Slope of NPV profile projects

[PassOrNothing]

I am trying to figure out how slopes of different projects would differ depending on when the cash flows comes in early or later in the life of the project. For example, if project 1 is expected to receive larger cash flows early in the life of the project while project 2 is expected to receive larger cash flows late in the life of the project, why is the slope of the NPV profile for project 1 compare to the slope of NPV profile for project 2 flatter?
I understand that the delay of cash flows for project 2 will be more sensitive to changes in the discount rate. But I cannot make the connection to the slope of the NPV profile.
Thanks so much!

 

[S2000magician]

If you have a cash flow of $10,000 in one year and you change the discount rate from 4% to 5%, the PV of that cash flow changes from $9,615 to $9,524, a change of -$91, or -0.95%.
If you have a cash flow of $10,000 in ten years and you change the discount rate from 4% to 5%, the PV of that cash flow changes from $6,755 to $6,139, an change of -$616, or -9.13%.
The farther out a large cash flow is, the greater the change (in dollars and in percent) when the discount rate changes; this makes sense: you’re discounting it for many years, so you’re compounding the change in the discount rate for many years.

 

[PassOrNothing]

I understand what you said, but I cannot make the connection to the slope of the different projects. Could you elaborate on that? Thanks!

 

[S2000magician]

The graph has NPV on the vertical axis and discount rate on the horizontal. Projects with large cash flows early will have their NPV change not very much when the discount rate changes (-$91 per 1% change, above), so their graphs will have a small (i.e., flatter) slope. Projects with large cash flows late will have their NPV change quite a lot when the discount rate changes (-$616 per 1% change, above), so their graphs will have a large (i.e., steeper) slope.