Lesson 5 of 9
Net present value
NPV is one subtraction, and the rule it produces is simple. What it quietly assumes about reinvestment and certainty is not.
By Javier Sanz
Founder of ValueScreener. Built Ninety Nine, a securities services provider, and sold it to a tier-one global bank; ran operations and internationalization at Alpaca, the brokerage API.
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In one minute
- NPV is the present value of what comes in, minus what you pay, at a rate you chose.
- A positive NPV means the stream beats the rate you tested it against. It says nothing about any other rate.
- The number carries two silent assumptions: that cash can be reinvested at the same rate, and that the forecast is the expected case.
Take a dated stream of cash, discount every item to today, and subtract the cost. What remains is the net present value. It is the cleanest decision rule in finance and the one most often quoted without its conditions.
An investment of 1,000 returning 300 a year for five years
- Tested at 8 percent$197.81
- Tested at 15 percent$5.65
- Tested at 20 percent−$102.82
The calculation
Discount each inflow with the factor for its own year, add them, subtract the outlay. Cash paid today is not discounted, because it is already today.
1,000 out, then 300 a year for five years, at 8 percent
- Year 1: 300 multiplied by 0.9259 equals 277.78.
- Year 2: 300 multiplied by 0.8573 equals 257.20.
- Year 3: 300 multiplied by 0.7938 equals 238.15.
- Year 4: 300 multiplied by 0.7350 equals 220.51.
- Year 5: 300 multiplied by 0.6806 equals 204.17.
- Add the five: 1,197.81. Subtract the 1,000 outlay. NPV is 197.81.
The shortcut is the annuity formula from the last lesson: 300 multiplied by 3.99271 is 1,197.81 in one step, and the six lines above exist so you can see what the one line did.
What a positive number means
It means this stream is worth more than 1,000 today, given the rate you used. That qualifier is the whole lesson. The same stream is worth 5.65 more than its cost at 15 percent and 102.66 less at 20 percent. NPV is not a property of the investment. It is the result of a comparison you specified.
The two silent assumptions
Reinvestment. Discounting every inflow at r is arithmetically identical to assuming each one, once received, earns r until the horizon. If the cash comes back and sits in a deposit account at a lower rate, the realised outcome is below what the NPV implied. The formula never mentions this and it is built in.
Certainty. An NPV computed on a single forecast treats that forecast as the expected case. A forecast built from a plan, a target or a management guide is usually the good case rather than the average one, and discounting the good case at an ordinary rate double counts the optimism.
How to use it without being used by it
Compute the NPV at both ends of the rate you would defend, not at the middle of it. If the sign is the same at both ends, the rate argument does not need settling. If the sign flips, the rate argument is the analysis, and reporting one number from the middle of the range hides that.
Check yourself
An NPV computed at 10 percent is positive. What have you established?
- The investment will make money
- The stream is worth more than its cost when discounted at 10 percent
- The internal rate of return is 10 percent
Show the answer
Answer: B.
NPV is a comparison against the rate you supplied. A different defensible rate can change the sign, which is why the rate belongs beside the answer.What does discounting every inflow at r implicitly assume about cash once it is received?
- That it is spent immediately
- That it earns r until the horizon
- That it earns the risk-free rate
Show the answer
Answer: B.
Discounting at r and compounding at r are the same operation read in opposite directions, so the reinvestment rate inside an NPV is the discount rate itself.Two analysts compute NPVs of opposite sign on identical cash flows. What is the most likely explanation?
- One of them made an arithmetic error
- They used different discount rates
- NPV is not a reliable method
Show the answer
Answer: B.
The cash flows are identical, so the only free input left is the rate. This is the normal case and it is why the rate is reported with the answer.Sources
- Williams, The Theory of Investment Value, 1938: value as the discounted stream, which NPV compares against a price.
- Damodaran, Investment Valuation: the reinvestment assumption embedded in a discount rate, and the treatment of a forecast as an expected case.
What this leaves out
- Every figure in this lesson is invented arithmetic chosen so the reader can recompute it. No company is described.
- NPV assumes the cash flows are the expected case. It contains no adjustment for a forecast that is systematically optimistic.
- Nothing here is advice about any security.