategy that achieves a Reward-to-Drawdown ratio of at least 2 with the least maxDD (of 16.86%) is the Pure CatRisk strategy. Even this level of drawdown may be unacceptable to a fund manager, even though it has ARTD   >   2. By following a mixed strategy in which two SRC metrics are applied simultaneously, one may obtain improved portfolio performance. This is demonstrated next using tracking control and Cat risk control together. Tracking risk aversion coefficient is applied at five different levels: Very Low, Low, Moderate, High, and Very High. For each of these levels, Cat risk is varied to plot portfolio performance, as given in Figures 8 and 9. In each of these two plots, the upper envelope may be thought of as the appropriate frontier in the two-dimensional risk metric space. Furthermore, Drawdown Duration to Drawdown Recovery time ratio, or DDDR ratio, for the mixed strategy (with moderate tracking aversion coefficient) remained between 0.33 and 0.55. A DDDR ratio of greater than 1 is desirable. DDDR for the pure (Markowitz) variance risk metric was zero because the portfolio never recovered from its maximum drawdown, for all ranges of variance risk aversion coefficients, during the invested period.

Fig. 8 Mixed SRC risk (1 day horizon): Sharpe vs. Std Deviation.

Fig. 9 Mixed SRC risk (1 day horizon): Reward-to-Drawdown vs. max Drawdown.

Under two-dimensional integrated SRC, the rebalancing strategy that achieves ARTD just over 2.0, with the lowest maxDD%, corresponds to a run of “Low” tracking risk aversion with CatRisk of $10,000 a day. This run has Return-to-Drawdown ARTD   =   2.04 with maxDD   =   6.07%, AShR   =   1.44, and ARoR   =   16.40%. The portfolio value over time for this run is depicted in Figure 10. In the same figure, two other graphs are plotted: the run that corresponds to the maximum ARTD of 4.89 (with a maxDD of 23.32%, AshR   =   2.59, and ARoR   =   118.14% for the pair “Very Low” tracking risk aversion and daily CatRisk   =   $60,000) and the run under “moderate” tracking risk aversion and daily CatRisk   =   $50,000 which yields the highest Sharpe ratio (AShR   =   1.63, with maxDD   =   14.53%, ARTD   =   1.79 and ARoR   =   30.01%). For these three cases, portfolio daily standard deviations are in Figure 11 and the daily portfolio betas are in Figure 12.

Fig. 10 Portfolio trajectories for selected mixed SRC strategies.

Fig. 11 Portfolio standard deviations for selected mixed SRC strategies.

Fig. 12 Portfolio daily betas for selected mixed SRC strategies.

6.2 Comparison with multistage models

Two-stage models are specified with a day-long first stage and a second stage of 4 days, for a future horizon of 5 days. Three stage models have a third stage of 5 days, for a 10 day-horizon. Multistage models, generally, are better able to trade-off transaction costs and control drawdown. Multistage models too are rolled over the 1450 trading days. Given that both Cat risk and Tracking risk in combination provided better risk control in single stage models, the multistage runs have this dual risk control settings. Portfolio variance as a risk metric (i.e., Markowitz metric) performs poorly even in the case of 2-stage models. To illustrate, the 2-stage model is run with pure variance (MARKOW) risk metric, and it is compared with a non-MARKOW model with SRC/DRC metrics, but with the same value for objective (quadratic) risk aversion coefficients. The non-MARKOW model had CatRisk   =   $40,000 and drawdown risk metric with . Portfolio trajectories and portfolio standard deviations are shown in Figures 13 and 14, respectively.

Fig. 13 Comparison of proposed risk metrics and variance risk metric—portfolio values.

Fig. 14 Comparison of proposed risk metrics and variance risk metric—Standard deviations.

A comparison of risk-return trade-off between 1- and 2-stage models is presented next, where the effects of transactions/market impact costs are also pursued. The per-share cost rate , set at 2% in the experiments so far, is increased to 5%. The two model versions have the exact same settings, except for the number of model periods, where the 2-period model is specified with (95-dimensional) return scenarios for the 5 day period. This analysis reveals two important properties in rebalancing. First, at a fixed transactions cost rate, 2-stage models perform better than 1-stage models. Second, as the transactions costs increase, the advantages of multistage models over single period models become increasingly prominent. These effects are captured in Figures 15-18. As the last figure indicates, 2-stage model performance is quite robust in the event trading costs increase.

Fig. 15 Comparison of 1- and 2-stage models at trading cost .

Fig. 16 Comparison of 1- and 2-stage models at trading cost .

Fig. 17 Performance of 1-stage model with increasing trading costs.

Fig. 18 Performance of 2-stage model with increasing trading costs.

Portfolio rebalancing using models with 3 periods yields significantly better performance with regard to drawdown characteristics. This is due to applying the drawdown risk (DDR) control, see (40), over 3 period-scenarios of the model. The drawdown duration to recovery ratios (DDDR) are compared across 1-, 2-, and 3-stage models with allowable relative drawdown set at 10% and specifying a very high drawdown risk aversion coefficient. These runs correspond to a “moderate” tracking risk aversion, for various levels of CatRisk control. As can be seen from Figure 19, these multistage models have rather impressive improvements over the single period counterpart.

Fig. 19 Drawdown recovery comparison of 1-, 2-, and 3-stage models.