When evaluating investment projects, a popular approach is to calculate the net present value (NPV). A positive NPV suggests carrying out the project is better than not doing that, in light of the project's risk. But is "not doing that" the same as doing nothing? It seems the NPV of doing nothing is zero: all cash flows are zero at all time points, and thus their discounted sum is also zero. But doing nothing is clearly worse than, say, investing risk free as long as the risk-free rate is positive. However, I think the NPV of a risk-free investment should also be zero, as the appropriate discount rate is the risk-free rate. So zero is worse than zero... How do I reconcile these things and show that doing nothing is a negative NPV endeavor (actually, a non-endeavor) - if it is?
NPV just looks at actual cashflows so NPV of doing nothing is 0.
What you are thinking about is the concept of opportunity cost from economics. Every action/inaction has a cost that you need to forgo your next best alternative.
From economic perspective thus you are correct doing nothing when you have option to have risk free investment is creating economic cost for you. However, textbook NPV only looks at accounting revenues and costs. Any implicit costs, though economically important are disregarded in NPV calculation. You could modify NPV to include this implicit costs and in fact in economic model when we have dynamic optimization problem of firm, a firm would discount economic profits that include opportunity cost, but I think unless you specify what you are doing most people would think you are making mistake in your NPV calculation so if you want to modify it, you should let your readers know.