Applied Corporate Finance Series - Time Value of Money

This is one of the basic and foundational concepts in finance. 

 

The essence of the time value of money concept is that a rand today is worth more than a rand in the future. If you have the rand today, you can invest it and earn a return on it so that on any future date you will have more than a rand. This is the future value of the rand. If you have PV₀ rand today and expect to be able to earn a return of r₁ by investing it for one period, then the future value FV₁ of your investment at the end of the period will be:

 

 

If you consider investing it for several more periods and in successive periods expect it to earn returns of r₂, r₃, and so on, then at the end of n periods the future value of your investment will be:

 

 

This process of calculating future value is called compounding because it includes earning returns on returns: in every period, you are earning a return not just on your original investment but also on all returns you have earned until then.

 

So far we have not imposed any requirement that the periods be equal, just that each return be appropriate for the length of the corresponding period. The first period, then, may be a year, in which case r₁ will have to be an annual return, the second period may be a month, in which case r₂ must be a monthly return, and so forth. This is the general formula for calculating future value over time, where the length of each period and the rate of return for it can be different.

 

If we assume that all periods are equal in length (for example, one year) and all the expected returns are the same (r), then we can simplify the equation to:

 

 

Assume that instead of asking what a certain amount of money today will be equal to at some point in the future, we ask what a certain amount of money in the future is equal to today. We then have to reverse the calculations, and the general equation for calculating the present value will be:

 

 

And if we assume that all the periods are equal in length and all the expected returns are the same, then the equation for present value becomes:

 

 

The process of calculating present value is called discounting, which is the inverse of compounding. This also involves earning a return on return, although it is not easy to see it here as it was in the case of compounding.

 

Calculating present and future values can also be viewed as the process of moving an amount of money forward or backward through time. The amounts of money involved are called cash flows because they involve cash as opposed to some accounting measures like earnings. We can write the present value of a cash flow that will occur tperiods from now as:

 

 

It is easy to see that if we are anticipating several cash flows over time, we can calculate the present value of each and then add them together to get the total present value of all cash flows.

 

Here are the key points to keep in mind about calculating present and future values and doing time value of money problems:

 

•   The time value of money concept applies only to cash flows because we can earn returns or have to pay returns only on cash we invest or borrow. We cannot calculate present values or future values for net income, operating income, and so on, because they do not represent cash.

•   Only cash flows taking place at the same point in time can be compared to one another and combined together. If you are dealing with cash flows that take place at different points in time, you have to move them to the same point in time, that is, calculate their present or future values at the same point in time before comparing or combining them. For such calculations, most of the time we either present value all cash flows to today or future value them to the farthest point in time the problem involves. However, if it is more convenient in a specific situation, we can move all cash flows to any other point in time as well. Whenever you calculate a present or future value (especially using Excel function) make sure you know which point in time they relate to.

•   The simpler formulas we derived, as well as most Excel functions, can be used only when all the periods are of equal length and the rate of return is the same for all periods. Otherwise you have to use the longer period-by-period formulas.

•   We use the term compounding when we calculate future values or move earlier cash flows to a later point in time, and the term discounting when we calculate present values or move cash flows to an earlier point in time. However, we often use the term discount rate to refer to the rates of return in both cases.

•   The most important thing to remember about the discount rate you choose to apply to one or a series of cash flows is that it must reflect the risk of the cash flows. It is easy to understand that the discount rate should be higher for more risky cash flows and lower for less risky cash flows. However, estimating the risk of a cash flow and deciding what the appropriate discount rate for it should be is one of the knottiest problems in finance. In the models in this lesson we will assume we know what the appropriate discount rate is.