Methods and apparatus employing hierarchical conditional variance to minimize downside risk of a multi-asset class portfolio and improved graphical user interface

Abstract

The traditional Markowitz mean-variance-optimization (MVO) framework that uses the standard deviation of the possible portfolio returns as a measure of risk does not accurately measure the risk of multi-asset class portfolios whose return distributions are non-Gaussian and asymmetric. A scenario-based conditional value-at-risk (CVaR) approach for minimizing the downside risk of a multi-asset class portfolio is addressed that uses Monte-Carlo simulations to generate the asset return scenarios. These return scenarios are incorporated into a modified Rockafellar-Uryasev based convex programming formulation to generate an optimized hedge. One example addresses hedging in an equity portfolio with options. Testing shows that a hierarchical CVaR approach generates portfolios with better predicted worst case loss, downside risk, standard deviation, and skew.

I claim: 
     
         1 . A computer-implemented method for interactively comparing performance of a plurality of investment portfolios within a window of a graphical user interface, the method comprising:
 electronically receiving by a programmed computer a plurality of return distributions corresponding to the plurality of investment portfolios wherein each return distribution comprises pairs of return and frequency values;   displaying a graphical representation of the return distribution for each investment portfolio within the window of the graphical user interface on a computer screen in a display order such that each return distribution is displayed over and potentially obscures any previously displayed return distribution;   monitoring, by a processor, a location of a user pointer indication to detect when the user pointer indication is located and hovering within the window of the graphical user interface displaying the return distributions; automatically determining an indicated return value corresponding to the return value of the displayed return distributions wherever the user pointer indication is located and hovering within the graphical user interface displaying the return distributions;   automatically altering, by the processor, the display order in which the return distributions are displayed so that, at the indicated return value corresponding to the user pointer indication, no return distribution completely obscures any other return distribution.   
     
     
         2 . The method of  claim 1  wherein the display order in which the return distributions are displayed is determined such that the frequency values of the return distributions at the indicated return value are in a decreasing order. 
     
     
         3 . The method of  claim 1  wherein the processor constantly monitors the activation of a second user indication capable of indicating a selection of a preferred return distribution and a corresponding preferred investment portfolio;
 automatically electronically outputting the preferred investment portfolio selection whenever the second user indication is activated. 
 
     
     
         4 . The method of  claim 3  wherein each of the plurality of investment portfolios is constructed to minimize a conditional value at risk (CVaR) estimate at a plurality of confidence limits. 
     
     
         5 . The method of  claim 4  wherein at least one confidence interval is between 90% and 99%. 
     
     
         6 . The method of  claim 4  wherein the CVaR minimization employs a regularized Rockafeller-Uryasev methodology modified to utilize returns lying within an elliptical uncertainty set. 
     
     
         7 . The method of  claim 4  wherein a second window in the graphical user interface displays a table of each confidence limit and the CVaR at each confidence limit for each investment portfolio. 
     
     
         8 . The method of  claim 7  wherein the table in the second window in the graphical user interface also displays a budget for each investment portfolio. 
     
     
         9 . The method of  claim 8  wherein the processor constantly monitors the table within the second window for changes in individual confidence limits or budgets;
 whenever a change in the confidence limits or budget is detected for an investment portfolio, the investment portfolio and return distributions are recomputed using those changes, and the displayed return distributions and CVaR values in the graphical user interface are automatically updated. 
 
     
     
         10 . The method of  claim 1  wherein at least one of the plurality of investment portfolios is an existing portfolio and at least one other investment portfolio is a hedge to reduce a risk estimate of the existing portfolio. 
     
     
         11 . A computer-implemented method for interactively comparing performance of a plurality investment portfolios within windows of a graphical user interface on a computer screen and then selecting a preferred investment portfolio, the method comprising:
 electronically receiving by a programmed computer a plurality of investment portfolios;   electronically generating a set of simulated returns for each investment portfolio;   electronically receiving by the programmed computer an initial graphing style and graphing order for displaying each set of simulated returns;   displaying a graphical representation of the simulated returns for each investment portfolio within a first window of a graphical user interface on a computer screen in the initial graphing style and graphing order such that each set of simulated returns is graphed on top of the previously graphed simulated returns;   displaying in a second window of the graphical user interface on the computer screen a first user indicator that sets a modified graphing style or graphing order;   monitoring the first user indication of the modified graphing style or graphing order;   automatically redisplaying the graphical representation of the simulated returns for each investment portfolio within the graphical user interface on the computer screen in the graphing style and graphing order specified by the first user indication;   displaying in a third window of the graphical user interface on the computer screen a second user indication that selects a preferred investment portfolio from the plurality of investment portfolios;   monitoring the second user indicator selecting a preferred investment portfolio;   automatically electronically outputting the preferred investment portfolio whenever the second user indication is activated to select a preferred investment portfolio.   
     
     
         12 . The method of  claim 11  wherein the graphical representations of the returns of the plurality of investment portfolios are automatically updated based on real-time data inputs from an electronic trading system. 
     
     
         13 . The method of  claim 12  wherein the preferred investment portfolio is output to the electronic trading system for execution. 
     
     
         14 . A computer-implemented method for interactively engineering and selecting a preferred portfolio of investments within windows of a graphical user interface on a computer screen connected to a programmed computer, the method comprising:
 displaying within a first window of the graphical user interface on the computer screen a data set that defines a set of portfolio construction parameters needed to compute an investment portfolio that minimizes a conditional variance of a portfolio at a set of prescribed confidence levels, where the set of portfolio construction parameters include:   a first user indicator in the first window of the graphical user interface of the computer screen a set of potential investment opportunities that are eligible to be included as part of the investment portfolio;   a set of general portfolio construction requirements that all eligible portfolios are to satisfy;   a data library for the set of investment opportunities containing data needed to model each potential investment opportunity mathematically on the programmed computer;   a tolerance for the differences in conditional value-at-risk;
 using the programmed computer and the data library to generate a sequence of simulated returns for each potential investment opportunity; 
 electronically receiving by a second user indication a set of two or more confidence levels at which a conditional value at risk is to be minimized; 
 sequentially calculating using the programmed computer a set of investment portfolios that minimizes the conditional value-at-risk for the portfolio at each confidence limit prescribed by the second user indication such that the conditional value-at-risk for the portfolio is within a tolerance of its minimum possible value; 
   displaying within a second window of a graphical user interface a distribution of returns for the portfolio of investment;   interactively changing the second user indication to specify different sets of two or more confidence limits and redisplaying the distribution of returns until a preferred investment portfolio and return distribution is obtained;   automatically electronically outputting the preferred investment portfolio whenever the second user indication is modified.   
     
     
         15 . The method of  claim 14  wherein the sequence of confidence limits includes at least 95% and 90%. 
     
     
         16 . A computer implemented method of estimating risks of an optimized portfolio when assets with nonlinear and asymmetric return distributions are included in the optimized portfolio, the computer implemented method comprising:
 specifying assets that may be included in the optimized portfolio;   employing a Monte-Carlo pricing engine to generate asset return scenarios for said assets;   specifying at least two confidence levels for conditional value at risk (CVaR) estimation;   specifying a CVaR tolerance;   employing a Rockafeller-Uryasev methodology modified to utilize return scenarios lying in an elliptical uncertainty set to produce the optimized portfolio whose CVaR is within the CVaR tolerance of the minimum possible CVaR at each confidence level; and   outputting the optimized portfolio and distributions of returns.   
     
     
         17 . The computer implemented method of  claim 16  wherein said assets comprise multi-asset class investments. 
     
     
         18 . The computer implemented method of  claim 16  wherein for N said assets and S asset return scenarios, the Monte-Carlo pricing engine produces an N by S matrix of asset return scenarios. 
     
     
         19 . The computer implemented method of  claim 16  further comprising repeating the method for additional sets of confidence levels to produce alternative optimized portfolios and distributions of returns. 
     
     
         20 . The computer implemented method of  claim 19  wherein said outputting step further comprises displaying at least two distributions of returns on a computer display screen, each corresponding to a repeated performance of the method;
 evaluating an order in which the distributions of returns are displayed; and 
 revising said order if it is determined a revised order more clearly displays overlaid data to a user. 
 
     
     
         21 . A computer implemented method of employing a hierarchical conditional value at risk (HCVaR) comprising:
 specifying at least two confidence levels for conditional value at risk (CVaR) estimation;   constructing a first portfolio that minimizes CVaR at a first confidence level (CVaR 1 );   storing the minimal CVaR for the first confidence level (CVaR 1 );   constructing a second portfolio that minimizes CVaR at a second confidence level subject to a constraint that CVaR at the first confidence level is less than CVaR 1  times a predetermined amount.