Inbreeding of cattle
Thanks for your question!
Inbreeding is defined as the mating of animals that are more closely related than the average of the breed. There are multiple consequences of inbreeding, both good and bad.
For example (all hypothetical for illustration purposes):
Let's say we choose to breed a bull to his half-sister. It is possible that the grandpa (the relative in common) carries an uncommon trait for aborting calves in the third trimester. As his offspring, the bull and his half sister now have that uncommon trait in their genetics. If they were mated, there is a possibility that the uncommon trait (abortion) could be expressed and, therefore, affect your percentage of live calves.
Inbreeding causes the offspring to inherit genes that are very similar since the parents are related. It is possible for the offspring to have identical genes inherited from both parents called homozygous. If the homozygous genes are for a positive trait, say growth, then it could have a positive effect.
However, in many cases, homozygous genes carry negative effects. In non-inbreeding matings, the negative effect would be hidden since it is less likely for the offspring to be homozygous in that trait. But in an inbreeding situation, the traits are more likely to become evident.
The decline in performance due to inbreeding is called inbreeding depression. In general, inbreeding depression has the greatest effect on reproductive traits, followed by growth traits. Inbreeding depression seems to have little or no effect on carcass traits. Specific traits where inbreeding depression has been noticed is percentage weaned, birth weight, pre-weaning gain, weaning weight, post-weaning gain, and final weight.
In regards to your choice to breed a bull to his half-sister, I would suggest deciding what you intend to do with the offspring. Will they go into the food chain? Or will they stay in the herd to continue passing on their genetic material?
I have included a link to a great publication from Oklahoma State University on "Inbreeding in Cattle." It gives more detail about inbreeding and other possible effects.