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In my notes for 'Financial Maths I', under the topic of 'Present value analysis', we have the following definitions:

Suppose we can borrow and lend money at nominal rate $r$ per annum. If we borrow $(1+r)^{-i}V$ pounds now and put it in the bank, then at the end of year $i$ we will have $(1+r)^{-i}V\cdot(1+r)^i=V$ pounds in the bank.

Then we define the present value of $V$ pounds at the end of year $i$ to be:
$$PV(V)=(1+r)^{-i}V$$

Furthermore, we can generalise the notion of present value to a cash flow system $a=(a_1,a_2,...,a_n)$, that is, a sequence of payments, where $a_i$ is the payment made at the end of year $i$

My question is: In this context, is cash flow system a vector, or is it a function of $a_i$?

YinWai Tse
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4 Answers4

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Sure, a sequence of reals is a vector when viewed in an appropriate way. It is also a value if a function from an interval of integers into $\mathbb R^n$, and you could also see it as a function of the $a_i$, though it is a rather boring function (the identity map).

A vector space over a field of scalars (here the real numbers) is a set $V$ with an addition operation (commutative, associative, with an identity $0$) and a scalar multiplication operation such that for each scalar $a$ and elements $u,v$ in $V$ we have $$a(u+v)=au+av$$ We also require that $1u=u$ for all $u$. That's it! Here the scalar multiplication operation multiples each term in the sequence by the scalar, and the addition is done term by term.

Matt Samuel
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    In this context, does it make more sense to call it a sequence or a function? From my understanding, viewing it as a function is not very useful, since it doesn't do much for us as an identity (in the case of finance, anyway) – YinWai Tse Aug 12 '17 at 14:56
  • @YinWai I agree it's more common to view it as a sequence, but they're not mutually exclusive. A sequence is a function. – Matt Samuel Aug 12 '17 at 15:00
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The object $a=(a_1,a_2,\ldots, a_n)$ is certainly not a "system": In mathematics the word "system" is reserved for more complicated things. It is, however, an $n$-tuple, array, or data vector, whereby we tend to use the word "vector" for such an array only if it would make sense to consider $\lambda a$ or $a+b$ under the given circumstances as well.

Christian Blatter
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  • There's no harm in calling it a system, and in finance it is of fundamental importance. The terms in the sequence can also be stochastic. – Matt Samuel Aug 12 '17 at 14:52
  • In financial mathematics it certainly does make sense to consider $\lambda a$ and $a + b$ because we build portfolios of cash flows from other cash flows. – Mark Aug 12 '17 at 14:53
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It is always a function of $a_i$, but it can be written with vectors or with a sigma sign.

1) $$PV(\textbf a,\textbf r)=\textbf a^T\cdot \textbf r,$$

where $a^T =\begin{pmatrix}{} a_1&a_2&a_3&\ldots & a_n \end{pmatrix}, \textbf r^T=\begin{pmatrix}{} \frac1{1+r}&\frac1{(1+r)^2}&\frac1{(1+r)^3}&\ldots & \frac1{(1+r)^n} \end{pmatrix}$

2) $$PV(a_1, a_2, a_3, \ldots, a_n, r)=\sum_{i=1}^{n} \frac{a_i}{(1+r)^i}$$

callculus42
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The correct terminology is cash flow stream, not system.

A cash flow stream may be finite, countable, or even continuous.

Your OP concerns a finite cash stream that (presumably) occurs at equal intervals of time. You can think of it as a vector, but it is important to keep track when the payments materialise. So most people would conceptualise *a$ as a real-valued function on $T = { 1, 2, \ldots, n}$.

mlc
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