# Does the traditional NPV formula of a cashflow double count risk?

Consider a cash flow stream of a single payment (1 period away). Its net present value is typically presented as

$$\text{NPV} = {\text{EV}(\text{Cash Flow}) \over 1 + d} \tag{1}$$

Here $d$ is supposed to be the "risk-adjusted discount rate" which, as I understand it, can be broken down as

$$d = t + r$$

where $t$ is the pure time value of money and $r$ is the "extra factor" for how risky the asset class is.

Question: Aren't we double-counting the risk of an asset by including the expected value in the numerator (which factors in the probability that a cash flow could be low) and the risk factor $r$ in the denominator? That is, shouldn't the NPV of an asset be either

$$\text{NPV} = {\text{EV}(\text{Cash Flow}) \over 1+t} \tag{2}$$

or

$$\text{NPV} = {\text{Cash Flow} \over 1+t + r} \tag{3}?$$

In (2) we factor in the risk of the asset by using an expected cash flow in the numerator, so that if the cash flow is really risky it will be weighted down. In (3) we factor in the risk of the asset by discounting by $1+t + r$ instead of just $t$. Crucially, we do one or the other; doing both -- as in (1) -- seems to double count the risk of an asset. How is this wrong?

• Two comments: (i) Your formula (2) implies that the expected return of the S&P500 index should be the risk free rate. Is that correct? (ii) If $\mathit{CashFlow}$ is a random variable and $\mathit{NPV}$ is a scalar, how is your formula (3) going to make any sense? Commented Feb 7, 2018 at 21:14
• You are correct that formula (2) is treating the discount rate as the risk free rate. In formula (3), $\text{Cash Flow}$ can be viewed as a scalar. Commented Feb 7, 2018 at 21:16
• So you're using two different senses of $\mathit{CashFlow}$? Is what you have in mind something like $\mathit{CashFlow}_2 = \mathbb{E}[\mathit{CashFlow}_1]$? Commented Feb 7, 2018 at 21:18
• No, more like $\text{Cash Flow}_1$ is the "plausible best case outcome" for $\text{Cash Flow}_2$. As in: suppose someone presented a business idea and said in year 2 we project to make \$100. In this case, you could either think "OK, they claim they're going to make$\$100$ but in reality their expected cash flow is $\$50$", or you could think "sure -- I'll assume they make$\$100$, but then I'm going to discount by $t + r$ to factor in the risk of this cash flow. Either of these seem fine to me. But what I don't think is appropriate is doing both. Am I mistaken? Commented Feb 7, 2018 at 21:21
• No -- the only thing that matters for valuation (it seems to me) is either (i) the expected cash flow discounted by the risk-free rate $t$, or (ii) the "plausible best case" outcome discounted by $t + r$. The third option: taking the expected cash flow and discounting by $t+r$ seems to penalize risky cash flows too much. Commented Feb 7, 2018 at 21:23

That formula is algebraically equivalent to saying different, stochastic assets can have different expected returns.

$$\mathbb{E} \left[ R_i \right] = r_f + \gamma_i$$

### Some simple algebra

Let $X_i$ be a random variable denoting a risky cash flow, $p_i$ be today's price of that risky cash flow, $r_f$ be the risk free rate, and $\gamma_i$ be some risk premium specific to asset $i$.

The formula you're objecting to is:

$$p_i = \frac{\mathbb{E} [X_i]}{r_f + \gamma_i}$$

Asset $i$'s return is given by $R_i = \frac{X_i}{p_i}$ By simple algebra you get $\mathbb{E} \left[ R_i \right] = r_f + \gamma_i$.

So all that formula is saying is that the expected return of asset $i$ is the risk free rate plus some risk premium $\gamma_i$. Without the $\gamma_i$ term (which is $d$ in your notation), every asset would have to have an expected return of the risk free rate, which is obviously wrong.

• Which formula do you mean with "that formula"? The common definition of NPV, or one of the adjusted ones by the author? Commented Jun 17, 2021 at 15:47
• how is Xi/E[xi] = 1? cuz the E[xi] for the current period is Xi I guess but Ri is more like a return rate for period i; not an absulute return cash flow value anyhow your explanation seems to solve the question asked here. No double counting of risk. Commented Jun 17, 2021 at 15:58

“You can't compensate for risk by using a high discount rate." - Warren Buffett at the 1998 Berkshire Hathaway Shareholder Meeting

The simple answer to your question is, “yes, many implementations of discounted cash flow analyses which adjust the discount rate for risk are double counting”. This practice is pervasive in academia, but has no basis in the time value of money principle.

I presume this practice comes from the incorrect interpretation of the Capital Asset Pricing Model, which itself may be interpreted as an incorrect interpretation of the Modigliani-Miller postulate on the irrelevance of capital structure.

Even those who realize this continue in this practice for heuristic reasons as it approximates the intuition that NPV must be downwardly adjusted for higher risk. It also allows for the heuristic valuation of negative expected cash flows, which is intractable in a deterministic context. Moreover, adjusting for asymmetric aversion to downside risk — as laid out in Prospect Theory — is mathematically and computationally inconvenient.

To my knowledge, there is no generally accepted way to discount an annuity under a conditional probability measure, which you represent as EV(*), even when that is taken to be a real world measure (vice a risk neutral measure). The most comprehensive works in this area are by Daniel Dufresne.

Anyhow, in order to avoid going down this rabbit hole, it might be wise to continue doing things the way your supervisor or professor expects.

Addendum, please note that @MatthewGunn 's answer is not wrong in the realm of quantitative finance in which price is assumed to be equal to the discounted expectation. I interpreted the question as a valuation/actuarial problem in which the author intends to discover fair value independently of price.

• I take it that you're referring to a practice common in venture capital of applying huge discount rates as an ad-hoc correction for absurd projections? For example take a projection of \$100 million, then apply a 900% discount rate to get a \$10 million valuation? Commented Feb 8, 2018 at 22:19
• @MatthewGunn That's the most extreme example I've ever seen. I find less extreme examples of that are pervasive in project finance and valuations. E.g., let's value companies by EBITDA and just add 10% to the discount rate. Or even as subtle as: stock "A" has a higher index beta than stock "B", therefore I'll discount stock A's cash flows at a higher rate. I know it's academic canon, but it still just doesn't feel right. Commented Feb 8, 2018 at 22:27
• Ahhh ok. Yeah, that approach is all over the place. The flaws are twofold: (1) the CAPM model doesn't work (2) even if you used a reasonable factor model, your beta estimates on those factors for individual companies are quite imprecise. (And my VC example isn't real... I exaggerated to an absurd level.) Commented Feb 8, 2018 at 22:33
• Does that mean that Matthew Gunn's response is only true if p_i can be calculated for period i somehow by reversing the formula? But if this does not hold, then well, there isn't a well known way for accounting in monetary terms for risk in the future? Commented Jun 17, 2021 at 16:14

No, but I can tell you why it feels like you are double counting.

Consider a cash flow $\tilde{x}=\tilde{x}(t,\mu,\sigma^2)$ to be received in the future. While many cash flows lack a first moment and so no defined mean or variance, let us assume at least the second moment is defined to make the discussion simple. Implicitly, your assumption of an expectation would require a first moment to exist. Indeed to make this easier, let us assume normality.

If $t$ is time; $\mu$ the center of location; and, $\sigma^2$ the scale parameter, then we can talk about a rate. From the formula $$\rho=\frac{\mathbb{E}(\tilde{x}(\mu(t),\sigma^2(t)))}{1+d(\mu(t),\sigma^2(t))}.$$ Your assumption of additivity is problematic, while it is often used as an approximation, if you think about it for a second you will see why. Instead, I am defining $$d=(1+r(\mu_0,\sigma^2_0))(1+\tau(t))$$ because I am using $t$ for time instead of risk. $\tau(t)$ is the function that maps the premium as a function of time to discount a certain cash flow.

We will adopt a Frequentist interpretation of probability to make this simple. Using multiplication all the constants are on the left and the expectation of the random variables are on in the center when we rearrange it as $$\rho(1+d(\mu(t),\sigma^2(t))=\mathbb{E}(\tilde{x}(\mu(t),\sigma^2(t)))=\rho\mu(t).$$

The present value cannot be stochastic as it is known by observation. Since $\mu(t)$ and $\sigma^2(t)$ are constants by definition in the Frequentist interpretation of probability nothing on the left or right contains any randomness at all. Only the center is random. That randomness is averaged out over the sample space so that only the point is left.

You could logically take this one step further and argue that $\mu(t)=\mu(t,\sigma^2(t))$. It feels like it is double counted because the mean is a function of the variance and the cash flow is a function of the mean and the variance. The rate is a function of the risk-adjusted mean, so it is a function of the mean and the variance.

You could dissolve the mean and convert it into a pure function of variance and time, and then only the scale parameter would exist in the numerator and the denominator.

$$\rho=\frac{\mathbb{E}(\tilde{x}(\sigma^2(t),t))}{1+d(\sigma^2(t),t)}$$

You are also missing the observation that $\mu(t,\sigma^2(t))$ is similar to an expenditure function and not merely a center of location.

If you step back one more unit of time, to before time zero, then $\rho$ becomes stochastic as well because $\tilde{\rho}=\rho$ if and only if $\mathbb{E}[\mathcal{U}(\tilde{x})]>\mathcal{U}(\tilde{\rho}=0)$, where $\mathcal{U}()$ is a utility function.