How to Calculate External Rate of Return

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External Rate of Return (ERR) Calculator

Calculate investment efficiency using external reinvestment rates

Results

External Rate of Return: 0%

What is External Rate of Return (ERR)?

The External Rate of Return (ERR) is a financial metric used in engineering economics and capital budgeting to evaluate the profitability of an investment. Unlike the Internal Rate of Return (IRR), which assumes project cash flows are reinvested at the project's own internal rate, the ERR assumes that positive cash flows are reinvested at an external rate, typically the company's Minimum Attractive Rate of Return (MARR) or a market reinvestment rate.

The ERR Formula

To calculate the External Rate of Return, we solve for the discount rate that equates the present worth of costs to the future worth of benefits. The mathematical expression is:

PW(Costs) * (1 + ERR)^n = FW(Benefits at Reinvestment Rate)

Where:

  • PW(Costs): The present value of all expenses or outflows.
  • FW(Benefits): The future value of all inflows, compounded at the reinvestment rate to the end of the project life.
  • n: The number of periods (years).

Step-by-Step ERR Calculation Example

Suppose you have an investment with the following data:

  • Initial Cost: 10,000
  • Annual Benefit: 3,000
  • Project Life: 4 Years
  • Reinvestment Rate: 7%
  1. Calculate FW of Benefits: Use the annuity formula: 3,000 * [((1 + 0.07)^4 – 1) / 0.07] = 13,319.83.
  2. Set up the ERR equation: 10,000 * (1 + ERR)^4 = 13,319.83.
  3. Solve for ERR: (1 + ERR)^4 = 1.331983 → 1 + ERR = 1.0743 → ERR = 7.43%.

Why use ERR instead of IRR?

The IRR method often overstates the potential return of a project because it assumes cash inflows can be reinvested at the same high rate the project itself generates. ERR provides a more realistic view by allowing the user to input a conservative market rate for reinvestment, making it a superior tool for decision-making in capital-constrained environments.

function calculateERR() { var cost = parseFloat(document.getElementById('initialInvestment').value); var benefit = parseFloat(document.getElementById('annualBenefit').value); var salvage = parseFloat(document.getElementById('salvageValue').value); var rate = parseFloat(document.getElementById('reinvestmentRate').value) / 100; var n = parseFloat(document.getElementById('projectYears').value); var resultContainer = document.getElementById('errResultContainer'); var output = document.getElementById('errOutput'); var analysis = document.getElementById('errAnalysis'); if (isNaN(cost) || isNaN(benefit) || isNaN(n) || isNaN(rate) || n <= 0 || cost <= 0) { alert("Please enter valid positive numbers for Investment, Benefits, and Years."); return; } // Calculate Future Worth of Benefits var fwBenefits = 0; if (rate === 0) { fwBenefits = (benefit * n) + salvage; } else { // FW of Annuity formula: A * [((1+i)^n – 1) / i] fwBenefits = (benefit * (Math.pow(1 + rate, n) – 1) / rate) + salvage; } // ERR Formula: (FW_Benefits / PW_Costs)^(1/n) – 1 var ratio = fwBenefits / cost; if (ratio (rate * 100)) { analysis.innerHTML = "This project is favorable as the ERR exceeds your reinvestment rate."; } else { analysis.innerHTML = "The project's external rate of return is lower than your reinvestment rate; evaluate with caution."; } }

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