## Abstract

The multi-period portfolio selection problem is formulated as a Markowitz mean–variance optimization problem in terms of time-varying means, covariances and higher-order and intertemporal moments of the asset prices. The crux lies in expressing the number of shares of any particular asset to be transacted on any future trading date, which is a non-anticipative process, as a polynomial of the changes in the discounted prices of all the risky assets. This results in the expected return of the portfolio being dependent on not only the means of the asset prices, but also the higher-order and intertemporal moments, and its variance being dependent on not only the second-order moments, but also the higher-order and intertemporal moments. As illustrations, we study the portfolio selection problems including the discrete version of the Merton problem. It is shown numerically that the efficient frontier obtained from Markowitz's discrete multi-period formulation coincides with that from Merton's continuous-time formulation when the number of rebalancing periods is ‘large’. The effects of dynamic trading, in particular volatility pumping, in comparison with a static single-period model are measured by a non-dimensional number, Dyn(*P*) (*P*, number of trading periods), which quantifies the relative gain due to dynamic trading. It is sufficient to rebalance the portfolio a few times in order to get more than 95% of the gain due to continuous trading.

## 1. Introduction

In his seminal work, Markowitz (1952, 1987) illustrates how the mean–variance principle is used to generate the admissible subset of investments for risk-averse individuals by eliminating any investment that has a lower mean and higher variance than a member of the given set of investment alternatives. The mean–variance approach has maintained its popularity because it is typically much more economical to trace out a mean–variance efficient set than to maximize the expected utility of terminal wealth. For example, depending on the utility function and the form of the joint distribution for the assets assumed, maximization of the expected utility of terminal wealth for a few securities can be as expensive as tracing out a mean–variance efficient set for thousands of securities. The crux of the mean–variance principle lies in the fact that, when linear inequality constraints are present, the optimal solution is a piecewise linear function of the parameter that defines risk aversion, known as the critical lines, which is the solution to certain affine portfolio selection problems related to the original problem. Karush–Kuhn–Tucker conditions are invoked to find the bounds for the parameter that defines risk aversion. Corresponding to each interval for the parameter that defines risk aversion, the affine portfolio selection problem depends upon the inequality constraints that are active. For the general mean–variance theory and its numerical implementation, we refer to Markowitz (1987) and Best (1996, 1999). A number of researchers have extended and refined the original model to include transaction costs, trading size and turnover constraints, sensitivity analysis to changes in mean return of the risky assets and other practical requirements (Perold 1984; Markowitz & Perold 1989; Best & Grauer 1991; Konno & Yamazaki 1991; Markowitz *et al.* 1993; Grinold & Hahn 2000; Konno & Wijayanayake 2001; Steinbach 2001). The mean–variance principle makes use of only the first two moments that characterize the random behaviour of the risky assets, and it addresses the investor's asset allocation problem for an investment horizon of only one period. The objective of this paper is to address these drawbacks.

Pioneering work has been carried out by Tobin (1965), Mossin (1969), Samuelson (1969), Fama (1970) and others on discrete-time multi-period models. There are three general frameworks for finding optimal decisions over the planning horizon: (i) the stochastic optimal control theory and the martingale approach (Merton 1969, 1971, 1973, 1996; Harrison & Kreps 1979; Karatzas *et al.* 1986; Pliska 1986; Ingersoll 1987; Cox & Huang 1989; Brennan *et al.* 1997; Campbell & Viceira 2002), (ii) the capital growth theory (Hakansson 1970; Fernholz & Shay 1982; Hakansson & Ziemba 1995) and (iii) the stochastic programming approach (Mulvey & Vladimirou 1992; Zenios 1993; Carino *et al.* 1994; Birge & Louveaux 1997; Maranas *et al.* 1997; Carino & Ziemba 1998; Consigli & Dempster 1998; Kipcak 2001; Darius *et al.* 2002; Dempster & Thompson 2002; Dempster *et al.* 2003*a*,*b*; Mulvey & Simsek 2005). By maximizing the growth rate of a portfolio that consists of many risky assets that are generated by correlated geometric Brownian motions, Luenberger (1998) unveils the dramatic effects of volatility pumping in the dynamic set-up. Hakansson & Ziemba (1995) review the capital growth literature. Using network models in the multi-period asset allocation problem, Mulvey & Simsek (2005) report that the traditional approach of rebalancing assets to a fixed strategic benchmark (which is given exogenously) generates higher returns when assets possess increased volatility. Ziemba & Vickson (1975) have collected numerous classic articles on both static and dynamic models of portfolio selection. Recently, Ziemba & Mulvey (1998) have edited a collection of articles on asset and liability modelling. Mulvey & Ziemba (1998) overview all these approaches and discuss the issues in asset and liability management. While the stochastic programming approach, the stochastic optimal control theory and the martingale approach address the modelling limitations of the mean–variance principle, the number of assets in the portfolio, the need for the optimal trading strategy for different values of the risk-aversion parameter and the nature of completeness of the market limit these approaches.

The objective of this paper is to formulate the multi-period portfolio selection problem as a Markowitz mean–variance optimization problem in terms of time-varying means, covariances and higher-order and intertemporal moments of the asset prices. The goal is to obtain approximate analytical solutions for the optimal trading strategies when there are no restrictions on short sales. In §2, the dynamic asset allocation problem is formulated. In §3, an approximate solution is obtained for the optimal trading strategy when the portfolio consists of one risky asset and one risk-free asset. The crux lies in expressing the number of shares to be transacted on any future trading date, which is a non-anticipative process, as a polynomial of the change in the discounted price of the risky asset. As illustrations, we study the discrete versions of the allocation problems when the portfolio consists of one risk-free asset and one risky asset whose dynamics follow: (i) a geometric Brownian motion, (ii) a generalized geometric Brownian motion in which the conditional rate of return is a known deterministic function of time, and (iii) a geometric Ornstein–Uhlenbeck process. For small values of risk, we show numerically that the efficient frontier obtained from Markowitz's discrete multi-period formulation coincides with that obtained from Merton's continuous-time formulation when the number of rebalancing periods is ‘large’. The effects of dynamic trading, in particular volatility pumping, in comparison with a static single-period model are measured by a non-dimensional number, Dyn(*P*) (*P*, number of trading periods), which quantifies the relative gain due to dynamic trading. Section 4 deals with the Markowitz mean–variance formulation when the portfolio consists of many risky assets. As an illustration, we study the discrete version of the Merton problem that contains one risk-free asset and many risky assets whose dynamics are generated by correlated geometric Brownian motions. Section 5 briefly deals with a trading strategy that incorporates the observed state variables driving the economy. Section 6 concludes the paper. In appendix A, we derive the efficient frontier for the Merton continuous-time portfolio selection problem and its linear approximation. In appendix B, we derive the efficient frontier for the Merton problem when the market parameters are known deterministic functions of time.

## 2. Formulation

The assets are bought at time *t*=*t*_{0}=0 with the available wealth *W*(*t*_{0}). This leads to the budget constraint given by (table 1)(2.1)We assume that rebalancing takes place instantaneously at discrete times, *t*_{j}, for *j*=1, 2, …, *M*. The value of the portfolio *W*(*t*_{j}) at *t*=*t*_{j} before the transactions is given by(2.2)and the value of the portfolio *W*′(*t*_{j}) at *t*=*t*_{j} after the transactions is given by(2.3)The conservation of wealth at any trading date *t*=*t*_{j} leads to(2.4)At the end of the investment period *t*=*t*_{M+1}, the value of the portfolio *W*(*t*_{M+1}) is given by(2.5)The time-zero prices of the assets *S*_{i,0}, *i*=0, 1, …, *N*, are deterministic and future prices of the assets *S*_{i,j}, *i*=0, 1, …, *N*, *j*=1, 2, …, *M*+1, are random from the time-zero perspective. The number of shares to be bought at time zero, denoted by *U*_{i,0}, *i*=0, 1, …, *N*, is deterministic, whereas the number of shares to be held at future trading dates, denoted by *U*_{i,j}, *i*=0, 1, …, *N*, *j*=1, 2, …, *M*, is random from the time-zero perspective.

We assume that short sales are allowed on all assets, and the portfolio consists of a risk-free asset identified by the index *i*=0. The price behaviour of the risk-free asset, {*S*_{0,0}=1, *S*_{0,1}, …, *S*_{0,M+1}}, is known deterministically and these values act as reference prices. The terminal wealth *W*(*t*_{M+1}), given by equation (2.5), is written as(2.6)where is the value of the portfolio discounted back to time zero. Define the rate of net profit of the investment discounted back to time zero, *R*, by(2.7)Let *α*_{i,j} denote the number of shares of the *i*th risky asset transacted at time *t*=*t*_{j}, i.e.(2.8)If the *i*th risky asset is bought (sold) at a trading date *t*=*t*_{j}, then *α*_{i,j} is positive (negative). With this notation, the rate of profit *R* can be written as(2.9)

The goal of an investor is to find the trading strategy,which maximizes the expected value of the portfolio return, *R*, for a fixed variance of *R*, or minimizes the variance of *R* for a fixed expected return.

Draviam & Chellathurai (2002) formulate the Markowitz mean–variance principles assuming that the number of shares invested in each risky asset is deterministic at all future trading dates. In reality, the number of shares of the *i*th asset to be transacted at future trading dates, *α*_{i,j} for *j*>0, must be a non-anticipative process of the underlying random processes of the asset prices and the observed state variables that drive the economy. When the trading strategy is dynamic and dependent on the asset prices, it is possible to ‘buy low and sell high’ by using the volatility of the stocks in a pumping action (Fernholz & Shay 1982; Luenberger 1998; Mulvey & Simsek 2005). This is not the case in the single-period model, as we will see in subsequent sections.

In the subsequent sections, the number of shares to be transacted on future trading dates is expressed as a function of the asset prices in an appropriate fashion so that the model avoids looking into the future. This feedback policy (Kendrick 1981) enhances the performance of the portfolio, as will be shown in subsequent sections. In §3, we study the portfolio selection problem when the portfolio consists of one risky asset and in §4 the asset allocation problem when the portfolio consists of many risky assets.

## 3. Markowitz mean–variance principle (one risky asset and one risk-free asset)

In this section, we study the portfolio selection problem when the portfolio consists of one risky asset and one risk-free asset. When *N*=1, equation (2.9) is written as(3.1)The number of shares, *α*_{1,j}, to be transacted at future time *t*=*t*_{j}, *j*=1, 2, …, *M*, is a non-anticipative process, and we write *α*_{1,j} as a polynomial of the difference between asset prices at times *t*=*t*_{j} and *t*_{j−1}, i.e.(3.2)where are the unknown deterministic parameters to be determined and *K* is the highest degree of the polynomial used to fit *α*_{1,j}. The number of shares transacted at time *t*=*t*_{j}, as shown in equation (3.2), allows us to select different strategies depending upon whether the discounted asset price at time *t*=*t*_{j}, , is greater or less than the discounted asset price at time *t*=*t*_{j−1}, (Samuelson 1991). The trading strategy, *α*_{1,j}, depends on prices at times *t*=*t*_{j} and *t*_{j−1} only and is otherwise path independent.

The nature of the functional form for the number of shares traded depends upon the complexity of the problem. The functional form for the trading strategy, given by equation (3.2), implicitly assumes that the trading strategy is not dependent on the state variables that drive the dynamics of the asset price. Merton (1973), in the continuous-time set-up, has shown that the solution to a long-term portfolio choice problem can be different to that of a short-term problem, if investment opportunities are varying over time. Long-term investors should include intertemporal hedging portfolios that protect from adverse changes in state variables. This is elaborated further in §5 in the multi-risky assets context.

The zeroth order, *K*=0, corresponds to the state-independent dynamic trading strategy for *α*_{1,j}; *K*=1 corresponds to the linear-state-dependent dynamic trading strategy for *α*_{1,j}; *K*=2 corresponds to the quadratic-state-dependent dynamic trading strategy for *α*_{1,j}; and so on. Using equations (2.8) and (3.2), we get(3.3)Note that the number of shares held in the risky asset at time *t*=*t*_{m}, denoted by *U*_{1,m}, is a function of asset prices up to time *t*=*t*_{m} only. Let(3.4)and(3.5)Let , *j*=1, 2, …, *M*, be a (*K*+1)-dimensional random vector, whose components are functions of discounted asset prices. Let , *j*=1, 2, …, *M*, be a (*K*+1)-dimensional vector of unknown parameters. Let be a random vector of dimension *M*(*K*+1)+1 and .

With this notation, the rate of profit *R* given by equation (3.1) is written as(3.6)Taking the expected value of *R*, equation (3.6) leads to(3.7)Similarly, the variance of *R* is written as(3.8)where *C*=*E*{[*Q*−*E*(*Q*)][*Q*−*E*(*Q*)]^{T}} is an *M*(*K*+1)+1-dimensional square matrix. Defining, for *j*=0, 1, …, *M*, *m*=0, 1, …, *M*,(3.9)the matrix *C* is written as(3.10)where the general covariance matrix *C* is composed of a scalar Cov(*Q*_{0}, *Q*_{0}); *K*+1-dimensional row vectors Cov(*Q*_{0}, *Q*_{j}), *j*=1, 2, …, *M*, and their transposes; and block matrices Cov(*Q*_{j}, *Q*_{m}), *j*=1, 2, …, *M*, *m*=1, 2, …, *M*, of dimension (*K*+1)×(*K*+1). In addition, Cov(*Q*_{j}, *Q*_{m})={Cov(*Q*_{m}, *Q*_{j})}^{T} and *C* is symmetric and positive semi-definite. The matrix *C* contains intertemporal (linear and higher-order) correlations in addition to the moments of different orders at all future trading dates.

Note that the expected return, *E*(*R*), depends on not only the means of the asset prices, but also the higher-order moments. The variance, Var(*R*), depends on the second-order moments and also on the higher-order moments. The order of the polynomial depends on the available information about moments of the asset prices. Only second-order, including intertemporal, moments are used when *K*=0; up to fourth-order moments are used when *K*=1; up to sixth-order moments are used when *K*=2; and so on. This remarkable dependence of the expected return of the portfolio and its variance on the (higher-order) moments of the asset prices is due to the possibility of multi-period trading. This aspect is totally absent in the single-period Markowitz model. In this context, it is useful to note the comments of Luenberger (1998. p. 417): ‘Conclusions about multi-period investment situations are not mere variations of single-period conclusions—rather they often reverse those earlier conclusions’.

In terms of the Markowitz mean–variance principle, the goal is to minimize(3.11)where *θ*(>0) denotes the non-dimensional parameter that defines risk tolerance of the investor. The first-order necessary condition for optimality leads to(3.12)If *C* is positive definite, the optimal solution, *A*_{opt}, is given by(3.13)The expected value of optimal *R*, *E*(*R*_{opt}), and its variance, Var(*R*_{opt}), are given byThese lead to the efficient frontier(3.14)where the non-dimensional number, called the Markowitz number, defined by(3.15)characterizes the slope of the efficient frontier and it depends on the time-varying moments of the risky asset. The efficient frontier is a straight line in (Std(*R*_{opt}), *E*(*R*_{opt})) space, where Std(*R*_{opt}) denotes the standard deviation of the optimal return.

There are no analytical solutions for the portfolio selection problems when state variables that influence the individual returns are used to describe the time-varying nature of the economy. However, when the investment opportunity set is independent of state variables, there are analytical solutions for selected portfolio selection problems (Merton 1969, 1971) and these are used as benchmark problems to test and illustrate the multi-period Markowitz formulation. In the following, we assume that the portfolio consists of one risk-free asset and one risky asset, and the price dynamics of the risky asset is generated by: (i) a geometric Brownian motion, (ii) a generalized geometric Brownian motion in which the conditional rate of return is a known deterministic function of time, and (iii) a geometric Ornstein–Uhlenbeck process, so that the price process is Gaussian.

### (a) Discrete version of the Merton asset allocation problem

We study the discrete version of the Merton dynamic portfolio selection problem (Merton 1969, 1971). In appendix A, the Merton continuous-time portfolio selection problem with many risky assets is briefly described. The efficient frontier is derived in terms of the expected return, *E*(*R*_{opt}), and the standard deviation of the optimal portfolio, Std(*R*_{opt}). It is shown that the efficient frontier depends on a non-dimensional number, called the Merton number, Mert, which is a function of all market parameters, and *T*, the planning horizon of the investor. In the neighbourhood of (Std(*R*_{opt})=0, *E*(*R*_{opt})=0), the efficient frontier is approximated by a straight line whose slope is solely determined by the square root of the Merton number (see appendix A). In this section, we assume that the portfolio consists of only one risk-free asset and one risky asset (see appendix A for notational details). For small values of risk, we show numerically that the efficient frontier obtained from Markowitz's discrete formulation coincides with that from Merton's continuous-time formulation when the number of rebalancing periods is large.

To calculate the moments required for the multi-period Markowitz mean–variance principle, the analytical solution for the dynamics of the risky asset given by(3.16)is used. The prices of the risky asset and the risk-free asset at any discrete time *t*=*t*_{j} are given byrespectively. Using the properties of the Brownian motionthe moments are calculated usingwhere *ξ*^{T}=(*m*_{1}, *m*_{2}, …, *m*_{J}) is a vector of exponents and *D* is the *J*×*J* covariance matrix whose (*i*,*j*)th entry is given by *D*(*i*,*j*)=min(*t*_{i},*t*_{j}).

Numerical results are presented for a test problem with two planning horizons, *T*=*t*_{M+1}−*t*_{0}=1 (year) and *T*=*t*_{M+1}−*t*_{0}=2 (years). We assume the following values for the parameters:We consider four values for the degree of the polynomial (*K*=0, 1, 2, 3) used to fit the number of shares transacted, *α*_{1,j}, at time *t*=*t*_{j}. Table 2 shows the slopes of the Markowitz efficient frontiers, , for different values of the order of the polynomial used, *K*, and the number of trading periods, *M*+1. For example, the slope of the Markowitz efficient frontier is 0.0941, when *M*+1=1, *T*=1 and *K*=0 and is 0.1252 when *M*+1=1, *T*=2 and *K*=0. The last column in table 2 gives the corresponding slopes of the continuous-time Merton efficient frontiers, , for the two planning horizons. The slope of the Markowitz efficient frontier is enhanced when the linear state-dependent trading strategy (*K*=1) is considered over the no-state-dependent trading strategy (*K*=0). The third- and fourth-order moments improve the slope of the Markowitz efficient frontier; the improvement is marginal when higher-order state-dependent strategies are considered. The slopes of the Markowitz efficient frontiers almost coincide with those of Merton with a finite number of trading dates (*M*+1=256) and *K*=3.

#### (i) Comparison between Markowitz and Merton efficient frontiers

When the portfolio consists of one risky asset and one risk-free asset, the slope of the continuous-time Merton efficient frontier (see equations (A 9) and (A 11) of appendix A) is given by(3.17)The objective is to see whether this relation holds good in the case of multi-period Markowitz mean–variance principles, when the number of trading dates is large. To do this, we select parameters such that the slope of the Merton efficient frontier, , is constant, say, . The number of trading periods is 128 and the order of the polynomial used to fit the trading strategy is two, i.e. *M*+1=128 and *K*=2.

Table 3 shows the slopes of the Markowitz efficient frontiers, , for different values of the volatility of the risky asset (*σ*_{1}) and the interest rate (*r*), with a fixed planning horizon *T*=1 (year). The conditional rate of return, *μ*_{1}, is selected such that it satisfies equation (3.17) with . Similarly, table 4 shows the slopes of the Markowitz efficient frontiers, , for different values of the volatility of the risky asset (*σ*_{1}) and planning horizons (*T*), with a fixed interest rate *r*=0.03(1/year). Again, the conditional rate of return, *μ*_{1}, is selected such that it satisfies equation (3.17) with . The slopes of the discrete-time Markowitz efficient frontiers almost coincide with the slope of the continuous-time Merton efficient frontier with a finite number of trading dates (*M*+1=128).

#### (ii) Effects of dynamic trading and volatility pumping

Multi-period models provide advantages over single-period approaches (Carino *et al.* 1994; Consigli & Dempster 1998; Mulvey & Ziemba 1998; Dempster *et al.* 2003*a*,*b*). To measure the effects of dynamic trading in comparison with static approaches, we define a non-dimensional number(3.18)where denotes the slope of the Markowitz efficient frontier when the number of trading periods is *P*. The non-dimensional number, Dyn(*P*), quantifies the relative gain due to dynamic trading over its static counterpart when the parameters of the problem are unchanged.

Table 5 shows the slopes of the Markowitz efficient frontiers for different values of the volatility of the risky asset and different numbers of trading periods. The values of the other parameters are: the planning horizon *T*=2(year), the conditional rate of return *μ*_{1}=0.09(1/year), the interest rate *r*=0.05(1/year) and the order of the polynomial used to fit the trading strategy *K*=2. The slopes of the continuous-time Merton efficient frontiers () are also shown. The last row in table 5 shows the effects due to dynamic trading, quantified by Dyn(256). Gains due to dynamic trading increase as the volatility increases. The gain is achieved by using the volatility of the stock in a pumping action. As Luenberger (1998) notes, volatility is an opportunity and is not the same as risk in the multi-period context. Dynamic investment strategies enhance the performance, especially for long-term investors. To save space, the details are not shown.

### (b) Conditional rate of return is a known deterministic function of time

In this section, we consider the discrete version of another continuous-time portfolio selection problem with a time-varying deterministic opportunity set. The portfolio consists of a risk-free asset with interest rate, *r*, and a risky asset whose price dynamics is generated by(3.19)where *S*(*t*) denotes the price of one share of the risky asset at time *t* and the conditional rate of return, *μ*(*t*), is a known deterministic function of time. The dynamics of the risky asset with a deterministic, time-varying expected return is implausible in reality. However, in the continuous-time set-up, this is one of the problems that can be solved analytically and hence can be calibrated with. Assume the conditional rate of return of the form(3.20)where *μ*_{1}(1/year) and *μ*_{d}(1/year) are the known deterministic constants and *ω* is the frequency of the conditional rate of return. For this selection of the conditional rate of return, the slope of the Merton efficient frontier is given by (see equation (B 5) of appendix B)(3.21)The slope of the Merton efficient frontier, , is independent of the frequency, *ω*, of the conditional rate of return.

#### (i) Comparison between Markowitz and Merton efficient frontiers

In the following, we show numerically that the slope of the efficient frontier obtained from the discrete mean–variance formulation, , coincides with the slope of the continuous-time Merton efficient frontier, , when the number of trading periods is large. We use the analytical solution for the dynamics of the risky asset given by(3.22)whereFor the particular selection of the conditional rate of return given by equation (3.20), we haveTable 6 shows the slopes of the Markowitz efficient frontiers for different values of the frequency (*ω*) of the conditional rate of return and the number of trading periods (*M*+1). The values of other parameters of the test problem are , *K*=2,the values are selected so that the slopes of the Merton efficient frontiers, .

The dramatic improvement in the slopes of the Markowitz efficient frontiers takes place only when the number of trading periods (*M*+1) coincides with twice the frequency (2*ω*) of the conditional rate of return. For *M*+1>2*ω*, the improvement in the slopes of the Markowitz efficient frontiers is marginal and with 128 trading periods, the slopes of the Markowitz efficient frontiers almost coincide with those of Merton. This behaviour is independent of the frequency, as shown by the continuous-time Merton analytical result (see equation (3.21)).

#### (ii) Effects of dynamic trading and volatility pumping

Table 7 shows the slopes of the Markowitz efficient frontiers for different values of the volatility of the risky asset (*σ*_{1}) and different numbers of trading periods (*M*+1). The values of other parameters of the test problem areThe effect of dynamic trading is dramatic, as can be seen from the values of Dyn(128), which increases as the volatility of the risky asset increases, indicating the nature of volatility pumping. The effect of volatility pumping is profound when the number of trading periods is relatively small and gradually decreases as the number of trading periods increases. It is sufficient to rebalance the portfolio a few times (eight times, in this case) to get more than 95% gain due to continuous trading.

### (c) Risky asset follows geometric Ornstein–Uhlenbeck process

Time-varying expected returns and intertemporal correlations between asset returns, due to changing business conditions, generate some predictability in asset returns (Summers 1986; Fama & French 1988; Lo & MacKinlay 1988; Poterba & Summers 1988; Lo 1991; Campbell *et al.* 1997). The variances of stock price changes do not grow linearly with time as they would if a true random walk held. Instead, in short periods they grow a bit faster than time showing weak positive autocorrelation. For longer periods of time, the autocorrelation becomes significantly negative. In this mean-reversion case, returns revert to an equilibrium value. If an asset is priced above its equilibrium value, it will be more inclined to decrease than to increase. Conversely, if an asset is priced below its equilibrium value, it will be more likely to increase than to depreciate further. Quoting Tobin, Samuelson (1991, p. 198) writes: ‘The market is not necessary macroefficient; instead it undergoes self-fullfilling waves above and below fundamentalist values’. Bessembinder *et al.* (1995) conduct analysis using data from financial, metal and agricultural markets. They conclude that the observed mean reversion is large for agricultural commodities and crude oil, small for metals relative to oil and agricultural markets and very weak for financial assets. Fama & French (1988) report that predictable variation is estimated to be approximately 40% of 3–5 year return variances of portfolios of small firms and falls to approximately 25% for portfolios of large firms.

This section considers a portfolio selection problem that consists of a mean-reverting risky asset and a risk-free asset with interest rate *r*. The dynamics of the risky asset is governed by(3.23)where *λ*(1/year) is the parameter that quantifies the rate of mean reverting and *β*_{0} and *β*_{1}(1/year) are the parameters that characterize the trend of the asset price dynamics. This geometric Ornstein–Uhlenbeck process, defined by equation (3.23), has been used by Merton (1969, 1996) to solve the continuous-time portfolio selection problem for a constant absolute risk aversion investor. In reality, there is less empirical evidence of univariate mean reversion in expected returns, but there is more evidence of multivariate mean reversion, which suggests that the drift of the return process depends on a vector of observed state variables. Another interesting and plausible case is that in which volatility is stochastic. As the complexity of the problem increases, to apply the Markowitz mean–variance principles, the time-varying means, covariances and higher-order and intertemporal moments need to be generated by Monte Carlo methods. However, to eliminate problems due to Monte Carlo methods, we consider the univariate geometric Ornstein–Uhlenbeck process, equation (3.23), for the risky asset so that the time-varying moments are calculated from the exact solution given by(3.24)where(3.25)The process *ϕ*(*t*) is Gaussian with mean(3.26)and covariance, for *s*<*t*,(3.27)We present the numerical results when the trend parameters, *β*_{0} and *β*_{1}, are fixed, say *β*_{0}=1 and *β*_{1}=0. This leads to, for the fixed planning horizon *T*, the slope of the Markowitz efficient frontier dependent only on the interest rate (*r*), the volatility of the risky asset (*σ*_{1}) and the rate of mean reverting (*λ*).

Figure 1 shows the surface plot of the slope of the Markowitz efficient frontier as a function of the volatility of the risky asset (*σ*_{1}) and the rate of mean reverting (*λ*). The values of the other parameters are: the planning horizon *T*=1.0(year), the interest rate *r*=0.05(1/year), the number of discrete-time trading periods *M*+1=64 and the order of the polynomial used to fit the trading strategy *K*=2. The slope of the Markowitz efficient frontier is V-shaped, and takes least values at approximately *λ*=0.05. The slope of the efficient frontier decreases as the volatility decreases, except near *λ*=0.05 (approx.).

Figure 2 shows the surface plot of the slope of the Markowitz efficient frontier as a function of the interest rate (*r*) and the rate of mean reverting (*λ*). The values of the other parameters are: the planning horizon *T*=1.0(year), the volatility of the risky asset , the number of discrete-time trading periods *M*+1=64 and the order of the polynomial used to fit the trading strategy *K*=2. Again, the slope of the Markowitz efficient frontier is V-shaped, and takes least values at *λ*=*r* (approx.). The slope of the Markowitz efficient frontier is not monotone with respect to the rate of mean reverting and the constant interest rate.

When the linear trend parameter, *β*_{1}, is not equal to zero, the slope of the Markowitz efficient frontier is a skewed V-shaped surface. The magnitude of the slope of the Markowitz efficient frontier is large when it is compared with its counterpart with no linear trend. The qualitative nature of the slope of the Markowitz efficient frontier does not change much, except that the skewed nature of the surface is reversed when there is a negative linear trend. To save space, the details are not shown here.

## 4. Many risky assets

In this section, we study the portfolio selection problem when the portfolio consists of many risky assets. For simplicity, we consider the linear state-dependent dynamic trading strategy, i.e.(4.1)where *V*_{i,j} and *i*=1, 2, …, *N*, *j*=1, 2, …, *M*, *k*=1, 2, …, *N*, are unknown parameters to be determined. We assume that the trading strategy is not dependent on the state variables and relax this assumption in §5 by incorporating the state variables into the trading strategy.

Substitution of (4.1) into (2.9) yields(4.2)Letbe an *N*-dimensional random vector and *U*_{0}=(*U*_{1,0},*U*_{2,0}, …, *U*_{N,0})^{T}∈*R*^{N}. Letbe an *N*+1-dimensional random vector whose components are functions of asset prices. Let , *i*=1, 2, …, *N*, *j*=1, 2, …, *M*, be a *N*+1-dimensional vector. Let and denote *N*(*N*+1)-dimensional vectors. Let and denote *N*+*NM*(*N*+1)-dimensional vectors. With this notation, equation (4.2) is written as(4.3)Taking the expected value of *R*, we get(4.4)Similarly, the variance of *R* is written as(4.5)where the ‘covariance matrix’,is given by(4.6)The general covariance matrix *C* is composed of block matrices: Cov(*P*_{0},*P*_{0}) is a matrix of dimension *N*×*N*, Cov(*P*_{0},*P*_{j}), *j*=1, 2, …, *M*, has *N* rows and *N*(*N*+1) columns and Cov(*P*_{j},*P*_{m}), *j*=1, 2, …, *M*, *m*=1, 2, …, *M*, has *N*(*N*+1) rows and *N*(*N*+1) columns. Cov(*P*_{j},*P*_{m})={Cov(*P*_{m},*P*_{j})}^{T}, *j*=0, 1, …, *M*, *m*=0, 1, …, *M* and *C* is symmetric and positive semi-definite. The matrix *C* contains intertemporal (linear and higher-order) correlations between assets. The vector *E*(*P*) contains moments of first and second order, and the covariance matrix *C* contains moments of order from two to four. In terms of the Markowitz mean–variance principle, the goal is to minimize(4.7)where *θ* denotes the parameter that defines risk tolerance of the investor. The first-order necessary condition for an interior minimum leads to(4.8)If *C* is positive definite, the optimal solution is given by(4.9)The optimal trading strategy is linear in the parameter that defines risk tolerance and the initial wealth.

As in §3, the efficient frontier is given by(4.10)where the non-dimensional number, called the Markowitz number,(4.11)characterizes the slope of the efficient frontier and depends on the time-varying moments of the asset prices.

### (a) Discrete version of the Merton asset allocation problem with many risky assets

In this section, we study the discrete version of the dynamic portfolio selection problem due to Merton (1969, 1971) with one risk-free asset and *N* risky assets. The Merton continuous-time portfolio selection problem with *N* risky assets is briefly described in appendix A, and the efficient frontier is derived in terms of the expected return, *E*(*R*_{opt}), and the standard deviation of the optimal portfolio, Std(*R*_{opt}). As stated in §3, in the neighbourhood of (Std(*R*_{opt})=0, *E*(*R*_{opt})=0), the efficient frontier is approximated by a straight line whose slope is solely determined by the square root of the Merton number (see appendix A). In the following, for small values of risk, we show numerically that the efficient frontier obtained from Markowitz's discrete-time formulation coincides with that obtained from Merton's continuous-time formulation when the number of rebalancing periods is large. In order to compute the moments of the risky assets, we use the analytical solution of the correlated geometric Brownian motions given by(4.12)where *i*=1, 2, …, *N*, *j*=1, 2, …, *N*.

#### (i) Two risky assets

This section presents the numerical results for the portfolio selection problem when the portfolio consists of two risky assets that are generated by correlated Brownian motions, with identical conditional rates of return (*μ*_{1}=*μ*_{2}) and volatilities (*σ*_{1}=*σ*_{2}), and the correlation coefficient between Brownian motions, *ρ*_{12}. In the continuous-time case, the slope of the Merton efficient frontier is given by (see equation (A 9) of appendix A)(4.13)The values of the parameters are

Table 8 shows the slopes of the Markowitz efficient frontiers for different values of the correlation between the two Brownian motions and the different number of trading periods. The values of the other parameters are: the rates of return *μ*_{1}=*μ*_{2}=0.09(1/year) and volatilities , the interest rate *r*=0.05(1/year) and the planning horizon *T*=1 (year). The slopes of the continuous-time Merton efficient frontiers are also shown. The slopes of the Markowitz efficient frontiers almost coincide with those of Merton when the number of trading periods is large (128 trading periods, in our case). The effect of dynamic trading is the greatest when there is negative correlation (see the values of Dyn(128) in table 8). It is also observed that the non-dimensional number Dyn(*P*) increases as the volatility increases. To save space, the details are not shown here.

#### (ii) Five risky assets

This section presents the numerical results for a hypothetical test problem when the portfolio consists of five risky assets with identical conditional rates of return (*μ*_{i}=0.09(1/year), *i*=1, 2, …, 5) and volatilities (*σ*_{i}=*σ*, *i*=1, 2, …, 5). The instantaneous correlation between Brownian motions is given by(4.14)

Table 9 shows the slopes of the Markowitz efficient frontiers for different values of the volatility of the risky assets and different numbers of trading periods. The values of the other parameters are: the conditional rates of return *μ*_{i}=0.09(1/year), *i*=1, 2, …, 5 and volatilities *i*=1, 2, …, 5, the interest rate *r*=0.05(1/year) and the planning horizon *T*=4(year). The slopes of the continuous-time Merton efficient frontiers are also shown. The effect of volatility pumping, demonstrated by the values of Dyn(32), is dramatic as the instantaneous correlation matrix between Brownian motions contains negative entries. It is possible to buy low and sell high leading to high volatility pumping.

## 5. Trading strategy incorporating the state variables

In §3, we have stated that the randomness displayed by asset prices can be attributed to a number of state variables, say, *X*_{1}, *X*_{2}, …, *X*_{L}. Merton (1973), in the continuous-time set-up, obtains mutual fund theorems and the equilibrium structure of expected returns when the dynamics of the investment opportunity set is described by many stochastic state variables, which are used to hedge against time-varying economic events. The state variables are generic in nature and Merton (1973) does not identify them. One approach to solving this identification problem is generally associated with the applications of Ross's arbitrage pricing theory (Burmeister *et al.* 2003). For a thorough discussion about the selection of state variables and factors, we refer to Chen *et al.* (1986) and Burmeister *et al.* (2003). Brennan *et al.* (1997) solve the strategic asset allocation problem, using stochastic optimal control theory, when the structure of expected returns is described by three state variables, namely the risk-free interest rate, the yield on the console bond and the dividend yield on the equity portfolio. The number of state variables is a limiting factor in the stochastic optimal control problem.

The observed state variables are not explicitly taken into account in the characterization of the trading strategy (see equations (3.2) and (4.1)). We can incorporate the observed state variables into the trading strategy just like the asset prices. For example, equation (4.1) can be modified to(5.1)where *X*_{n,j}, *n*=1, 2, …, *L*, are the observed state variables at time *t*=*t*_{j} and are the unknown deterministic parameters to be determined. This helps to hedge against time-varying economic events associated with the state variables. The additional terms that correspond to the state variables act as components of intertemporal hedging portfolios for long-term investors.

## 6. Conclusions

By formulating the multi-period portfolio selection problem as a Markowitz mean–variance optimization problem in terms of time-varying moments (of any order) of the asset prices, approximate analytical solutions for the optimal trading strategies are obtained. The crux lies in expressing the number of shares of any particular asset to be transacted on any future trading date, which is a non-anticipative process, as a polynomial of the changes in the discounted prices of all the risky assets and the changes in the state variables that drive the economy. This results in the expected return of the portfolio being dependent on not only the means of the asset prices at discrete times, but also the higher-order and intertemporal moments, and its variance being dependent on not only the second-order moments at discrete times, but also the higher-order and intertemporal moments. The order of the polynomial depends on the available information about the moments. If just the second-order, including intertemporal, moments on asset prices are given, a state-independent dynamic trading strategy is used. If up to fourth-order moments on asset prices are given, a linear state-dependent dynamic trading strategy is used. It has been shown that the optimal trading strategy is linear in the parameter that defines the risk tolerance, and the efficient frontier is a straight line in the expected return–standard deviation of the portfolio space.

To illustrate the efficacy of the multi-period Markowitz mean–variance principle, we have studied the discrete version of the continuous-time Merton portfolio selection problem. For small values of risk, we have shown numerically that the efficient frontier obtained from Markowitz's discrete multi-period formulation coincides with Merton's continuous-time formulation when the number of rebalancing periods is large. The second-order moments dominate the optimal strategy and the higher-order moments, characterized by dynamic and state-dependent trading strategies, enhance the performance of the portfolio. The effects of dynamic trading in comparison with a static single-period model are measured by a non-dimensional number, Dyn(*P*) (*P*, number of trading periods), which quantifies the relative gain due to dynamic trading. The effect of dynamic trading is profound when the number of trading periods is relatively small, and it gradually dampens down as the number of trading periods increases. It is sufficient to rebalance the portfolio a few times in order to get more than 95% of the gain due to continuous trading, and the actual number of rebalancings is problem dependent. The non-dimensional number Dyn(*P*) increases as the volatility of the risky asset increases, indicating the nature of volatility pumping; this pumping is dramatic when the portfolio contains assets that are generated by negatively correlated Brownian motions.

As another illustration, we have studied the discrete version of the portfolio selection problem when the portfolio consists of one risk-free asset, and one risky asset whose price dynamics follows a geometric Ornstein–Uhlenbeck process. It is found that the slopes of the Markowitz efficient frontiers are not monotone with respect to the rate of mean reverting and the constant interest rate.

## Acknowledgments

The authors are grateful to Prof. Harry M. Markowitz for his critical comments on an earlier draft of this paper. The authors dedicate this paper, with love and appreciation to their friends, Mr S. Peter Suganthan, Kavalkinaru, Tamil Nadu, India, and Ms Myriam Nia, Waterloo, Ontario, Canada, for their moral support and continuous encouragement.

## Footnotes

↵† Present address: Credit Risk Analytics, Treasury and Risk Management, Canadian Imperial Bank of Commerce, Toronto, Ontario, Canada M5L 1A2.

- Received February 3, 2007.
- Accepted December 13, 2007.

- © 2008 The Royal Society