Internet-of-things technologies will enable collecting vast amounts of mobility data from car owners. Such connected car services can be value-adding but also create new privacy hazards. This paper studies whether and how privacy concerns of car owners can be compensated by offering monetary benefits. We study the case of usage based car insurance services for which the insurance fee is adapted to measured mileage and driving behaviour. A conjoint experiment shows that consumers prefer their current insurance products to usage based car insurance. However, when offered a minor financial compensation, they are willing to give up their privacy to car insurers. Consumers find privacy of behaviour and action more valuable than privacy of location and space. The study is a first to compare different forms of privacy in the acceptance of connected car services. Hereby, we contribute to more fine-grained understanding of privacy concerns in the acceptance of digital services, which will become more important than ever in the upcoming Internet-of-things era.