Portfolio Optimization

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Worst-case scenario optimization

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are known precisely. In practice, however, statistical estimates of parameters may have significant estimation error. In particular it is relevant for the sample estimate  of vector  that represents the excess part of

. Ignorance of estimation error in usually leads to unwarrantably extreme values of portfolio weights and dramatic shifts in portfolio structure when previous estimates are modified with recent historical data.

One way to avoid the mentioned above undesirable properties of an optimal portfolio is to include the estimation error explicitly in optimization process. Most practically sound results in this direction were obtained in

1. The set of possible values for the true vector  is defined.
   
2. The optimal portfolio, corresponding to relative risk aversion coefficient , is obtained by solving the following Min-Max problem:
   
   

In other words, the investor seeks for portfolio with the best performance under the worst-case scenario. Inner minimization over expected instantaneous excess returns in the above expressions can be regarded as reflection of investor’s uncertainty aversion. A convenient measure for uncertainty aversion degree is confidence level , used below to define . As approaches 1, investor’s aversion to uncertainty increases to infinity.

Two versions of set are considered below.

Separate Confidence Intervals

Let be the number of portfolio components and be sample counterpart of . For every  corresponding confidence interval for  in the form of is obtained by solving the following equation:

where
– sample volatility in -th asset
– average number of observations in one year
– available number of observations in -th asset
– inverse function of Student’s t-distribution with degrees of freedom: it returns value  such that , where  is -distributed random variable with degrees of freedom.
Set , corresponding to Separate Confidence Intervals for , is defined as -dimensional rectangular parallelepiped .
Then, the stated Min-Max problem, corresponding to

, reduces to

where portfolio worst-case excess Mu is defined in the following way:

In case of uncorrelated assets the probability of falling into  is . In the general case calculation of the probability of falling into is more complicated. It could be desirable to explicitly state the probability of falling into than defining it implicitly by means of confidence intervals for individual assets. This is done in the next section.

Joint Confidence Interval

Suppose all portfolio components possess the same number of observations . Consider the quantity , which is a sample estimate of squared

, corresponding to portfolio excess Mu equal to . This quantity is a convenient measure for the distance between  and .

Joint Confidence Interval  for is defined in the following way:

Solution for has the following form:

where is an inverse function of -distribution with  and degrees of freedom: it returns value such that , where  is -distributed random variable with  and degrees of freedom.
Set , corresponding to the above joint confidence interval for , is defined as follows:

The Min-Max problem, corresponding to risk aversion coefficient , simplifies to

where portfolio worst-case excess Mu is defined as

Correspondingly, the Min-Max problem, corresponding to the target shortfall probability criterion, simplifies to the following: