Discounting a finite series of costs at unknown times

[smhaladuick]

Consider a finite series of repeated costs C that occur at a series of times ti. Is there a solution to discount these costs by interest rate r to account for time value of money, i.e. solve for S? The times ti of each cost are unknown, but the number of costs n is known, and the average time (E[t]) is known.

$S=C \sum_{i=1}^{n}\frac{1}{({1+r})^{{t}_{i}}}$

 

[jvicens]

Hello,

Thank you for bringing up this interesting question. I can confirm that there is indeed a solution to discount these costs by interest rate r to account for the time value of money. The equation you have provided, $S=C \sum_{i=1}^{n}\frac{1}{({1+r})^{{t}_{i}}}$, is known as the discounted cash flow (DCF) formula and is commonly used in financial analysis to account for the time value of money.

To explain this further, let's break down the equation. The variable S represents the present value of all the future costs, C is the amount of each cost, and n is the total number of costs. The exponent in the denominator, ${t}_{i}$, represents the time at which each cost occurs. By raising the interest rate, r, to the power of the time, we are essentially discounting the future costs to their present value.

The DCF formula takes into account the fact that money has a time value, meaning that receiving money in the present is more valuable than receiving the same amount in the future. This is because money can be invested and earn interest over time. Therefore, by discounting the future costs, we are taking into account the opportunity cost of not having that money available to invest now.

To solve for S, we would need to know the values of C, r, and the times at which each cost occurs, ${t}_{i}$. However, as you mentioned, the times ti of each cost are unknown. In this case, we can use the average time, E[t], to estimate the present value of the future costs. This is not a perfect solution, but it can provide a rough estimate.

In summary, there is indeed a solution to discount these costs by interest rate r to account for the time value of money. The DCF formula is a useful tool for calculating the present value of future costs, taking into account the time value of money. However, it is important to note that this is just one tool and there may be other factors to consider in a more comprehensive financial analysis.

I hope this helps to answer your question and provide some insight into the concept of discounted cash flow. Let me know if you have any further questions.