# Time Value of Money

Money has a time component i.e. money has a time unit identifying when it is received/paid.

The reason is that there is an opportunity cost associated with money in the future; we miss out on the opportunity to invest it today. This cost is determined by the nature of the investment:

* If we invest the money in something relatively safe, then the opportunity cost is low
* If we invest the money in something relatively risky, then the opportunity cost is high

Opportunity cost is basically the return on investment.

As such, $100 today is worth more than $100 in the future because having $100 today will keep us from missing out on the opportunity to invest it.

Since money has a time unit, we cannot aggregate money across different points in time directly. To do so, we need the following tools:

* Timeline
* Discount Factor

## Timeline

![](/files/-M5-0_RzH7HqS0i9PJwd)The **timeline** is used to represent **cash flows** (the flow of money in/out) at specific points in time.\
0 denotes the current period/timestamp. 1, 2, 3... are future periods/timestamps.\
$$CF\_i$$ denotes the cash flow at timestamp i.

It is important to note that we must **never aggregate (add/subtract) cash flows at different timestamps!**

Just as we cannot add/subtract different currencies before converting them to a common base using an exchange rate, we also cannot aggregate cash flows at different timestamps without bringing them to a common base first. To do so, we need some sort of exchange rate for time.

## Discount Factor

The **discount factor** serves as an exchange rate for time. It is given by:

$$(1+R)^t$$

where:

* R is the **rate of return** offered by investment alternatives in the capital markets of equivalent risk. It is also known as **discount rate**, **hurdle rate** or **opportunity cost of capital**  \
  For simplicity, we assume that R is constant over time
* t is the **number of time periods** into the future (t>0) or past (t<0) to move CFs


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