Decentralized Portfolio Management

We use a mean-variance model to analyze the problem of decentralized portfolio management. We find the solution for the optimal portfolio allocation for a head trader operating in n different markets, which is called the optimal centralized portfolio. However, as there are many traders specialized in different markets, the solution to the problem of optimal decentralized allocation should be different from the centralized case. In this paper we derive conditions for the solutions to be equivalent. We use multivariate normal returns and a negative exponential function to solve the problem analytically. We generate the equivalence of solutions by assuming that different traders face different interest rates for borrowing and lending. This interest rate is dependent on the ratio of the degrees of risk aversion of the trader and the head trader, on the excess return, and on the correlation between asset returns.


Introduction.
The optimal allocation of investment resources to different assets is one of the most well-known problems in finance and has been extensively studied.The most frequent approach to solve this problem is the mean-variance analysis, which assumes that the investor is concerned with only two parameters of the probability distribution of total returns on investment: the mean and the variance.Markowitz [1959] and Tobin [1958] were the first to put the trade-off between risk and return on a solid analytic footing.Sharpe [1964] and Lintner [1965] extended the mean-variance theory to an equilibrium theory.
Mean-variance models solve the problem of optimal portfolio allocation for a head trader operating in n different markets.The solution to this problem is called the optimal centralized allocation.It is interesting to notice that if investors are von Neumann-Morgenstern expected utility maximizers, then a mean-variance analysis could be justified either by assuming quadratic utility functions or that the distribution of asset returns is multivariate normal.We could solve the problem analytically for a quadratic utility function, but, as Huang and Litzenberger [1988] observed, the quadratic utility function exhibits increasing absolute risk aversion, which implies that risky assets are inferior goods.This unappealing property induced the use of a negative exponential utility function.
In practice, most investment firms operate with a head trader leading a number of traders specialized in different markets.The head trader gives general guidelines.One of those guidelines is a common benchmark rate, the interest rate that each trader faces when drawing resources from the firm to invest in risky assets.Traders, however, solve their problem for optimal allocation in a decentralized way.The aggregation of the investment decisions by the specialized traders is called the optimal decentralized allocation.In general, the optimal decentralized allocation is different from the centralized one for a number of reasons: different risk aversion, different probability distributions of assets etc. Therefore the mean-variance analysis is of limited use for big investment firms that have many specialized traders.
Decentralization of the investment process has not been a major theme in the finance literature for the past decades.Sharpe [1981] has been one of the first researchers to notice this gap, stating that "more research on this subject is needed".
According to Sharpe [1981], the rationale for having multiple managers in charge of a single portfolio is the need for specialization and diversification.Specialization would be justified by the need of superior skills for the analysis of certain types of industries and diversification would be a natural way to hedge against bad decisions taken by a single manager.Barry and Starks [1984] analyzed the phenomenon of multiple managers using the principal-agent theory approach.They developed a model to show that risk-sharing considerations alone are sufficient to produce a decision to employ multiple agents (even in the absence of specialization and diversification).
In this paper, we are not concerned with the reasons for the existence of multiple managers.We investigate the conditions under which the centralized and decentralized solutions are equivalent.This is important because the mean-variance model could then be implemented by firms with decentralized investment decisions, as most large trading firms are.
We solve the problem with different, increasing levels of generalization.For all levels, we solve the optimal centralized and decentralized portfolio problem for n risky assets assuming that each agent has a negative exponential utility function and that rates of returns of risky assets are multivariate normally distributed, so the problem can be solved via a deterministic equivalent.The interest rate for borrowing or lending by the traders is used as control variable.We use Stevens' [1998] characterization of the inverse of the covariance matrix to get a closed-form optimal solution.
In the basic case, we suppose each trader trades in one risky asset and one riskless asset.Allowing the interest rate for the riskless asset to be different for different traders, we generate the equivalence of the centralized and decentralized solutions.Furthermore, we show that the trader's interest rate depends on the ratio of the degree of risk aversion of the trader and the head trader, on the excess return, and on the correlations between assets.
In the first generalization, we allow each trader to trade in m different risky assets, but there is an empty intersection between subsets of risky assets managed by traders.In the second generalization, we assume that traders manage the same subset of risky assets.Finally, we allow each trader to manage any subset of risky assets.
In Section 2 we find the optimal solutions for the head trader and the traders, and establish the equivalence condition for the basic case.In Section 3 we perform comparative-static analysis.Section 4 presents the generalizations.
In Section 5 we present a second best solution to the decentralization problem.We impose a general condition for global optimality, that the proportion of money invested in risky assets to the proportion of money invested in the risk-free asset be the same across managers and the head trader.In this case, managers can invest freely in risk assets according to their beliefs on the expected returns and covariance matrix, but the head trader defines the proportion of money invested in risky assets.The allocations defined in this section are similar to the ones presented in previous sections as the riskfree interest rate depends on risk aversion, expected returns, risk and correlations among assets in the economy.Nonetheless, in this particular case it is possible to solve the problem by using wealth instead of the risk-free interest rate, which seems to be more in line with current practice of trading firms.Section 6, summarizes the results and concludes.

The Equivalence of Centralized and Decentralized
Solutions: The Basic Case.
In this section we consider the case where each trader manages one risky asset and one risk-free asset.First we calculate the head trader centralized solution, where he manages all assets, making no use of specialized traders.Then we calculate the decentralized solution, where specialized traders manage risky assets.Since each trader is concerned only with the risk of one asset when making his investment decision, the traders' aggregate solution will usually not be compatible with the Markowitz (head trader) solution for the portfolio of all risky assets.We then calculate the conditions for the centralized and decentralized solutions to coincide.

Head Trader Centralized Solution.
We consider a head trader having an initial wealth W 0 to invest in nrisky assets a a risk-free asset f over a finite investment period.Let rj be the stochastic return of risky asset j, r f be the return of the risk-free asset, α j be the proportion of wealth invested in the j − th risky asset, and α f be the fraction of wealth invested in the risk-free asset during the investment period α f = 1 − n j=1 α j .Then (1 + rj ) α j W 0 is the return of the investment in the j − th risky asset at the end of the investment period.the head trader wishes to maximize the expected utility of his stochastic wealth: To have a closed form solution, we assume that returns on risky assets are multivariate normally distributed with mean vector r and covariance matrix .We solve the optimal portfolio problem for the head trader with a negative exponential utility function, U (W ) = 1 − e aW , where a is the strictly positive absolute risk aversion.In this case the head trader objective function can be written as: where f ( W ) is a normal density function.Under the assump-tions, the above function (2.1) has a certainty equivalent formulation given by * : max where α ′ = (α 1 , . . ., α n ) is a 1×n vector of risky assets weights, r = (r 1 − r f , r 2 − r f , . . ., r n − r f ) is a 1 × n vector of expected excess return, and The first-order conditons are: 3) we get an n-equation system: where r is the n × 1 vector of excess returns.

Traders' Decentralized Solution.
We assume that there are n traders; each one specialized in a different risky asset.The i − th trader has initial wealth W 0,i , allocated to him by the head trader, to invest in a risky asset i.Let α i,i be the fraction of his initial wealth invested in the risky asset, a i be his coefficient of risk aversion, r f,i be the risk-free rate the i − th trader pays the firm to allocate wealth in his risky asset.In other words, r f,i is the benchmark of the i − th trader.The use of different benchmarks for different traders is the instrument the head trader will use to make traders' decentralized solutions compatible with Markowitz's in the aggregate.The objective function for the i − th trader is: The first condition is: and the optimal solution for the risky asset is Expression (2.8) shows that the proportion of initial wealth invested in the risky asset is decreasing in wealth, an inconvenient byproduct of the hypothesis that traders have exponential utility functions.

Equivalence of Centralized and Decentralized Solutions.
The condition for the decentralized optimal choices (of the n traders) to be equivalent to the optimal centralized choice of the head trader is: (2.9) Solving for the risk-free rate for the i−th trader, we obtain Using Stevens [1998] characterization of the inverse of the covariance matrix we derive: where R 2 i and β ij are the coefficient of determination and the coefficients in the multiple regression of the excess return of the i−th asset on the excess returns of the other assets.The factor σ 2 i 1 − R 2 i is the part of the variance of the i − th return that cannot be explained by the regression on the other risky excess returns, which is equivalent to the estimate of the variance of the regression residual.So the denominator in this expression is the part of asset i's variance that cannot be diversified away times the absolute risk aversion.The numerator is proportional to the difference between asset i's excess return and the sum of other risky assets' excess returns weighted by the respective coefficient in the multiple regression.Substituting (2.11) in (2.10) we have where a H represents the head trader's coefficient of risk aversion.If the head trader and all traders have the same coefficient of risk aversion, the above expression simplifies to The head trader can use (2.12) (or (2.13)) to control traders in their optimal allocation problems.
In the calculation of trader i's optimal solution we have made the assumption that he maximizes the expected utility of the wealth he manages, E U W = E 1 − e −a i , W .However, trader i should maximize the expected utility of his own payoff.If his payoff is a fixed proportion δ i of the wealth he manages he should maximize the function , where η i is his true coefficient of risk aversion † .If we define a i = δ i η i we can easily see that the problem does not change.
Further generalizations of the traders' payoff function could be made.An interesting case arises when traders have unlimited liability.For example, their payoffs can be the excess of final wealth over a benchmark B, represented as Wi = τ i W − B , where the benchmark is non-stochastic and τ i is the fraction of the excess wealth that trader i receives.
However, more general compensation schemes may not work within this approach.An interesting example is the introduction of options in the compensation scheme.Traders receive a fixed fee f and, in addition, an option based on the excess wealth that they have generated.In this case, Wi = f i + τ i max 0, W − B .
One further simplification of expression (2.12) obtains when we assume that assets returns are independent.In this case, (2.12) becomes: Also, (2.11) simplifies to: (2.15) In this case, if the head trader imposes W 0,i = W 0 and r f,i = r f , that is to say, gives all wealth to the i − th trader and uses as benchmark rate the interest rate that the trading firm faces, then the decentralized solution is equivalent to the centralized one as (2.15) gives the same allocation as (2.8) if the i − th trader and the head trader have the same risk aversion coefficient.

Comparative-static Analysis.
Equation (2.12) gives us a sufficient condition for the decentralized and centralized problems to coincide in the sense that the dollar amounts invested in each risky asset will be the same in both problems.The dollar amounts invested in the riskless asset, on the other hand, are different.However, this is irrelevant because traders' investments in riskless assets are only notional.It is the firm paying interest to itself.
If we look at equations (2.8) and (2.11) we can say that, ceteris paribus, if the expected return of the i−th asset increases, both head trader and traders would invest more in this risky asset.But (2.12) shows that, for them to increase the investment in the risky asset i by the same amount, it is necessary that the benchmark for trader i changes according to: The sign of (3.1) depends on the ratio of risk aversion coefficients.The term that pre-multiplies this ratio is greater than one.If the coefficients of risk aversion are similar and the return on asset i is highly correlated with other risky assets' returns (high R 2 i ), equation (3.1) implies that trader i's benchmark should be reduced, leading to further investment in risky asset i by trader i.This is so because the head trader would reduce investments not only in the riskless asset but also in the other risky assets correlated with asset i.Since trader i does not invest in other risky assets, he has to draw relatively more money from his riskless asset.Off course, the risk-free rate of other traders that trade risky assets positively correlated with the i − th trader would increase to compensate the fast increase in the reduction of investment in the riskless asset made by trader i (this is illustrated by equation (3.2) below).
However, if the head trader is much more risk averse than trader i and risky assets are not very correlated, so that the ratio of risk averse coefficient times the inverse of 2 1 − R 2 i is less than one, trader i's benchmark should increase in order to counterbalance the increase in i − th asset expected return.
Otherwise, differences in risk aversion would generate different centralized and decentralized allocations.
If the return of the i − th risky asset increases is positively correlated with the return of the j − th risky asset, an increase in the expected return of the i − th risky asset would require an increase in the j − th trader's benchmark.This is because the head trader solution would require taking investments from risky assets correlated with the i − th risky asset to the i − th risky asset.The instrument the head trader have to induce the j − th trader to reduce his investments in his risky asset is by increasing his benchmark.This is captured by equation (3.2) bellow: When the head trader increases the risk-free rate for traders he induces a risk exposure reduction.If β ji is negative then the correlation between the i − th and j − th asset returns is negative.In this case, investment in the j − th asset increases with the increase in expected return in the i − th asset because it can be used to hedge an increase of investment in the i − th asset.

The Equivalence of Centralized and Decentralized
Solutions: a Generalization.
One important generalization would be to allow traders to trade in more than one market.Actually, most firms have traders specialized in a few assets.In this section we allow traders to trade in subsets of risky assets and find the equivalence condition for the optimal portfolio allocation.To make it easier to understand, we do this generalization in four steps.First, we study a typical trader solution.Second, we generalize the equivalence result where traders trade in non-intersecting subsets of risky assets.Third, we study the case where traders trade in the same subset of risky assets.Finally, we allow traders to trade in any subset of risky assets.

Trader i's solution.
We use an amount of wealth and one benchmark for each risky asset, so if the i − th trader manages m risky assets there will be m risk-free rates to control his optimal decisions.This i − th trader has initial wealth W k,i , allocated to him by the head trader, to invest in the k−th risky asset.He has a different wealth available for each asset.Let α k,i be the fractions of initial wealth invested in the k − th risky asset, by the i − th trader.Let r f k ,i be the risk-free return the trader pays to the firm to use wealth in his k − th risky asset investment.The objective function for the i − th trader is: The first order conditions are given by: Observing that when a trader has exponential utility the amount he invests in the risky assets is independent of wealth, we can make, without loss of generality, the simplifying assumption that W k,i = W j,i ∀i, j and W 0,i ≡ W k,i .Then the first-order conditions become Thus we have a system with m equations to solve.Let The first-order conditions give Using the same approach as in section 2 we derive the proportion to be allocated in the k − th risky asset by the i − th trader.
where R 2 k,i and R 2 k are the R-squared for the multiple regression of the excess return of the k − th asset on the excess returns of m risky markets for the i − th trader and the R-squared for the multiple regression of the excess return for the head trader of all the n risky assets, respectively.Solving this equation for the risk-free rate for the k − th asset we derive: Thus we have m equations that must be determined simultaneously for r f k ,i and r f j ,i , for j = 1, . . .m, j = k.

4.2
The case where traders trade in non-intersecting subsets of risky assets.
If there are many traders, each with a subset of the full asset basket, without intersection between these subsets, then all that the head trader has to do is to solve these systems separately for each trader.We are assuming that traders are not specialized in the same assets, but instead that all traders take decisions regarding different assets.
We can write (4.5) in matrix notation.Assuming that has full rank, we have .
If we assume that assets returns are independent then it does not matter whether the traders solve the problem in a decentralized or centralized way.The optimal solution will be the same.

4.3
The case where traders trade in the same subsets of risky assets.
Another generalization would be the case were there are l traders making decisions in m risky markets and the head trader's asset basket is composed of n risky assets, where n > m.We assume that all traders observe the same m risky assets and make allocation decisions in these markets based on their inferences about expected return and risk of the m risky assets and on their own degrees of risk aversion.To generate equivalence we need: and r f k ,i = r f k ,j ∀i, j, so that the riskless rate would be the same for different traders that make decisions on the same asset.Also, let and Rearranging terms to solve for the risk-free rates, we get from (4.6) that: We have the following system to solve, in matrix notation: where b m×m is the same as defined before, r f m×1 is a vector of risk-free rates for the i − th trader, r m×1 is a vector of risky rates of returns and

The general case.
The final generalization would be the case where each trader's portfolio is any subset of the global portfolio.Assuming that an allocation made by the head trader is a vector in R \ , we would have traders' portfolios in subspaces R ⊆ R \ , where l ≤ n.Different traders can build portfolios in such a way that the intersection of assets between portfolios is not always an empty set.This would be the most general case.Let α j,i be the allocation made by the i − th trader in the j − th asset.Then, we have The allocation in the j − th asset depends on the expected returns, the covariance matrix on assets traded by the i-th trader, on the risk-free interest rates for each asset, on his risk aversion coefficient and on his initial endowment.
If we assume that there are l traders and n assets, then equivalence can be achieved if the condition below is satisfied: The solution would be the same as found in the previous case with a slight difference ‡ : traders trade in subsets of the global portfolio, so by construction some α j,i are zero (for some asset j and trader i).This means that the matrices used before would have some zero elements.
Another way to look at condition (4.10) would be as a system with n equations and n unknown parameters (risk-free interest rates for each asset).Since this is a linear system, if the determinant of the system is different from zero (which it generally is), we always have a solution for it that can be implemented numerically.
To do that we need to estimate the risk aversion coefficients of each trader, which is not an especially difficult task.Then the implementation of the methodology presented in this paper in a real world situation could be easily done by allowing traders to trade with some predetermined amount of wealth and benchmark for each asset he trades.

A second-best solution.
A problem with the solutions presented so far is that the managers (traders) choose exactly the same portfolio that the head trader does.They supply the entries for a matrix with expected returns and expected variance-covariance for all the relevant assets, but they are induced to invest the same amounts the head trader does.
A second best solution could be the case when the head trader induces each trader to have the same ratio of the investment in risky assets on the risk-free asset.This is a much weaker condition for equivalence, which allows much more flexibility for traders, and where decentralization plays a significant ‡ As before we can assume that the benchmark on the j−th asset is the same for all traders who trade on that asset.
role.Another interesting result for this second best solution is that the instrument the head trader has to use to induce traders to reach the second best solution is closer to the one used currently: a common benchmark for each trader, the same for all risky assets.
We develop this extension within the generalization with each trader specialized in different subsets of the assets of the economy, with no intersection between them.Our condition is that the ratio of the investment in risky assets on the riskfree asset should be the same for both the head trader and a particular trader, say trader i.Then thus condition can be written as: where m is the number of risky asset traded by trader i and n is the total number of risky assets the firm trades.The term in the right of this equality is to be considered a desired constant for the Head Trader, for example H, the optimal ratio of investment in risky assets to the investment in the risk-free asset, implied by maximizing it's own utility function.Thus, we can rewrite this equality as We know from the equations derived previously that We can allow the risk free interest rate to be the same for all assets.A trader will have the same benchmark or opportunity cost, independent of the particular assets that he is trading on.However, different traders can have different benchmarks.The expression above can be rewritten as Then the optimal investment in risky asset k can be written as (5.5) and the sum of these investments can be written as where m k=1 F k,i ≡ F i and m k=1 L k,i ≡ L i Thus, the second best equivalence condition can be rewritten as Rearranging for the benchmark rate yields The benchmark interest rate depends on risk of all the m assets in this economy and also on the correlations between them.Differences from different traders will emerge depending from differences on their own characteristics (risk aversion) and from differences in expected returns and risk relations among assets they trade.
It is also possible to solve the second best solution using the amount of wealth make available of investment instead of benchmark as the instrument the head trader uses to induce the result.In the remaining of this section we develop this result.From our condition (5.2) and replacing (5.4) in (5.2) we have that Rearranging for the initial wealth of trader i yields Expression (5.13) is similar to (5.10) but we are using the initial wealth as control variable, while the risk free rate (our benchmark interest rate) is held constant and can be equal to the risk free interest rate that the head trader faces.This condition is much easier to implement and it seems as a good proxy of what trading firms or investors would do, in general.
Our solutions are similar to the first-best situation in that they depend on risk aversion of traders, risks and expected returns and on correlation of assets in the economy as well.Nonetheless, this solution seems to be more in line with current practice.

Summary and conclusion.
In this paper conditions were derived for the solutions to the centralized (head trader and decentralized (many traders) to be equivalent.This is important because the mean-variance model can be implemented by firms with decentralized decisions, as most large trading firms are.This equivalence is established for the case when traders have no better information than the head trader.The next step in our research agenda is to study the case when traders are better informed than the head trader.
We showed that if the head trader uses a different interest rate for each trader to borrow or lend as control variable, he can achieve the same allocation with the decentralized portfolio decision problem that he can with the centralized portfolio decision problem.Furthermore, we found that this interest rate depends on the ratio of the degree of risk aversion of the trader and the head trader, on excess returns, and on the correlation of assets.
We next allowed traders to trade in m markets and found that within this framework it is possible to achieve the same allocation in the decentralized portfolio decision problem as in the centralized one, if the interest rate to borrow or lend is used as a control variable.In this case the head trader must solve m simultaneous equations to find these interest rates simultaneously.The same problem was solved for the case where all traders trade the same subset of the global assets.Finally, a general solution was provided where traders are allowed to trade in any subset of risky assets.
We solve a second best problem by imposing that the proportion of risky assets to the risk-free interest rate be the same for the head trader and managers.Thus, managers can invest freely in their subset of risky assets provided that the proportion of risky assets is maintained constant.In this case the head trader can use two control variables.He can choose either controlling traders' investment by imposing a risk-free interest rate to each trader or by distributing the initial endowment among traders.Since the head trader can use wealth as a control vari-able this solution seems to be more in line with current practice in trading firms.As the solutions that we have derived previously, wealth available for traders are a function of traders risk aversion, expected returns, risk and correlation among assets in the economy.
Further generalizations would be dropping the CARA utility function assumption and using an intertemporal framework.Besides, allowing for short sales would be an interesting issue to examine.Sometimes traders may want to stay short in the market and the principal (head trader) could supply them with collateral (instead of money).
Submitted in May 2003. Revised in November 2003.