According to the PMBOK® Guide, specifically within the Control Costs and Control Schedule processes, the Schedule Performance Index (SPI) is a measure of schedule efficiency, expressed as the ratio of earned value to planned value.

The Formula: The formula for SPI is:

$$SPI = \frac{EV}{PV}$$

Where:

EV (Earned Value): The value of the work actually performed expressed in terms of the approved budget assigned to that work.

PV (Planned Value): The authorized budget assigned to scheduled work.

The Calculation:

Given the values from the question:

$EV = \$40,000$

$PV = \$50,000$

($BAC$ and $AC$ are provided but are not needed for the $SPI$ calculation)

$$SPI = \frac{40,000}{50,000}$$

$$SPI = 0.8$$

Interpretation:

An SPI value less than 1.0 indicates that less work was completed than was planned (the project is behind schedule).

An SPI of 0.8 means the project is progressing at only 80% of the planned rate.

Conversely, an SPI greater than 1.0 would indicate the project is ahead of schedule.

Comparison with Other Options:

A. 1: This would be the result if $EV = PV$ (e.g., $40,000 / 40,000$), indicating the project is exactly on schedule.

B. 0.4: This would be the result of $EV / BAC$ ($40,000 / 100,000$), which is not a standard performance index.

C. 0.5: This would be the result of $EV / AC$ ($40,000 / 80,000$), which is actually the Cost Performance Index (CPI) for this specific data set.

D. 0.8: This is the correct mathematical result for the Schedule Performance Index.