In a genomic evaluation, it is desirable to have low computing cost while retaining high accuracy of evaluation for young animals. When the population is large but only few animals have phenotypes, especially for low heritability traits, the convergence rate of BLUP or single-step genomic BLUP (ssGBLUP) can be very slow. While eliminating old pedigrees can seriously affect (G)EBV for old animals, usually only younger animals are candidates for selection. This study investigates the effect of pedigree truncation on convergence rate and accuracy of prediction for young animals. The data consisted of 216k, 221k, 722k, and 579k phenotypes on 4 traits (T1, T2, T3, T4) from a purebred pig line. Heritabilities were <0.1 for T1 and T2, and >0.2 for T3 to T4. A total of 2.4 million animals born from 1971 to 2016 were included in the complete pedigree. Genotypes were available for 33,502 animals and consisted of 60,003 SNP. A bivariate animal model was fit for T1–2, and T3–4, separately. Computations were done by BLUP or ssGBLUP, and were conducted with complete pedigree or different levels of pedigree depth (Pn), where n = 1, 2, 3, 4, 5. Pedigree depth n was defined as n ancestral generations from the animals with phenotypes. The number of pedigree animals for T1–2 (T3–4) varied from 226k (760k) for P1 to 228k (767k) for P5. Genomic relationship matrix was inverted either by a regular or the algorithm for proven and young (APY). GEBV between runs with the complete and pruned pedigrees for genotyped animals were correlated at >0.99 for P2 to P5. For T1–2 (T3–4), convergence required up to 7,381 (1,421) rounds with the complete pedigree; this number decreased for different levels of pedigree depth up to less than 1,730 (854) rounds for P2. Use of the APY inverse in ssGBLUP improved convergence up to 25% on average, without affecting accuracy. Pedigree pruning and the APY algorithm are important tools to reduce the computing cost of ssGBLUP without negatively impacting accuracy of predictions.