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Time Value of Money


Because we can earn interest on money received, there is a time value of money associated with cash flows. As a basic principle (Park 2009):

  • A nearby euro is worth more than a distant euro.


The value of a cash flow stream not only depends on the magnitude of the inflows and outflows, but also on the timing of the cash flows relative to each other.

Future Value


A deposit of P euros in a bank account at an interest rate i generates interest of the principal as well as the accumulated interest that has been added to the account as follows.

  • At the end of the first year the balance is P + iP = P(1+i).
  • At the end of the second year the balance is P(1+i)+iP(1+i) = P(1+i)(1+i).
  • ...
  • At the end of the tenth year the balance is P(1+i)...(1+i) (ten times).


Therefore, due to compound interest, the future value F of a principal P is

      (1)               F = P(1+i)k


If we wish to find the value after ten years, we insert k = 10. The equation applies to an ideal bank account, that accrues interest year by year at no administration costs.

Each cash flow in a cash flow stream thus has a future value. Assume we take each cash flow and deposit it in an ideal bank as it arrives (every year). If the cash flow is negative we cover it by taking out a loan. The final balance in the account is a combination of the individual cash flows.

Consider the cash flow stream (x0, x1, ..., xK). After the Kth year the initial cash flow x0 will have grown to x0(1+i)K. The next cash flow, x1, received after the first year, will at the end of the final year have been in the account for only K - 1 years, and hence it will have a value of x1(1+i)K - 1. And so on. The final cash flow xK will not collect any interest.

In summary, the future value of the cash flow stream is (Luenberger 1998)

      (2)               F = x0(1+i)K + x1(1+i)K - 1 + ... + xK - 1(1+i) + xK


All cash flows are included, both positive and negative, which explains why it is called the net future value (NFV) of the cash flow stream.


Example (net future value) Consider the cash flow stream (-10, 9, 1, 1) when the interest rate is 5%. If you just add the values, the final value is 1 (= -10 + 9 + 1 + 1). If you include the interest rate, the net future value is

      NFV = -10 * (1.05)3 + 9 * (1.05)2 + 1 * (1.05) + 1 = 0.396



Present Value


Cash flows are worth less and less the longer we look toward the time horizon, because we can earn interest in the meantime.


Example (present value) Imagine you have to choose between two alternatives:

  1. You will receive 104 EUR in one year, or
  2. you receive 100 EUR now and deposit it in the bank for one year at 4% interest rate.


Clearly the result is the same after one year; you will receive 104 EUR. Therefore, receiving 104 EUR in one year has the same value as receiving 100 EUR now when the interest rate is 4%. We say that the 104 EUR to be received in one year from now has a present value of 100 EUR.



The present value of a future cash flow is less than the face value of the cash flow. The previous formulas can be reversed in time in order to calculate the present value of a future cash flow.

The present value of a future amount F, when the interest rate is i, is given by Equation (1) also, but we have to invert it to find P:

      (3)               P = F/(1+i)k


The present value of a cash flow stream is a combination of the cash flows. Given a cash flow stream (x0, x1, ..., xK) and interest rate i per period, the present value of the stream is (Luenberger 1998)

      (4)               P = x0 + x1/(1+i) + x2/(1+i)2 + ... + xK/(1+i)K


All cash flows are included, both positive and negative, and therefore it is called the net present value (NPV) of the cash flow stream. Net present value is the present value of the benefits minus the present value of the costs. It is a single number, and it must be positive for an investment to be considered worthy; the more positive the better.


Example (net present value) Take the same cash flow stream (-10, 9, 1, 1) as in the previous example. The net present value is

      NPV = -10 + 9/(1.05) + 1/(1.05)2 + 1/(1.05)3 = 0.342


However, if we take the cash flow stream (-10, 1, 9, 1), where the second and the third element have swapped places, then

      NPV = -10 + 1/(1.05) + 9/(1.05)2 + 1/(1.05)3 = -0.021


Although the direct sums of the two cash flow streams (their simple future values) are equal, the net present values are unequal. Furthermore, the latter NPV is even negative, which indicates a loss.




The example shows that the timing of the cash flows plays a role. We would prefer that the large inflows arrive as early in the stream, if at all possible.

Two cash flow streams are equivalent if their net present values are equal -- for a given interest rate. And vice versa: If two cash flows have the same net present value, they are equivalent -- for a given interest rate. The net present value is a single number, which is both necessary and sufficient to characterise a cash flow stream. Therefore the cash flow stream may be transformed, by a bank for instance, in a variety of ways as long as the net present value remains the same. A customer can thus use the bank to tailor the stream to another, more desirable shape.

Relationship Between NPV and NFV


Equation (2) above defines the future value F of a cash flow stream. Likewise Equation (4) defines the present value P of a cash flow stream. Now take for instance the equation for F (Equation 2) and divide both sides by (1+i)k. The resulting right hand side is the same as the right hand side defining P. If we substitute NPV for P and NFV for F, just to emphasise there are both negative and positive cash flows, we can find one from the other by the following relationships,

      (5)               NFV = NPV(1+i)k


and

      (6)               NPV = NFV/(1+i)k


That is, we are assured that the previous Equations (1) and (3) also apply to whole cash flow streams. We thus interpret the NPV (NFV) as an equivalent cash flow stream consisting of just one cash flow.

  1. Wikipedia Time value of money



Created by system. Last Modification: Wednesday 02 April 2014 14:05:19 CEST by jj.