# Estimating Risk under Interval Uncertainty: Sequential and Parallel Algorithms

## Publication Date

12-2008

Technical Report: UTEP-CS-08-43

Short version published in Proceedings of the 2nd Annual Econometric Conference, Chiang Mai, Thailand, January 5-6, 2009, pp. 47-60; full paper published in International Journal of Intelligent Technologies and Applied Statistics, 2010, Vol. 3, No. 1, pp. 57-70.

## Abstract

In traditional econometrics, the quality of an individual investment -- and of the investment portfolio -- is characterized by its expected return and its risk (variance). For an individual investment or portfolio, we can estimate the future expected return and a future risk by tracing the returns x1, ..., xn of this investment (and/or similar investments) over the past years, and computing the statistical characteristics based on these returns. The return (per unit investment) is defined as the selling of the corresponding financial instrument at the ends of, e.g., a one-year period, divided by the buying price of this instrument at the beginning of this period. It is usually assumed that we know the exact return values x1, ..., xn. In practice, however, both the selling and the buying prices unpredictably fluctuate from day to day -- and even within a single day. These minute-by-minute fluctuations are rarely recorded; what we usually have recorded is the daily range of prices. As a result, we can only find the range of possible values of the return xi. In this case, different possible values of xi lead, in general, to different values of the expected return E and of the risk V. In such situations, we are interested in producing the intervals of possible values of E and V.

In the paper, we describe algorithms for producing such interval estimates. The corresponding sequential algorithms, however, are reasonably complex and time-consuming. In financial applications, it is often very important to produce the result as fast as possible. One way to speed up computations is to perform these algorithms in parallel on several processors, and thus, to speed up computations. In this paper, we show how the algorithms for estimating variance under interval uncertainty can be parallelized.

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