Discrete-Time Dynamic Portfolio Optimisation with Trading costs.

Discrete-Time Dynamic Portfolio Optimisation with Trading costs. J.A. Sefton∗ Imperial College Business School Imperial College London South Kensington campus London SW7 2AZ June 23, 2010

Abstract Given investors risk-return preferences can be represented by a Constant Relative Risk Aversion (CRRA) Utility function, the problem of maximizing the total return to a portfolio with regular rebalancing over a given horizon can be rewritten as an optimal risk-sensitive (or H∞) control problem. Further if the dynamic evolution of the forecasts to the equity assets can be written as linear stochastic system — which can encompass a simple representation of trading transaction costs as in Engle, Ferstenberg (2007) and Almgren, Chriss (2000) — then the optimal portfolio can be written in terms of the solution to a matrix Riccati equation. It is shown that the optimal dynamic portfolio can be rewritten as the optimal static mean-variance portfolio plus a weighted sum of Merton (1973) hedging portfolios. This solution procedure is applied to both the forecast horizon and to the finite-horizon dynamic asset allocation problem discussed in Barberis (2000).

Very Preliminary Draft for Imperial Business School Conference on Portfolio Optimisation. Future revisions can be obtained from [email protected]. Apologies for all typos and grammatical errors. Keywords: Portfolio Optimisation, Transaction Costs, Hedging Portfolios

∗

The author would like to thank Robert Kosowski and Sid Browne for helpful discussions

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1

Introduction

This paper presents a discrete-time explicit solution to the dynamic portfolio construction problem over both finite and infinite time horizons when markets are incomplete, trading is costly and the investor faces linear portfolio constraints. The analysis is kept tractable by restricting the framework to CRRA utility, log-normal returns and lineargaussian state processes - (often labelled as the exponential gaussian benchmark). We illustrate the approach with respect to the ‘forecast horizon problem’; the problem of combining two investment strategies - .such as value and momentum - that have very different return horizons and implementation costs. We first demonstrate that the problem of maximizing the total return to a portfolio over a given horizon can be written as an optimal risk-sensitive (or H∞) robust control problem, see Whittle (1990) and Iglesias and Glover (1991). This paper is not the first to suggest this approach to solving portfolio allocation problems. Fleming (1995) used risk-sensitive methods to obtain asymptotic results for a portfolio management problem. Bielecki and Pliska (1999), Bielecki, Chanceliery, Pliska, and Sulem (2004), Fleming and Sheu (2002) and Nagai and Peng (2002) also recast the dynamic portfolio optimisation as a continuous time risk-sensitive control problem in order to develop solution procedures. In a series of papers, Schroder and Skiadas (1999), Schroder and Skiadas (2003) and Schroder and Skiadas (2005) develop the related ‘utility gradient approach’ of Cox and Huang (1989), He and Pearson (1991) and Duffie and Skiadas (1994). Though starting from a different angle, this approach is closely related to the robust control problem. In incomplete markets, the stochastic discount factor (SDF) that supports prices is not unique. The max-min problem of finding the SDF that minimizes the maximum attainable utility over all SDFs, is equivalent ‘in some sense’ to the robust control problem of maximising utility in the face of a set of disturbances. However this paper, to our knowledge, is the first to use these approaches to derive explicit solutions to the discrete time dynamic portfolio construction problem with trading costs. We believe the discrete-time form is particularly suited to investment applications, though the expressions can be more involved. To rephrase the dynamic portfolio problem within a state-space framework, we pay particular attention on how to model both transaction costs and portfolio constraints. Once the problem is in this form, we are able to derive an explicit solution to the general dynamic portfolio problem over finite horizon using only a minor extension to the results in Whittle (1990). We discuss the limit of this solution as the investment horizon tends towards infinity. The optimal portfolio will be a weighted combination of a mean variance portfolio and a number of hedging portfolios. For two special cases, when all transaction costs are zero (so there are only the classical Merton intertermporal hedging demands), and when returns are uncorrelated with innovations to future expected returns (so there are only hedgeing demands induced be the serial correlation of trading) we derive an expression for these hedging portfolios. In the general case, the expressions are more involved. The dynamic portfolio problem was first studied by both Samuelson (1969), Merton (1969)and Merton (1971). They gave conditions for when dynamic optimal portfolio was identical to the single period (or myopic) optimal portfolio. These conditions are: firstly, that investors preferences are described by a power utility function — implying portfolio

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allocations are independent of wealth; and secondly that returns are independently distributed over time. Since these seminal papers, there has been an immense research effort to relax some of these assumptions. Attempts to generalise the form of investors preferences have generally concentrated on solving the problems for the more general set of Epstein-Zin preferences. This work is very elegantly summarised in Brandt (Forthcoming). Epstein-Zin preferences separate out an investor’s appetite for risk from their motive to smooth consumption over time. Campbell and Viceira (2002) solve a dynamic portfolio problem over an infinite horizon assuming these preferences. They demonstrate that the optimal portfolio allocations are very sensitive to the investor’s risk appetite, but only weakly related to their desire to smooth consumption. Of more direct relevance, to this paper, is the body of work investigating the asset allocation decision when the investment opportunities vary over time. This work has generally set up an asset allocation problem with a few well-chosen assets and solved the dynamic portfolio problem numerically, Brennan, Schwartz, and Lagnado (1997) is an early example of this work. They derive an expression for the optimal portfolio in continuous time, and solve the resultant PDE using finite difference methods. More commonly, the approach adopted has been to solve the associated dynamic programming problem directly. General approaches to solving these problems are discussed in detail in Judd (1998) and Carroll (2008). Examples of the use of these techniques applied to the problem of dynamic portfolio optimisation are Lynch and Balduzzi (2002), Barberis (2000) and Dammon, Spatt, and Zhang (2001). These approaches all suffer from ‘the curse of dimensionality’ in that the complexity of the problem increases with the power of the number of states. To circumnavigate this problem, Rust (1997) suggested using a simulation approach. In a recent series of papers, Brandt, Goyal, Santa-Clara, and Stroud (2005), van Binsbergen and Brand (2007) and Koijen and Werker (2007) have applied this idea to the dynamic portfolio optimisation. More recently, Sangvinatsos and Wachter (2005) and Cvitanic, Lazrak, Martellini, and Zapatero (2006) have used the ’utility gradient approach’ of Schroder and Skiadas (2003) to derive a explicits solution in continuous time to dynamic portfolio problems. However, the approaches closest to this paper are Campbell and Viceira (2002) and Sangvinatsos and Wachter (2005). Campbell and Viceira (2002) derive an explicit solution to an asset allocation problem over an infinite horizon. This paper builds on their work. It generalise the work to the study of finite horizon problems in discrete time. It also shows how the solution to these asset allocation problems relate to the solution of a risk-sensitive Riccatti equation, making the approach viable even for a large number of assets. Finally it incoporates a model of trading costs within the framework. The costs of trading equity are often broken down into two components. Firstly, there are the real costs of trading - often further divided into order processing, monopoly rents and inventory costs. Secondly, there are the informational or implicit costs of trading; often referred to as the price impact of the trade. Freyre-Sanders, Guobuzaite, and Byrne (2004), Stoll (2000) are recent summaries of the large body of research on modelling these trading costs and frictions. However, as Hasbrouck (2007) and Almgren (2008) discuss, technological innovation is driving changes to the organisation of equity markets. Firstly, exchange markets are becoming more order and less quote driven; and secondly there is a gradual increase in the percentage of trades executed on crossing networks or electronic communication networks (ECNs) relative to exchange floors. These changes have not only driven down total costs, Huang and Stoll (1997), Domowitz, Glen, and Madhavan (2001) and Cooper, Gutierrez Jr., and Marcum (2005). They have also effected effected the relative magnitude of the different cost components with greater competition driving down 3

real costs relative to the implicit ones, see Domowitz, Glen, and Madhavan (2001) and the Trade Reports in ?. This, perhaps, explains the slight change in focus of the research work on incorporating trading costs into the dynamic portfolio optimisation problem. Earlier work on portfolio optimization in the presence of transaction costs, such as Kyle (1985), Demsetz (1968) and O’Hara and Oldfield (1986), assumed costs were a constant proportion of the value traded; in effect ignoring the informational costs. In contrast, the more recent literature has focused on the price impact of a trade; for asMadhaven (2000) notes, ‘while most researchers recognize that quoted spreads are small, implicit trading costs can actually be economically signicant because large trades move prices’. Lo and Bertsimas (1998), Bertsima, Lo, and Hummel (1999), Almgren and Chriss (2000) and Engle and Ferstenberg (2006) are all examples of this work. They investigate the optimal execution of a large trade, often called the pre-trade analytics. As Engle and Ferstenberg (2006) stress, the optimal execution problem can be viewed as trade-off between risk and return. Executing the trade in one large block increases costs, whereas breaking down the trade and executing it over time as series of smaller trades reduces the impact cost but increases the variance of the (average) execution price. Though these papers do discuss non-linear specifications, their analytical results are for a linear specification of the price impact function. However, Huberman and Stanzl (2004) show that when the price impact of a trade is permanent, only linear price-impact functions rule out quasi-arbitrage and thus support viable market prices. This paper also assumes that the price impact of a trade is linear in the size of the trade. However by embedding, what Engle and Ferstenberg (2006) refer to as AlmgrenChriss dynamics, it differentiates between the temporary and permanent price impact of a trade; the temporary impact being the immediate price impact of the trade, and the permanent being the long run impact1 . This paper contributes to this literature in a number of ways. Firstly, investors’ actions are consistent with CRRA preferences defined on end-ofperiod wealth net of transaction costs. Secondly, the approach allows for any intertemporal correlation in returns induced by the trading. Finally, it embeds the execution problem as part of a tactical asset allocation problem. This is useful as implicit costs are economically large enough to reduce the notional or paper gains from an investment strategy, see the discussion in Madhaven (2000). Hence the optimal combination of assets within a dynamic strategy is likely to be materially affected by the costs of trading these assets; the faster the turnover of the strategy, the more crititcal these costs will become. The structure of the paper is as follows; in section 2, we argue that costs to trading induce serial correlation between net portfolio returns at time t and future expected returns. Thus trading costs will generate a demand for intertemporal hedging portfolios in a very similar way to the Merton’s classical intertemporal CAPM. Of the previous papers looking at portfolio construction in the presence of linear trading costs, only Engle and Ferstenberg (2006) mention these intertemporal hedging demands; they do not, however, solve the problem in this case. In section 3 we set up the general dynamic portfolio problem. Section 4 states the solution to the general dynamic portfolio problem and discusses the limit of this solution 1

In the market micromarket literature, the permanent price impact of the trade is associated with the impounding of any infomation in the trade into prices. The temporary impact is associated with the increase in demand for liquidty which dissapates over time.

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as the time horizon tends towards infinity. In section 5, we use the solution technique to investigate two intertemporal problems. In the first one, we derive an explicit solution to the strategic asset allocation problem discussed in Barberis (2000). We also derive expressions for the welfare loss (in return units) to pursuing a myopic portfolio optimisation approach relative to the full dynamic one. In the second example we look at the tactical asset allocation problem of combining two investment strategies of different half-lives in the presence of transactions costs. We show that optimal dynamic portfolio is equivalent to a mean-variance portfolio on a smoothed forecast of returns. The final setion concludes and discusses how the approach can be extended to problems that involve signal extraction or learning as well as portfolio construction.

2

Time Diversification of Risk

In Merton’s intertemporal CAPM, expected returns are time varying. In any period t, if asset returns are correlated to the innovation to expected returns, then asset returns will also be correlated over time. It is this correlation which induces the intertemporal hedging demands. Trading costs will also induce serial correlation in returns. An innovation to expected returns, will cause an investor to trade his portfolio, buying into the assets with a higher future expected returns (and selling assets with lower future expected returns). However, if trading costs are significant, the act of buying those assets will impact on the returns to those assets today. Thus trading induces serial correlation in asset returns. In this respect both problems are similar; they both create a demand for intertemporal hedging portfolios. We now illustrate this idea using a simplified two period model. It also serves to introduce the notation. All functional forms are chosen for simplicity; a full discussion of the model is given in section 3. Denote the log return to n assets over the period t to t+1 by the n-vector rt+1 , and the expected return to these assets conditional on all information available at time t − k as Et−k (rt+1 ) = rˆt+1|t−k

for all k ≥ 0

for short. Let the time evolution of the conditional expected return be described by the AR(1) equation rt+1|t + η t+1 (1) rˆt+2|t+1 = Aˆ where A ∈ Rn×n is referred to as the state transistion matrix. Equation (1) is alternatively referred to as either the state transition equation or forecasting equation depending on the circumstances. It is assumed the innovation ηt+1 is normally distributed with zero mean and covariance matrix Σηη . For the moment, we shall assume a simple portfolio construction rule. The investor choses portfolio wt at time t to be a linear function of next period expected returns wt = K rˆt+1|t

(2)

where K ∈ Rn×n . In this two period model, the portfolio is rebalanced once at the end of period t + 1. We shall model the price impact of this trade as a linear function of the change in portfolio positions, δ 0 ∆wt+1 , where ∆wt+1 := wt+1 − wt and δ 0 ∈ Rn×n is the

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multiplier. Returns over the period t to t + 1are equal to expected returns plus any the price impact due to trading plus an innovation, εt+1 , rt+1 = rˆt+1|t + δ 0 ∆wt+1 + εt+1

(3)

where εt+1 is normally distributed with zero mean and covariance matrix Σεε . We shall refer to equation (3) as the output equation. We note that the price impact of trade is uncertain. This uncertainty is subsumed within the innovation εt+1 and is not modeled separately. Given this set up, we can write down the returns to our portfolios in each of the two periods in terms of variables known at time t and the innovations. The return in period t + 1 is
 T T wt rt+1 = rˆt+1|t K T (I + δ 0 K (A − I)) rˆt+1|t + δ 0 Kηt+1 + εt+1 and in period t + 2 is


T
 T rt+2 = Aˆ rt+1|t + ηt+1 K T Aˆ rt+1|t + ηt+1 + εt+2 wt+1

The return in the second period is a function of the innovation to expected returns, ηt+1 , in the first period. Hence the returns in the two periods are correlated if: 1. Cov(η t+1 , εt+1 )= Σηε .=0; if innovations to future expected returns, ηt+1 , are correlated with return innovations εt+1 , portfolio returns across periods are correlated. It is this correlation that induces Merton’s original intertemporal hedging motives. 2. Σηη = 0 and δ 0 = 0; if investment opportunities are time-varying, transaction costs will also induce correlation across periods in portfolio net returns. Our dynamic portfolio construction problem can be understood as maximizing a multiperiod mean-variance type objective in the presence of correlation in portfolio returns over time.

3

The Dynamic Optimisation Problem

In this section we set up the multi-period portfolio optimisation problem and demonstrate that it can be recast as a risk-sensitive control problem. We then discuss our model of both implicit and explicit transaction costs, and in particular how this model relates to the recent work of Engle and Ferstenberg (2006) and Almgren and Chriss (2000). Finally we set up the general state-space model, which embeds the transaction cost model with both the forecasting and output equations. Assume a universe of n securities, and denote the return to these securities between time t and time t + 1 - which we shall refer to as period t + 1 - by the n-vector Rt+1 . We shall assume Rt+1 is lognormally distributed and denote the log returns by the lower case rt+1 , i.e. rt+1 = log(Rt+1 ). Our investor chooses a portfolio of these assets at the beginning of each period. We shall denote his portfolio at time t by the n−vector of weights
wt . Therefore if  his wealth at time t is Wt , his wealth at time t + 1 is given by Wt+1 = Wt wtT Rt+1 ; where ·T denotes the transpose of a matrix or vector. Our investor forms his expectations of future returns, Rt+1 , based on his available information a time t. This information set includes, as well as all past return information, data on other financial and economic variables. We shall refer to this data as the conditioning variables, and denote the values of these varaiables at time t by the vector st . 6

The dynamic portfolio problem is to choose the portfolio weights, wt , in each period so as to maximise a given performance criterion over the investment horizon, t = 0, 1..., T . Throughout this paper, we shall assume that the dynamic process describing the evolution of returns is known to the investor and does not vary with time. However it would be a trivial generalisation to allow the process to vary as long as this variation is pre-determined and not stochastic. It would amount to little more than indexing the relevant matrices by t. We do not consider the problem described in Barberis (2000), where the investor knows the form of the return generating process but must learn about the parameter values from observed returns. However we do discuss in the conclusion how this approach can be generalised to investigate this problem too.

3.1

The Optimisation Criterion

Our investor chooses his portfolios over time so as to maximize a CRRA function of his terminal wealth,   Ut = Et WT1−γ (4) P

Writing the compound return to the portfolios over the entire investment period as Rp = er , then log of utility is

 (1 − γ) 1 log Ut = log (Wt ) + E rP + V ar rP + ..... 1−γ 2

(5)

where the higher order terms are the higher order cumulants2 of rp . So if RP is lognormally distributed the higher order terms are zero and the objective function is equivalent a meanvariance criterion. If γ > 1 the investor is risk averse with higher values of the parameter associated with greater risk aversion. With γ = 1, the investor is risk neutral and behaves so as to maximise the log of terminal wealth- the Kelly Criterion. Finally if γ < 1, the investor is risk-seeking. In this paper we shall assume that γ ≥ 1. Substituting into equation (4) the portfolio returns for each period, the objective can be rewritten as  1−γ  T  T  wi−1 eri Ut = Wt1−γ Et  (6) i=t+1

We now use Campbell, Chan and Viceira (2001, Appendix A) log-linear approximation for the portfolio returns in any period. If we denote Cov(rt ) = Σ and the diagonal elements of the covariance matrix by the n-vector σ2 = diag(Σ), equation (6) can be well approximated by  1−γ  T  1 T T 2  Ut ≈ Wt1−γ Et  ewi−1 ri + 2 wi−1 (σ −Σwi−1 ) 2

i=t+1  1 T T 2 (1−γ) T i=t+1 wi−1 ri + 2 wi−1 (σ −Σwi−1 )

 = Wt1−γ Et e

(7) (8)


 The cumulant generating function of a random variable x is the function K(t) = log E etx = tκ1 +

t2 κ 2! 2

+

t3 κ 3! 3

+ ... . The term κi is referred to as the ith cumulant of x.

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In the limit, as the time interval becomes shorter, the approximation holds exactly. Alternatively, if there is costless trading and assets are traded continuouly then relationship also holds exactly. The second term in the exponential is often referred to as the excess growth rate in stochastic portfolio theory, an overview is Karatzas and Fernholz (2009). We have therefore rewritten our objective as the expected value of the exponential of the sum of period cost functions. We shall refer to these period cost functions as Ci where  1 T
2 T ri + wi−1 σ − Σwi−1 (9) Ci := wi−1 2 which are quadratic in the control or choice variables wi and normally distributed random variables ri+1 . Maximizing an objective function of this form is termed a risk-sensitive control problem in the literature, Whittle (1990). Whittle derived an analytical solution to this class of control problems.

3.2

Modelling Transaction Costs

Preserving analytical tractability places considerable restrictions on how general we can make the model of trading costs. However, we attempt a stylised model of trading costs along the lines of Engle and Ferstenberg (2006) and Almgren and Chriss (2000). Though inevitably schematic, these models do capture some essential features of trading costs. On trading a security, the price tends to move against a trade; the larger the trade, the greater the likely price movement. However, only a percentage of this impact tends to be permanent and the price is likely to revert partially after the trade. Modelling these implicit trading costs or ‘slippage’ is active area of research. A recent review of the literature is Freyre-Sanders, Guobuzaite, and Byrne (2004). In order to model the costs, they are broken down into two components; the real costs which include order processing and inventory costs and the informational costs. For small trades, the order processing costs are the most significant. However as trade size increases the informational and inventory costs become ever more significant. As we shall focus on these two elements of costs, our model is best understood as a representation of the cost of trading a sizeable portfolio. The initial rise in the price can explained by models of both adverse selection and inventory control. Given information asymmetry in the market, an investor wishing to trade may have private information. The movement in price during the trade reflects the market pricing in this information. This effect will be permanent. However, the trade may also require a liquidity provider to take a significant temporary position in the stock. To compensate them for this extra risk, they are likely to widen their spreads increasing further the price impact of the trade. In contrast, this effect is likely to be temporary, as spreads will revert once liquidity providers have restored their positions to their preferred levels. Figure 1 is a schematic representation of our model of transaction costs. At time t + 1, the investor observes returns rt+1 and the value of any conditioning variables st+1 . Based on this information, he may decide to rebalance his portfolio from wt to wt+1 . This trade instantaneously moves the price in the direction of the trade by a factor δ 0 ∆wt+1 , where δ 0 ∈ ℜn×n . Though this specification does allow for cross effects, they are likely to be small. A period later, at time t + 2, the prices partially revert by a factor −δ 1 ∆wt+1 ,where δ 1 ∈ ℜn×n . Thus at any time t+1 the observed returns to the n securities after price impact is

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Time

t

t+1

t+2

T-1

T

Wealth

Wt

Wt+1

W t+2

WT-1

WT

Portfolio weights wt

wt+1

w t+2

wT-1

wT

Stock returns Portfolio returns

rt+1

rt+2

rT

wt rt+1

wt+1rt+2

wT-1rT

Temporary impact

Price Impact due change in portfolio at time T+1: ∆wT+1

Impact on returns due to change in portfolio at time T+1: ∆wT+1

Permanent impact

δ0∆wt+1 δ1∆wt+1

Figure 1: Timing Diagram of the permanent and transistory components of a trade.

rt+1 = rˆt+1|t + εt + δ 0 ∆wt+1 − δ 1 ∆wt

(10)

Thus if the investor wishes to increases his position in a security, the price moves against him whilst trading into the security. In the ensuing period, the price partially reverts and so returns are lower than otherwise. The permanent price impact of the trade is (δ 0 − δ 1 )∆wt+1 and therefore captures the informational costs, the term δ 1 ∆wt models any inventory costs. The technical challenge is to solve the portfolio construction in the presence of the instantaneous price impact δ 0 ∆wt+1 . In the terminology of the control literature, it transforms the control problem from a state feedback to a full information problem. We will discuss this in detail later when we outline the solution. Remark: Equivalence of our model of transaction costs to that of Engle and Ferstenberg (2006) and Almgren and Chriss (2000). Their model is expressed in prices or dollar terms rather than returns. We use their notation for the scope of this remark. These authors differentatiate between the ‘fair value’ price for the n assets at time t, denoted by p t ( this can be understood as the mid-point between the bid-ask spread) and the transaction or trade price pt 3 . Their transaction cost model is then defined by the two equations pt − p t = S∆xt ∆ pt = L∆xt + µ + εt

(11) (12)

where ∆xt is the value change in the portfolio weights or shares, µ is the conditional expected price return (µ/ pt is rˆt|t−1 in our notation), S is the temporary or short run market impact 3 Actually in their notation the notation is pt is the ‘fair price’ and p t is the trade price. We have reversed this notation, for consistency with the rest of the paper.

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of the trade, and L is the permanent or long run price impact. As they point out, Huberman and Stanzl (2004) implies that the permanent effect must be linear and time-invariant to avoid arbitrage opportunities. These equations can be rearranged to imply ∆pt+1 = ∆ pt+1 + S∆xt+1 − S∆xt = µ + εt + (L + S) ∆xt+1 − S∆xt

(13) (14)

If we divide this equation through by the traded price, pt , so as to write the relationship in returns rather than dollar terms, the equation expresses the same relationship as equation 10. The temporary impact matrix, S, is associated with the matrix δ1 (scaled by prices) and the permanent impact matrix L with a scaled version of the matrix (δ 0 − δ 1 ).


Remark: On the assumption that the price impact is proportional to the size of the trade. Some work on portfolio optimization in the presence of transaction costs model the price impact as a signed constant - with the constant being postive for buys and negative for sells - and being calibrated to be a proportion of the observed bid-ask spread (this model could be considered as focussing predominatly on order processing costs). Atkinson and Mokkhavesa (2004) and Leland (2000) are recent examples of this work. The solution to these trading problems have a common form. Around the desired portfolio allocation, there are ’no-trade’ zones. As long as the portfolio allocation remains within the confines of these zones, it is optimal not to trade. If the allocation moves outside, then the minimum size trade is executed to bring it back to the edge of the zone. However calculating these no-trade zone is numerically intensive even in simple problems. In contrast, Lo and Bertsimas (1998), Bertsima, Lo, and Hummel (1999), Almgren and Chriss (2000) and Engle and Ferstenberg (2006) all model the price impact as proportional to the size of the trade. Though undoubtably part of the motivation is for reasons of tractability, it is also motivated by the observation of Madhaven (2000) that ‘while most researchers recognize that quoted spreads are small, implicit trading costs can actually be economically signicant because large trades move prices’. To illustrate this idea, split the single trade ∆xt (in the previous remark) into n equally sized smaller trades to be executed at times t ,t + 1, . . . , t + n − 1. Now rewrite equation (11) as pt − p t = S.sgn(∆xt ) ∆ pt+1 = αS.sgn(∆xt ) + εt

(15) (16)

where sgn() is the sign function. In this reformulation, S can be seen as half the bid-ask spread. If α = 0 there is no permanent impact on the price from the trade; for 1 ≥ α > 0 there is some information transfer and the mid or ‘fair’ price moves with the trade. The average trade price for the block trade ∆xt is the just average price of the n individual trades   i=n−1 i=n−1  (n − 1) 1  pt+i − p t = 1 + εt+i . (17) α S.sgn(∆xt ) + n 2 i=0

i=0

If α > 0, then for large trades, i.e. large n, the second term on the right- hand side dominates. In this case, the price impact is roughly proportional to the size of the trade and the cost of the trade will be signifcantly larger. Which of these two models is a better model of trading costs will depend both on the size of the portfolio and what proportion of trading costs are informational. Our model is 10

more likely to be the better model for large portfolios (i.e. large n) and when informational costs are significant (i.e. large α). What proportional of trading costs are informational? Researchers have tried to estimate this in a number of ways. The predominant approach is to try estimate the proportion of the price change following a trade that is permanent, Huang and Stoll (1997), Stoll (2000) are recent examples. On intra-day data Stoll (2000) present a number of estimates of this proportion that range from a high of 46.2% to a low of 0%. Similalry Huang and Stoll (1997) on daily data presents estimates again in the range from a high of 45% to a low of 5%4 . The likely reason for the wide range is how the different approaches account for the way large trades are generally executed as a number of small trades5 . Almgren, Thum, Hauptmann, and Li (2005) is the only published work based on a trade database that explcitly records how large trades were broken down and executed. Their model is non-linear and depends explictly on the immediacy of the execution, so it is not possible to quote a single estimate for informational costs. But for trades that are roughly 10% of daily volumes and executed smoothly over half a day, they find that information costs account for just about 50% of total costs. For larger trades or when less immediacy is required, the proportion is higher still.
We have kept the implicit trading cost model simple, by not introducing any extra dynamics. Though the immediate price impact δ0 ∆wt+1 must always be instantaneous, one could easily introduce dynamics into the reversion process. For example if we define n new state variables, dt by the difference equations, dt+1 = Ad dt + (I − Ad )δ 1 ∆wt+1

(18)

rt+1 = rˆt+1|t + εt + δ 0 ∆wt+1 − dt

(19)

where Ad ∈ ℜn×n is any stable matrix6 , then we can amend equation (??) as The long run impact on prices remains (δ 0 − δ 1 )∆wt+1 but now the impact is spread over a number of periods. The closer the eigenvalues of Ad are to 1, the slower the reversion process. If A = 0 then the system collapses to equation 10. In this paper, we stick with the specification 10 as it does not require inflating the state space, but recognise that 18 and 19 maybe a closer representation of observed market dynamics. To complete our model of transactions costs, we also incorporate a term to represent any explicit costs of trading. This term includes any explicit commission or tax charges (such as Stamp Duty in the UK) for trading. As an explicit trading charge, it is a charge on trading but has no influence on the returns to the assets. The return to the portfolio gross of explicit costs in period t is given by equation 9. We modify this equation to include explicit trading costs. The period cost function, Ci , now refers to log portfolio returns net of explicit costs, where

4

 1 T
2 T Ci := wi−1 ri + wi−1 σ − Σwi−1 − ∆wiT T ∆wi 2

(20)

On the NYSE exchange, Stoll (2000) estimates the effective spread to be 5.56 cents and the traded spread to be 3.1 cents, thus the informational proportion of costs is (5.56-3.1)/5.56= 45%. 5 Huang and Stoll (1997) make this point explicitly. In Stoll (2000), his first approach based on the difference between the traded and effective spreads implicitly adjusts for large trades. This approach delivers the high estimates. In contrast the results from the impact regression treats large and small trades equivalently. It delivers a far lower estimates. 6 By stable we refer to the condition that all its eigenvalues are less than 1.

11

and T ∈ ℜn×n is a symmetric, non-negative definite matrix. This quadratic representation of explcit costs ensures that costs are always positive.

3.3

Portfolio Investment Constraints and Rebalancing

In each period, wealth must be fully invested in the portfolio of assets, i.e. the portfolio postitions sum to 1. In addittion, there may be other investment restrictions imposed on the portfolio too. These may be imposed so as to limit risk exposures, for example it might be required that the portfolio positions at some level of aggregation be market neutral. We shall sssume there are p ≥ 1 independent restrictions; the first of these being the fully-invested constraint. Given that these restrictions are enforced throughout the investment horizon, we shall represent them as, P T wt = c

(21)

and P ∈ ℜn×p and c ∈ ℜp . Without loss of generaility, we shall assume the columns of P are orthogonal7 . Then there exists a n × (n − p) unitary dilation matrix P ⊥ that satisfies the equation T    P P ⊥ = In P P⊥ This parametrisation implies the portfolio weights can be written as    c  ⊥ wt = P P ut

(22)

where ut is a (n − p)− vector. Given the constraints (21), the investor only has the freedom to choose weights within the (n − p) dimensional space spanned by the columns of P ⊥ . We have parameterised this choice as ut . We now consider the portfolio rebalancing equation. At the beginning of every period the investor chooses portfolio weights that satisfy the constraints. Even if there is no active trading, positions will change over any period due to the returns in that period. Given no trading, portfolios positions next period are related to positions in the previous period as, wt+1 = wt + wt .

(rt+1 − rp,t+1 ) (1 + rp,t+1 )

where ‘.’ denotes element-by-element multiplication of the vectors and rp,t+1 denotes the total return to the portfolio, rp,t+1 = wtT rt+1 . Hence simply maintaining positions constant over time will require some trading. However, the costs incured by this induced trading are generally small; especially when considered as a percentage of total trading costs for an actively managed equity fund. As this induced turnover is non-linear in the states, it is not possible to model them in our quasi- linear framework. We shall therefore assume that, unless there is some active trading, positions remain constant over time wt+1 = wt . This assumption effectivly assumes that the portfolio can be costlessly rebalanced back to the postions in the previous period; and is equivalent to ignoring the costs from induced 7

The requirement that the columns of P are orthogonal is only a normalisation that will prove useful later. It does not place any restriction on the form of the constraints. For if the constraints are written in ˜ 1/2 be a Cholesky factor of P˜ T P˜ , then the matrices P˜ ˜T a non normalised   form −1P wt = c˜ , let the matrix P −1 1/2 1/2 P˜ and P˜ c˜ satisfy the normalised equation .

12

trading. The approximation therefore only effects the modelling of transaction costs, and plays no role, for example, in Section 4.2 when we look at the dynamic portfolio problem in the absense of trading costs. Given that we have assumed portfolio postions remain constant in the absence of active trading, and that the constraints in equation (21) hold in every period, then the change in portfolio postions is ∆wt = P ⊥ ∆ut . In a minor abuse of notation, we shall let the investor’s choice variable be denoted by ∆ut+1 too and define ut+1 = ut + ∆ut+1 .

(23)

with the portfolio held during any period given by the equation (22). We have simplified the problem by assuming the same constraints are imposed in every period. One could relax this assumption and and allow the constraints to vary over time. This would of course induce some turnover in the portfolio weights but would not alter the nature of the problem. Far more difficult would be to allow inequality rather than equality constraints. In this case, it would only be possible to preserve the analytical tractability of the solution procedure if the active set of constraints at any point in time was independent of the state-space. This is unlikely, however, to be the case in all but the most trivial examples.

3.4

The State-Space Model

We are now in a position to write down the state transition equation or return forecasting equation. The ns states of the forecasting model at time t are denoted st ; these states are the conditioning variables of asset returns. For example in Brennan, Schwartz, and Lagnado (1997) and Campbell and Viceira (2002), these states include the dividend yield, the long and short rates of interest and lagged excess returns to stocks; whereas the dividend yield is the only state in Barberis (2000). The linear dynamic process describing the evolution of these states is, st+1 = As st + Bs η t+1 + µs

(24)

where the innovation ηt+1 is normally distributed with zero mean and covariance matrix Σηη . Gross expected returns in the next period, i.e. gross of transaction costs, are a linear function of these conditioning variables, rˆt+1|t = Cs st + κs

(25)

Gross realised returns are equal toexpected  returns plus the innovation, ǫt where the coηt+1 is variance of the innovations ξ t+1 = εt+1
 Cov ξ t+1 := Cov



η t+1 εt+1



=



Σηη Σηε ΣTηε Σεε



=: Σ

Hence, if Σηǫ = 0 then the innovations to investment opportunities are correlated with return innovations; it is this correlation that will induces Merton’s intertemporal hedging motives. 13

In order to model transaction costs, we need to augment the state-space in equation (24 with both the changes to and the level of the portfolio control variables ut , 

           st+1 0 µs As 0 0 st Bs 0   ut+1  =  0 I 0   ut + 0 0  η t+1 + I  ∆ut+1 + 0  (26) εt+1 ∆ut+1 ∆ut I 0 0 0 0 0 0

We shall call this equation the state-transition equation.

The output equation maps the states and control variables onto the individual variables that appear in the within period cost function Ci . The output equation can be written as        st rt+1 Cs 0 −δ 1 P ⊥ 0 I  η t+1 ⊥   ut  +  0 0   wt  =  0 P 0 + εt+1 0 0 0 ∆wt+1 ∆ut 0 0     ⊥ δ0 P κs  0  ∆ut+1 +  P c  0 P⊥ 

(27)

By augmenting the state-space with the change in portfolio positions ∆ut , it is possible to make returns rt+1 net of both the instanteous trading costs, δ 0 P ⊥ ∆ut+1 , and any price reversion due to the trade in the previous period, −δ 1 P ⊥ ∆ut . The within period cost function Ct+1 in equation (9) is a quadratic function of the output variables in equation (27),

Ct+1



 T     T  1 rt+1 0 rt+1 rt+1 0 0 2I =  wt   12 I − 12 Σεε 0   wt  − 2  − 14 σ2   wt  (28) 0 0 −T ∆wt+1 ∆wt+1 0 ∆wt+1

These equations describe the system. However, need to introduce a more compact notation. To this end will define the states of the system, xt , and the outputs, zt , where     st rt+1 xt =  ut  zt =  wt  ∆ut ∆wt+1 respectively. The equations (26) and (27) can now be written more succinctly as xt+1 = Axt + B1 ξ t+1 + B2 ∆ut+1 + µ zt = Cxt + D1 ξ t+1 + D2 ∆ut+1 + κ

(29)

by making the obvious definitions. Similarly the cost function (28) can be written Ct+1 = ztT Rzt − 2S T zt . by again making the obvious definitions

14

(30)

3.5

The Optimisation Problem

We now state the full dynamic portfolio optimisation problem. The investor, at time t, observes both the states and the innovations. Thus the information, yt , available at time t is     xt−1 xt−1   yt = ηt (31) =: ξt εt

As the investor observes, both the states, xt−1 , and the innovations, ξ t , he effectively observes the expected returns for period t + 1 and gross realised returns at time t. Based on this information and past signals, Yt = {yt , yt−1 , yt−2 , . . . .} - he rebalances his portfolio subject to any constraints (i.e. he chooses ∆ut ). Net returns are then equal to the observed realised returns minus any costs resulting from this rebalancing. The purpose of describing the system in this precise form, stems from this desire that costs should be instantaneously expressed in net returns.

If we denote the class of all such feasible strategies by the set π, the dynamic portfolio construction problem is to find the strategy that maximises the CRRA function of end of period wealth   T (32) Vt = max Et e(1−γ) i=t+1 Ci π

As far as possible we have stated the problem as a risk sensitive control problem. However there are a number of differences to the problem as stated in Whittle (1992). The most critical of these is that the portfolio rebalancing rule at time t is allowed to be function of the innovations ξ t . This is necessary if we are to allow for the instantaneous price impact of the trade. Inoring this particular aspect of trading costs, would enable us to write the problem in the form of Whittle (1992), as a state-feedback problem. This generalisation, as we shall shortly show, does not change the general approach to solving for the optimal strategy but does change the form of the solution significantly. It is an example of the discrete-time full information control problem discussed in Iglesisas and Glover (1991). Other minor generalisations to the work of Whittle (1992) are that we do not impose the conditions that either S = 0 or D1 = 0. The former is necessary is we are to model the log-linear approximation term wtT σ2 and the latter if we are to model either instantaneous implicit or explicit trading costs. Finally we do not require our cost matrix R to be negative definite, though this has little impact on the solution over finite periods.

3.6

Some Remarks on the Problem Set Up

There are many examples in the academic literature that investigate return predicability using a VAR representation as their forecasting model; Brennan, Schwartz, and Lagnado (1997), Barberis (2000), Campbell and Viceira (2002), Kandel and Stambaugh (1987) and Kandel and Stambaugh (1996) are prominent examples. Of these, Campbell and Viceira (2002) is perhaps the closest in treatment to the one presented here. In Campbell and Viceira (2002), the investor chooses an allocation between nominal T-bills, nominal bonds and an equity index. The states are the short real and nominal bill rates, excess returns to equity and bonds, the dividend-price ratio on equity and a measure of the term spread (roughly 5 year minus 1year yields). They estimate a first order VAR 15

model on both annual and quarterly data. Like Campbell and Shiller (1988) and Fama and French (1988) and many authors since, they find that the dividend-price predicts future equity returns. They also estimate that the innovations to the dividend-price are negatively correlated (-0.733) to innovations to equity. The first of these findings implies that the share of equity in the optimal portfolio will be a function of the current dividend-price ratio. The second implies that the risk-averse investor will have a significant hedging motive. Barberis (2000), in a slightly earlier paper, looked at a similar but simpler model. His states are just the equity excess return and the dividend-price ratio. He also finds similar levels of predicability and a srong negative correlation in the innovations. Later in this paper, we will derive an analytical solution to his model. Practioneers have focussed more on forecasting the cross-sectional return of assets. This approach to active management is explained in Grinold and Kahn (1999). Recently among practioneers, there has been more interest in the dynamic problem, particularly focussing on transaction costs; e.g. Grinold (2006). Here the problem can be understood as how to combine forecasts of stock returns over different horizons. For example, a forecast based on the book-price ratio of individual stocks evolves fairly slowly over time, whereas a momentum forecast based on past recent returns is likely to evolve relatively faster. If trading costs are significant, postitions taken on the basis of the two forecasts will depend on the relative horizon of the forecasts. We illustrate this idea in the second example of the paper. In these models, the states are not observed variables such as returns as in Campbell and Viceira (2002), but the conditional forecasted return to the assets rˆt+1|t . In the next section, we show that the two formulations are related by a state-space co-ordinate transformation.

3.6.1

The choice of the state-space coordinates

One common approach to estimating a forecasting model, is to estimate a Vector AutoRegressive (VAR) model on a sample data set. To illustrate this, assume the following VAR has been estimated on past asset returns and some financial conditioning variables, vt ; this is the case in Campbell and Viceira (2002).          ˜r rt+1 A˜r A˜r,v rt B µ ˜r = + ˜ ηt+1 + (33) vt+1 vt µ ˜v 0 A˜v Bv under the structural assumption that asset returns do not Granger cause the financial variables, vt 8 . In this model the states st are the vector of returns and conditioning variables, st = [rtT vtT ]T . Given this data generating process (DGP), expected asset returns over period t + 1 given information at time t are     rt Et (rt+1 ) = rˆt+1|t = A˜r A˜r,v +µ ˜r . vt We now rewrite equation (33) in terms of transformed states space co-ordinates rˆt+1|t and vt by multiplying through equation (33) on the left by the matrix   A˜r A˜r,v 0 I

and, with a little rearrangement, this gives 8

We only impose this condition as it simplifies the following expressions. It can be relaxed.

16







A˜r A˜r A˜r,v 0 A˜v



rˆt+1|t vt





A˜r A˜r,v 0 I



˜r B ˜v B





 µ ˜r = + ηt+1 + µ ˜v (34) This is an equivalent representation of the forecasting model but now the states are the expected returns rather than the realised returns. An active investment process will usually rˆt+2|t+1 vt+1

I A˜r,v 0 I



be built around an explicit forecasting model. These models forecast next period’s expected return. Hence a VAR model estimated directly on these forecasts will be in the form of (34) rather than (33). However, as we have shown, the two are equivalent through a state-space co-ordinate transformation. 3.6.2

Including price impact dynamics into the system equations

Section 3.2 discussed possible reasons for including a more detailed model of the dynamics of the implicit price impact of trade. Here we simply outline how these dynamics can be included within the problem framework. Augmenting the system to include equations (18) and (19) implies that now equation (26) must be written            µs As 0 0 st Bs 0  st+1 0  ut+1  =  0 I 0   ut + 0 0  ηt+1 +  ∆ut+1 + 0  I εt+1 ⊥ 0 0 0 Ad 0 0 dt+1 dt+1 (I − Ad ) δ 0 P (35) and equation (27) becomes 

           δ0P ⊥ rt+1 st κs Cs 0 −I 0 I   wt  =  0 P ⊥ 0   ut + 0 0  η t+1 + 0  ∆ut+1 + P c  . εt+1 ∆wt+1 dt 0 0 0 0 0 0 P⊥ (36) If there are p investment constraints, this augmentation has inflated the state-space by p dimensions. This is unavoidable unless the following condition, P T Ad δ 0 P ⊥ = 0 is satisfied. This condition requires that a feasible trade, i.e. a trade consistent with the constraint P T wt = c, does not impact prices in the portfolio space that is constrained. If this condition is satisfied then there exists a matrix A˜d such that δ 0 P ⊥ A˜d = Ad δ 0 P ⊥ . The states can then be redefined as 

d˜t+1 = A˜d d˜t + (I − A˜d )∆ut+1 .

(37)

where d˜t now has dimension (n − p). Equations (35) and (36) must be modified in line with this new definition. 3.6.3

Simplifying the problem when transactions costs are ignored

The problem reduces in dimension if the some of the transaction costs can be ignored. If we assume that any price impact is permanent, i.e. δ1 = 0, then some of the states (those related to the change in the portfolio weights, ∆ut ) become unobservable. As these unobservable states are stable, the state space can be reduced to xt = [sTt uTt ]T . The statetransition equation can now be written as

17



st+1 ut+1



=



As 0 0 I



st ut



and the output equation as



      Bs 0  µs 0 ηt+1 + 0 0  + ∆ut+1 +  0  I εt+1 0 0 0

(38)

            Cs 0 0 I  rt+1 δ0P ⊥ κs  wt  =  0 P ⊥  st + 0 0  η t+1 + 0  ∆ut+1 + P c  (39) ut εt+1 0 0 ∆wt+1 0 0 P⊥ 0 

If the explicit costs are set to zero, T = 0, then the output vector can be reduced in size and the the output equation (39) can be reduced to 

rt+1 wt



=



Cs 0 0 P⊥



st ut



+



0 I 0 0



ηt+1 εt+1



+



δ0P ⊥ 0



∆ut+1 +



κs Pc



(40)

with the obvious changes being made to the cost matrices R and S, i.e. the relevant rows and columns of zeros are removed. Finally if the permanent price impact is also set to zero δ 0 = 0 then the problem reduces to a state feedback problem. To see this, observe that equations (38) and (40) can be seen as a cascade of two systems. The first is an integrator transforming the inputs ∆ut+1 to the output ut , and the second a reduced state-space problem where the state-transition equation is     ηt+1 st+1 = As st + Bs 0 + µs (41) εt+1

and the output equation is          0 I κs η t+1 0 rt+1 = Cs st + + ut + 0 0 P⊥ wt εt+1 Pc

(42)

One can now solve this reduced state problem directly with the only difference beting that the investor choice variables are ut rather than ∆ut+1 .

4

Solution to the Dynamic Programming Problem

In this section we write down the solution to the dynamic programming problem. We shall first write down the general solution that allows for both intertemporal hedging and portfolio transaction costs simultaneously. We shall then look at two special cases; the first when transaction costs are zero. In this case we shall show that the optimal portfolio can be broken down into two components, the single period optimal mean-variance portfolio plus the intertemporal hedging portfolios. In the second case, we shall assume there zero correlation between the forecast innovations and the return innovations, Σηǫ = 0, and show that again the optimal portfolio has two components. The first is again the mean-variance portfolio and the second can also be seen as hedging portfolio against future expected transaction costs. We defined the value function in equation 32 as   T Vt = max Et e(1−γ) i=t+1 Ci (43) π

18

As far as possible so far, we have stated the problem as a risk sensitive control problem. However it will prove more convenient to work with the following transformation of the 1 log Vt . This transformed value function satisfies the backward value function, Q∗t = 1−γ recursion,
 exp ((1 − γ) Q∗t ) = max Et exp (1 − γ) Ct+1 + (1 − γ) Q∗t+1 ∆ut+1

(44)

In the appendix, we prove that the transformed value function Q∗t is a quadratic function of the states Q∗t = xTt Πt xt − 2ΦTt xt + φt

(45)

where the matrix Πt is non-negative definite. It will prove useful to partition the matrix Πt , and the vector Φt conformally with the state vector xt . Thus we rewrite (45) as T     T   Π11,t Π12,t Π13,t st+1 Φ1,t st+1 st+1 Q∗t =  ut+1   ΠT12,t Π22,t Π23,t   ut+1  −2  Φ2,t   ut+1  +φt (46) ∆ut+1 ΠT13,t ΠT23,t Π33,t ∆ut+1 Φ3,t ∆ut+1 

The following theorem states the main result of the paper. The proof is in the Appendix.

Theorem 1 At time T, the parameters Πt , Φt and φt of the value function in equation (45) are initialised to zero. If we define the following matrices at time t + 1,  T   1   T      B1 D1 ((γ − 1) Σ)−1 0 2 Vt+1 = R D1 D2 + Πt+1 B1 B2 + D2T B2T 0 0   V11,t+1 V12,t+1 (47) = : T V12,t+1 V22,t+1   T     T  K1 B1 D1 −1 (48) = −Vt+1 RC1 + Πt+1 A Kt+1 = K2 D2T B2T   T   T    D1 B1 L1 −1 Lt+1 = = Vt+1 (S − Rκ) + (Φ − Π µ) (49) t+1 t+1 L2 D2T B2T then for t < T , Πt , Φt and φt are determined by the backward recursion. T Πt = AT Πt+1 A + C1T RC1 − Kt+1 Vt+1 Kt+1 T T T Φt = C1 (S − Rκ) + A (Φt+1 − Πt+1 µ) + Kt+1 Vt+1 Lt+1 |2 (γ − 1) Σ| |Vt+1 | 1 log − κT (2S − Rκ) − φt = φt+1 + 2 (γ − 1) |V22,t+1 |

(50)

µT (2Φt+1 − Πt+1 µ) − LTt+1 Vt+1 Lt+1

Further the optimal portfolio allocation at time t can be expressed in terms of these matrices as    ⊥ −1
T 1 ⊥T
T ⊥ wt = P c − P δ 1 + δ 1 + 2T P Π12,t st − Φ2,t − Π22,t − P 2    1 ⊥T
1 ⊥T
T T δ 1 + δ 1 + 2T P c + P δ 1 + δ o + 2T wt−1 P (51) 2 2 19

This theorem proposes an analytical solution to dynamic portfolio construction when an investor as CRRA preferences and returns are lognormally distributed. The Gordian Knot in solving these type of stochastic dynamic programming problems is to calculate the expectation in equation (44). In this case it is possible, because of a number of assumptions. Firstly, we assume that the state equation is linear which implies that the function to be integrated remains quadratic in the innovation. Secondly, because the innovation is lognormally distributed, we are able to integrate the function by a straightforward substitution. We first find the value of the innovation that ‘completes the square’. This innovation has been termed the extremising value of the innovation. Then we shift the mean of the innovation by this value, which removes all cross terms in the control variables, states and the innovation. The subsequent integration is now a simple exercise. The extremising innovation can be interpreted as the worst-case innovation within in a ball of allowable innovations. The radius of this ball is linearly related to the parameter of risk aversion γ - the larger γ the larger the radius of the ball. Thus one can see the risk-averse portfolio problem as the choice of portfolio weights in the face of a ‘worst-case’ innovation. The more risk-averse the investor becomes, the larger the ball from which this ‘worst-case’ innovation can be chosen. Glover and Doyle (1990) were the first to formalise the connection between this risk-sensitive control problem and the solution to a max-min control problem; that of finding the control that maximises performance in the face of a worst case perturbation. In this analysis, the Risk-Sensitive Riccati equation (50) plays a fundamental role. They show that the existence of a stable limit to this equation as t → −∞ is equivalent to the dynamic linear system described by the state equations having an L∞ norm less than γ − 1.

4.1

Two other results

1. In this section we look at two related results. 2. Welfare cost if other optimal portfolio i.e. the mean variance portfolio is used. 3. The long run solution - Campbell and Vicieria - as the Ricatti solution tends to a stable solution. This section is as yest uncompleted. Though a derivation of 2. is in the appendix. We now look at two special cases of the general result. In the first, we assume there are no transaction costs. Therefore the only serial correlation in net returns is due to the correlation between the return and forecast innovations This gives rise to Merton intertemporal hedging motives. We show that in this case the optimal portfolio in 1 can be rewritten as the the single-period mean variance portfolio plus these hedging portfolios.

4.2

Case 1: No Transactions Costs

The equation for the optimal portfolio, equation (51), has a very familiar form. For the scope of this section, we shall assume that all transactions costs are zero, i.e. T = δ 0 = δ 1 = 0, then equation (51) reduces to
T  wt = P c − P ⊥ Π−1 22,t Π12,t st − Φ2,t .

(52)

The term, Π22,t is an adjusted covariance matrix and the term in brackets is the vector of adjusted expected returns. The following corollary makes this clear.

20

Corollary 2 Set all transaction costs to zero,T = δ 0 = δ 1 = 0, and make the following definitions T ΠC,t+1 = Π11,t+1 − Π12,t+1 Π−1 22,t+1 Π12,t+1

ΦC,t+1 = Φ1,t+1 − Π12,t+1 Π−1 22,t+1 Φ2,t+1

and define the transformed covariance matrix as    1 T   T  T Σηη Σηε 0 Bs ΠC,t+1 Bs 0 Bs ΠC,t+1 Bs 0 2 Bs ΠC,t+1 Bs ˜ = Σ + (γ − 1) ΣTηε Σεε 0 Σεε 0 I 0 I   ˜ ˜ ηε Σ Σ (53) = : ˜ ηη T ˜ εε Σηε Σ then we can write the optimal portfolio in equation (51) as either as .

or

.    −1 ⊥ ˜ −1 ˜ ˜ εε − Σ ˜ Tηε Σ wt = P c − P ⊥ P ⊥T Σ Σ P P ⊥T ηη ηε    1 2 ˜ T ˜ −1 ˜ T ˜ −1 T ˜ ˜ rˆt+1|t + σ − Σεε − Σηε Σηη Σηε P c − Σηε Σηη Bs (ΠC,t+1 (Ast + µ) − ΦC,t+1 ) 2

    −1 −1 1 P ⊥T rˆt+1|t + σ2 − γΣεε P c −(γ − 1) P ⊥T γΣεε P ⊥ P ⊥T ΣTηε wtH wt = P c+P ⊥ P ⊥T γΣεε P ⊥ 2 (54) H where the weights, wt , on the ns hedging portfolios are  −1  −1
T  H ⊥ ⊥T ˜ ⊥ ⊥T ˜ T ˜ ˜ P Σηε wt = Bs ΠC,t+1 Bs Σηη − Σηε P P Σεε P      −1 1 2 ˜ T ⊥ ⊥T ˜ ⊥ ⊥T ˜ ˜ Bs (ΠC,t+1 (Ast + µ) − ΦC,t+1 ) − Σηε P rˆt+1|t + σ − Σεε P c − Σηε P c P P Σεε P 2 Hence if either γ = 1 or Σηǫ = 0 then the solution reduces to the standard mean-variance portfolio.

Corollary 1 states that in the case of the no transactions costs, the optimal portfolio can rewritten as equal to the solution to the single period mean-variance problem    −1 1 1 e wtM V = P c − P ⊥ P ⊥T Σεε P ⊥ P ⊥T rt+1|t + σ2 − Σεε P c (55) γ 2 plus a weighted sum of ns hedging portfolios −1 (γ − 1)  ⊥T P Σεε P ⊥ P ⊥T ΣTηε (56) γ These portfolios hedge investors against future changes in the opportunity set. If there is a negative shock η t to future expected returns, then these portfolios will be expected to deliver a positive returns so as to smooth performance over the longer horizon. The higher the investor’s risk aversion the greater the investor’s position in these portfolios. Of note is that that both the mean-variance portfolio and the hedging portfolios are independent of the the horizon of the problem. However the weights held in the hedging portfolios wtH does vary over time. 21

4.3

Case 2: No correlation between forecast and return innovations

In this section, we assume away any intertemporal hedging motives resulting from the correlation forecast and the return innovations, i.e. Σηǫ = 0. Now the only auto-correlation in net returns over time is induced by trading. We show that in a similar way as to the previous section, we can re-express the optimal portfolio as equal to the mean-variance portfolio plus the weighted some of the hedging portfolios. Corollary 3 Assume no correlation between forecast and the return innovations, Σηǫ = 0, and make the following definitions −1 ΠC,t+1 = Π11,t+1 − Π12,t+1 V22,t+1 ΠT12,t+1   1 ⊥T T −1 ΦC,t+1 = Φ1,t+1 − Π12,t+1 V22,t+1 Φ2,t+1 − P (δ 0 − δ 1 ) P c 2

H = (γ − 1) BsT ΠC,t+1 Bs + Σ−1 ηη 
T 1 T˜ = δ + δ1 + T 2 0

Then the matrix V22,t is equal to

V22,t = P ⊥T γΣεε + δ 0 + δT0 + 2T

 −1 −1 −1 P ⊥T T˜ + 2 (γ − 1) T˜ T P ⊥ V22,t+1 ΠT12,t+1 Bs H −1 BsT Π12,t+1 V22,t+1 P ⊥T T˜ P ⊥ 2T˜T P ⊥ V22,t+1

and the optimal portfolio as   −1 ⊥T −1 −1 MV RN H wt = P c+P ⊥ V22,t P + T˜T P ⊥ V22,t+1 rˆt+1|t + (γ − 1) T˜T P ⊥ V22,t+1 ΠT12,t Bs H −1 BsT rˆt+1|t rˆt+1|t (57) where

VM V,t = P ⊥T γΣεε + δ 0 + δ T0 + 2T P ⊥   −1 VRN,t = VtM V + 2P ⊥T T˜T P ⊥ V22,t+1 P ⊥T T˜ P ⊥ −1 −1 VRA,t = VtRN + 2 (γ − 1) P ⊥T T˜T P ⊥ V22,t+1 ΠT12,t+1 Bs H −1 BsT Π12,t+1 V22,t+1 P ⊥T T˜P ⊥

1 MV rˆt+1|t = rˆt+1|t + σ2 + T˜ut−1 − γΣεε P c 2   1 RN rˆt+1|t = ΠT12,t (As st + µs ) − Φ2,t+1 − P ⊥T (δ0 − δ 1 )T P c 2

H = ΠC,t+1 (As st + µs ) − ΦC,t+1 rˆt+1|t

and the optimal portfolio as   −1 −1 −1 MV RN H wt = P c+P ⊥ VRA,t P ⊥T rˆt+1|t + T˜T P ⊥ V22,t+1 rˆt+1|t + (γ − 1) T˜T P ⊥ V22,t+1 ΠT12,t Bs H −1 BsT rˆt+1|t (58) which can be rewritten as wt = wM V,t + wRN,t + wRA,t

22

(59)

where −1 −1 P ⊥T T˜T P ⊥ V22,t+1 ΠT12,t+1 Bs wRA,t = P ⊥ VRN,t  −1 −1 −1 −1 P ⊥T T˜P ⊥ VRN,t P ⊥T T˜T P ⊥ V22,t+1 ΠT12,t+1 Bs H − BsT Π12,t+1 V22,t+1   −1 −1 P ⊥T T˜P ⊥ VRN,t P ⊥T BsT Π12,t+1 V22,t+1

Now we can use the matrix inversion lemma to prove the following identity

 −1 −1

   P ⊥T X − B T Y −1 B P ⊥ P ⊥T x − B T Y −1 y = P ⊥T XP ⊥ P ⊥T x+   −1  −1  −1  −1  ⊥T ⊥ ⊥T T ⊥ ⊥T ⊥ ⊥T T ⊥ ⊥T ⊥ ⊥T P B P B P x−y P XP P XP Y − BP BP P XP and we cope with the constraint matrix by the simple arrangement below

 
 x − X − B T Y −1 B P c − B T Y −1 y = (x − XP c) − B T Y −1 (y − BP c)

(60)

This identity can be used repeatedly to write the expression as equal to the mean-variance portfolio plus hedging portfolios.

5

Two Illustrative Examples

We shall now set up two example problems. Later we will be able to give an analytical solution to both these examples. The first, taken from Barberis (2000), is long horizon asset allocation problem. It abstracts away from any transaction costs, but is good example of how even a small amount serial correlation in returns can induce a large shift in portfolio allocations over long horizons. This problem was solved in Barberis (2000); using numerical techniques we shall derive an analytical solution using the approach described in this paper. The second example is motivated by problems that arise in active portfolio management. In this stylised example, the manager has two forecast signals. The first signal decays relatively quickly, the second more slowly. The problem is how to weight together these two signals together, taking into account both expected returns, transaction costs and the forecast horizon.

5.1

Example 1: Investing for the Long Run - Barberis (2000)

The following VAR model was estimated by Barberis (2000) on monthly US data between 1986 and 1995. In this example rt refers to the excess returns on the value-weighted index of stocks traded on the NYSE as calculated by the Center for Security Prices (CRSP). The conditioning variable st is the aggregate 12 month historic dividend yield, st = dt /pt . The VAR model is 

        2  η t+1 µs ηt+1 σ η σηε = + + where Cov = εt+1 µr εt+1 σηε σ 2ε (61) where we also write σηε = ρση σε in terms of the correlation coefficent. The parameter values that were estimated on his data sample were as = 0.9577, asr = 1.0919, µs = 0.0013, µr = −0.0303, σ2η = 2.6E −6 , σ 2ε = 0.0019 and ρ = −0.9323. st+1 rt+1





0 as 0 asr



st rt

23

In this model dividend yields are slowly mean-reverting, with high yields predicting higher expected returns. Further, the innovations to the dividend yield are highly negatively correlated to the return innovation suggesting a strong hedging motive for holding equity. Given this data generating process (DGP) for stock returns, the dynamic portfolio problem is to choose the optimal portfolio of stock and T-Bills so as to maximise a power function of end of period wealth T periods later. The return to T-Bills is assumed to be constant and denoted as rf = 0.0036. We shall denote the return to equity as rte where rte = rt + rf and the respective weights of these assets in our portfolio as wte and wtf where wt = [wte wtf ]T . The only investment constraint is that the portfolio is fully invested or that wte + wtf = 1. We can therefore write the portfolio constraint matrix as     1 1 1 ⊥ =√ P P 2 1 −1 √ where the fully-invested constraint is the requirement that P T wt = 1/ 2. Given this notation, we can now write the problem in in the state-space form used throughout this paper. The full state-space and output equations are respectively

xt+1

           st 0 st+1 µs as 0 0 1 0  η t+1 +  1  ∆ut+1 +  0  =  ut+1  =  0 1 0   ut  +  0 0  εt+1 1 0 0 0 0 0 ∆ut+1 ∆ut 0 (62) 

and 

    zt =    

e rt+1 f rt+1 wte wtf e ∆wt+1 f ∆wt+1





        =      

asr 0 0 0 √1 0 2 0 − √12 0 0 0 0

0 0 0 0 0 0





     st     ut +      ∆ut  

0 0 0 0 0 0

and the cost function is Ct+1 = ztT Rzt − 2S T zt where   1 0 0 0 0 0 2 1  0 0  0  2 0 0  2  1  σε   R =  2 01 − 2 0 0 0  and  0 2 0 0 0 0     0 0 0 0 0 0  0 0 0 0 0 0

1 0 0 0 0 0





       ηt+1   +   εt+1      

    S=   

0 0 2 − σ4ε 0 0 0

0 0 0 0 √1 2 − √12





 µr + rf    rf     1    2    ∆ut+1 + 1     2    0  0 (63)

        

(64)

In this problem all transactions costs have been set to zero. The state-space of the problem can therefore be reduced along the lines described in section 3.6.3. The statetransition equation for this reduced problem becomes     ηt+1 (65) + µs xt+1 = st+1 = as st + 1 0 εt+1 24

with the relevant rows and columns in the output an cost equations (63) and (64) deleted as described in section 3.6.3. The problem can be further simplified as one of the assets is risk free. Map the outputs zt to z t = Zzt where     1 −1 0 0 rt+1  T   0 1 0 0   rf  Z11 0   Z = and therefore z t =  −1  0 0 1 0 =  wte  0 Z11 0 0 1 1 1

This transformation maps the assets to a linearly equivalent set of assets, and performs the inverse map on the weights. Under the transformation, the cost matrices R and S are

= (Z −1 )T RZ −1 and S = SZ −1 respectively. We have chosen the mapping so mapped to R that two elements of the output vector, z t , are now constants. These rows can therefore be deleted and the constant incorporated into the cost function directly. The output equation for this reduced problem is            0 asr rt+1 0 1 µr ηt+1 u s = + z t = + + t t 1 √1 wte 0 εt+1 0 0 2 2

zt − 2S T z t + rf where with the cost function written, Ct+1 = z tT R     1 0 0 2

= 2 and S = R σ 2ε 1 − − σ4ε 2 2

5.1.1

(66)

Solution to Investing in the Long Run

The model has a single state, and is therefore relatively tractable. We first solve for the parameters Πt , and Φt in the transformed value function Q∗t . These are derived from the iterative equations, (50). Given Πt+1 , denote the conditional variance term σ 2η|ε,t+1 as9 σ2η|ε,t+1

   (γ − 1) 2 1 2 = Πt+1 1 + 2 (γ − 1) Πt+1 ση 1 − ρ 2 γ

then Πt , and Φt are calculated by the iterative relationships   ση 2 1 a2sr 1 Π2t+1 (γ − 1) Πt = asr ρ + as − 2 γσ2ε 2 σ2η|ε,t+1 γ σε   1 asr σ2ε Φt = − + µ r 2 γσ2ε 2     ση ση (γ − 1) (γ − 1) σ 2ε 1 Πt+1 − 2 asr ρ Πt+1 ρ as − (Πt+1 µs − Φt+1 ) − µr + 2 ση|ε,t+1 γ σε γ σε 2 and the optimal portfolios is  asr st + µr + wte = γσ2ε 9

σ 2ε 2



(γ − 1) ση ρ − γ σε



Πt+1 2σ2η|ε,t+1



H rt+1

 −1 ˜ ηη − Σ ˜ ηε P ⊥ P ⊥T Σ ˜ εε P ⊥ ˜ Tηε . In the notation of corollary 2 σ 2η|ε,t+1 = Σ P ⊥T Σ

25

(67)

where H rt+1

     (γ − 1) ση σ2ε = Πt+1 (as st + µs ) − asr st + µr + − Φt+1 ρ γ σε 2

The structure of the optimal portfolio is clear. The expected excess return to equity 2 e ) = a s + µ + σ ε . Hence the first term in equation (67) is the next period is E(rt+1 sr t r 2 H can mean-varance optimal portfolio. The second term is the hedging term. The term rt+1 be seen as the expected change in future returns given this period return. The effective H divided by its change in today’s expected return due to changes in future returns is rt+1 covariance and multiplied by the covariance term σηε . This term is now expressed in units of this period expected return.

5.2

Example 2: The Forecast Horizon Problem

We shall assume again a two asset universe. The first asset is a self-financing portfolio with a non-zero expected return, mt . We shall label this portfolio the ‘momentum portfolio’. The defining characteristic of this portfolio is that the expected return is assumed to decay relatively quickly to zero, or alternatively the term structure of expected returns decays quickly to zero. The other portfolio is also self-financing but its expected return, vt , is assumed to decay far more slowly. We shall label this portfolio the ‘value portfolio’. The evolution of the expected returns to each of these portfolios is described by a first order AR(1) process. Thus         ηm,t+1 mt+1 mt 0 (1 − λm ) λm 0 = + (68) vt+1 vt ηv,t+1 0 (1 − λv ) 0 λv where the forecast innovations are independently distributed with variance σ2η and 1 > λm > λv > 0. Returns to the two portfolios, in the absence of trading, are equal to the expected return plus an innovation       rm,t+1 mt εm,t+1 = + (69) rv,t+1 vt εv,t+1 where again we shall assume that both the innovations are independently distributed with variance σ2ε and uncorrelated with the innovations η t . Now if there were no transaction costs, the optimal dynamic portfolio would be the sequence of single period mean-variance optimal portfolios. We shall assume, however, there is a cost to trading these portfolios. To keep it simple, the only cost will be due to the permanent market impact of the trade. Let the the portfolio weights in the two assets be denoted wtm and wtv respectively, and let wt = [wtm wtv ]T , and assume are no constraints imposed on the portfolio (P ⊥ is therefore the identity matrix). Then the state-transition equation can be written

26

xt+1

  (1 − λm ) mt+1 0  vt+1   0 (1 − λv ) = =  m    wt+1 0 0 v 0 0 wt+1     ηm,t+1 λm 0 0 0  0 λv 0 0   ηv,t+1     +  0 0 0 0   εm,t+1   εv,t+1 0 0 0 0 

and the output equation is    1 0 rm,t+1  rv,t+1   0 1 = zt =  m   0 0  wt+1 v wt+1 0 0

As always, the cost function  0  0  R= 1  2 0

0 0 1 0

 0 mt   0   vt 0   wtm wtv 1

 

0   0 +   0 0

0 0 0 0

1 0 0 0

0 0 1 0 0 0 1 0

   mt 0  t  0   +   vm  0  wt  1 wtv  0   m 0   ∆wt+1 v 0  ∆wt+1 1

 η m,t+1 0   1   ηv,t+1 0   εm,t+1 0 εv,t+1

(70)

(71)

 δm 0   m   0 δ v  ∆wt+1  + v   0 0  ∆wt+1 0 0  

is Ct+1 = ztT Rzt − 2S T zt where this time    1 0 0 0 2 1  0   0 0    2 2 S =  σ2ε   and σε 0 −2 0  −  4  σ 2ε σ 2ε 1 0 − − 2 2 4

(72)

Though there are two assets in this problem, the behaviour of the two assets are independent. The problem could therefore be decoupled into two smaller ones, the optimal holding in the momentum portfolio and the optimal holding in the value portfolio. Each of these two smaller problems has effectively only 2 states. With this simplification we are able to obtain some algebraic expressions. We shall focus on momentum subsystem and solve for the for the optimal portfolio weight on the momentum asset. The state-transition and output equations for this sub-system are

xm,t+1 =



mt+1 m wt+1



=



(1 − λm ) 0 0 1



mt wtm



+



λm 0 0 0



ηm,t+1 εm,t+1



+



0 1



and the output equation is           η m,t+1 mt 1 0 0 1 δm rm,t+1 m zm,t = ∆wt+1 = + + 0 1 0 0 wtm wtm εm,t+1 0

m ∆wt+1

(73)

(74)

T z where R and S are the obvious submaand the cost function is Ct+1 = ztT Rm zt − 2Sm t m m trices of R and S respectively. The expressions for the subsytem relating to the value asset are the same certeris paribis.

5.2.1

Solution to the Forecast Horizon Problem

In this subsection we keep the notation consistent with Section 4.3 and in particular Corollary 3. All the terms relate to the subsystem in equations (73) and (74). We first define a

27

number of quantities in terms of the parameters Πt+1 , and Φt+1 . First note that because T = 0, V22,t+1 = Π22,t+1 and Π11,t+1 Π22,t+1 − Π212,t+1 |Πt+1 | = Π22,t+1 Π22,t+1 Π12,t+1 = Φ1,t+1 − Φ2,t+1 Π22,t+1

ΠC,t+1 = ΦC,t+1

Ht+1 = (γ − 1) BsT ΠC,t+1 Bs + Σ−1 ηη = (γ − 1) T˜ =

λ2m |Πt+1 | 1 + 2 Π22,t+1 2ση

δm 2

and these definitions imply that λ2m σ2η Π22,t+1

−1 T Bs = Bs Ht+1

1 2 Π22,t+1

+ (γ − 1) λ2m σ2m |Πt+1 |

The matrices Πt , and Φt are calculated by the backward iteration     δ m (1−λm ) Π12,t+1 2 |Πt+1 | 1 ) − (1 − λ m Π11,t Π12,t Π 2 Π22,t+1   2 2 − =  1 δm (1−λm ) 22,t+1 Π σ δ 2m 12,t+1 ε Π12,t Π22,t 2 − 2 Π22,t+1 − γ 2 + δ m + 4Π22,t+1   t+1 |  (1 − λm ) Π|Π22,t+1 λ2m σ2η Π22,t+1 t+1 | (1 − λm ) Π|Π22,t+1 (γ − 1) 1 2 2 δm Π12,t+1 Π + (γ − 1) λ σ |Π | − t+1 m η 2 22,t+1 2 Π22,t+1 and 

Φ1,t Φ2,t





= 

 (1 − λm ) Φ1,t+1 −

(γ − 1)

2

− σ4ε −

Π12,t+1 Π22,t+1 Φ2,t+1

δm

2Π22,t+1 Φ2,t+1

λ2m σ2η Π22,t+1 1 2 Π22,t+1

 

−

+ (γ − 1) λ2m σ2η |Πt+1 |





t+1 | (1 − λm ) Π|Π22,t+1

Φ1,t+1 −

Π

− δ2m Π12,t+1 22,t+1

Π12,t+1 Φ2,t+1 Π22,t+1

These equations are difficult to simplify further maintaining full generality. However we can solve for the steady-state solution when the investor is risk neutral, γ = 1. The steadystate solution for risk averse investors can then be shown to have a similar parametrisation. To this end, denote steady-state solution of the equations by the subscript ss in place of the subscript t, and the risk neutrality by the superscript RN. If we define the parameter   2 σ 4δ m σ

2ε = ε 1 + 1 + 2 2 σε

then the steady state solutions are 

Π11,ss Π12,ss Π12,ss Π22,ss

RN



 =

2

(1−λm )2 (δ m + σ2ε )

((1−λm )2 −1)(δm + σ2ε ) σ2ε ) 1 (δm + 2 (δm + σ2ε ) 28

2

σε ) 1 (δ m + 2 (δ m + σ 2ε )

− 12




  2 δ m λm + σ

ε



Π

− δ2m Π12,t 22,t

and



Φ1,ss Φ2,ss

RN



=

σ ε )σ ε 1 (1−λm )(δm + 4 λm (δ m + σ 2ε )σ

2ε 1 2 −4σ

ε 2

2

 

and the optimal portfolio weight can be written in the form          σ

2ε σ 2ε σ

2ε mt 1 1 σ2ε RN,m RN,m + 1 − 2 wt−1 = 2 wt 2 2 + 2 + 1− 2 2 σ

ε δ m λm + σ

ε σ

ε δ m λm + σ

ε σ

ε

or more succintly
 

 wtRN,m = 1 − χShrink P ortf olio 1 − χShrink Alpha wMV,t + χShrink Alpha (wLR ) + χShrink P ortf olio w where   σ2 χShrink W eight = 1 − ε2 σ

ε   σ

2ε 1− χShrink Alpha = δ m λm + σ

2ε

Thus in the presence of transaction costs, the optimal portfolio is weighted average of first the conditional mean-variance portfolio, wM V , and the unconditional long run portfolio, wLR , and then a weighted average of this portfolio and last period portfolio. Thus the shortrun mean-variance portfolio is first shrunk back to the unconditional long run optimal portfolio - where the degree of shrinking is increasing in both speed of decay and the size of the costs. Then this portfolio is shrunk back to last periods portfolio but in proportion to only to costs of trading. If can be show (Apologies - still to be typed out),but in the risk averse case, γ > 1, the optimal portfolio structure has a similar structure 


 m 1 − χShrink Alpha wM V,t + χShrink Alpha (wLR ) +χShrink P ortf olio wt−1 wtm = 1 − χShrink P ortf olio but now

wMV,t wLR

  1 mt 1 = + γ σ 2

2ε   1 = 2γ

In Figure 2 below, we plot value of the shrinkage parameters as risk aversion varies. We have in mind here a typical short term reversal strategy implemented on daily data - the signal has a relatively rapid decay. We compare three cost models. 1. All the trading costs are due to permanent price impact: λm = 0.1. and daily volatility, σε = 0.63% (10% annualised) and ση = 0.13% (2% annualised).. Trading costs are kept relatively low at δ 0 = 10bp. 2. In this example, all price impact is temporary: λm = 0.1. and daily volatility, σε = 0.63% (10% annualised) and ση = 0.13% (2% annualised), δ 0 = δ 1 = 10bp/ and T = 0. 3. In this example, all costs are explicit and there is no price impact: λm = 0.1. and daily volatility, σε = 0.63% (10% annualised) and ση = 0.13% (2% annualised), δ 0 = δ 1 = 0% and T = 0.05. 29

0 .9 0 .8 P ortfo lio S h rin ka g e

0 .7

A lp h a S hrin ka g e

0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Figure 2: Plot of the the variation of the portfolio and alpha shrinkage values as the risk aversion varies, γ. The value of the other parameters are roughly calibrated to daily data, λm =0.1, σ ε =0.63% (10% annualised), ση =0.13% (2% annualised), δ 0 =0.1% and δ 1 = T = 0.

1

Portfolio Shrinkage 0.8

Alpha Shrinkage

0.6 0.4 0.2 0 1

3

5

7

9

11

13

15

17

19

Figure 3: Plot of the the variation of the portfolio and alpha shrinkage values as the risk aversion varies, γ. The value of the other parameters are roughly calibrated to daily data, λm =0.1, σε =0.63% (10% annualised), ση =0.13% (2% annualised), δ 0 =0.1%, δ 1 = 0.1% and T = 0.

30

1 0.8 0.6

Portfolio Shrinkage

0.4

Alpha Shrinkage 0.2 0 1

3

5

7

9

11

13

15

17

19

Figure 4: Plot of the the variation of the portfolio and alpha shrinkage values as the risk aversion varies, γ. The value of the other parameters are roughly calibrated to daily data, λm =0.1, σε =0.63% (10% annualised), ση =0.13% (2% annualised), δ 0 = δ1 = 0.0% and T = 0.05%. Though a basic example, the three simulations illustrate of number of points. Firstly the shrinkage parameters decline with risk aversion. As the mean variance portfolio weights decline at the rate 1/γ, implicit trading costs fall implying the need for less shrinkage. Secondly if the price impact is temporary rather than permanent, there is less shrinkage. Thirdly if all trading costs are explicit, rather than implicit, then there is no alpha shrinkage. These simulations are therefore in line with the observations that only smaller funds run strategies based on signals with a fast decay. As a fund size rises, implicit costs become a higher proportion of their trading costs, implying alpha shrinkage rises until it will be shrunk away, effectively implying the signal is ignored.

A

Appendix

The Appendix contains proofs of the results in the text. We start of by stating a couple of Lemmas. Lemma 4 Given following quadratic maximization problem, maxu Q(x, u), where Q=



x u

T 

V11 V12 T V12 V22



x u



−2



T

W1

W2T

 x  u

and V22 < 0, then  T  −1 u∗ = arg max Q(x, u) = −V22 V12 x − W2 =: Kx + L u    T  T T ∗ T −1 T −1 T −1 Qu = max (x, u) = x V11 − V12 V22 V12 x − 2 W1 − W2 V22 V12 x − W2 V22 W2 u

T

= x V11 x − 2W 1 x − (Kx + L)T V22 (Kx + L) T

We now apply this lemma in the following Lemma. It is a similar to Lemma 6.1.2 in Whittle (1992) but the for the case when the innovations ξ are observed; the full information case. 31

Lemma 5 Define the following quadratic maximization problem, maxu Q(x, u), where       T  V11 V12 V13 x x x   2 T T T   θ   T     ξ  V12 V22 V23 ξ + θ W1 W2 W3 Q = − log Eξ exp − ξ θ 2 T T u u u V13 V23 V33 (75) ξ ∼ N (0, Σ), V22 < 0 and u can be a function of both x and ξ . Then the optimal control    x  T T −1 ∗ (76) − W3 u = arg max (x, u) = −V33 V13 V23 u ξ and the attained maximum value is "  " T T 1 T " −1 T " Q∗ = log "1 + θΣ V22 − V23 V33 V23 "+x V11 x−2W1 x−(Kx + L) θ where the feedback matrices are defined as  −1  T    V22 + (θΣ)−1 V23 V12 and K=− T T V13 V23 V33

 

V22 + (θΣ)−1 T

V23



V23

V33 (77)



(Kx + L)

   −1   V22 + (θΣ)−1 V23 W2 L= T W3 V V33 23

(78)

Proof : Equation (76) follows immediately from Lemma 4. This can be substituted back into equation (75) and rewriting the expectations as an integral gives  #   T        1 V12 V11 θ x 2 V13 T T −1 − V33 exp − Q= − log T V13 V23 V23 θ 2 ξ V12 V22 + (θΣ)−1 (2π)n/2 |Σ|1/2       x  θ T T T T T T −1 −1 + W3 V33 W3 dξ +θ V13 V23 W1 W2 − W3 V33 ξ 2

The trick is to complete the square in the innovation ξ by the substitution ξ = ξ ∗ + ψ, where  −1  T    −1 T −1 T −1 ξ ∗ = − V22 + (θΣ)−1 − V23 V33 V23 V13 x − W2 − V23 V33 W3 V12 − V23 V33 The integral can now be rewritten as

     −1  T θ T  −1 T −1 T −1 T −1 T V11 − V13 V33 V12 − V13 V33 exp − x V13 − V12 − V13 V33 V23 V22 + (θΣ)−1 − V23 V33 V23 V23 2    T  −1  T  T T T −1 T −1 T −1 T −1 T +θ W1 − W3 V33 V13 − W2 − W3 V33 V23 V22 + (θΣ)−1 − V23 V33 V23 V23 V12 − V13 V33 x #

+

 −1  T T θ T θ T T T −1 T −1 T −1 T −1 W2 − W3 V33 V23 V22 + (θΣ)−1 − V23 V33 V23 V23 + W3 V33 W W2 − W3 V33 2 2    θ −1 T − ψ T V22 + (θΣ)−1 − V23 V33 V23 ψ dψ 2

32

All but the last term are independent of ψ and so the expression can be integrated and rearranged as " "  1 " −1 T " V23 " + Q∗ = log "1 + θΣ V22 − V23 V33  θ    −1  T  T −1 −1 T −1 T −1 T −1 T V11 − V13 V33 V13 − V12 − V13 V33 V23 V22 + (θΣ) − V23 V33 V23 V12 − V13 V33 V23 x x    T  −1  T  T T T −1 T −1 T −1 T −1 T V13 − W2 − W3 V33 V23 V22 + (θΣ)−1 − V23 V33 V23 V23 V12 − V13 V33 −2 W1 − W3 V33 x    −1  T T T T T T −1 −1 T −1 T −1 T −1 W2 − W3 V33 V23 + W3 V33 W3 − W2 − W3 V33 V23 V22 + (θΣ) − V23 V33 V23

It is possible to simplify this expression by noting the matrix identity that  −1     −1    T −1 −1 V22 + (θΣ)−1 I 0 V23 I −V23 V33 V23 0 V22 + (θΣ)−1 − V23 V33 = −1 T T −1 0 I V23 I −V33 V23 V33 0 V33

so equation (??) can be rewritten     −1  T  " −1 "    1 V22 + (θΣ) V23 T V12  " −1 T " Q∗ = log "1 + θΣ V22 − V23 V33 V23 "+x V11 − V12 V13 x T T θ V13 V23 V33     −1  T  −1   V22 + (θΣ) V23 T V12  T T x −2 W1 − W2 W3 T T V13 V23 V33    −1   −1   V + (θΣ) V23 W2 T T 22 − W2 W3 T W3 V23 V33 A rearrangement of this equation produces (77).

A.1




Proof of Theorem 1

Assume for the moment the function Q∗t+1 has the quadratic form Q∗t+1 = xTt+1 Πt+1 xt+1 − 2ΦTt+1 xt+1 + φt+1

(79)

We shall first prove that if it has this form at time t+1 then it has this functional form at time t. Then, as it clearly has the form at time T, it follows by the standard induction argument that it must have this form for all t < T . Given this assumption, we can write the expression on the right hand side of equation (44) by substituting in from the state-space equations (29) and (30)   T    xt A C1T     Ct +Q∗t+1 =  ξ t+1   D1T  R C1 D1 D2 +  B1T  Πt+1 A B1 B2   ξ t+1  ∆ut+1 D2T B2T ∆ut+1   xt  
 

 ξ t+1  A B1 B2 2 S T − κT R C1 D1 D2 + ΦTt+1 − µT Πt+1 ∆ut+1 
T 
T T T − 2S − κ R κ − 2Φt+1 − µ Πt+1 µ + φt+1 

xt

T 

33



The problem of maximising Et exp (1 − γ) Ct + Q∗t+1 with respect to ∆ut is an application of Lemma 5 earlier. Given the definitions in equations (47)-(49) of the theorem statement, an application of Lemma 5 implies that the maximand can be written

 1 log max Et exp (1 − γ) Ct + Q∗t+1 ∆ut+1 (1 − γ) "  " −1 " " −1 T log "1 − 2 (1 − γ) Σ V11,t+1 − (2 (1 − γ) Σ)−1 − V12,t+1 V22,t+1 = V12,t+1 " 2 (1 − γ)  

  
T +xTt AT Πt+1 A + C1 RC1 xt − 2 ΦTt+1 − µT Πt+1 A + S T − κT R C1 xt

 − (Kt+1 xt + Lt+1 )T Vt+1 (Kt+1 xt + Lt+1 ) − 2S T −κT R κ − 2ΦTt+1 −µT Πt+1 µ + φt+1

Q∗t =

Please note that as far as possible we have kept the notation consistent between Lemma 5 and Theorem 1. However the matrix V is defined slightly differently in this theorem, and in particular the (1, 1) block. Thus Q∗t is quadratic in the states xt . Equating terms of the same order in xt gives the iterative relations in (50). We also used the matrix identity that " " " " −1 T − V V V "V11,t+1 12,t+1 22,t+1 12,t+1 " |V22,t+1 | = |Vt+1 | to simplify the expression slightly. Lemma 5 also gives an expression for the control, ∆ut+1 , that achieves the optimum. However this is an expression for the change in portfolio weights and will be in terms of the innovation ξ t+1 . It will therefore prove simpler to derive this expression for the optimal portfolio from the first order conditions. Returning to the Bellman equation (44), the first order condition is 0=



∂Ct ∂∆ut+1

T

+



∂xt+1 ∂∆ut+1

T

∂Q∗t+1 ∂xt+1

(80)

which after substituting out for the partial derivatives from equations (29), (30) and (45) gives B2T (Πt+1 xt+1 − Φt+1 ) + D2T (Rzt − S) = 0

(81)

Expanding this equation in terms of the partition of Π and Φ in equation (81) and substituting for the definitions of B2 , D2 , R and S we derive the equation     Π Π Π s Φ 11,t+1 12,t+1 13,t+1 t+1 1,t+1    0 I I  ΠT12,t+1 Π22,t+1 Π23,t+1   ut+1  −  Φ2,t+1 + 0 ΠT13,t+1 ΠT23,t+1 Π33,t+1 ∆ut+1 Φ3,t+1 

1 ⊥T T δ0 2P

−P ⊥T T

(82) Due to the structure of our problem, some of the terms in the partitioned matrices Π and Φ are  constant.  Multiplying both sides of the Riccati equation (50) on the left by the matrix 0 0 I implies 

0 0 I



Πt+1 =



0 0 I


T  −1 T A Πt+2 A + C1T RC1 − Kt+2 Vt+2 Kt+2

As the third column of the partitioned matrices A and Kt+2 are zero, most of the terms drop out and we have    T    Π13,t+1 ΠT23,t+1 Π33,t+1 = 0 0 I C1T RC1 = 0 − 12 P ⊥T δ T1 P ⊥ 0 (83) 34



 

rt+1 wt ∆wt+1

Similarly from the second line in (50) we prove that    T   T ˜ −1 C1 (S − Rκ) + AT (Φt+2 − Πt+2 µ) − Kt+2 Vt+2 Lt+2 0 0 I Φt+1 = Φ3,t+1 = 0 0 I =

1 ⊥T T P δ1 P c 2

(84)

Substituting (83) and (84) into (82) and rearranging gives the following expression for the control at time t + 1,      Π12,t+1 0 Π11,t+1 Φ1,t+1 st+1    Π22,t+1 − 12 P ⊥T δ1 P ⊥   ut+1  −  Φ2,t+1 0 = 0 0 I  ΠT12,t+1 1 ⊥T T 1 ⊥T T ⊥ ∆ut+1 δ1 P c 0 − 2 P δ1 P 0 2P   rt+1   1 ⊥T T ⊥T  + 0 2 P δ 0 −P T wt  ∆wt+1    ⊥ 
1 ⊥T
T 1 1 T δ 1 + δ 1 + 2T P ut+1 + P ⊥T δ 1 + δT0 + 2T P ⊥ ut + P ⊥ = Π12,t+1 st+1 − Φ2,t+1 + Π22,t+1 − P 2 2 2 If we note that wt = P ⊥ ut + P c, then the expression above can be rearranged to give ut+1

    ⊥ −1 
1 ⊥T
T 1 = − Π22,t+1 − P δ 1 + δ 1 + 2T P ΠT12,t+1 st − Φ2,t+1 − P ⊥T δ T1 + δ 1 + 2T P c 2 2   1 ⊥T
T δ 1 + δ o + 2T wt + P 2

Lagging this expression by a period and rewriting in terms of the portfolio weights wt using equation (21), we derive the final expression in Theorem 1.

A.2

Proof of Corollary 1

To ease the notational burden we shall partition some of the matrices conformally with the states and make the following additional definitions         Bs 0 0 I B D1,1 1,1     B1 = D1 = 0 0 = 0 0 = B1,2 D1,2 0 0 0 0

and make the following definitions (the first two are a restatement of some definitions in the corollary statement) T ΠC,t+1 = Π11,t+1 − Π12,t+1 Π−1 22,t+1 Π12,t+1

ΦC,t+1 = Ft+1 =

Φ1,t+1 − Π12,t+1 Π−1 22,t+1 Φ2,t+1

(85) (86)

1 T ((γ − 1) Σ)−1 + B1,1 ΠC,t+1 B1,1 2

The first section of the proof is to demonstrate that the optimal portfolio can be written in the form

w−P c = −



0 P ⊥T

T 

1 T ⊥ Ft+1 2 D1,1 P 1 ⊥T D1,1 − 12 P ⊥T Σεε P ⊥ 2P

35

−1 

1 T T (Π B1,1 C,t+1 (As t + µ) − ΦC,t+1 ) + 2D1,1 P c
1 ⊥T rˆt+1|t + 12 σ2 − Σεε P c 2P (87)



When there are no transaction costs, the states corresponding to the portfolio weights do not enter the cost function Ci . We can therefore remove theserows from the output vector zt . Now these states are unobservable from the output and so the state-space can be reduced. This reduced problem can be rewritten as a state-feedback problem; with the only states being the conditioning variables st . It can be shown, relatively easily, that the value function for this reduced state problem is Q∗t = sTt ΠC,t st − 2ΦTC,t st + φt

(88)

where ΠC,t , ΦC,t are defined above. Further the optimal state feedback to this problem is given in equation (87). Rather than redefining the problem in this simpler state-feedback form, we shall derive these relations rather mechanically, but a little tediously, from our current model set-up. The reason for this, is that we need most of the intermediate relations in the proof of Corollary 2 which does not lend itself to this simplification. Substituting in from equations (83) and (84), and multiplying out the expression for Vt+1 in equation (47) gives

Vt+1 =



−1 1 T Π T Π B1,1 B1,1 t+1 B1,1 + 2 ((γ − 1) Σ) 
t+1 Πt+1 B1,1 Π22,t+1 − 12 P ⊥T δ T1 + δ 1 + 2T P ⊥



=:



V11,t+1 V12,t+1 T V12,t+1 V22,t+1 (89)

Similarly straightforward matrix multiplication implies that

Kt+1 Lt+1 AT Πt+1 A + C1T RC1 and

 1 T T Π T Π ⊥ B1,1 0 B1,1 11,t+1 As 12,t+1 +
2 D1,1 P  , (90) = ΠT12,t+1 As Π22,t+1 + 12 P ⊥T δ T0 − δ T1 P ⊥ 0     T T Pc − Π11,t+1 µs ) − 12 D1,1 L1 −1
B1,1 (Φ1,t+1 
 = Vt+1 = (91) L2 Φ2,t+1 − ΠT12,t+1 µs − 12 P ⊥T δ T0 − δ T1 P c   ATs Π11,t+1 As ATs Π12,t+1 + 12 CsT P ⊥ 0 ⊥  =  ΠT12,t+1 As + 12 P ⊥T Cs Π22,t+1 − 12 P ⊥T Σεε P ⊥ − 12 P ⊥T δ 1 P (92) 1 ⊥T T ⊥ 0 − 2 P δ1 P 0 −1 −Vt+1





 − 12 CsT P c + ATs (Φ1,t+1 −
Π11,t+1 µs ) 
 C1T (S − Rκ)+AT (Φt+1 − Πt+1 µ) =  −P ⊥T 14 σ2 + 12 κs − 12 Σεε P c + Φ2,t+1 − ΠT12,t+1 µs  1 ⊥T T δ1 P c 2P (93) To go any further we must assume no transaction costs. In this case, the matrix inversion lemma implies that the inverse of Vt+1 can be written

−1 Vt+1

=



 −1 −1 T Ft+1 −Ft+1 B1,1 Π12,t+1 Π−1 22,t+1 −1 −1 −1 −1 T −1 T T −Π−1 22,t+1 Π12,t+1 B1,1 Ft+1 Π22,t+1 + Π22,t+1 Π12,t+1 B1,1 Ft+1 B1,1 Π12,t+1 Π22,t+1 (94)

36



Now substituting these into the recursive relations in equation (50) implies that   1 ⊥T 1 −1 T Σεε + D1,1 Ft+1 D1,1 P ⊥ Πt,22 = − P 2 2 1 1 −1 T ΠTt,12 = B1,1 ΠC,t+1 As P ⊥T Cs − P ⊥T D1,1 Ft+1 2 2   1 ⊥T 1 2 σ + κs − Σεε P c Φ2,t = − P 2 2   1 T 1 ⊥T −1 T − P D1,1 Ft+1 B1,1 (ΦC,t+1 − ΠC,t+1 µs ) − D1,1 P c 2 2

Substituting these relations into equation (51) for the optimal portfolio implies

   −1 1 −1 T ⊥T ⊥ wt = P c + P P Σεε + D1,1 Ft+1 D1,1 P 2      1 1 T −1 T (ΠC,t+1 (As st + µs ) − ΦC,t+1 ) + D1,1 Pc P ⊥T Cs st + κs + σ2 − Σεε P c − P ⊥T D1,1 Ft+1 B1,1 2 2 ⊥

This expression can be rearranged to give equation (87). To derive the final expressions in the corollary statement, it is necessary to invert the matrix −1 Ft+1 . To this end, define the matrix   1 −1 1 T Σ + (γ − 1)Bs ΠC,t+1 Bs Ht+1 = (γ − 1) 2 ηη

Now we can write the inverse in terms of this matrix  
−1 T −1 
1 1 −1 + Σ−1 Σ T Σ−1 Σ Σ − Σ Σηε Σηη + BsT ΠC,t+1 Bs − 2(γ−1) Σ−1 Σηε Σεε − ΣTηε Σ ηε εε ηε ηη ηη ηε ηη ηη 2(γ−1) −1 Ft+1 = 
−1 T −1
1 1 T −1 − 2(γ−1) Σηε Σηη Σεε − ΣTηε Σ−1 ηη Σηε 2(γ−1) Σεε − Σηε Σηη   −1 −1 −1 H Σ Σ Ht+1 ηε ηη t+1
 = −1 1 T −1 T −1 −1 −1 ΣTηε Σ−1 ηη Ht+1 2(γ−1) Σεε − Σηε Σηη Σηε + Σηε Σηη Ht+1 Σηη Σηε   −1 ˜ −1 ˜ Ht+1 Σ ηη Σηε = ˜T Σ ˜ Tηε Σ ˜ −1 ˜ −1 ˜ 2(γ − 1)Σεε − 2Σ Σ ηη ηε ηη Σηε where we have used the matrix inversion lemma to invert Σ in the second line, and the definitions in equation (53) to simply the expression in the third line. Substituting this into the expression for the optimal portfolio in equation (??) gives

   −1 ⊥ ˜T Σ ˜ −1 ˜ wt = P c + P ⊥ P ⊥T γΣεε − Σ ηε ηη Σηε P      1 2 ⊥T ⊥T ˜ T ˜ −1 T ˜ ηε P c P Cs st + κs + σ − γΣεε P c − P Σηε Σηη Bs (ΠC,t+1 (As st + µs ) − ΦC,t+1 ) − Σ 2

A final application of the Matrix Inversion Lemma implies that

   −1  −1 ⊥ ⊥T ˜ ⊥ ˜ −1 ˜ ˜ Tηε Σ P ⊥T γΣεε − Σ Σ Σ P = P P + ηε εε ηη  −1 −1  −1  −1  ⊥T ˜ ⊥ ⊥T ˜ T ⊥ ⊥T ˜ ⊥ ⊥T ˜ T ˜ ˜ ηε P ⊥ P ⊥T Σ ˜ εε P ⊥ ˜ P Σηε Σηη − Σηε P P Σηε P Σεε P Σ P Σεε P

We can substitute be substituted back to give the final expression in the corollary. 37

A.3

Proof of Corollary 2

We start the proof of this corollary from equations (89) to (93). Under the assumption the Σηǫ = 0, the equation (89) reduces to

Vt+1



1 BsT Π11,t+1 Bs + 2(γ−1) Σ−1 ηη  0 = T Π12,t+1 Bs

0 1 −1 2(γ−1) Σεε

0

 BsT Π12,t+1   V11,t+1 V12,t+1  0 = T V12,t+1 V22,t+1  ⊥
T 1 ⊥T Π22,t+1 − 2 P δ 1 + δ 1 + 2T P

As before we shall invert this matrix. If we redefine the matrices ΠC,t+1 and Ht+1 as −1 ΠT12,t+1 ΠC,t+1 = Π11,t+1 − Π12,t+1 V22,t+1 1 −1 Ht+1 = Σ +(γ − 1)BsT ΠC,t+1 Bs 2 ηη

And so the inverse can be written as 

−1 −1 T −1 0 −(γ − 1)Ht+1 Bs Π12,t+1 V22,t+1 (γ − 1)Ht+1 −1 0 2(γ − 1)Σεε 0 Vt+1 = −1 −1 −1 −1 −1 T −1 −(γ − 1)V22,t+1 ΠT12,t+1 Bs Ht+1 0 V22,t+1 + (γ − 1)V22,t+1 ΠT12,t+1 Bs Ht+1 Bs Π12,t+1 V22,t+1 (95) We shall substitute this expression into the expression for Kt+1 and Lt+1 in equations (90) and (91) respectively. However we can simplify the expressions by defining

and noting that

T˜ = δ T0 + δ 1 + 2T 
T$ = P ⊥T δ T0 + δ 1 + 2T P ⊥


 1 1 Π22,t+1 + P ⊥T δ T0 − δ T1 P ⊥ = V22,t+1 + T$ 2 2 Some rather tedious algebra then implies that

Kt+1



−1 −1 T −1 (γ − 1)Ht+1 0 −(γ − 1)Ht+1 Bs Π12,t+1 V22,t+1 0 2(γ − 1)Σεε 0 = − −1 −1 −1 −1 −1 T T T −(γ − 1)V22,t+1 Π12,t+1 Bs Ht+1 0 V22,t+1 + (γ − 1)V22,t+1 Π12,t+1 Bs Ht+1 Bs Π12,t+1 V   T T Bs Π11,t+1 As Bs Π12,t+1 0 1 ⊥  0  0 2P 1$ T V22,t+1 + 2 T 0 Π12,t+1 As  −1 T −1 T −1 $ Bs ΠC,t+1 As − (γ−1) (γ − 1)Ht+1 2 Ht+1 Bs Π12,t+1 V22,t+1 T  ⊥ 0 (γ − 1)Σεε P  = −  −1 ΠT12,t+1 As −  V22,t+1 −1 −1 −1 T I + 12 V22,t+1 + (γ − 1)V22,t+1 ΠT12,t+1 Bs Ht+1 Bs Π −1 −1 T (γ − 1)V22,t+1 ΠT12,t+1 Bs Ht+1 Bs ΠC,t+1 As

38

and T Kt+1 Vt+1 Kt+1

and

 0 ATs Π12,t+1 ATs Π11,t+1 Bs 1 ⊥T =  ΠT12,t+1 Bs V22,t+1 + 12 T$T  2P 0 0 0  −1 T −1 T −1 T$ Bs ΠC,t+1 As − 12 Ht+1 Bs Π12,t+1 V22,t+1 Ht+1  ⊥ 0 (γ − 1)Σεε P      −1 T V22,t+1 Π12,t+1 As −  −1 −1 −1 T I + 12 V22,t+1 + V22,t+1 ΠT12,t+1 Bs Ht+1 Bs Π12,t+ −1 −1 T ΠT12,t+1 Bs Ht+1 Bs ΠC,t+1 As V22,t+1       T −1 T 1 T $ A Π + T V As ΠC,t+1 Bs Ht+1 Bs ΠC,t+1 As + 12,t+1 22,t+1 2 s  −1 −1 T 1 T ΠT12,t+1 As ATs Π12,t+1 V22,t+1  2 As ΠC,t+1  Bs Ht+1 Bs Π12      (γ−1) ⊥T −1 =  ΠT12,t+1 As Σεε P ⊥ + V22,t+1 + 12 T$T V V22,t+1 + 12 T$T V22,t+1  2 P  −1 −1 T −1 T 1 $T −1 T  − 12 T$T V22,t+1 ΠT12,t+1 Bs Ht+1 Bs ΠC,t+1 As 4 T V22,t+1 Π12,t+1 Bs Ht+1 Bs Π12,t+1 V 0 



(γ

−1 − 1)ATs ΠC,t+1 Bs Ht+1

0



−1 − ATs Π12,t+1 V22,t+1 −1 T Π B (γ − 1)A s C,t+1 s Ht+1  −1 −1 I + 12 T$T V22,t+1 + (γ − 1)V22,t+1 ΠT12 0

  T Vt+1 Lt+1 = − (γ−1) $T −1 Kt+1 −1  − 2 T V22,t+1 ΠT12,t+1 Bs Ht+1 (γ − 1)Σεε P ⊥ 0 0   T Bs (Φ1,t+1 − Π11,t+1 µs ) 1   2Pc 

−  T T 1 ⊥T T Φ2,t+1 − 2 P δ 0 − δ 1 P c − Π12,t+1 µs     −ATs ΠC,t+1 0 −1 T ⊥T Σ P c  + (γ − 1)  1 T $T V −1 ΠT  Bs Ht+1 Bs (ΦC,t+1 − ΠC,t+1 µs ) + =  (γ−1) εε 22,t+1 12,t+1 2 2 P 0 0   T    −As Π12,t+1   1 ⊥T
T   −1 T T 1 $T P T Φ − V + − − δ µ δ P c − Π V   22,t+1 22,t+1 2,t+1 0 1 12,t+1 s 2 2 0 We can now expand the relevant terms in expression (51). We first look at the (2,1) and (2,2) block in the Ricatti equation T Vt+1 Kt+1 Πt = AT Πt+1 A + C1T RC1 − Kt+1

which on substitution gives

1 ⊥T 1 (γ − 1) $T −1 −1 −1 T ΠT12,t+1 As + Bs ΠC,t+1 As T V22,t+1 ΠT12,t+1 Bs Ht+1 P Cs − T$T V22,t+1 2 2 2
1 1 γ (γ − 1) $T −1 −1 T −1 −1 T V22,t+1 ΠT12,t+1 Bs Ht+1 T$− T$T V22,t+1 T$− P ⊥T δT0 + δ = − P ⊥T Σεε P ⊥ − Bs Π12,t+1 V22,t+1 2 4 4 2

ΠT12,t+1 = Π22,t

We now do the same for the (2,1) block of the parameter Φt . This gives us iteration of

39

the 
 (γ − 1) ⊥T 1 1 2 1 σ + κs − Σεε P c + Φ2,t+1 − ΠT12,t+1 µs + P Σεε P c 4 2 2 2 (γ − 1) $T −1 −1 T + T V22,t+1 ΠT12,t+1 Bs Ht+1 Bs (ΦC,t+1 − ΠC,t+1 µs ) 2       1 $T 1 ⊥T
T −1 T T δ 0 − δ 1 P c − Π12,t+1 µs − V22,t+1 + T V22,t+1 Φ2,t+1 − P 2 2  
 1 2 1 = − P ⊥T σ + κs − γΣεε P c− δ T0 − δ T1 P c 2 2 (γ − 1) $T −1 −1 T T V22,t+1 ΠT12,t+1 Bs Ht+1 + Bs (ΦC,t+1 − ΠC,t+1 µs ) 2     1 ⊥T
T 1 $T −1 T T δ 0 − δ 1 P c − Π12,t+1 µs Φ2,t+1 − P − T V22,t+1 2 2

Φ2,t = −P ⊥T



Substituting these expansions into our expression for the optimal portfolio in equation (51) gives    1
T ⊥T δ + δ1 + T P ⊥ V22,t = Π22,t − P 2 1 γ (γ − 1) $T −1 1 −1 T −1 −1 T V22,t+1 ΠT12,t+1 Bs Ht+1 = − P ⊥T Σεε P ⊥ − Bs Π12,t+1 V22,t+1 T$− T$T V22,t+1 T$− 2 4 4  
  1
T 1 − P ⊥T δ T0 + δ 0 + 2T P ⊥ − P ⊥T δ1 + δ1 + T P ⊥ 2 2 1 γ (γ − 1) $T −1 −1 T −1 −1 T V22,t+1 ΠT12,t+1 Bs Ht+1 T$− T$T V22,t+1 T$−T$T − T$ = − P ⊥T Σεε P ⊥ − Bs Π12,t+1 V22,t+1 2 4 4  (γ − 1) T ⊥ −1 1 1 ⊥T −1 T −1 − T P V22,t+1 ΠT12,t+1 Bs Ht+1 Bs Π12,t+1 V22,t+1 P ⊥T T + T T P ⊥ V22 = − P γΣεε + T T + T + 2 2 2

40

and

      1
1 ⊥T
T T ⊥T T ⊥ P ⊥T wt−1 = + P δ0 − δ1 P c+ P δ1 + δ o + T P 2 2  1 ⊥T 1 $T −1 (γ − 1) $T −1 1 ⊥T 1 2 −1 T T T P Cs st − T V22,t+1 Π12,t+1 As st + σ + κs − γΣεε P T V22,t+1 Π12,t+1 Bs Ht+1 Bs ΠC,t+1 As st + P 2 2 2 2 2    (γ − 1) $T −1 1 $T −1 1 ⊥T
T −1 T T T T V22,t+1 Π12,t+1 Bs Ht+1 Bs (ΦC,t+1 − ΠC,t+1 µs ) + T V22,t+1 − Φ2,t+1 − P δ 0 − δ 1 P c − ΠT1 2 2 2       1 ⊥T
T 1
T ⊥T T ⊥ + P δ 0 − δ 1 P c+ P δ1 + δo + T P P ⊥T wt−1 2 2   1 ⊥T 1 2 (γ − 1) $T −1 −1 T = P Bs (ΠC,t+1 (As st + µs ) − ΦC,t+1 ) T V22,t+1 ΠT12,t+1 Bs Ht+1 Cs st + κs + σ − γΣεε P c + 2 2 2       ⊥  1 ⊥T
T 1 ⊥T
1 $T −1 T T T δ0 − δ1 P c + δ 1 + δ o + 2T P − T V22,t+1 Π12,t+1 (As st + µs ) − Φ2,t+1 − P P P ⊥T wt− 2 2 2   1 1 −1 −1 T Cs st + κs + σ2 − γΣεε P c + (γ − 1)T T P ⊥ V22,t+1 = P ⊥T ΠT12,t+1 Bs Ht+1 Bs (ΠC,t+1 (As st + µs ) − ΦC,t+1 2 2      1 ⊥T
T T ⊥ −1 T T ⊥ ⊥T

δ0 − δ1 P c +T P P wt−1 −T P V22,t+1 Π12,t+1 (As st + µs ) − Φ2,t+1 − P 2     1 1 −1 −1 T Cs st + κs + σ2 − γΣεε + T P c + (γ − 1)T T P ⊥ V22,t+1 = P ⊥T ΠT12,t+1 Bs Ht+1 Bs (ΠC,t+1 (As st + µs ) − 2 2      1 ⊥T
T T ⊥ −1 T T

−T P V22,t+1 Π12,t+1 (As st + µs ) − Φ2,t+1 − P δ0 − δ1 P c +T wt−1 2 ΠT12,t+1 st −Φ2,t

A.4

Calculating the performance of a linear portfolio construction rule

As in the proof of Theorem 1, we shall assume that the acheived utility has the following quadratic form   

t

t xt − 2Φ

Tt xt + φ Ut = Wt1−γ Et exp (1 − γ) xTt Π = Wt1−γ Et exp ((1 − γ)Qt )

(96) (97)

where the tildes are used to differentiate the parameters from those associated with the value function. Again we can use the time-separability of the utility to write this expression as Ut = Wt1−γ Et exp ((1 − γ) (Ct + Qt+1 ))

(98)

where the tildes are used to differentiate the parameters from those associated with the value function. We can now re write the expression on the right hand side of equation (98)

41

by substituting in from the state-space equations (29) and (30)   T    xt A C1T    

t+1 A B1 B2   ξ t+1  Ct +Qt+1 =  ξ t+1   D1T  R C1 D1 D2 +  B1T  Π T T ∆ut+1 D2 B2 ∆ut+1   xt 
   T   T T T

 2 S − κ R C1 D1 D2 + Φt+1 − µ Πt+1 ξ t+1  A B1 B2 ∆ut+1   
T

Tt+1 − µTt+1 Π

t+1 µ + φ − 2S − κT R κ − 2Φ t+1 

xt

T 

as before. We have assumed that the portfolio construction rules is linear in the states and thus can be written as   x  t

1 K

2 +L ∆ut+1 = K ξ t+1

Now define the following transformed output matrices

1 = C1 + D2 K

1 C

= A + B2 K

1 A

κ

= κ + D2 L

Substituting this into the  T  xt Ct + Qt+1 = ξ t+1  

T R − 2 ST − κ

1 = D1 + D2 K

2 D

1 = B1 + B2 K

1 B

µ

= µ + B2 L

expression above implies that         x 

T

T C A t 1

t+1 A

1 D

B

1

1 + R C Π T T

ξ D1 B1 t+1      x  t

Tt+1 − µ

t+1

1 D

B

1

1 + Φ

T Π C A ξ t+1    

t+1 µ

Tt+1 − µ

T R κ

− 2Φ

T Π

+ φt+1 − 2S T − κ

We can now proceed in a very similar way to the proof of Lemma 5. Define the following matrices             0

T

11 V 12

T 0 C V A 1



V = T V

22 = D

T R C1 D1 + B

T Πt+1 A B1 + 0 1 ((γ − 1)Σ)−1 V 12 2 1 1        

t+1

Tt+1 − µ % =

1 D

1 + Φ

B

1 %2 = S T − κ %1 W

T R

T Π W C A W and define the affine transformation of the the innovation ξ t+1 as   −1 T %2 + ψ ξ t+1 = −V 22 V12 xt − W

The integral can now be rewritten as Et exp ((1 − γ) (Ct + Qt+1 )) = 

T %2 V −1 V 12 %1 − W −2(1−γ) W 22



#

1

   T −1 −1 T V22 V12 x exp (1 − γ)x V 11 − V 12

(2π)n/2 |Σ|1/2     %2T − 2S T − κ

t+1 µ %2 V −1 W

Tt+1 − µ x+(1−γ)W

+

T R κ

− 2Φ

T Π 22  T

φ t+1 + (1 − γ)ψ V22 ψ dψ 42

All but the last term are independent of ψ and so the expression can be integrated and rearranged as before. We can therefore derive the following iterative procedure for calculating

t

t and φ

t, Φ the parameters Π

T

t = A

t+1 A

+C

1T RC

T Π

1 − V 12 V −1 V 12 Π 22      −1 T

t = C

1T (S − R

T Φ

t+1 µ

1T Φ

t+1 µ

t+1 − Π

t+1 − Π Φ κ) + A

− V 12 V 22 κ) + B

D1 (S − R " " 1 " "

log |2 (γ − 1) Σ| "V 22 " − κ κ) −

T (2S − R φt = φt+1 + 2 (γ − 1)  

t+1 µ %2 V −1 W %2T

t+1 − Π

−W µ

T 2Φ 22

This can be substituted back into the orginal expression (96) to derive the acheived performance of this portfolio construction rule.

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