Internal Rate of Return (Single In/Out Cash Flow)
Internal Rate of Return (Single In/Out Cash Flow)
Internal rate of return (IRR) is the discount rate at which an investment's net present value is exactly zero — in other words, the annualized growth rate the investment actually delivered (or must deliver) over its holding period. For the simple case of one cash outflow followed by one cash inflow years later, IRR reduces to a direct compound-growth-rate formula.This is the same math behind "what annual return did my investment actually earn?" — a single lump sum invested and later cashed out has an implied compound annual growth rate that lets it be compared apples-to-apples against other investment options quoted as an annual percentage.
For a single investment growing from CF_0 to CF_1 over n years, the implied rate of return is IRR = (CF_1/CF_0)^(1/n) − 1. where CF_0i is the initial investment, CF_1i is the amount it grows to, n_irr is the number of years, and IRR_calc is the resulting annualized rate of return, all monetary amounts as plain numbers (dollars implied, no currency unit).
Inputs
CF_0i := 50000=50000
CF_1i := 80000=80000
n_irr := 5=5
Take the ratio of final to initial value and raise it to the 1/n power to "un-compound" the total growth back into an annual rate, then subtract 1 to express it as a growth rate rather than a growth multiple.
IRR_calc := (CF_1i/CF_0i)^(1/n_irr) - 1=0.1
Results
Growing $50,000 to $80,000 over 5 years works out to an annualized return in roughly the 9-10% range — a healthy but not extraordinary result for a multi-year investment. This single-cash-flow formula is a simplification; a real investment with interim cash flows (dividends, partial withdrawals) needs the more general IRR calculation that finds the rate solving the full NPV-equals-zero equation across every cash flow, not just a single beginning and ending value.
Assumptions & limitations
- Single cash outflow followed by a single cash inflow — no interim cash flows.
- Return compounds annually at a constant rate.
- No taxes, fees, or inflation adjustment included.
- Monetary amounts are plain numbers (dollars implied).