<abstract><p>The South African financial market is developing with periods of high and low volatility. Employing an adequate volatility model is essential to manage market risk. This research study was designed to investigate the effectiveness of the fractionally integrated asymmetric power autoregressive conditional heteroskedasticity contrasted with long-memory GARCH-type models, such as the fractionally integrated generalized autoregressive conditional heteroskedasticity and the hyperbolic generalized autoregressive conditional heteroskedasticity for producing the measure of market risk known as the value at risk. These long-memory GARCH-type models assume that the distributions of the index returns follow normal, student-$ t $, skewed student-$ t $ and generalized error distributions. The historical closing price time series of the Johannesburg Stock Exchange all share, the mining and the banking indices are considered. The value at risk and its backtesting for short and long trading positions on the different confident levels are computed and they correspond to the right and left quantiles of the return distributions, respectively. The results reveal that FIAPARCH with a standard student-$ t $ distribution is an appropriate model for producing a robust value at risk in the context of mining and banking indices. Alternatively, FIGARCH with the assumed skewed student-$ t $ distribution model is a good fit to produce a value at risk for the Johannesburg Stock Exchange All Share Index.</p></abstract>