## 2.1 Cash flows

As indicated in the previous chapter, a basic application of actuarial mathematics is to model the transfer of money. Insurance companies, banks and other financial institutions engage in transactions that involve accepting sums of money at certain times, and paying out sums of money at other times.

To construct a model for describing this situation, we will first fix a time unit. This can be arbitrary, but in most applications it will be taken as some familiar interval of time. For convenience we will assume that time is measured in years, unless we indicate otherwise. We will let time 0 refer to the present time, and time *t* will then denote *t* time units in the future. We also select an arbitrary unit of capital. In this chapter, we assume that all funds are paid out or received at integer time points, that is, at time 0, 1, 2, …. The amount of money received or paid out at time *k* will be called the *net cash flow* at time *k* and denoted by *c*_{k}. A positive value of *c*_{k} denotes that a sum is to be received, whereas a negative value indicates that a sum is paid out. The entire transaction is then described by listing the sequence of cash flows. We will refer to this as a *cash flow vector*,

where *N* is the final duration for which a payment is made.

For example, suppose I lend you 10 units of capital now and a further 5 units a year from ...