Finance-Theory · Unit 21 · Video 7 · Interactive Practice

Descartes' Rule of Signs for IRR, and the Politics NPV Leaves Out

IKey Formulas

Formula	Name	What it says
$\text{NPV} = \sum_{t=0}^{k} CF_t\,x^t,\quad x = \dfrac{1}{1+r}$	NPV as a polynomial	Degree $k$ in $x$ ; every IRR is a zero of it
$r > -100\% \iff x > 0$	Admissible rates	IRRs above $-100\%$ are the positive real zeros in $x$
$\#\text{IRRs} = V - 2m \ge 0,\quad m = 0, 1, 2, \dots$	Descartes' rule of signs	$V$ = number of sign changes in $CF_0, CF_1, \dots, CF_k$ : the count of IRRs above $-100\%$ is $V$ or fewer by an even number (a repeated zero counted by its multiplicity)
$V = 1 \;\Rightarrow\; \text{exactly one IRR}$	One sign change	Holds for a conventional stream ( $-$ then $+$ ) and a loan-type stream ( $+$ then $-$ ) alike

Key Insight: Signs bound the number of IRRs; magnitudes decide which count occurs. Outside the four conditions there is no systematic set of cases, and the bottom line is NPV.

IISame Signs, Different Counts

The stream −$200, +$870, −$1,213, $+CF_3$ keeps the pattern $-\,+\,-\,+$ throughout, yet its IRR count changes.

CF₃ $546

💡 Here the first later flow, $870, is opposite in sign to the initial −$200 and larger in magnitude: the feature every earlier example in the unit shared, and it settles nothing about the count.

IIISign Changes Against Crossings

Descartes' rule caps the IRR count at the number of sign changes; one sign change forces exactly one IRR.

CF₀ −$300 CF₁ $50 CF₂ $150 CF₃ $200

💡 Challenge: build a stream with two sign changes and two IRRs, then one with two sign changes and no IRR at all.

IVQuiz Questions

Problem 1 · Counting Sign Changes

Given: a project with cash flows −$60, +$170, −$175, +$90, −$20 at dates 0 to 4. What does Descartes' rule of signs allow for the number of IRRs above $-100\%$ ?

✅ Correct! The signs

-\,+\,-\,+\,-

change four times, so the count is

4

or fewer by an even number.

❌ Not quite. The number of sign changes is an upper bound, not the count: it can fall short by any even number, because the remaining zeros of the polynomial may be complex.

❌ Not quite. Five cash flows give four adjacent pairs, so at most four sign changes; count the changes between neighbouring flows, not the flows.

❌ Not quite. The count drops from the number of sign changes only in steps of two: complex zeros of a real polynomial come in conjugate pairs.

❌ Not quite. Count the sign changes between neighbouring flows, then allow fewer by an even number.

Show solution

With $x = 1/(1+r)$ the NPV is the quartic $-60 + 170x - 175x^2 + 90x^3 - 20x^4$ , and $r > -100\%$ exactly when $x > 0$ .

Sign sequence: $-\,+\,-\,+\,-$ . Every neighbouring pair flips sign, so $V = 4$ .

Descartes' rule: the number of IRRs above $-100\%$ is $4$ , $2$ or $0$ .

The signs alone cannot say which. Solving numerically, this stream happens to have two: $r \approx -39.5\%$ and $r \approx 38.2\%$ . The negative one counts too, since it lies above $-100\%$ .

Problem 2 · The Shared Feature Is Not a Rule

Given: cash flows −$400, +$1,400, −$1,629, +$630 at dates 0 to 3. A colleague notes that the first later flow is opposite in sign to the initial one and larger, as in every earlier example of the unit, and concludes the project has exactly one IRR. Check the rates $5\%$ , $20\%$ and $25\%$ : how many IRRs above $-100\%$ does it have?

✅ Correct! All three rates make NPV exactly zero. The rule allows at most three, so these are all of them.

❌ Not quite. That feature is what the earlier examples happened to share; it is not a rule. Only a single sign change guarantees one IRR, and this stream has three.

❌ Not quite. Three sign changes allow three IRRs or one, never none. Plug in

20\%

: the NPV is exactly zero.

❌ Not quite. With three sign changes Descartes' rule allows

3

1

; a count of two is impossible.

❌ Not quite. Evaluate the NPV at each of the three rates.

Show solution

Multiply the NPV equation by $(1+r)^3$ and write $g = 1 + r$ :

-400g^3 + 1400g^2 - 1629g + 630 = 0.

Expand the candidate factorisation:

(g - 1.05)(g - 1.20)(g - 1.25) = g^3 - 3.5g^2 + 4.0725g - 1.575,

and $-400$ times this is $-400g^3 + 1400g^2 - 1629g + 630$ ✓. So $5\%$ , $20\%$ and $25\%$ are all IRRs. A direct check at $20\%$ :

-400 + \frac{1400}{1.2} - \frac{1629}{1.2^2} + \frac{630}{1.2^3} = \frac{-4800 + 14000 - 13575 + 4375}{12} = 0.

The signs $-\,+\,-\,+$ change three times, so three is the maximum: the stream has exactly three IRRs. The opposite-and-larger first flow decided nothing.

Problem 3 · One Sign Change, Read the Right Way

Given: a financing deal pays the firm $205 today, and the firm then pays $36, $72 and $216 at dates 1, 2 and 3. The cost of capital is $25\%$ . Find the IRR count and the decision.

How many IRRs above $-100\%$ , and where?

At a $25\%$ cost of capital, should the firm take the deal?

✅ Correct! One sign change gives exactly one IRR,

20\%

; for loan-type flows the rule reverses, and the NPV at

25\%

is positive.

❌ Not quite. The signs are

+\,-\,-\,-

: only one change, so the rule gives exactly one IRR.

❌ Not quite. Loan-type flows have an IRR; what changes is how the rule reads it. One sign change guarantees exactly one.

❌ Not quite.

25\%

is the cost of capital, not the IRR. At

25\%

the payments are worth

28.8 + 46.08 + 110.592 = 185.472

, less than the $205 received, so

25\%

does not make the NPV zero.

❌ Check the rate. Discount each payment at the candidate rate and compare with the $205 received.

❌ Not quite. "Accept if IRR exceeds the hurdle" is the rule for investment-type flows. Here the firm receives money first, so the IRR is its borrowing rate: accept when it is below the cost of capital.

❌ Not quite. Compute it:

205 - 28.8 - 46.08 - 110.592 = 19.528 > 0

❌ Not quite. Compute the NPV at

25\%

Show solution

Count. The signs are $+\,-\,-\,-$ : one change, so exactly one IRR above $-100\%$ .

Locate it. At $20\%$ the payments are worth

\frac{36}{1.2} + \frac{72}{1.2^2} + \frac{216}{1.2^3} = 30 + 50 + 125 = 205,

exactly the $205 received, so the IRR is $20\%$ , and by the rule it is the only one.

Decide. At the $25\%$ cost of capital,

\text{NPV} = 205 - \frac{36}{1.25} - \frac{72}{1.25^2} - \frac{216}{1.25^3} = 205 - 28.8 - 46.08 - 110.592 = 19.528,

about +$19.53. Accept. The deal is a loan at $20\%$ for a firm whose capital costs $25\%$ ; for loan-type flows the IRR rule reverses to "accept if IRR is below the cost of capital".

Problem 4 · What the Models Leave Out

Given: a chief executive hires you as a new division manager to turn the division around, and in your first quarter you propose a specific restructuring. Nobody yet knows whether the proposal is good or bad.

What is the most likely outcome?

Would an investor find such political considerations already reflected in a listed company's stock price?

✅ Correct! Both answers turn on politics, which sits outside P&L, NPV, the cost of capital and risk adjustment, yet shapes real decisions inside firms and in markets.

❌ Not quite. The prediction is made without knowing whether the proposal is smart or stupid, so no NPV or IRR figure can drive it. What drives it is the new hire's standing.

❌ Not quite. The lack of a track record works the other way: someone who has just hired you to do a job has to give you room before starting to overrule you.

❌ Not quite. Ask what the chief executive knows: not the proposal's quality, only that you were just hired to make changes.

❌ Not quite. That is the traditional assumption the video questions. In Andrew Lo's view, it is not clear any economic explanation accounts for letting Lehman fail; a political one may.

❌ Not quite. The video's answer is not a flat no: politics played a very big role in 2008, even for private investors.

❌ Not quite. Recall the 2008 contrast between Bear Stearns and Lehman Brothers.

Show solution

Inside the firm. The outcome can be predicted without judging the proposal: the chief executive most likely agrees, because someone who has just hired you to turn a division around has to give you the benefit of the doubt for a while. That prediction has nothing to do with P&L, NPV, the cost of capital or risk adjustment; it is what "political considerations" means, and capital budgeting has to factor them in.

In the market. The answer is yes and no. Bear Stearns was steered into a merger with JPMorgan in March 2008, with lending from the Federal Reserve Bank of New York; Lehman Brothers, in some respects bigger and more intertwined, filed for bankruptcy on 15 September 2008. Lo's conjecture, offered as speculation, is that after the backlash to the Bear rescue a second rescue was politically untenable. Politics played a very big role even for private investors.

Maximizing NPV remains the course's objective, but it is an approximation to a much more complex reality.

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