# Why does NPV correspond to "cash in our pockets now" for risky investments?

For a positive NPV project with risk free cash flows and assuming access to a competitive money market, it is trivial to show (by appropriately borrowing or lending at the risk-free rate) that the magnitude of the NPV is precisely equivalent to money you could have in your pocket right now. My textbook (Corporate Finance by Berk and DeMarzo) seems to suggest that this property is still true for positive NPV projects when the cash flows are risky. In what sense is this true? For such risky positive NPV projects, the cash flows in question are of course expected cash flows discounted at a rate appropriate to the risk of the cash flows. Because the cash flows are not guaranteed, how can we similarly get the magnitude of the NPV as cash into our pocket right now?

"Because the cash flows are not guaranteed, how can we similarly get the magnitude of the NPV as cash into our pocket right now?"

"For such risky positive NPV projects, the cash flows in question are of course expected cash flows discounted at a rate appropriate to the risk of the cash flows."

To elaborate: First, let's define what "NPV = 0" actually means:

Assume your risk free rate is 4%, and you have \$1,000 to invest. Outside of earning interest, assume you could invest in the stock market with an expected return of 10%. You know the stock investment will be more volatile, and you know that the expected return of 10% is only higher than the risk-free rate, as fair compensation for that risk. What is the NPV of a stock market investment?

The answer: 0. The stock market investment is returning exact compensation for the risk associated. Receiving \$1,100 after a year of stock investments is considered financially identical under these assumptions, as receiving \$1,040 after a year of cash deposit saving. Neither option returns more than their respective risk rating would imply.

So what would a + NPV imply? A + NPV implies that your return would be higher than expected based on the risk rating you have used to discount your future cashflows. The simplest example: Investing in your company's matched stock purchase plan. Say that your company allows you to invest \$1,000 / year by purchasing company stock, and that it will match half of your purchase with an extra \$500. So exact same as the above stock investment plan, except it will return an extra \$550 at the end of the year, representing the extra amount invested by your employer with a 10% assumed earnings rate.

So, your investment of \$1,000 has an expected return in 12 months of \$1,650, when a 10% discount rating would imply a return of only \$1,100. Remember that given the 10% discount rating, your future \$1,650 in value is worth only \$1,500 today, and it would only cost you \$1,000 of your own money. In this scenario, your NPV of investing in your company's matched stock plan is \$500, which is intuitively the same as just getting an extra \$500 bonus from your employer.

Any other scenario with a positive NPV basically means the same thing - you are getting compensated higher than your assumed discount rating would imply. And if the NPV is 0, the project is basically "exactly as worth it as any other type of investment with the same risk profile". Any compensation above 0, is effectively valued the same as just extra cash today.

• Thank you very much for your answer. If possible, are you able to elaborate specifically on how we can make that \$500 be "cash in our pocket" right now, as we can do in the risk-free case?
– EE18
Commented Dec 5, 2023 at 0:14
• @EE18 It has the same economic value as \$500 cash in your pocket; you can't actually convert into literal cash in your pocket. If it was a product that you could sell to someone else (ie: if you could transfer your employment benefits to another person), then fair compensation would be the NPV of \$500. Commented Dec 5, 2023 at 14:04

This is kinda just the definition of "a competitive money market" and "net present value". A competitive money market (an idealized version of the market with no transaction costs, information asymmetry, or other such complications) is by definition one in which a person who currently is entitled to future cash flows can sell the right to those future cash flows for a consistent, well defined value, and that value is, by definition, the net present value.

Because the cash flows are not guaranteed, how can we similarly get the magnitude of the NPV as cash into our pocket right now?

Because people in the money market are willing to exchange a small amount of certain money now for a large amount of uncertain money in the future.

The NPV for a risky project reflects investors' willingness to take the risk. Taking the risk means that the investors are prepared to give you cash today in exchange for (a share of) the future payments.

Just as there is an amount investors will pay for future risk-free cash flows (what you term a competitive money market), there is a market for future risky cash flows. A "risk premium" in returns is demanded, which corresponds to the higher discount rate in the NPV. This allows converting between risky projects and today's cash.

Stock and bond markets are examples (stock dividends are obviously risky; corporate bond payouts are risky because of the possibility of default). Every time a company issues shares (private or public offering), its future risky payoff is being priced and realized in cash.

It might not be practical to solicit investment bids for any particular project (especially a small one), but in principle such a market exists.

• Thank you for your answer. If it's possible, I'd like to reiterate the same question as I did to the other answerer:are you able to elaborate specifically on how we can make any positive NPV be "cash in our pocket" right now, as we can do in the risk-free case? Are you saying the following (for example)? Suppose we have a project with a certain risk profile and NPV of +500. Are you saying that (we assume) there is a market to sell this set of cash flows, and we can then buy an a set of cash flows with the same risk profile in order to cover this sale...
– EE18
Commented Dec 5, 2023 at 1:57
• ...where the "coverage" is in the sense of expected value (and hence this is not an arbitrage opportunity as it would be if the cash flows were risk-free)? But we nevertheless have \$500 in our pocket right now?
– EE18
Commented Dec 5, 2023 at 1:58
• @EE18 Every IPO and every corporate acquisition is an example of what you describe -- a company (or its owner) offers future risky cash flows in exchange for cash now. Commented Dec 5, 2023 at 6:44

the magnitude of the NPV is precisely equivalent to money you could have in your pocket right now.

"Equivalent" does not mean that it can actually happen in reality.

A positive NPV means that the expected future value of the project is the same as if you did have that money in cash now and could invest it at the discount rate. It does not mean that just taking on the project magically creates that much cash in your pocket now. The only way to "turn it into cash now" is for someone to buy the project from you; then they get the same expected (not guaranteed) return as putting the money in the bank at that interest rate.

Both are true for risk-free and risky projects. The difference for a risky project is that the future value is not guaranteed, but is the average of all possible returns, weighted by their probability.