Present Value as the Discounted Sum of Future Cash Flows

# Understanding Hamilton's formula for present value


# Understanding Hamilton's formula for present value












I am a mathematician with almost no knowledge in economy and econometrics trying to read Hamilton's Time series analysis.

At the very beginning of the book, Hamilton considers an order-$1$ difference equation given by $\forall n \in \mathbb{N}, y_n = \phi y_{n-1} + w_n$. I understand this equation in the following way: each period of time, the money in my bank account gets multiplied (the $\times \phi$ part), and I take out money from it or add some money to it (the $+w_n$ part).

He then defines the present value of a future stream $y_t, y_{t+1}, \cdots$ by the formula $\sum^{\infty}_{i = 0} \frac{y_{t+i}}{(1+r)^i}$ where $r$ denotes the interest rate, and I don't understand this formula.

Of course, I understand that if today, I have $y_t$, if I put all my money in a machine that each day multiplies my money by $(1+r)$, after $i$ days, I will have $y_t(1+r)^i$; so having $y_t$ today and having $y_t(1+r)^i$ in $i$ days are, in some sense, the same thing.

However, I don't understand what is represented by the sum over all future days. While the money is in the $(1+r)$-multiplying machine, I can't use it, and I think it makes no sense to sum two things, one of which is unavailable while the other is available! Moreover, let us assume $r = 0$. If I have $1$ dollar today, and the interest rate is zero, the sum is infinite. Of course, the dollar that is summed an infinite number of times is the same dollar!

Something like $\frac{1}{N}\sum^{N-1}_{i=0} \frac{y_{t+i}}{(1+r)^i}$ would have made more sense to me (it would be something like the expected present value of the money after a random uniform number of days between $0$ or $N-1$)...

PS: Feel free to recommend me any time series analysis that is mathematically-oriented!

## Answer by Bob Jansen (score 1)

https://quant.stackexchange.com/a/74469

You have to think of the $y_t$ as one cash flow at time $t$ you will receive. This stream of cash flows is expected to be finite so the sum never goes to infinity as a finite amount of $y_t$ are non-zero. Take for example a zero risk 10 year coupon bond. This formula would give you the present value of the 10 cash flows generated by the bond for a given $r$.

Regarding your post scriptum: I don’t think it makes sense to have an econometrics that is completely math oriented. Econometrics wants to be applied.