# Net present value

de:Kapitalwert fr:valeur actuelle nette pl:NPV Net present value is a form of calculating discounted cash flow. It encompasses the process of calculating the discount of a series of amounts of cash at future dates, and summing them. It is a time consuming process, but not difficult at all.

For example: X corporation during capital budgeting is trying to decide whether or not to proceed with a new product line. The new product will have startup costs, operational costs, and incoming cash flows increasing over time.

X corporation's CFO has declared that all new projects must have an NPV of more than zero and an internal rate of return of more than the weighted average cost of capital. Which just means that the project must pay the company back within five years, and must return more than its normal short-term (money market) rate of return. The weighted average cost of capital is 10% per annum (year).

This project will have a cash outlay (up-front cost to purchase machinery and train employees) of $100,000, and the personnel and maintenance costs will be$5,000 per year. It will return nothing the first year, but is projected to earn $31,250 per year after that. The NPV is calculated for each cash flow: $-100,000 today = $-100,000 / 1.10^0 =$-100,000.

 $-5,000 today =$-5,000 / 1.10^0 = $-5,000.  $-5,000 in 1 yr = $-5000 / 1.10^1 =$-4,545.45

 $-5,000 in 2 yr =$-5000 / 1.10^2 = $-4,132.23  $-5,000 in 3 yr = $-5000 / 1.10^3 =$-3,756.57

 $-5,000 in 4 yr =$-5000 / 1.10^4 = $-3,415.07  $31,250 in 1 yr = $31,250 / 1.10^1 =$28,409.09

 $31,250 in 2 yr =$31,250 / 1.10^2 = $25,826.45  $31,250 in 3 yr = $31,250 / 1.10^3 =$23,478.59

 $31,250 in 4 yr =$31,250 / 1.10^4 = \$21,344.17


Then all of the discounted (current) values are added together to find the NPV:

 -100,000.00
-  5,000.00
-  4,545.45
-  4,132.23
-  3,756.57
-  3,415.07
28,409.09
25,826.45
23,478.59
21,344.17
-----------
-21,791.02


Net Present Value can thus be calculated by the following formula, where t is the amount of time (usually in years) that cash has been invested in the project, N the total length of the project (in this case, five years), i the weighted average cost of capital and C the cash flow at that point in time.

[itex]\mbox{NPV} = \sum_{t=0}^N \frac{C_t}{(1+i)^t}[itex]

If you add up the original cash flows without discounting them, you find that the money spent is entirely bought back. This particular example is why discounted cash flow methods of valuation are superior to ones not based on time value of money.

The above example is based on a constant rate being used for future interest rate predictions and works very well for small amounts of money or short time horizons. Any calculations which involve large amounts or protracted time spans will use a yield curve to give different rates for the various time points on the calculation. So, the rate for 1 year may be the 10% - the (money market) rate while the rate for 2 years will be 12% and that for 3 years 12.5%.

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