NPV & IRR Formula — PV10, Payout, Breakeven for Oil & Gas

Overview

NPV discounts future cash flows back to today's dollars at a chosen rate. IRR is the rate at which NPV equals zero. PV10 is NPV at exactly 10% discount, used in SEC reserve reporting (Regulation S-X 4-10 and ASC 932). Payout is the time it takes to recover the initial investment from undiscounted (or discounted) cash flow. Breakeven price is the commodity price at which NPV equals zero, holding all other assumptions constant.

In upstream economics, the discount rate is typically applied monthly because production declines monthly and royalty/severance accrue monthly. IRR is sensitive to the shape of the cash flow profile — a front-loaded recovery delivers a higher IRR than a back-loaded recovery with the same total NPV. Profitability index (PI) is the ratio of present value of inflows to absolute value of outflows, used when capital is constrained.

Theory

The basic NPV calculation discounts each period's cash flow by (1 + r)^-t, where r is the periodic discount rate and t is the period index. The SEC PV10 convention uses 10% annual discount applied monthly: r_monthly = (1 + 0.10)^1/12 - 1. IRR is the rate r such that NPV(r) = 0; Newton-Raphson typically converges in 5–10 iterations from a sensible starting guess (e.g., r_0 = 0.20 / period).

Formulas

Net Present Value

NPV(r) = sum_{t=0}^{N} CF_t / (1 + r)^t

CF_t = net cash flow in period t
r    = periodic discount rate
N    = total number of periods

For monthly cash flow at 10% annual rate: r_monthly = (1.10)^(1/12) - 1 = 0.007974.

PV10 (SEC Convention)

PV10 = sum_{t=0}^{N} CF_t / (1 + 0.10)^t          (annual cash flow)

PV10 = sum_{t=0}^{N} CF_t / (1 + r_m)^t           (monthly cash flow)
where r_m = (1 + 0.10)^(1/12) - 1 = 0.007974

CF_t is AFTER-tax revenue net of royalty, severance, ad valorem,
LOE, and CAPEX, BEFORE federal income tax.

Internal Rate of Return (Newton-Raphson)

Find r such that NPV(r) = 0

f(r)  = sum_t CF_t * (1+r)^-t
f'(r) = sum_t -t * CF_t * (1+r)^(-t-1)

Iteration: r_{n+1} = r_n - f(r_n) / f'(r_n)
Convergence when |f(r_n)| < 1e-6 or |r_{n+1} - r_n| < 1e-8

Starting guess: r_0 = 0.20 / period
Max iterations: 100 (typically converges in 5-15)

Payout (Undiscounted)

Payout = smallest T such that  sum_{t=0}^{T} CF_t >= 0

Cumulative_CF(T) = sum_{t=0}^{T} CF_t
Linear interpolation between sign-changing periods:
  T_payout = T - 1 + |Cumulative_CF(T-1)| / CF_T

Discounted Payout

Disc_Payout = smallest T such that  sum_{t=0}^{T} CF_t / (1+r)^t >= 0

Breakeven Price (bisection)

Solve P_oil such that NPV(P_oil, all_other_inputs) = 0

Bracket: [P_lo, P_hi]
Bisection until |NPV| < tolerance

Alternative: secant method using two NPV evaluations
to project the zero-crossing.

Profitability Index

PI = PV_inflows / |PV_outflows|

PV_inflows  = sum of positive CF_t discounted at r
PV_outflows = sum of negative CF_t discounted at r

Decline-Parameter Sensitivity

Arps hyperbolic forecast: q(t) = qi / (1 + b * Di * t)^(1/b)

Sensitivity matrix:
  qi  +/- 10%, 20%
  Di  +/- 10%, 20%
  b   +/- 0.1, 0.2

For each (qi, Di, b) variant, recompute monthly production,
revenue, NPV, IRR, payout, breakeven. Report delta from base.

Royalty / NRI / Severance Handling

Gross_revenue   = q * P_realized
Royalty         = Gross_revenue * (1 - NRI)
Severance_tax   = Gross_revenue * sev_rate
Ad_valorem      = Gross_revenue * adval_rate

NRI  = working_interest * (1 - royalty_rate)
       (with overrides, ORI, NPI as additional deductions)

Net_revenue     = Gross_revenue - Royalty - Severance - Ad_valorem
Net_cash_flow   = Net_revenue - LOE - CAPEX - ARO

Key Symbols

Worked Example

Given: Single horizontal well. CAPEX = -$8,000,000 at t=0. Net cash flow stream (years 1–5, in $millions): 4.2, 2.8, 1.7, 1.0, 0.6. Discount rate = 10% annual.

Step 1 — PV10 (annual, SEC convention):

PV10 = -8.0 + 4.2/1.10 + 2.8/1.21 + 1.7/1.331 + 1.0/1.4641 + 0.6/1.6105
     = -8.0 + 3.818 + 2.314 + 1.277 + 0.683 + 0.373
     = $0.465 MM

Step 2 — IRR (Newton-Raphson):

Starting guess r = 0.10
NPV(0.10) = 0.465  (positive, raise r)
NPV(0.15) = -0.413 (negative, IRR between)
NPV(0.125) = 0.014 (very close)
NPV(0.127) ~ 0

IRR ~ 12.7%

Step 3 — Undiscounted payout:

Cumulative CF:
  t=0: -8.0
  t=1: -3.8
  t=2: -1.0
  t=3: +0.7

Sign change between t=2 and t=3:
T_payout = 2 + 1.0 / 1.7 = 2.59 years

Step 4 — Profitability Index:

PV_inflows = 3.818 + 2.314 + 1.277 + 0.683 + 0.373 = 8.465
PV_outflows = 8.000

PI = 8.465 / 8.000 = 1.058

Valid Ranges & Pitfalls

When the formula is not enough

• Multi-well portfolios with shared infrastructure — allocate capital and overhead per well using a defensible rule.
• Hedged positions — blend hedge price into the realized-price series period by period.
• Royalty trusts, ORI, and NPI — layer-by-layer revenue deductions need explicit modeling, not a single NRI.
• Tax shields and depletion allowances — switch to after-federal-tax cash flow with depletion and IDC modeled.
• Real options (drill / defer / abandon) — deterministic NPV understates the value of flexibility; use Monte Carlo or LSM.

References

1. SEC — Regulation S-X, Rule 4-10 (Financial Accounting and Reporting for Oil and Gas Producing Activities): https://www.law.cornell.edu/cfr/text/17/210.4-10
2. FASB Accounting Standards Codification (ASC) 932 — Extractive Activities: Oil and Gas.
3. Society of Petroleum Engineers (SPE) — Petroleum Resources Management System (PRMS), 2018.
4. Brealey, R.A., Myers, S.C., Allen, F. (2019). Principles of Corporate Finance, 13th ed. McGraw-Hill. Chapters 5–6 (NPV and IRR).
5. Mian, M.A. (2011). Project Economics and Decision Analysis, Volume 1: Deterministic Models, 2nd ed. PennWell.
6. Newendorp, P.D. & Schuyler, J. (2000). Decision Analysis for Petroleum Exploration, 2nd ed. Planning Press.