In this paper the two dimensional model of the investment in shares is presented. The shares prices from five different world stock exchanges (New York, London, Frankfurt, Honk Kong, and Sydney) are examined. The copula functions are used to model the risk of investment. The Hurst threshold exponent derived from the local Detrended Fluctuation Analysis is used to determine the safe investment portfolios with no extreme drops in shares prices. The most important result states that the threshold value is not universal for different markets, however, it is influenced by the subsequent level of market freedom. It was shown, that the level, relatively larger in US, UK, and Australia than in Germany and China, affects the Hurst exponent threshold value.