Genetic parameters were estimated for protein yield (PY), clinical mastitis (CM), somatic cell score, number of inseminations (NI) and days from calving to first insemination (CFI) in first-parity Swedish Red cows by series of tri-variate linear animal models. The heritability of PY was moderate (0.34 ± 0.004), and the heritabilities of the functional traits were all low (0.014 ± 0.001–0.14 ± 0.004). The genetic correlation between CM and CFI (0.38 ± 0.05) was stronger than the correlation between CM and NI (0.05 ± 0.06), perhaps because CM and CFI usually are observed in early lactation when the cow is likely to be in negative energy balance, whereas NI generally is recorded when the cow is not in negative energy balance any more. The genetic correlation between NI and CFI was very close to zero (−0.002 ± 0.05), indicating that these two fertility traits have different genetic backgrounds. All genetic correlations between PY and the functional traits were moderate and unfavourable, ranging from 0.22 ± 0.02 to 0.47 ± 0.03. In addition, the effect of including genetic and phenotypic correlations between the trait groups milk production, udder health and female fertility on the accuracy of the selection index was quantified for a heifer, a cow and a proven bull. The difference between the accuracy obtained by multi-trait and single-trait evaluations was largest for the cow (0.012) and small for the heifer and the bull (0.006 and 0.004) because the phenotype of the cow for one trait could assist in predicting the Mendelian sampling term for a correlated trait.