Modern portfolio theory1 is generally regarded to have started with the mean-variance (M-V) analysis of Harry M. Markowitz [1]. Using the means and variances (or equivalently expected returns and standard deviations) of asset returns as the criteria for portfolio management, Markowitz showed how to create a frontier of investment portfolios over a single holding period [0,T] such that each of them has the greatest possible expected return, given their level of standard deviation or risk. Under the M-V assumptions, we obtain the two-fund separation theorem2 [4,5] which states that an investor, based on his degree of risk averseness, decides at initial time 0 to hold a certain combination of the risk-free asset and the market portfolio3 so as to maximize his expected utility at terminal time T. For example, at time 0, a conservative investor will allocate a smaller percentage of his funds to the market portfolio whereas an aggressive investor will allocate a larger percentage to the market portfolio. Once the allocation of funds has been made at time 0 between the two assets, M-V analysis dictates that no further trading will be done by the investor until time T.

M-V analysis presupposes that the rate of return on the risk-free asset and the expected returns and standard deviations of risky assets do not change over the holding period. It is not too incorrect to assume that they remain unchanged for short holding period (e.g., three or six months), but it is inaccurate to assume that they remain unchanged for long holding period (e.g., five years or longer). For long period, an M-V model under such presupposition is obviously an unreasonable approximation to the actual securities market. In fact, numerous empirical studies have attested the dynamic and stochastic nature of interest rates [10-12] and asset prices [13-19]. For example, many studies [13,15,17,18] have found marked negative long-term serial correlation in stock returns, which may result from a combination of a changing expected return and expected return reverting to its mean over time. Hence, asset management based on static M-V models is bound to err for long holding period.

In this study, we employ stochastic differential equations to characterize the dynamic and stochastic nature of the prices of the underlying assets in a securities market. In addition, we use stochastic control to derive a formula for the optimal fraction (i.e., optimal control) invested in each of the assets at each time over the holding period. To adequately depict the stochastic nature of the securities market, we assume that it is driven by the following three assets4: a risk-free asset (a short bond), a risky asset (a market index), and another risky asset (a long bond). For the risk-free asset, we assume that its underlying risk-free rate or short rate follows the Vasicek [24] interest rate model. The reason for using the Vasicek model is its mean-reverting property. Under the Vasicek model, the short rate will tend to be pulled back to some longrun average level over time when it is either too high or too low. For each of the two risky assets, we assume that the drift of its price is made up of the short rate plus an appropriate risk premium (we will estimate the two risk premiums in Section 3) whose magnitude depends on the riskiness of the asset. Such characterization of the prices of the two risky assets is consistent with many empirical findings [13,15,17,18] for asset prices.

To sum up, in M-V models, an investor, based on his degree of risk averseness, can decide ONLY at initial time 0 how to divide his funds between the risk-free asset and the market portfolio (often an index fund in practice) to maximize his expected utility at time T. In our threeasset model, an investor, based on his degree of risk averseness, can determine at EACH time over the holding period the optimal allocation between the three assets (i.e., short bond, long bond, and market index) to maximize his expected utility at terminal time T. As such, our model is superior to M-V models in that ours is dynamic but theirs is static.

The rest of the paper proceeds as follows. In Section 2, we set up a dynamic model driven by three assets and use stochastic control to derive an explicit formula for the optimal fraction of wealth invested in each of the three assets. In Section 3, we use maximum likelihood method to estimate the relevant parameters of our model. Section 4 shows how to implement our model numerically using implicit finite difference method. In Section 5, we report the optimal percentage invested in each of the three assets at times t = 2 and 8 over a holding period of ten years for two types of investors (one type is less riskaverse and the other more risk-averse). Section 6 concludes this study.

Our three-asset model involves a risk-free asset, a risky asset, and another risky asset. Let be the price of the risk-free asset, be the price of the first risky asset, and be the price of the second risky asset. We assume that the instantaneous rate of the risk-free asset is the short rate, the expected rate of return on the first risky asset is, and the expected rate of return on the second risky asset is

. Hence, we have . Accordingly, we describe the dynamics of the short rate and the three asset prices by the following stochastic differential equations:

where, , and are standard Brownian motions with, , and. In the following, we form a three-asset portfolio such that we invest a fraction of our wealth in the first risky asset, in the second risky asset, and the rest in the riskfree asset. Then, the dynamics of wealth can be written as

Substituting, , and of (2), (3), and (4)

into (5) and simplifying the equation, we have

where. Letting be a strictly concave-shaped utility [4,25] defined over wealth and assuming that the investor allocates his wealth between the three assets so as to maximize his expected utility at terminal time T, we obtain the maximized utility function at time such that

subject to the budget constraint of (6). The fact that the utility function is strictly concave implies that the investor is risk averse. In this study, we assume power utility

(where is the risk averseness parameter) for the investor. There are two reasons for using power utility. One is that an explicit solution can be obtained for our model with power utility [26-29], but not with other utility functions (e.g., exponential or quadratic utility). The other is that power utility leads to an optimal solution which is independent of wealth and thus will greatly simplify the derivation. In fact, empirical evidence [4,25] suggests that the typical utility of an investor is characterized by decreasing absolute risk averseness and constant relative risk averseness. These properties are consistent with power utility.

With no money added or withdrawn from our threeasset portfolio at any time, the maximized utility function at time is

Letting and expanding

by applying Taylor’s theorem, we obtain

where, , ,

, and.

Substituting, , , , , , , , , , , , , and into (9), we get

Taking expectation of (10) and noting, we have

Substituting (11) into (8) and dividing each side by dt, we obtain the following Hamilton-Jacobi-Bellman (HJB) equation:

To solve the HJB equation, we first let

, where is any positive function. With no consumption until time T, we have

.

Thus, we have. For simplicity, we let

. Given, we have

, , ,

, , ,

, , ,

, ,

, ,

,.

After replacing J-related terms by L-related terms in (12) and simplifying the equation, we obtain

where. Collecting terms involving and in (13) and rearranging, we obtain

where

.

We note that does not involve and. Furthermore, by setting, we obtain the first-order conditions to maximize equation in (14) such that

Solving (15) and (16) for and, we obtain the following optimal fractions and invested in the first and second risky assets, respectively, at time:

Correspondingly, the optimal fraction invested in the risk-free asset is.

As mentioned in Section 1, the securities market is assumed to be driven by the following three assets: a risk-free asset (a short bond), a risky asset (a market index), and another risky asset (a long bond). In this section, we estimate the parameters in (1), (3), and (4) by using the maximum likelihood (ML) method [30]. ML estimation is based on large sample asymptotic estimators, which are asymptotically normally distributed. Since we use 30 years of daily price data for estimation, the ML estimates so obtained should be accurate.

Three sets of data, a total of 7826 daily price observations from 2 January 1978 to 31 December 2007, are used for estimation. One set is the 3-month US Treasury bill rate (retrieved online from the database of the US Federal Reserve Bank of St. Louis) and the other two sets are the S&P 500 Index value and the 10-year US government bond price (both retrieved from the Datastream database). We use the 3-month Treasury bill rate to proxy for the short rate, the S&P 500 Index⁵ for the price of the first risky asset, and the 10-year US government bond for the price of the second risky asset.

The stochastic differential equation (SDE) in (1) can be expressed in discrete form as follows:

where is a standard normal deviate and

is distributed as

. With a sample size of n, the natural logarithm of the likelihood function, considered as a function of, , and, can be written as

The SDE in (3) can be expressed in discrete form as follows:

where is a standard normal deviate. Given that

and, the logarithm of price relatives is normally distributed with mean and variance =. With a random sample size of n, the natural logarithm of the likelihood function, considered as a function of and, can be written as

The SDE in (4) can be expressed in discrete form as follows:

where is a standard normal deviate. Given that

and, the logarithm of price relatives is normally distributed with mean = and variance =. With a random sample size of n, the natural logarithm of the likelihood function, considered as a function of and, can be written as

ML estimates are found by maximizing (19) with respect to, , and using the 3-month Treasury bill rates, by maximizing (21) with respect to and using the S&P 500 Index value, and by maximizing (23) with respect to and using the 10-year US government bond prices. The estimates for the seven parameters are as follows:, , , , , , and.

In this study, we use a 10-year holding period. For implementation, we first transform the domain of our continuous problem into a discretized domain. Hence, we partition our 10-year holding period [0,T] into m equal intervals of length such that there are

trading times indexed by. At each time (where), we construct a three-dimensional grid with three axes representing, , and, where ranges from 0.01 to 0.12, from 0.03 to 0.20, and from 0.03 to 0.015. That is, point on the grid at time corresponds to, , and , where, , and. With the above setup, we determine numerically the optimal percentages invested in each of the three assets at each point on the three-dimensional grid at each time based on (17).

To implement, we use implicit finite difference method to compute numerically the partial derivatives of. Specifically, the partial derivatives are first converted into a set of difference equations and these difference equations are then computed iteratively backward in time. Accordingly, the firstand second-order partial derivatives of are approximated by the following finite difference operators [31-33] in which, for simplicity, we suppress the argument t of:

The algorithm begins at time and works backward in time. For each point on the threedimensional grid at time, we compute the value of from the value of, where at time T for all values of, , and. The values of and

are computed from the values of using (17). At time, we compute the value of from the value of and then the values of and are computed from the values of. This algorithm is done iteratively until we reach time. To ensure satisfactory convergence, we use a small value of 0.001 for. Hence, for a 10-year holding period, the number of intervals is

.

As pointed out in Section 2, we assume that the investor has a power utility, where is the risk averseness parameter. Note that an investor with a larger means that he is more risk averse. Copeland et al. [4] used a power utility with and Brennan et al. [34] used a power utility with. Hence, we set for less risk-averse investors and for more risk-averse investors. In the following, we use Figures 1-4 to show the optimal percentages for the former in Subsection 5.1 and Figures 5-8 to show those for the latter in Subsection 5.2. In each figure, fixing r = 0.02 and then r = 0.08, we examine how their optimal percentages in market index (panel A) and long bond (panel B) change for different and.

In Figures 1-8, note that (1) we use MU for and NU for and (2) = optimal percentage in short bond, where = optimal percentage in market index and = optimal percentage in long bond. For example, in Figure 1,

and when. Hence the optimal percentage in short bond = , given t = 2, r = 0.02, and.

Over the last 40-plus years, the mean-variance (M-V) analysis has been widely used by investment practioners for asset management. M-V analysis assumes that the means and variances of asset returns do not change over the holding period. However, in a fast-changing securities market, it is inaccurate to assume that they remain unchanged—especially for long holding period. As such, this study sets up a dynamic model driven by three assets to depict the stochastic nature of the market and uses stochastic control to derive an explicit formula for the optimal fraction invested in each of the three assets. Using implicit finite difference method, we determine numerically the optimal percentages invested in the three assets (i.e., short bond, long bond, and market index) at each time over the holding period for less risk-averse and more risk-averse investors. In general, at each time over the holding period, more risk-averse investors would allocate a larger percentage of their wealth to short bond than less risk-averse investors.