riaology
Learning path 5 · Topic 21 of 30

Find the zero-worth interest rate

IRR is the rate that makes net equivalent worth zero. It is not the annual dollar profit divided by investment for a multi-period project.

Change a scenario. Watch the consequence.
012345Receipts (+)Expenses (−)

Period 0: −$50,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.

-24.0k0524.692525.0k50Present worth ($)Discount rate (%)
● Current scenarioPresent worth
IRR in percent · %0=−I+A(P/A,i∗,N),i∗=15.238237%0=-I+A(P/A,i^*,N),\quad i^*=15.238237\%

The conventional independent investment meets MARR.

Follow the reasoning

IRR is a rate that makes equivalent net worth zero, not a dollar return.

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

A conventional investment has an initial negative flow and later nonnegative receipts, at least one positive. Its PW decreases strictly with i>−1, so it has one root in that domain; the root is positive only when total undiscounted receipts exceed investment.

02

Build the calculation

For a conventional independent investment, compare IRR with MARR. This rule does not automatically apply to borrowing-type sequences, nonconventional flows, or ranking mutually exclusive investments. Reinvestment interpretations differ across methods; examine external-rate assumptions when using ERR.

0=∑t=0NCFt(1+i∗)t0=\sum_{t=0}^N\frac{CF_t}{(1+i^*)^t}
Practice with feedback

Set up. Calculate. Explain.

Answer unit: %

Rebuild the starting scenario

Investment: 50000 $; Annual net receipt: 15000 $/year; Number of years: 5 years; MARR: 10 % per year. Calculate the IRR in percent.

Setup hint

IRR is a rate that makes equivalent net worth zero, not a dollar return.

Open the visual solution0=−I+A(P/A,i∗,N),i∗=15.238237%0=-I+A(P/A,i^*,N),\quad i^*=15.238237\%
  1. IRR is a rate that makes equivalent net worth zero, not a dollar return.
  2. This conventional investment has one initial outflow and later positive receipts: PW decreases monotonically for i>−1.
  3. An IRR may be negative if undiscounted receipts fail to recover the investment. Compare a unique investment IRR with MARR; do not rank mutually exclusive projects by IRR alone.
012345Receipts (+)Expenses (−)

Period 0: −$50,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.

-24.0k0524.692525.0k50Present worth ($)Discount rate (%)
● Current scenarioPresent worth
IRR in percent · %0=−I+A(P/A,i∗,N),i∗=15.238237%0=-I+A(P/A,i^*,N),\quad i^*=15.238237\%

The conventional independent investment meets MARR.

Visual solution

IRR is a rate that makes equivalent net worth zero, not a dollar return.

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.

Answer unit: %

Transfer the idea to a new scenario

Invest $24,000 and earn $7,000 net at each of five year-ends. Find IRR; compare it with MARR 10%.

Setup hint

IRR is a rate that makes equivalent net worth zero, not a dollar return.

Open the visual solution0=−I+A(P/A,i∗,N),i∗=14.054713%0=-I+A(P/A,i^*,N),\quad i^*=14.054713\%
  1. IRR is a rate that makes equivalent net worth zero, not a dollar return.
  2. This conventional investment has one initial outflow and later positive receipts: PW decreases monotonically for i>−1.
  3. An IRR may be negative if undiscounted receipts fail to recover the investment. Compare a unique investment IRR with MARR; do not rank mutually exclusive projects by IRR alone.
012345Receipts (+)Expenses (−)

Period 0: −$24,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.

-11.8k0-421.812511.0k50Present worth ($)Discount rate (%)
● Current scenarioPresent worth
IRR in percent · %0=−I+A(P/A,i∗,N),i∗=14.054713%0=-I+A(P/A,i^*,N),\quad i^*=14.054713\%

The conventional independent investment meets MARR.

Visual solution

IRR is a rate that makes equivalent net worth zero, not a dollar return.

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

For a conventional investment with PW(MARR)>0, which relation holds?

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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