According to the PMBOK® Guide, specifically in the Control Costs process within the Project Cost Management knowledge area, the Cost Performance Index (CPI) is the specific metric used to measure the cost efficiency of a project.

Definition of CPI: CPI is a measure of the cost efficiency of budgeted resources, expressed as the ratio of earned value ($EV$) to actual cost ($AC$). The formula is:

$$CPI = \frac{EV}{AC}$$

Efficiency Indicator: Because it is an index (a ratio), it tells you how much value you are getting for every dollar spent.

A CPI of 1.0 indicates the project is exactly on budget (spending $1 to get $1 of work).

A CPI greater than 1.0 indicates that the work is being performed with better efficiency than planned (under budget).

A CPI less than 1.0 indicates that the work is being performed inefficiently (over budget).

Importance: CPI is considered the most critical EVM metric as it influences the calculation of the Estimate at Completion (EAC). It provides a clear snapshot of how efficiently the project team is using the financial resources allocated to the project.

Why other options are incorrect:

Option A: Cost variance (CV): While CV also relates to cost performance, it is expressed as a currency value ($CV = EV - AC$) rather than a ratio. It shows the magnitude of the deviation from the budget, but not the " efficiency rate " or " percentage " of efficiency.

Option C: To-complete performance index (TCPI): TCPI is a measure of the cost performance that must be achieved with the remaining resources to meet a specific goal (like the original BAC or a new EAC). It describes the efficiency required for the future, not the efficiency of the work already completed.

Option D: Variance at completion (VAC): VAC is a projection of the final budget deficit or surplus ($VAC = BAC - EAC$). It is a forecasting metric used to see where the project will end up, not a measure of current work efficiency.