External rate of return

The external rate of return (ERR) — a close cousin of IRR — handles cash-flow streams with multiple sign changes that would otherwise produce the multiple-IRR pathology. ERR assumes that interim cash flows are reinvested externally at the MARR rather than internally at the project’s own IRR.

The reinvestment assumption is what makes ERR more realistic in many cases. Pure IRR implicitly assumes that every positive interim cash flow gets reinvested at the project’s own (high) IRR — usually unrealistic. ERR’s assumption — that you can reinvest at the more conservative MARR — is closer to reality for most firms.

There are two flavours.

Precise ERR

Track the project’s cash balance at every period. When the balance is positive, it grows at MARR; when negative, it grows at the trial ERR (the unknown we’re solving for). Iterate by trial-and-error or Newton’s method until the final balance is zero. The ERR that produces a zero terminal balance is the precise ERR.

The procedure:

1. Initialise cash balance to zero.
2. For each period, add that period’s cash flow, then grow the balance:
   • if positive, multiply by $(1+\text{MARR})$.
   • if negative, multiply by $(1+\text{ERR})$.
3. The final balance should be zero. Iterate on ERR to make it so.

Precise ERR tracks the actual timing of positive and negative balances over the project life. It’s the most accurate but also the most computationally involved.

Approximate ERR

A simpler procedure that doesn’t track interim balances:

1. Move all positive cash flows forward to time $N$ at the MARR (compound them as if they earn MARR until project end).
2. Move all negative cash flows forward to time $N$ at the (unknown) ERR.
3. Set the two totals equal and solve for ERR.

In equation form:

${\sum }_{t:C_t>0}{C}_t(1+\text{MARR}{)}^{N-t}={\sum }_{t:C_t<0}|C_t|(1+\text{ERR}{)}^{N-t}.$

Approximate ERR doesn’t care about the order in which positives and negatives occur — only their totals. It’s much simpler to compute and works fine for accept/reject decisions, where exact ERR isn’t required.

Decision rule (both flavours):

$\text{Accept if }\text{ERR}>\text{MARR},\text{ else reject.}$

For the simpler well-behaved case, see Internal rate of return. For the underlying PW approach, see Present worth method. ERR is essentially IRR’s safety net for irregular cash flows.