Managing a budget, at its simplest, is tedious. At its worst, budget management is cumbersome and time-consuming. Budgets and their management are not specifically addressed in the Agile Manifesto. However, if the role of the Scrum master ensures comprehension and enablement of the Scrum principles, and the role of the product owner is to help the Scrum team deliver value to the business, I'm not entirely sure budget management falls into the category of "delivering business value."
Is it important to know how much you're spending and where funds could be used more efficiently? Absolutely. Is this something the Scrum master or product owner should concern himself or herself with when his or her focus needs to be on delivering business value within the sprint? I'm not sure. This is where a project manager’s skill set can help a Scrum team. Allowing the project manager to be responsible for the budget frees time for the Scrum master and/or product owner to focus on the Scrum team and serving their needs. By taking over the task of managing the budget, a project manager can add value to the Scrum team, while the Scrum team adds value to the business.
Risk management is a key part of any project or product implementation. While risk management is clearly the domain of the Scrum master for the individual Scrum team, what about on a large project, where there are multiple Scrum teams? Whose responsibility is it then? This is another task that can be assumed by a project manager, should one be deemed necessary by the organization.
The identifying, tracking and resolving of risks across multiple Scrum teams, when handled by a project manager in agile, allows the Scrum master to focus exclusively on the needs of his or her team. If this was to be handled by the Scrum master, especially when coordinating with other Scrum teams, focus could be divided, negatively impacting the delivery of business value. By allowing a project manager to assume that task, the Scrum master can simply be kept in the loop, rather than an active participant in the risk resolution process.
As anyone who's worked across project teams knows, trying to coordinate the efforts of multiple project or product teams is like herding cats. When multiple Scrum teams are working on the same product, having a project manager coordinate across teams for task dependencies, communication updates and release planning can be invaluable.
As the Agile Manifesto clearly states, delivering business value is the core of Agile software development. Delivering business value and working increments of software is completely possible without the assistance of a project manager. Agile organizations can and do function very well without the role of a project manager.
However, it's my opinion that by potentially keeping the project manager role in an Agile project, specifically to perform the cumbersome, day-to-day tasks, allows the Scrum team to focus on delivering business value. By shielding the Scrum team from these responsibilities, the Scrum master can better facilitate the team’s activities and assist them in continually adding value. Maintaining the project manager role is an option to give the Scrum master the time and space to effectively meet the team’s needs, all while maintaining focus on delivering business value.