Methods and apparatus employing hierarchical conditional value at risk 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.

What is claimed is: 
     
         1 . A computer-implemented method comprising:
 specifying a set of pairs of confidence levels for conditional value at risk (CVaR) estimation utilizing a window in a graphical user interface;   constructing a set of possible investment portfolios that minimizes CVaR at a first confidence level of each pair of the set of pairs, subject to a constraint that CVaR at a second confidence level of each pair of the set of pairs is less than a predetermined amount;   computing a set of return distributions corresponding to the set of possible investment portfolios;   displaying a graphical representation of the set of return distributions for the set of possible investment portfolios as lines within the window of the graphical user interface so that, apart from when one line crosses another in the graphical representation, the line corresponding to one return distribution of the set of return distributions does not obscure the lines of any other return distribution of the set of return distributions, regardless of the ordering in which the lines are displayed;   dynamically adjusting an ordering in which the lines are displayed on top of previously displayed lines within the window of the graphical user interface based at least in part on interaction with the graphical representation of the set of return distributions for negative returns within the window of the graphical user interface; and   electronically outputting at least one possible investment portfolio of the set of possible investment portfolios and its associated confidence pair based at least in part on the graphical representation of the set of return distributions for the set of possible investment portfolios.   
     
     
         2 . The method of  claim 1  wherein the graphical representation of the set of return distributions for the set of possible investment portfolios is automatically updated based on real-time data inputs from an electronic trading system. 
     
     
         3 . The method of  claim 1  wherein said at least one possible investment portfolio and its associated confidence pair is output to an electronic trading system for execution. 
     
     
         4 . The method of  claim 1  wherein said at least one possible investment portfolio and its associated confidence pair has a smaller left tail distribution than one or more other return distributions in the set of return distributions. 
     
     
         5 . The method of  claim 1  wherein, for a given one of the set of pairs of confidence levels, the first confidence level is greater than 90% and less than 99%. 
     
     
         6 . The method of  claim 1  wherein, for a given one of the set of pairs of confidence levels, the first confidence level is 90% and the second confidence level is 95%. 
     
     
         7 . The method of  claim 1  further comprising displaying an additional graphical representation of the set of return distributions for the set of possible investment portfolios, wherein each return distribution in the set of return distributions for the set of possible investment portfolios in the additional graphical representation comprises is drawn as a bar graph of a different solid color within the window of the graphical user interface, the bar graphs having a display order such that the bar graph of each return distribution in the set of return distributions is displayed over and without obscuring bars of any other return distribution for return values less than a left-tail return value of that return distribution since a corresponding bar displayed on top of a previous bar is never taller than any previously displayed bars. 
     
     
         8 . The method of  claim 7  further comprising dynamically switching the window of the graphical user interface between the graphical representation and the additional graphical representation. 
     
     
         9 . The method of  claim 1  wherein a given possible investment portfolio in the set of possible investment portfolios is a hedge to reduce a risk estimate of an existing investment portfolio. 
     
     
         10 . The method of  claim 1  wherein said at least one possible investment portfolio and its associated confidence pair is associated with a given return distribution of the set of return distributions that is determined to be more advantageous than one or more other return distributions in the set of return distributions. 
     
     
         11 . The method of  claim 10  wherein the given return distribution is determined to be more advantageous than the one or more other return distributions in the set of return distributions by interactively altering the set of pairs of confidence levels and redisplaying the set of return distributions. 
     
     
         12 . The method of  claim 1  wherein dynamically adjusting an ordering in which the lines are displayed on top of previously displayed lines comprises automatically determining the ordering in which the lines are displayed based at least in part on a location of a cursor in the window of the graphical user interface. 
     
     
         13 . The method of  claim 12  wherein automatically determining the ordering in which the lines are displayed comprises:
 detecting the location of a cursor in the window of the graphical user interface; 
 generating an indication line corresponding to a return value at the location of the cursor in the window of the graphical user interface; and 
 selecting an ordering for display of the lines corresponding to the set of return distributions for the set of possible investment portfolios based at least in part on determining coordinates where the lines corresponding to the set of return distributions for the set of possible investment portfolios intersect the indication line. 
 
     
     
         14 . The method of  claim 13  wherein detecting the location of the cursor in the window of the graphical user interface comprises detecting hovering of the cursor, wherein detecting hovering of the cursor comprises detecting that the cursor remains in place for a predetermined time. 
     
     
         15 . An apparatus comprising:
 at least one processing device comprising a processor coupled to a memory;   the at least one processing device being configured:
 to specify a set of pairs of confidence levels for conditional value at risk (CVaR) estimation utilizing a window in a graphical user interface; 
 to construct a set of possible investment portfolios that minimizes CVaR at a first confidence level of each pair of the set of pairs, subject to a constraint that CVaR at a second confidence level of each pair of the set of pairs is less than a predetermined amount; 
 to compute a set of return distributions corresponding to the set of possible investment portfolios; 
 to display a graphical representation of the set of return distributions for the set of possible investment portfolios as lines within the window of the graphical user interface so that, apart from when one line crosses another in the graphical representation, the line corresponding to one return distribution of the set of return distributions does not obscure the lines of any other return distribution of the set of return distributions, regardless of the ordering in which the lines are displayed; 
 to dynamically adjust an ordering in which the lines are displayed on top of previously displayed lines within the window of the graphical user interface based at least in part on interaction with the graphical representation of the set of return distributions for negative returns within the window of the graphical user interface; and 
 to electronically output at least one possible investment portfolio of the set of possible investment portfolios and its associated confidence pair based at least in part on the graphical representation of the set of return distributions for the set of possible investment portfolios. 
   
     
     
         16 . The apparatus of  claim 15  wherein the at least one processing device is further configured to display an additional graphical representation of the set of return distributions for the set of possible investment portfolios, wherein each return distribution in the set of return distributions for the set of possible investment portfolios in the additional graphical representation comprises is drawn as a bar graph of a different solid color within the window of the graphical user interface, the bar graphs having a display order such that the bar graph of each return distribution in the set of return distributions is displayed over and without obscuring bars of any other return distribution for return values less than a left-tail return value of that return distribution since a corresponding bar displayed on top of a previous bar is never taller than any previously displayed bars. 
     
     
         17 . The apparatus of  claim 16  wherein the at least one processing device is further configured to dynamically switch the window of the graphical user interface between the graphical representation and the additional graphical representation. 
     
     
         18 . An article of manufacture comprising a non-transitory processor-readable storage medium having encoded therein executable code of one or more software programs, wherein the one or more software programs when executed by at least one processing device cause the at least one processing device:
 to specify a set of pairs of confidence levels for conditional value at risk (CVaR) estimation utilizing a user interface in a graphical user interface;   to construct a set of possible investment portfolios that minimizes CVaR at a first confidence level of each pair of the set of pairs, subject to a constraint that CVaR at a second confidence level of each pair of the set of pairs is less than a predetermined amount;   to compute a set of return distributions corresponding to the set of possible investment portfolios;   to display a graphical representation of the set of return distributions for the set of possible investment portfolios as lines within a window of the graphical user interface so that, apart from when one line crosses another in the graphical representation, the line corresponding to one return distribution of the set of return distributions does not obscure the lines of any other return distribution of the set of return distributions, regardless of the ordering in which the lines are displayed;   to dynamically adjust an ordering in which the lines are displayed on top of previously displayed lines within the window of the graphical user interface based at least in part on interaction with the graphical representation of the set of return distributions for negative returns within the window of the graphical user interface; and   to electronically output at least one possible investment portfolio of the set of possible investment portfolios and its associated confidence pair based at least in part on the graphical representation of the set of return distributions for the set of possible investment portfolios.   
     
     
         19 . The article of manufacture of  claim 18  wherein the one or more software programs when executed by the at least one processing device further cause the at least one processing device to display an additional graphical representation of the set of return distributions for the set of possible investment portfolios, wherein each return distribution in the set of return distributions for the set of possible investment portfolios in the additional graphical representation comprises is drawn as a bar graph of a different solid color within the window of the graphical user interface, the bar graphs having a display order such that the bar graph of each return distribution in the set of return distributions is displayed over and without obscuring bars of any other return distribution for return values less than a left-tail return value of that return distribution since a corresponding bar displayed on top of a previous bar is never taller than any previously displayed bars. 
     
     
         20 . The article of manufacture of  claim 19  wherein the one or more software programs when executed by the at least one processing device further cause the at least one processing device to dynamically switch the window of the graphical user interface between the graphical representation and the additional graphical representation.