riaology
Learning path 5 · Topic 23 of 30

Spot an ambiguous rate of return

Nonconventional cash flows can reverse sign more than once. Cleanup or overhaul expenses late in a project may produce multiple IRR roots or no real root.

Change a scenario. Watch the consequence.
012Receipts (+)Expenses (−)

Period 0: −$1,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.

-1.7k-50-839.05251.89100PW ($)Discount rate (%)
● Current scenarioPresent worth
present worth at MARR · $PW(i)=−1000+23001+i−1320(1+i)2,PW(15%)=1.890359PW(i)=-1000+\frac{2300}{1+i}-\frac{1320}{(1+i)^2},\quad PW(15\%)=1.890359

Admissible algebraic roots: 10.000%, 20.000%. Use PW at MARR to decide; multiple roots are ambiguous.

Follow the reasoning

The sign pattern −,+,− permits multiple positive roots but does not guarantee them.

Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.

01

What the model means

Count sign changes after removing zeros as a warning, not a guaranteed count of positive IRRs. A polynomial in x=1/(1+i) or in 1+i can reveal roots, but only financially admissible i>−1 values belong to the standard compound-interest domain.

02

Build the calculation

For flows −1000,+2300,−1320, both 10% and 20% satisfy PW=0. PW at a 15% MARR is positive even though a single selected-root comparison could suggest another decision. Use PW at MARR or an explicit external-return model instead.

0=−1000(1+i)2+2300(1+i)−13200=-1000(1+i)^2+2300(1+i)-1320
Practice with feedback

Set up. Calculate. Explain.

Money: enter dollars

Rebuild the starting scenario

Cleanup cost at year 2: 1320 $; MARR: 15 % per year. Calculate the present worth at MARR.

Setup hint

The sign pattern −,+,− permits multiple positive roots but does not guarantee them.

Open the visual solutionPW(i)=−1000+23001+i−1320(1+i)2,PW(15%)=1.890359PW(i)=-1000+\frac{2300}{1+i}-\frac{1320}{(1+i)^2},\quad PW(15\%)=1.890359
  1. The sign pattern −,+,− permits multiple positive roots but does not guarantee them.
  2. Multiply by (1+i)² and solve the resulting quadratic; the financial domain is i>−1.
  3. At cleanup cost $1,320, both 10% and 20% make PW zero. A single IRR ≥ MARR rule is unsafe.
012Receipts (+)Expenses (−)

Period 0: −$1,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.

-1.7k-50-839.05251.89100PW ($)Discount rate (%)
● Current scenarioPresent worth
present worth at MARR · $PW(i)=−1000+23001+i−1320(1+i)2,PW(15%)=1.890359PW(i)=-1000+\frac{2300}{1+i}-\frac{1320}{(1+i)^2},\quad PW(15\%)=1.890359

Admissible algebraic roots: 10.000%, 20.000%. Use PW at MARR to decide; multiple roots are ambiguous.

Visual solution

The sign pattern −,+,− permits multiple positive roots but does not guarantee them.

Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.

Checks allow 0.1% numerical rounding (minimum $0.05 or 0.001 in the stated nonmonetary unit). Interpolated factors and rates are approximations; keep table precision consistent.

Money: enter dollars

Transfer the idea to a new scenario

Use flows −$1,000 at year 0, +$2,300 at year 1, −$1,300 at year 2. Find PW at 12%.

Setup hint

The sign pattern −,+,− permits multiple positive roots but does not guarantee them.

Open the visual solutionPW(i)=−1000+23001+i−1300(1+i)2,PW(12%)=17.219388PW(i)=-1000+\frac{2300}{1+i}-\frac{1300}{(1+i)^2},\quad PW(12\%)=17.219388
  1. The sign pattern −,+,− permits multiple positive roots but does not guarantee them.
  2. Multiply by (1+i)² and solve the resulting quadratic; the financial domain is i>−1.
  3. At cleanup cost $1,320, both 10% and 20% make PW zero. A single IRR ≥ MARR rule is unsafe.
012Receipts (+)Expenses (−)

Period 0: −$1,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.

-1.6k-50-791.372517.27100PW ($)Discount rate (%)
● Current scenarioPresent worth
present worth at MARR · $PW(i)=−1000+23001+i−1300(1+i)2,PW(12%)=17.219388PW(i)=-1000+\frac{2300}{1+i}-\frac{1300}{(1+i)^2},\quad PW(12\%)=17.219388

Admissible algebraic roots: 0.000%, 30.000%. Use PW at MARR to decide; multiple roots are ambiguous.

Visual solution

The sign pattern −,+,− permits multiple positive roots but does not guarantee them.

Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.

Checks allow 0.1% numerical rounding (minimum $0.05 or 0.001 in the stated nonmonetary unit). Interpolated factors and rates are approximations; keep table precision consistent.

Explain the distinction

What is the best response to multiple IRR roots?

Can you defend your setup?

Can you place each amount on the timeline, match the interest period, calculate the equivalent quantity, and state the decision or limitation?

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