"I would gladly pay a dollar tomorrow for a hamburger today" can be formalized by an ODE

Photo credit to Valeria Boltneva

One dollar received today is more valuable than one dollar received in ten years, if for no other reason than the potential to invest that dollar today and allow its value to grow for 10 years. This is called the time discount value of money. The net present value calculation is the mathematical expression of the time discount value of money; it is the answer to the question "Would you rather have $50,000 today or $75,000 next year?" If you are thinking long term, the question is "How much would I be able to earn with $50,000 by next year and is that more than $75,000?" Because we may wish to compare payments at many different time points, it is standard practice to frame the question as "How much money would I need today to generate $75,000 next year and is that more than $50,000?" For a pile of cash received at a future time, the NPV is the size of the pile of cash required today to generate that future pile of cash at a given rate of return.

Discrete future sums

Here we will consider a single sum of \(V_t\) dollars at future time \(t\) In the next section we will discuss NPV calculations for rates of income.

$$ \begin{align} \frac{dV}{dt} &= r_rV \\ \frac{dV}{V} &= r_rdt\\ \int^{V_t}_{\text{NPV}} \frac{dV}{V} &= \int^t_0 r_rdt\\ \log\left(\frac{V_t}{\text{NPV}}\right) &= r_rt\\ \text{NPV} &= V_t e^{-r_rt} \end{align} $$

Continuous future income: income streams and value

This equation sums linearly for multiple discrete piles of cash at different times:

$$ \begin{align} \text{NPV} &= \sum_{i = 0,1,\dots} V_{t_i} e^{-r_rt_i} \end{align} $$

Now consider an income stream, \(F(t)\) where \(F\) is in dollars per time and a future value which is the rate of cash flow times a small time step, \(V_t = F(t)\Delta t\).

$$ \begin{align} \text{NPV} &= \sum _{i=0,1,\dots}^{i= t/\Delta t} e^{-r_ri\Delta t} F(i\Delta t)\Delta t\\ \text{NPV} &= \lim_{\Delta t \to 0} \sum _{i=0,1,\dots}^{i = t/\Delta t} e^{-r_ri\Delta t} F(i\Delta t)\Delta t\\ \text{NPV} &= \int^{t}_{0} e^{-r_r\tau} F(\tau) d\tau \\ \end{align} $$

For a constant source of income (rate independent of time):

$$ \begin{align} \label{eq:net_p} \text{NPV} &= F\int^{t}_{0} e^{-r_r\tau}d\tau \\ \text{NPV} &= \left.-\frac{F}{r_r} e^{-r_r\tau} \right\rvert^{t}_{0} \\ \text{NPV} &= \frac{F}{r_r} \left( 1- e^{-r_rt} \right) \\ \end{align} $$

This raises an initially counterintuitive point - the net present value of a limitless stream of continual income is finite. Even with the most conservative interest rates (1% per year certificate of deposit), the infinite continuous income stream is never worth more than 100x its annual value. Assuming a reasonable but still conservative stock market return of 4-5% the income stream is never worth more than 20-25x its annual value.

$$ \label{eq:NPV_inf} \begin{align} \left. \text{NPV}(F) \right\vert _{t \xrightarrow{} \infty} &=\lim_{t \xrightarrow{} \infty} \frac{F}{r_r} \left( 1- e^{-r_rt} \right) \\ \left. \text{NPV}(F) \right\vert _{t \xrightarrow{} \infty} &= \frac{F}{r_r} \end{align} $$

If this is still confusing, remember than a sufficiently large pile of cash today could generate an income greater than \(F\) for an equally indefinite period. Consider a limiting case, how much is $0.01 per year forever worth? A high yield savings account could provide that return on $1.00 (if it were not for the minimum balance requirements) so the $0.01 per year is not worth much.