Leveraged Return on Investment A general return on investment (ROI) formula is:

\begin{equation}
\text{ROI}=\frac{\text{Net Profit}}{\text{Total Investment}}.
\end{equation}

But if you think about an investment that requires some sort of down payment, thus it is deemed as leverated return on investment (LROI), what would be a sensible way to compute such return? Specifically, I am interested in having $\textit{down payment}$ in the denominator. Think of a construction project or home investment.

\begin{equation}
\text{LROI}=\frac{\text{?}}{\text{Down Payment}}.
\end{equation}

Any suggestions?
I am not sure I understand your question. In most projects, the bulk of the investment is a down payment, so the ROI is meant to capture precisely that. I.e. the average return that the money you put in the project will yield. This has in mind that the total investment is mostly done today and the returns come in the future.

If you restrict the denominator to only be the down payment, you seem to want to capture the average return of the money you initially put into the project (excluding the returns that are derived due to money put into the project after the down payment). The problem is how you attribute how much of the future profits are due to the initial down payment and how much due to the investments that are done in the middle of the project, etc.

This is why  I think  that the ROI is already capturing the notion you want. If the project will not require a lot of money after the down payment, then the denominators will not be so different anyway. However if the project might require a lot more money after the initial down payment, it is harder to claim that X amount of the net profits can be attributed exclusively to the initial down payment.

You are correct that the ROI does not provide any information about leverage, so maybe you want to complement it with some other ratio like (down payment)/(total investment) or another over-the-counter financial ratio. However, considering a measure like net profits/down payment will seem very deceiving to me.