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Rate-of-Return

Engineering Economics · FE Reference Handbook section

Engineering Economics
0 formulas
10 exam-style examples
~45 min
All Engineering Economics lectures

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

  • The minimum acceptable rate-of-return (MARR) is that interest rate that one is willing to accept, or the rate one desires to earn
  • on investments. The rate-of-return on an investment is the interest rate that makes the benefits and costs equal.

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

This section is conceptual; there are no equations to memorise.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1
Rate of return on an equipment investment, before and after tax — Rate-of-Return

A contractor invests $76,000 in equipment that returns $25,000 per year for 8 years with no salvage. Determine the rate of return, and estimate the after-tax return at a 24% tax rate on the net income.

Given

  • P = $76,000

  • A = $25,000/yr

  • n=8yrn = 8 yr
  • Taxrate=24Tax rate = 24%

Find

Before-tax rate of return and an after-tax estimate

Start with the thinking

  • The rate of return is the interest rate that makes present worth zero — solved by trial or by the calculator's IRR.
  • A quick after-tax screen scales the annual return by (1 − tax rate) and re-solves.

Step-by-step solution

  1. Formula

    0=−P+A(P/A,i,n)0 = -P + A(P/A, i, n)

  2. Set up

    (P/A,i,8)=P/A=76000/25000=3.0400(P/A, i, 8) = P/A = 76000/25000 = 3.0400

  3. Solve for i

    theratesatisfying[1−(1+i)−8]/i=3.0400isi=28.46the rate satisfying [1 - (1+i)^-8]/i = 3.0400 is i = 28.46%

  4. After-tax cash flow — A_at = A(1 − t) = $25,000(1 − 0.24) = $19,000

  5. After-tax (P/A) required — 4.0000

  6. Solve again — i_at ≈ 18.62%

Answer:

Before-tax ROR ≈ 28.46% per year

Why the other options are there

  • 163.2% (simple total return)
  • 32.9% (ignored the time value of money)

Reference: FE Reference Handbook — Engineering Economics → Rate-of-Return

Example 2
Rate of return on an equipment investment, before and after tax — Rate-of-Return (2)

A contractor invests $103,000 in equipment that returns $18,000 per year for 10 years with no salvage. Determine the rate of return, and estimate the after-tax return at a 34% tax rate on the net income.

Given

  • P = $103,000

  • A = $18,000/yr

  • n=10yrn = 10 yr
  • Taxrate=34Tax rate = 34%

Find

Before-tax rate of return and an after-tax estimate

Start with the thinking

  • The rate of return is the interest rate that makes present worth zero — solved by trial or by the calculator's IRR.
  • A quick after-tax screen scales the annual return by (1 − tax rate) and re-solves.

Step-by-step solution

  1. Formula

    0=−P+A(P/A,i,n)0 = -P + A(P/A, i, n)

  2. Set up

    (P/A,i,10)=P/A=103000/18000=5.7222(P/A, i, 10) = P/A = 103000/18000 = 5.7222

  3. Solve for i

    theratesatisfying[1−(1+i)−10]/i=5.7222isi=11.69the rate satisfying [1 - (1+i)^-10]/i = 5.7222 is i = 11.69%

  4. After-tax cash flow — A_at = A(1 − t) = $18,000(1 − 0.34) = $11,880

  5. After-tax (P/A) required — 8.6700

  6. Solve again — i_at ≈ 2.68%

Answer:

Before-tax ROR ≈ 11.69% per year

Why the other options are there

  • 74.8% (simple total return)
  • 17.5% (ignored the time value of money)

Reference: FE Reference Handbook — Engineering Economics → Rate-of-Return

Example 3
Rate of return on an equipment investment, before and after tax — Rate-of-Return (3)

A contractor invests $172,000 in equipment that returns $25,000 per year for 10 years with no salvage. Determine the rate of return, and estimate the after-tax return at a 24% tax rate on the net income.

Given

  • P = $172,000

  • A = $25,000/yr

  • n=10yrn = 10 yr
  • Taxrate=24Tax rate = 24%

Find

Before-tax rate of return and an after-tax estimate

Start with the thinking

  • The rate of return is the interest rate that makes present worth zero — solved by trial or by the calculator's IRR.
  • A quick after-tax screen scales the annual return by (1 − tax rate) and re-solves.

Step-by-step solution

  1. Formula

    0=−P+A(P/A,i,n)0 = -P + A(P/A, i, n)

  2. Set up

    (P/A,i,10)=P/A=172000/25000=6.8800(P/A, i, 10) = P/A = 172000/25000 = 6.8800

  3. Solve for i

    theratesatisfying[1−(1+i)−10]/i=6.8800isi=7.45the rate satisfying [1 - (1+i)^-10]/i = 6.8800 is i = 7.45%

  4. After-tax cash flow — A_at = A(1 − t) = $25,000(1 − 0.24) = $19,000

  5. After-tax (P/A) required — 9.0526

  6. Solve again — i_at ≈ 1.85%

Answer:

Before-tax ROR ≈ 7.45% per year

Why the other options are there

  • 45.3% (simple total return)
  • 14.5% (ignored the time value of money)

Reference: FE Reference Handbook — Engineering Economics → Rate-of-Return

Example 4
Rate of return on an equipment investment, before and after tax — Rate-of-Return (4)

A contractor invests $138,000 in equipment that returns $36,000 per year for 8 years with no salvage. Determine the rate of return, and estimate the after-tax return at a 24% tax rate on the net income.

Given

  • P = $138,000

  • A = $36,000/yr

  • n=8yrn = 8 yr
  • Taxrate=24Tax rate = 24%

Find

Before-tax rate of return and an after-tax estimate

Start with the thinking

  • The rate of return is the interest rate that makes present worth zero — solved by trial or by the calculator's IRR.
  • A quick after-tax screen scales the annual return by (1 − tax rate) and re-solves.

Step-by-step solution

  1. Formula

    0=−P+A(P/A,i,n)0 = -P + A(P/A, i, n)

  2. Set up

    (P/A,i,8)=P/A=138000/36000=3.8333(P/A, i, 8) = P/A = 138000/36000 = 3.8333

  3. Solve for i

    theratesatisfying[1−(1+i)−8]/i=3.8333isi=20.03the rate satisfying [1 - (1+i)^-8]/i = 3.8333 is i = 20.03%

  4. After-tax cash flow — A_at = A(1 − t) = $36,000(1 − 0.24) = $27,360

  5. After-tax (P/A) required — 5.0439

  6. Solve again — i_at ≈ 11.57%

Answer:

Before-tax ROR ≈ 20.03% per year

Why the other options are there

  • 108.7% (simple total return)
  • 26.1% (ignored the time value of money)

Reference: FE Reference Handbook — Engineering Economics → Rate-of-Return

Example 5
Rate of return on an equipment investment, before and after tax — Rate-of-Return (5)

A contractor invests $135,000 in equipment that returns $31,000 per year for 10 years with no salvage. Determine the rate of return, and estimate the after-tax return at a 31% tax rate on the net income.

Given

  • P = $135,000

  • A = $31,000/yr

  • n=10yrn = 10 yr
  • Taxrate=31Tax rate = 31%

Find

Before-tax rate of return and an after-tax estimate

Start with the thinking

  • The rate of return is the interest rate that makes present worth zero — solved by trial or by the calculator's IRR.
  • A quick after-tax screen scales the annual return by (1 − tax rate) and re-solves.

Step-by-step solution

  1. Formula

    0=−P+A(P/A,i,n)0 = -P + A(P/A, i, n)

  2. Set up

    (P/A,i,10)=P/A=135000/31000=4.3548(P/A, i, 10) = P/A = 135000/31000 = 4.3548

  3. Solve for i

    theratesatisfying[1−(1+i)−10]/i=4.3548isi=18.89the rate satisfying [1 - (1+i)^-10]/i = 4.3548 is i = 18.89%

  4. After-tax cash flow — A_at = A(1 − t) = $31,000(1 − 0.31) = $21,390

  5. After-tax (P/A) required — 6.3114

  6. Solve again — i_at ≈ 9.38%

Answer:

Before-tax ROR ≈ 18.89% per year

Why the other options are there

  • 129.6% (simple total return)
  • 23.0% (ignored the time value of money)

Reference: FE Reference Handbook — Engineering Economics → Rate-of-Return

Example 6
Rate of return on an equipment investment, before and after tax — Rate-of-Return (6)

A contractor invests $131,000 in equipment that returns $10,000 per year for 10 years with no salvage. Determine the rate of return, and estimate the after-tax return at a 21% tax rate on the net income.

Given

  • P = $131,000

  • A = $10,000/yr

  • n=10yrn = 10 yr
  • Taxrate=21Tax rate = 21%

Find

Before-tax rate of return and an after-tax estimate

Start with the thinking

  • The rate of return is the interest rate that makes present worth zero — solved by trial or by the calculator's IRR.
  • A quick after-tax screen scales the annual return by (1 − tax rate) and re-solves.

Step-by-step solution

  1. Formula

    0=−P+A(P/A,i,n)0 = -P + A(P/A, i, n)

  2. Set up

    (P/A,i,10)=P/A=131000/10000=13.1000(P/A, i, 10) = P/A = 131000/10000 = 13.1000

  3. Solve for i

    theratesatisfying[1−(1+i)−10]/i=13.1000isi=0.01the rate satisfying [1 - (1+i)^-10]/i = 13.1000 is i = 0.01%

  4. After-tax cash flow — A_at = A(1 − t) = $10,000(1 − 0.21) = $7,900

  5. After-tax (P/A) required — 16.5823

  6. Solve again — i_at ≈ 0.01%

Answer:

Before-tax ROR ≈ 0.01% per year

Why the other options are there

  • -23.7% (simple total return)
  • 7.6% (ignored the time value of money)

Reference: FE Reference Handbook — Engineering Economics → Rate-of-Return

Example 7
Rate of return on an equipment investment, before and after tax — Rate-of-Return (7)

A contractor invests $116,000 in equipment that returns $35,000 per year for 8 years with no salvage. Determine the rate of return, and estimate the after-tax return at a 25% tax rate on the net income.

Given

  • P = $116,000

  • A = $35,000/yr

  • n=8yrn = 8 yr
  • Taxrate=25Tax rate = 25%

Find

Before-tax rate of return and an after-tax estimate

Start with the thinking

  • The rate of return is the interest rate that makes present worth zero — solved by trial or by the calculator's IRR.
  • A quick after-tax screen scales the annual return by (1 − tax rate) and re-solves.

Step-by-step solution

  1. Formula

    0=−P+A(P/A,i,n)0 = -P + A(P/A, i, n)

  2. Set up

    (P/A,i,8)=P/A=116000/35000=3.3143(P/A, i, 8) = P/A = 116000/35000 = 3.3143

  3. Solve for i

    theratesatisfying[1−(1+i)−8]/i=3.3143isi=25.16the rate satisfying [1 - (1+i)^-8]/i = 3.3143 is i = 25.16%

  4. After-tax cash flow — A_at = A(1 − t) = $35,000(1 − 0.25) = $26,250

  5. After-tax (P/A) required — 4.4190

  6. Solve again — i_at ≈ 15.47%

Answer:

Before-tax ROR ≈ 25.16% per year

Why the other options are there

  • 141.4% (simple total return)
  • 30.2% (ignored the time value of money)

Reference: FE Reference Handbook — Engineering Economics → Rate-of-Return

Example 8
Rate of return on an equipment investment, before and after tax — Rate-of-Return (8)

A contractor invests $53,000 in equipment that returns $18,000 per year for 8 years with no salvage. Determine the rate of return, and estimate the after-tax return at a 26% tax rate on the net income.

Given

  • P = $53,000

  • A = $18,000/yr

  • n=8yrn = 8 yr
  • Taxrate=26Tax rate = 26%

Find

Before-tax rate of return and an after-tax estimate

Start with the thinking

  • The rate of return is the interest rate that makes present worth zero — solved by trial or by the calculator's IRR.
  • A quick after-tax screen scales the annual return by (1 − tax rate) and re-solves.

Step-by-step solution

  1. Formula

    0=−P+A(P/A,i,n)0 = -P + A(P/A, i, n)

  2. Set up

    (P/A,i,8)=P/A=53000/18000=2.9444(P/A, i, 8) = P/A = 53000/18000 = 2.9444

  3. Solve for i

    theratesatisfying[1−(1+i)−8]/i=2.9444isi=29.73the rate satisfying [1 - (1+i)^-8]/i = 2.9444 is i = 29.73%

  4. After-tax cash flow — A_at = A(1 − t) = $18,000(1 − 0.26) = $13,320

  5. After-tax (P/A) required — 3.9790

  6. Solve again — i_at ≈ 18.80%

Answer:

Before-tax ROR ≈ 29.73% per year

Why the other options are there

  • 171.7% (simple total return)
  • 34.0% (ignored the time value of money)

Reference: FE Reference Handbook — Engineering Economics → Rate-of-Return

Example 9
Rate of return on an equipment investment, before and after tax — Rate-of-Return (9)

A contractor invests $159,000 in equipment that returns $36,000 per year for 5 years with no salvage. Determine the rate of return, and estimate the after-tax return at a 27% tax rate on the net income.

Given

  • P = $159,000

  • A = $36,000/yr

  • n=5yrn = 5 yr
  • Taxrate=27Tax rate = 27%

Find

Before-tax rate of return and an after-tax estimate

Start with the thinking

  • The rate of return is the interest rate that makes present worth zero — solved by trial or by the calculator's IRR.
  • A quick after-tax screen scales the annual return by (1 − tax rate) and re-solves.

Step-by-step solution

  1. Formula

    0=−P+A(P/A,i,n)0 = -P + A(P/A, i, n)

  2. Set up

    (P/A,i,5)=P/A=159000/36000=4.4167(P/A, i, 5) = P/A = 159000/36000 = 4.4167

  3. Solve for i

    theratesatisfying[1−(1+i)−5]/i=4.4167isi=4.28the rate satisfying [1 - (1+i)^-5]/i = 4.4167 is i = 4.28%

  4. After-tax cash flow — A_at = A(1 − t) = $36,000(1 − 0.27) = $26,280

  5. After-tax (P/A) required — 6.0502

  6. Solve again — i_at ≈ 0.01%

Answer:

Before-tax ROR ≈ 4.28% per year

Why the other options are there

  • 13.2% (simple total return)
  • 22.6% (ignored the time value of money)

Reference: FE Reference Handbook — Engineering Economics → Rate-of-Return

Example 10
Rate of return on an equipment investment, before and after tax — Rate-of-Return (10)

A contractor invests $113,000 in equipment that returns $30,000 per year for 5 years with no salvage. Determine the rate of return, and estimate the after-tax return at a 20% tax rate on the net income.

Given

  • P = $113,000

  • A = $30,000/yr

  • n=5yrn = 5 yr
  • Taxrate=20Tax rate = 20%

Find

Before-tax rate of return and an after-tax estimate

Start with the thinking

  • The rate of return is the interest rate that makes present worth zero — solved by trial or by the calculator's IRR.
  • A quick after-tax screen scales the annual return by (1 − tax rate) and re-solves.

Step-by-step solution

  1. Formula

    0=−P+A(P/A,i,n)0 = -P + A(P/A, i, n)

  2. Set up

    (P/A,i,5)=P/A=113000/30000=3.7667(P/A, i, 5) = P/A = 113000/30000 = 3.7667

  3. Solve for i

    theratesatisfying[1−(1+i)−5]/i=3.7667isi=10.25the rate satisfying [1 - (1+i)^-5]/i = 3.7667 is i = 10.25%

  4. After-tax cash flow — A_at = A(1 − t) = $30,000(1 − 0.20) = $24,000

  5. After-tax (P/A) required — 4.7083

  6. Solve again — i_at ≈ 2.04%

Answer:

Before-tax ROR ≈ 10.25% per year

Why the other options are there

  • 32.7% (simple total return)
  • 26.5% (ignored the time value of money)

Reference: FE Reference Handbook — Engineering Economics → Rate-of-Return

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