Identifying a recommended portfolio of financial products for an investor based upon financial products that are available to the investor7016870Abstract A financial advisory system is provided. According to one aspect of the present invention, return scenarios for optimized portfolio allocations are simulated interactively to facilitate financial product selection. Return scenarios for each asset class of a plurality of asset classes are generated based upon estimated future scenarios of one or more economic factors. A mapping from each financial product of an available set of financial products onto one or more asset classes of the plurality of asset classes is created by determining exposures of the available set of financial products to each asset class of the plurality of asset classes. In this way, the expected returns and correlations of a plurality of financial products are generated and used to produce optimized portfolios of financial products. Return scenarios are simulated for one or more portfolios including combinations of financial products from the available set of financial products based upon the mapping. Claims What is claimed is: Description COPYRIGHT NOTICE
At this point it is important to point out that more, less, or a completely different set of factors may be employed depending upon the specific implementation. The factors listed in Table 1 are simply presented as an example of a set of factors that achieve the goal of spanning the range of investments typically available to individual investors in mainstream mutual funds and defined contribution plans. It will be apparent to those of ordinary skill in the art that alternative factors may be employed. In particular, it is possible to construct factors that represent functions of the underlying asset classes for pricing of securities that are nonlinearly related to the prices of certain asset classes (e.g., derivative securities). In other embodiments of the present invention, additional factors may be relevant to span a broader range of financial alternatives, such as industry specific equity indices. On a periodic basis, the financial product mapping module 315 maps financial product returns onto the factor model. In one embodiment, the process of mapping financial product returns onto the factor model comprises decomposing financial product returns into exposures to the factors. The mapping, in effect, indicates how the financial product returns behave relative to the returns of the factors. According to one embodiment, the financial product mapping module 315 is located on one of the servers (e.g., the financial staging server 120, the broadcast server 115, or the AdviceServer 110). In alternative embodiments, the financial product mapping module 315 may be located on the client 105. In one embodiment of the present invention, an external approach referred to as "returns-based style analysis" is employed to determine a financial product's exposure to the factors. The approach is referred to as external because it does not rely upon information that may be available only from sources internal to the financial product. Rather, in this embodiment, typical exposures of the financial product to the factors may be established based simply upon realized returns of a financial product, as described further below. For more background regarding returns-based style analysis see Sharpe, William F. "Determining a Fund's Effective Asset Mix," Investment Management Review, December 1988, pp. 59-69 and Sharpe, William F. "Asset Allocation: Management Style and Performance Measurement," The Journal of Portfolio Management, 18, no. 2 (Winter 1992), pp. 7-19 ("Sharpe [1992]"). Alternative approaches to determining a financial product's exposure to the factors include surveying the underlying assets held in a financial product (e.g. a mutual fund) via information filed with regulatory bodies, categorizing exposures based on standard industry classification schemes (e.g. SIC codes), identifying the factors exposures based on analysis of the structure of the product (e.g. equity index options, or mortgage backed securities), and obtaining exposure information based on the target benchmark from the asset manager of the financial product. In each method, the primary function of the process is to determine the set of factor exposures that best describes the performance of the financial product. The tax adjustment module 320 takes into account tax implications of the financial products and financial circumstances of the user. For example, the tax adjustment module 320 may provide methods to adjust taxable income and savings, as well as estimates for future tax liabilities associated with early distributions from pension and defined contribution plans, and deferred taxes from investments in qualified plans. Further, the returns for financial products held in taxable investment vehicles (e.g. a standard brokerage account) may be adjusted to take into account expected tax effects for both accumulations and distributions. For example, the component of returns attributable to dividend income should be taxed at the user's income tax rate and the component of returns attributable to capital gains should be taxed at an appropriate capital gains tax rate depending upon the holding period. Additionally, the tax module 320 may forecast future components of the financial products total return due to dividend income versus capital gains based upon one or more characteristics of the financial products including, for example, the active or passive nature of the financial product's management, turnover ratio, and category of financial product. This allows precise calculations incorporating the specific tax effects based on the financial product and financial circumstances of the investor. Finally, the tax module 320 facilitates tax efficient investing by determining optimal asset allocation among taxable accounts (e.g., brokerage accounts) and nontaxable accounts (e.g., an Individual Retirement Account (IRA), or employer sponsored 401(k) plan). In this manner the tax module 320 is designed to estimate the tax impact for a particular user with reference to that particular user's income tax rates, capital gains rates, and available financial products. Ultimately, the tax module 320 produces tax-adjusted returns for each available financial product and tax-adjusted distributions for each available financial product. The portfolio optimization module 340 calculates the utility maximizing set of financial products under a set of constraints defined by the user and the available feasible investment set. In one embodiment, the calculation is based upon a mean-variance optimization of the financial products. The constraints defined by the user may include bounds on asset class and/or specific financial product holdings. In addition, users can specify intermediate goals such as buying a house or putting a child through college, for example, that are incorporated into the optimization. In any event, importantly, the optimization explicitly takes into account the impact of future contributions and expected withdrawals on the optimal asset allocation. Additionally, the covariance matrix used during optimization is calculated based upon the forecasts of expected returns for the factors generated by the factor module 310 over the investment time horizon. As a result, the portfolio optimization module 340 may explicitly take into account the impact of different investment horizons, including the horizon effects impact from intermediate contributions and withdrawals. The simulation processing module 330 provides additional analytics for the processing of raw simulated return scenarios into statistics that may be displayed to the user via the UI 345. In the one embodiment of the present invention, these analytics generate statistics such as the probability of attaining a certain goal, or the estimated time required to reach a certain level of assets with a certain probability. The simulation processing module 330 also incorporates methods to adjust the simulated scenarios for the effects induced by sampling error in relatively small samples. The simulation processing module 330 provides the user with the ability to interact with the portfolio scenarios generated by the portfolio optimization module 340 in real-time. The annuitization module 325 provides a meaningful way of representing the user's portfolio value at the end of the term of the investment horizon. Rather than simply indicating to the user the total projected portfolio value, one standard way of conveying the information to the user is converting the projected portfolio value into a retirement income number. The projected portfolio value at retirement may be distributed over the length of retirement by dividing the projected portfolio value by the length of retirement. More sophisticated techniques may involve determining how much the projected portfolio value will grow during retirement and additionally consider the effects of inflation. In either event, however, these approaches erroneously assume the length of the retirement period is known in advance. It is desirable, therefore, to present the user with a retirement income number that is more representative of an actual standard of living that could be locked in for the duration of the user's retirement. According to one embodiment, this retirement income number represents the inflation adjusted income that would be guaranteed by a real annuity purchased from an insurance company or synthetically created via a trading strategy involving inflation-indexed treasury bond securities. In this manner, the mortality risk is taken out of the picture because regardless of the length of the retirement period, the user would be guaranteed a specific annual real income. To determine the retirement income number, standard methods of annuitization employed by insurance companies may be employed. Additionally, mortality probabilities for an individual of a given age, risk profile, and gender may be based on standard actuarial tables used in the insurance industry. For more information see Bowers, Newton L. Jr., et al, "Actuarial Mathematics," The Society of Actuaries, Itasca, Ill., 1986, pp. 52-59 and Society of Actuaries Group Annuity Valuation Table Task Force, "1994 Group Annuity Mortality Table and 1994 Group Annuity Reserving Table," Transactions of the Society of Actuaries, Volume XLVII, 1994, pp. 865-913. Calculating the value of an inflation-adjusted annuity value may involve estimating the appropriate values of real bonds of various maturities. The pricing module 305 generates the prices of real bonds used to calculate the implied real annuity value of the portfolio at the investment horizon. Referring now to the plan monitoring module 350, a mechanism is provided for alerting the user of the occurrence of various predetermined conditions involving characteristics of the recommended portfolio. Because the data upon which the portfolio optimization module 340 depends is constantly changing, it is important to reevaluate characteristics of the recommended portfolio on a periodic basis so that the user may be notified in a timely manner when there is a need for him/her to take affirmative action, for example. According to one embodiment, the plan monitoring module 350 is located on the AdviceServer 110. In this manner, the plan monitoring module 350 has constant access to the user profile and portfolio data. In one embodiment, the occurrence of two basic conditions may cause the plan monitoring module 350 to trigger a notification or alert to the user. The first condition that may trigger an alert to the user is the current probability of achieving a goal falling outside of a predetermined tolerance range of the desired probability of a achieving the particular goal. Typically a goal is a financial goal, such as a certain retirement income or the accumulation of a certain amount of money to put a child though college, for example. Additionally, the plan monitoring module 350 may alert the user even if the current probability of achieving the financial goal is within the predetermined tolerance range if a measure of the currently recommended portfolio's utility has fallen below a predetermined tolerance level. Various other conditions are contemplated that may cause alerts to be generated. For example, if the nature of the financial products in the currently recommended portfolio have changed such that the risk of the portfolio is outside the user's risk tolerance range, the user may receive an indication that he/she should rebalance the portfolio. Plan monitoring processing, exemplary real world events that may lead to the above-described alert conditions, and additional alert conditions are described further below. The UI module 345 provides mechanisms for data input and output to provide the user with a means of interacting with and receiving feedback from the financial advisory system 100, respectively. Further description of a UI that may be employed according to one embodiment of the present invention is disclosed in U.S. Pat. Nos. 5,918,217 and 6,012,044, both entitled "USER INTERFACE FOR FINANCIAL ADVISORY SYSTEM," the contents of which are hereby incorporated by reference. Other modules may be included in the financial advisory system 100 such as a pension module and a social security module. The pension module may be provided to estimate pension benefits and income. The social security module may provide estimates of the expected social security income that an individual will receive upon retirement. The estimates may be based on calculations used by the Social Security Administration (SSA), and on probability distributions for reductions in the current level of benefits. Core Asset Scenario Generation FIG. 4 is a flow diagram illustrating core asset class scenario generation according to one embodiment of the present invention. In embodiments of the present invention, core assets include short-term US government bonds, long-term US government bonds, and US equities. At step 410, parameters for one or more functions describing state variables are received. The state variables may include general economic factors, such as inflation, interest rates, dividend growth, and other variables. Typically, state variables are described by econometric models that are estimated based on observed historical data. At step 420, these parameters are used to generate simulated values for the state variables. The process begins with a set of initial conditions for each of the state variables. Subsequent values are generated by iterating the state variable function to generate new values conditional on previously determined values and a randomly drawn innovation term. In some embodiments, the state variable functions may be deterministic rather than stochastic. In general, the randomly drawn innovation terms for the state variable functions may be correlated with a fixed or conditional covariance matrix. At step 430, returns for core asset classes are generated conditional on the values of the state variables. Returns of core asset classes may be described by a function of a constant, previously determined core asset class returns, previously determined values of the state variables, and a random innovation term. Subsequent values are generated by iterating a core asset class function to generate new values conditional on previously determined values and a random draws of the innovation term. In some embodiments, the core asset class functions may be deterministic rather than stochastic. In general, the randomly drawn innovation terms for the core asset class functions may be correlated with a fixed or conditional covariance matrix. In alternative embodiments, steps 410 and 420 may be omitted and the core asset class returns may be generated directly in an unconditional manner. A simple example of such a model would be a function consisting of a constant and a randomly drawn innovation term. A preferred approach would jointly generate core asset class returns based on a model that incorporates a stochastic process (also referred to as a pricing kernel) that limits the prices on the assets and payoffs in such a way that no arbitrage is possible. By further integrating a dividend process with the other parameters an arbitrage free result can be ensured across both stocks and bonds. Further description of such a pricing kernel is disclosed in a copending U.S. patent application entitled "PRICING KERNEL FOR FINANCIAL ADVISORY SYSTEM," application Ser. No. 08/982,941, filed on Dec. 2, 1997, assigned to the assignee of the present invention, the contents of which are hereby incorporated by reference. Factor Model Asset Scenario Generation Referring now to FIG. 5, factor model asset scenario generation will now be described. A scenario in this context is a set of projected future values for factors. According to this embodiment, the factors may be mapped onto the core asset factors by the following equation: rit=αi+β1iST_Bondst+β2iLT_Bondst+β3iUS_Stockst+εt (EQ #1)
At step 510, the beta coefficients (also referred to as the loadings or slope coefficients) for each of the core asset classes are determined. According to one embodiment, a regression is run to estimate the values of the beta coefficients. The regression methodology may or may not include restrictions on the sign or magnitudes of the estimated beta coefficients. In particular, in one embodiment of the present invention, the coefficients may be restricted to sum to one. However, in other embodiments, there may be no restrictions placed on the estimated beta coefficients. Importantly, the alpha estimated by the regression is not used for generating the factor model asset scenarios. Estimates of alpha based on historical data are extremely noisy because the variance of the expected returns process is quite high relative to the mean. Based on limited sample data, the estimated alphas are poor predictors of future expected returns. At any rate, according to one embodiment, a novel way of estimating the alpha coefficients that reduces the probability of statistical error is used in the calibration of the factor model. This process imposes macroconsistency on the factor model by estimating the alpha coefficients relative to a known efficient portfolio, namely the Market Portfolio. Macroconsistency is the property that expected returns for the factor asset classes are consistent with an observed market equilibrium, that is estimated returns will result in markets clearing under reasonable assumptions. The Market Portfolio is the portfolio defined by the aggregate holdings of all asset classes. It is a portfolio consisting of a value-weighted investment in all factor asset classes. Therefore, in the present example, macroconsistency may be achieved by setting the proportion invested in each factor equal to the percentage of the total market capitalization represented by the particular factor asset class. At step 520, a reverse optimization may be performed to determine the implied factor alpha for each factor based upon the holdings in the Market Portfolio. This procedure determines a set of factor alphas that guarantee consistency with the observed market equilibrium. In a standard portfolio optimization, Quadratic Programming (QP) is employed to maximize the following utility function: ##EQU1##
Inputs to a standard portfolio optimization problem include E(r), C(r), and Tau and QP is used to determine X. However, in this case, X is given by the Market Portfolio, as described above, and a reverse optimization solves for E(r) by simply backing out the expected returns that yield X equal to the proportions of the Market Portfolio. Quadratic Programming (QP) is a technique for solving an optimization problem involving a quadratic (squared terms) objective function with linear equality and/or inequality constraints. A number of different QP techniques exist, each with different properties. For example, some are better for suited for small problems, while others are better suited for large problems. Some are better for problems with very few constraints and some are better for problems with a large number of constraints. According to one embodiment of the present invention, when QP is called for, an approach referred to as an "active set" method is employed herein. The active set method is explained in Gill, Murray, and Wright, "Practical Optimization," Academic Press, 1981, Chapter 5. The first order conditions for the optimization of Equation #2 are: ##EQU2##
At step 530, factor returns may be generated based upon the estimated alphas from step 520 and the estimated beta coefficients from step 510. As many factor model asset scenarios as are desired may be generated using Equation #1 and random draws for the innovation values εt. A random value fore is selected for each evaluation of Equation #1. According to one embodiment, e, is distributed as a standard normal variate. In other words εt is drawn from a standard normal distribution with a mean of 0 and a standard deviation of 1. Advantageously, in this manner, a means of simulating future economic scenarios and determining the interrelation of asset classes is provided. Financial Product Exposure Determination As discussed above, one method of determining how a financial product behaves relative to a set of factor asset classes is to perform returns-based style analysis. According to one embodiment, returns for a given financial product may be estimated as a function of returns in terms of one or more of the factor asset classes described above based on the following equation: rft=αft+Sf1r1t+Sf2r2t+ . . . +Sfnrnt+εt (EQ #4)
The financial product exposure determination module 315 computes the factor asset class exposures for a particular fund via a nonlinear estimation procedure. The exposure estimates, Sfn, are called style coefficients, and are generally restricted to the range [0,1] and to sum to one. In other embodiments, these restrictions may be relaxed (for example, with financial products that may involve short positions, the coefficients could be negative). Alpha may be thought of as a measure of the relative under or over performance of a particular fund relative to its passive style benchmark. At this point in the process, the goal is to take any individual group of assets that people might hold, such as a group of mutual funds, and map those assets onto the factor model, thus allowing portfolios to be simulated forward in time. According to one embodiment, this mapping is achieved with what is referred to as "returns-based style analysis" as described in Sharpe [1992], which is hereby incorporated by reference. Generally, the term "style analysis" refers to determining a financial product's exposure to changes in the returns of a set of major asset classes using Quadratic Programming or similar techniques. FIG. 6 is a flow diagram illustrating a method of determining a financial product's exposures to factor asset class returns according to one embodiment of the present invention. At step 610, the historical returns for one or more financial products to be analyzed are received. According to one embodiment, the financial product exposure module 315 may reside on a server device and periodically retrieve the historical return data from a historical database stored in another portion of the same computer system, such as RAM, a hard disk, an optical disc, or other storage device. Alternatively, the financial product exposure module 325 may reside on a client system and receive the historical return data from a server device as needed. At step 620, factor asset class returns are received. At step 630, QP techniques or the like are employed to determine estimated exposures (the S coefficients) to the factor asset class returns. At step 640, for each financial product, expected future alpha is determined for each subperiod of the desired scenario period. With regards to mutual funds or related financial products, for example, historical alpha alone is not a good estimate of future alpha. That is, a given mutual fund or related financial product will not continue to outperform/under perform its peers indefinitely into the future. Rather, empirical evidence suggests that over performance may partially persist over one to two years while under performance may persist somewhat longer (see for example, Carhart, Mark M. "On Persistence in Mutual Fund Performance." Journal of Finance, March 1997, Volume 52 No. 1, pp. 57-82). For example, future alpha may depend upon a number of factors, such as turnover, expense ratio, and historical alpha. Importantly, one or more of these factors may be more or less important for particular types of funds. For example, it is much more costly to buy and sell in emerging markets as compared to the market for large capitalization US equities. In contrast, bond turnover can be achieved at a much lower cost, therefore, turnover has much less affect on the future alpha of a bond fund than an equity fund. Consequently, the penalty for turnover may be higher for emerging market funds compared to large cap U.S. equities and bond funds. Improved results may be achieved by taking into account additional characteristics of the fund, such as the fact that the fund is an index fund and the size of the fund as measured by total net assets, for example. According to one embodiment of the present invention, a more sophisticated model is employed for determining future alpha for each fund: αt=αbase+ρt(αhistorical-αbase (EQ #5) where,
According to one embodiment, αbase=C+β1Expense_Ratio+β2Turnover+β3Fund_Size (EQ #6)
Portfolio optimization is the process of determining a set of financial products that maximizes the utility function of a user. According to one embodiment, portfolio optimization processing assumes that users have a mean-variant utility function, namely, that people like having more wealth and dislike volatility of wealth. Based on this assumption and given a user's risk tolerance, the portfolio optimization module 340 calculates the mean-variance efficient portfolio from the set of financial products available to the user. As described above, constraints defined by the user may also be taken into consideration by the optimization process. For example, the user may indicate a desire to have a certain percentage of his/her portfolio allocated to a particular financial product. In this example, the optimization module 340 determines the allocation among the unconstrained financial products such that the recommended portfolio as a whole accommodates the user's constraint(s) and is optimal for the user's level of risk tolerance. Prior art mean-variant portfolio optimization traditionally treats the problem as a single period optimization. Importantly, in the embodiments described herein, the portfolio optimization problem is structured in such as way that it may explicitly take into account the impact of different investment horizons and the impact of intermediate contributions and withdrawals. Further the problem is set up so that it may be solved with QP methods. Referring now to FIG. 7, a method of portfolio optimization according to one embodiment of the present invention will now be described. At step 710, information regarding expected withdrawals is received. This information may include the dollar amount and timing of the expected withdrawal. At step 720, information regarding expected future contributions is received. According to one embodiment, this information may be in the form of a savings rate expressed as a percentage of the user's gross income or alternatively a constant or variable dollar value may be specified by the user. At step 730, information regarding the relevant investment time horizon is received. In an implementation designed for retirement planning, for example, the time horizon might represent the user's desired retirement age. At step 740, information regarding the user's risk tolerance, Tau, is received. At step 750, the mean-variance efficient portfolio is determined. According to one embodiment, wealth in real dollars at time T is optimized by maximizing the following mean-variance utility function by determining portfolio proportions (Xi): ##EQU3##
The product of gross returns represents the compounding of values from year 1 to the horizon. Initial wealth in the portfolio is represented by contribution C0. Importantly, the financial product returns need not represent fixed allocations of a single financial product. Within the context of the optimization problem, any individual asset return may be composed of a static or dynamic strategy involving one or more financial products. For example, one of the assets may itself represent a constant re-balanced strategy over a group of financial products. Moreover, any dynamic strategy that can be formulated as an algorithm may be incorporated into the portfolio optimization. For example, an algorithm which specifies risk tolerance which decreases with the age of the user could be implemented. It is also possible to incorporate path dependent algorithms (e.g., portfolio insurance). According to Equation #8, contributions are made from the current year to the year prior to retirement. Typically, a contribution made at time t will be invested from time t until retirement. An exception to this would be if a user specifies a withdrawal, in which case a portion of the contribution may only be held until the expected withdrawal date. An alternative to the buy and hold investment strategy assumed above would be to implement a "constant mix" investment strategy or re-balancing strategy. For purposes of this example, it is assumed that the recommended fixed target asset-mix will be held in an account for each year in the future. Therefore, each year, assets will be bought and/or sold to achieve the target. Let fi be the fraction of account wealth targeted for the i-th asset, then the sum of the fractions must equal one. In the following "evolution" equations, nominal wealth aggregation is modeled for a single taxable account from the current time t=0 to the time horizon t=T. It is assumed that "N" assets are in the account, labeled by the set of subscripts {i=1, . . . , N}. The superscripts minus and plus are used to distinguish between the values of a variable just before, and just after, "settlement". The settlement "event" includes paying taxes on distributions and capital gains, investing new contributions, buying and selling assets to achieve the constant mix, and paying load fees. For example, W+(t) is the total wealth invested in all assets just after settlement at time "t". The evolution equations for the pre- and post-settlement values, the "dollars" actually invested in each asset, are: ##EQU5## In the above equation, the double-bar operator ∥ ∥ is equal to either its argument or zero, whichever is greater. From Eq.(19a), we see that the pre-settlement value at any time (after the initial time) is just the gross return on the post-settlement value of the previous time less the "positive-part" of any distribution, i.e. the "dividend". Here, ki(t) is the portion of the return of the i-th asset that is distributed, and Ri(t) is the total nominal return on the i-th asset in the one-year period [t-1, t]. We also assume that an initial, pre-settlement value is given for each asset. Eq.(19b) defines the post-settlement value in terms of the asset's constant mix and the total account value after settlement. Since we "cash-out" the portfolio at the time horizon, the final amount in each asset at t=T is zero. The pre- and post-settlement, total values are governed by the pair of equations: ##EQU6## In Eq.(19d), C(t) is the nominal contribution to the account at time "t", D(t) is the total of all distributed "dividends", L(t) is the "leakage", the total amount paid in loads to both rebalance and to invest additional contributions, and S(t) is the "shrinkage", the total amount paid in taxes on distributions and capital gains. We note that W+(T) is the final horizon wealth after all taxes have been paid. The value of D(t), the total of all distributed dividends, is the sum of the positive distributions: ##EQU7## Similarly, the "leakage" L(t) is the total amount of dollars paid in loads, and Li(t) is the number of dollars paid in loads on just the i-th asset. These individual loads depend on 1i, the front-end load fee (a rate) on the i-th asset. ##EQU8## If there is a short-term loss (negative distribution), the load fee paid on an asset's purchase is just a fixed fraction of the purchase price.i When there is a short-term gain (positive distribution), we can re-invest any part of it without load fees, and pay fees only on purchases in excess of the gain. Note that at the horizon, we "cash-out", and don't pay any load fees. iThe dollar amount of a load fee is proportional to the ratio l/(1-l). That's because our wealth variables are all measured as "net" loads. To see this, suppose we make a contribution c. After loads, we are left with W ##CHR1## (1-l)c. In terms of W, the amount we paid in loads is L=lc=[l/(1-l)]W. The equation for the "shrinkage" S(t), the total amount paid in taxes, has two terms. The first term is the tax on distributions and is multiplied by the marginal tax-rate; the second term is the tax on capital gains and is multiplied by the capital gains tax-rate. ##EQU9## In Eq.(19h), the capital gains tax depends on the basis Bi(t), the total of all after-tax nominal-dollars that have been invested in the i-th asset up to time "t". Note that there can be either a capital gain or loss. The double-bar operator ensures that capital gains are triggered only when there is a sale of assets. At the horizon, we sell all assets, and automatically pay all taxes. The basis Bi(t), evolves according to the following recursion equation: ##EQU10## Note that all new purchases are made with after-tax dollars, and add to the basis; all sales decrease the basis. Further, any load paid to purchase an asset adds to the basis. We assume that the initial basis Bi(0) of an asset is either given, or defaults to the initial, pre-settlement value so that the average basis is initially equal to one. A "constitutive" equation for ki(t) is needed to complete our system of equations. Short-term distributions depend on the "type" of asset; here we model mutual funds: ##EQU11## Often, we set the initial distribution to zero, and assume that the asset's initial pre-settlement value has already accounted for any non-zero, initial value. We note that the distribution is proportional to the amount of wealth at "stake" during the prior-period. For mutual funds, we assume that the distribution is a fraction κi of the prior-period's total return, and therefore is also proportional to Ri(t). Note that the distribution in Eq.(20a) can be a gain (positive) or a loss (negative). In contrast, the constitutive equation for stocks takes the form: ##EQU12## For stocks, the proportionality constant κi models a constant dividend "yield", and the distribution is always a gain (non-negative). For stocks (mutual funds), the distribution is proportional to the gross (simple) return. Before we leave this section, a word on 401(k) plans and IRA's (with no load funds). For these accounts, the loads and taxes are ignored, and there is no basis in the asset. At "settlement", the user just re-balances their account. The evolution equations for these accounts is trivial in comparison to the equations for a general taxable account: ##EQU13## At the time horizon T, the total wealth in a non-taxable account is just W+(T). This is a pre-withdrawal total value. When retirement withdrawals are made from a tax-free account, they are taxed at the client's average tax-rate, τa. Therefore, the "after-tax" equivalent value is equal to "pre-tax" wealth W+(T) times the tax factor (1-τa) How do we aggregate taxable and non-taxable accounts to get total portfolio wealth? We choose non-taxable accounts as a baseline. If all the funds in a non-taxable account were converted to an annuity, and the annuity payments were taken as withdrawals, then the withdrawals would mimic a salary subject to income taxes. This is precisely the client's pre-retirement situation. Before aggregating a taxable account, we scale its "after-tax" value to this baseline using the formula: ##EQU14## Essentially, the baseline equivalent is obtained by grossing up values using the average tax-rate. The evolution equation variables appear "implicitly" in the recursion relations. Hence, we need to "iterate" at each time step to solve for "explicit" variable values.ii We illustrate this process with an example. Consider the simple case where there are no distributions, contributions, or taxes; just loads, and a constant-mix strategy. Here, the evolution equations simplify to a single equation for the total, after-settlement wealth W+(t): ##EQU15## iiIn practice a robust root-finding algorithm may be used rather than iteration. Note, we only know W+(t) as an implicit function of W+(t-1), but given a guess for it's value, we can refine the guess by substituting it into the right-side of Eq.(23). It's instructive to re-write Eq.(23) as the pair of equations in terms of an "effective" return Re(t): ##EQU16## Eq.(24a) is the evolution equation for a single asset with the effective return. Eq.(24b) is an implicit equation for the effective return Re(t) in terms of the asset returns Ri(t). We solve for the effective return using iteration. When the loads are equal to zero, as expected, the effective return is just a weighted-average of the asset returns. Even when the loads are not zero, this average return is a good initial guess for the iteration procedure. In fact, using the average return as the initial guess and iterating once yields the following explicit approximation for the effective return: ##EQU17## Eq.(25b) is consistent with our intuition, and agrees well with higher order iterates. To determine the mutual fund input moments we must first calculate the kernel moments. This procedure calculates successive annual kernel moments and averages the result. The resulting mean and covariance matrix is then utilized by the reverse optimization procedure and also as an input into the optimization procedure. To calculate analytic core moments, first we must describe the wealth for each core asset for an arbitrary holding period. For each of the core assets, the resulting wealth from an arbitrary investment horizon can be written as: [Note, this is an approximation for equities] ##EQU18## Where:
Since X, Π, and δ are independent, we can deal with each of these expectations separately. For example, consider the contribution in the above equation from inflation. The summation can be rewritten as: ##EQU20## Next, we need to use iterated expectations to determine this expectation. We can write the expectation at time zero as the repeated expectation over the various innovations. For example, the equation for inflation can be rewritten as: ##EQU21## Assuming inflation follows a modified square root process: ##EQU22## Where ∥ ∥ denotes the Heaviside function ##EQU23## Now we recursively start taking the expectations over epsilon starting at the end and working backward. So: ##EQU24## Where the approximation is due to the Heaviside function. Combining this with the above equation yields: ##EQU25## In general for any time period t, an exponential linear function of Π has the following expectation: ##EQU26## The critical feature is that an exponential linear function of Π remains exponential linear after taking the expectation. This invariance allows for the backward recursion calculation. Only the constant (A) and the slope (B) are changing with repeated application of the expectation operator. The evolution of A and B can be summarized as ##EQU27## ##EQU28## In addition, the Bj coefficient has to be increased by (c+f) to account for the additional Πj term in the summation. To implement this recursive algorithm to solve for expected wealth, first define the following indicator variable: ##EQU29## Next, the following algorithm may be employed: ##EQU30## ##EQU31## The same technique applies to X since it is also a square root process. A similar technique can be used to create a recursive algorithm for the δ component. The only difference is that δ is an AR(l) process instead of a square root process. In particular, ##EQU32## For this AR(l) process, the expectation is of the following form. ##EQU33## The evolution of A and B is thus summarized as: ##EQU34## ##EQU35## The recursive relationship for δ is then: ##EQU36## ##EQU37## This framework for calculating expected wealth can also be used to calculate the variance of wealth for an arbitrary holding period. From the definition of variance, we have: ##EQU38## ##EQU39## ##EQU40## So the same technique can be used with a simple redefinition of the constants to be twice their original values. Similarly, the covariance between any two core assets can be calculated by simply adding corresponding constants and repeating the same technique. For the current parameter values, the constants for Bills, Bonds, and Equities are:
Above, a methodology was described for calculating core asset analytic moments for arbitrary horizons. This section describes how these moments are translated into annualized moments. The procedure described in this section essentially calculates successive annual moments for a twenty (20) year horizon and computes the arithmetic average of these moments. These 'effective' annual moments may then be used as inputs into the reverse optimization procedure and the individual optimization problem. For this calculation, first make the following definitions:
These expected returns and covariance are calculated using the formulas described above The effective annual expected return for asset j is then calculated as: ##EQU41## Similarly, the effective annual covariance between returns on asset i and returns on asset j are calculated as: (Note, the weights, ωt, are between zero and one, and sum to one.) ##EQU42## In one embodiment, this annualizing technique could be personalized for a given user's situation. For example, the user's horizon could specify T, and their level of current wealth and future contributions could specify the relevant weights. However for purposes of illustration, the relevant 'effective' moments for optimization and simulation are computed assuming a horizon of 20 years (T=20), and equal weights (i.e. 1/T). The techniques described in this section allow for the calculation of the following effective annual moments:
Plan Monitoring Exemplary conditions which may trigger an alert of some sort from the plan monitoring module 350 were described above. At this point, some of the real world events that may lead to those alert conditions will now be described. The real world events include the following: (1) a financial product's style exposure changes, (2) the market value of the user's assets have changed in a significant way, (3) new financial products become available to the user, (4) the risk characteristics of the user's portfolio have deviated from the desired risk exposure, or (5) the currently recommended portfolio no longer has the highest expected return for the current level of portfolio risk (e.g., the portfolio is no longer on the mean-variance efficient frontier). An efficient frontier is the sets of assets (portfolios) that provide the highest level of return over different levels of risk. At each point on the efficient frontier, there is no portfolio that provides a higher expected return for the same or lower level of risk. When a financial product's exposures change it may pull the user's portfolio off of the efficient frontier. That is, due to a shift in the investment style of a particular financial product, the portfolio as a whole may no longer have the highest expected return for the current level of risk. According to one embodiment of the present invention, if the inefficiency is greater than a predetermined tolerance or if the inefficiency will substantially impact one of the user's financial goals, such as his/her retirement income goal, then the user is notified that he/she should rebalance the portfolio. However, if the inefficiency is within the predefined tolerance then the plan monitoring module 350 may not alert the user. In one embodiment, the predefined tolerance depends upon the impact of the inefficiency on expected wealth. In addition, the tolerance could depend upon relevant transaction costs. A significant change in the market value of the user's assets may affect one or both of the probability of achieving a financial goal and the current risk associated with the portfolio. In the case that the user's portfolio has experienced a large loss, the portfolio may no longer be within a predetermined probability tolerance of achieving one or more financial goals. Further, as is typical in such situations, the risk associated with the portfolio may also have changed significantly. Either of these conditions may cause the user to be notified that changes are required in the portfolio allocation or decision variables to compensate for the reduction in market value of the portfolio. A large increase in the value of the user's portfolio, on the other hand, could trigger an alert due to the increase in the probability of achieving one or more financial goals or due to the altered risk associated with the newly inflated portfolio. When one or more new financial products become available to the user, the user may be alerted by the plan monitoring module 350 if, for example, a higher expected return may be possible at lower risk as a result of diversifying the current portfolio to include one or more of the newly available financial products. Having explained the potential effects of some real world events that may trigger alerts, exemplary plan monitoring processing will now be described with respect to FIG. 8. At step 810, the data needed for reevaluating the current portfolio and for determining a current optimal portfolio is retrieved, such as the user profile and portfolio data which may be stored on the AdviceServer 110, for example. Importantly, the user profile may include investment plan profile information stored during a previous session, such as the probability of reaching one or more financial goals, the risk of the portfolio, and the like. As described above, selected user information on the AdviceServer 110 may be kept up to date automatically if the financial advisory system 100 has access to the record-keeping systems of the user's employer. Alternatively, selected user information may be updated manually by the user. At step 820, a current optimal portfolio is determined, as described above. Importantly, changes to the user database and/or portfolio data are taken into consideration. For example, if one or more new financial products have become available to the user, portfolios including the one or more new financial products are evaluated. At step 830, the current portfolio is evaluated in a number of different dimensions to determine if any trigger conditions are satisfied. For example, if the increase in expected wealth, or the increase in the probability of reaching one or more investment goals resulting from a reallocation to the current optimal portfolio is above a predetermined tolerance, then processing will continue with step 840. Additionally, if the risk of the current portfolio is substantially different from the investment plan profile or if the probability of achieving one or more financial goals is substantially different from the investment plan profile, then processing continues with step 840. At step 840, advice processing is performed. According to one embodiment of the present invention, based upon the user's preference among the decision variables, the system may offer advice regarding which decision variable should be modified to bring the portfolio back on track to reach the one or more financial goals with the desired probability. In addition, the system may recommend a reallocation to improve efficiency of the portfolio. An alert may be generated to notify the user of the advice and/or need for affirmative action on his/her part. As described above, the alert may be displayed during a subsequent user session with the financial advisory system 100 and/or the alerts may be transmitted immediately to the user by telephone, fax, email, pager, fax, or similar messaging system. Advantageously, the plan monitoring module 350 performs ongoing portfolio evaluation to deal with the constantly changing data that may ultimately affect the exposure determination process and the portfolio optimization process. In this manner, the user may receive timely advice instructing him/her how to most efficiently achieve one or more financial goals and/or maintain one or more portfolio characteristics based upon the available set of financial products. In the foregoing specification, the invention has been described with reference to specific embodiments thereof. It will, however, be evident that various modifications and changes may be made thereto without departing from the broader spirit and scope of the invention. The specification and drawings are, accordingly, to be regarded in an illustrative rather than a restrictive sense.
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