The internal rate of return is the discount rate at which a project's net present value is zero. As the discount rate used increases, the net present value of a project decreases. Therefore, if the discount rate used is higher than the project's internal rate of return, the net present value of the project will be less than zero and thus will be a negative amount.
Use of a discount rate that is greater than the firm's cost of equity may result in a negative net present value for the investment, or it may result in a positive net present value for the investment. Which it will be depends upon the cash flows.
The risk-free rate is not relevant to a capital budgeting analysis using net present value.
The internal rate of return is the discount rate at which a project's net present value is zero. As the discount rate used decreases, the net present value of a project increases. Therefore, if the discount rate used is lower than the project's internal rate of return, the net present value of the project will be greater than zero.
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