Present Value Formulas

Present Value and Annuity Formulas

Type	Formula
1. Present Value of a Lump Sum	PV = FV / (1 + i)^n
2. Present Value of an Annuity	—
2.1 Present Value of an Ordinary Annuity (T = 0)	PV = PMT × (1 – (1 + i)^(-n)) / i
2.2 Present Value of an Annuity Due (T = 1)	PV = PMT × (1 – (1 + i)^(-n)) / i × (1 + i)
3. Present Value of a Growing Annuity (g ≠ i)	—
3.1 Ordinary Annuity (T = 0)	PV = PMT × (1 – ((1 + g)^n × (1 + i)^(-n))) / (i – g)
3.2 Annuity Due (T = 1)	PV = PMT × (1 – ((1 + g)^n × (1 + i)^(-n))) / (i – g) × (1 + i)
4. Present Value of a Growing Annuity (g = i)	—
4.1 Ordinary Annuity (T = 0)	PV = PMT × n / (1 + i)
4.2 Annuity Due (T = 1)	PV = PMT × n
5. Present Value of a Perpetuity	PV = PMT / i
6. Present Value of a Growing Perpetuity (g < i)	PV = PMT / (i – g)
7. Present Value of a Growing Perpetuity (g = i)	PV = PMT × (1 + i) / i
8. Present Value for Combined Future Value Sum and Cash Flow Annuity	—
8.1 Combined FV Sum + Ordinary Annuity (T = 0)	PV = FV / (1 + i)^n + PMT × (1 – (1 + i)^(-n)) / i
8.2 Combined FV Sum + Annuity Due (T = 1)	PV = FV / (1 + i)^n + PMT × (1 – (1 + i)^(-n)) / i × (1 + i)
9. Present Value for Combined FV Sum with Growing Annuity (g ≠ i)	PV = FV / (1 + i)^n + PMT × (1 – ((1 + g)^n × (1 + i)^(-n))) / (i – g)
10. Present Value for Combined FV Sum with Growing Annuity (g = i)	PV = FV / (1 + i)^n + PMT × n / (1 + i)
11. Present Value for Combined FV Sum and Annuity including Compounding, Time, and Rate	PV = FV / (1 + i/m)^(m×t) + PMT × (1 – (1 + i/m)^(-m×t)) / (i/m)
12. Present Value with Continuous Compounding (m → ∞)	—
12.1 FV Sum + Ordinary Annuity (T = 0)	PV = FV × e^(-i×t) + PMT × (1 – e^(-i×t)) / i
12.2 FV Sum + Annuity Due (T = 1)	PV = FV × e^(-i×t) + PMT × (1 – e^(-i×t)) / i × (1 + i)
13. Growing Annuity (g ≠ i) with Continuous Compounding	PV = PMT × (1 – e^(-(i-g)×t)) / (i – g)
14. Growing Annuity (g = i) with Continuous Compounding	PV = PMT × t × e^(-i×t)
15. Perpetuity with Continuous Compounding (t → ∞)	PV = PMT / i
16. Growing Perpetuity (g < i) with Continuous Compounding	PV = PMT / (i – g)
17. Growing Perpetuity (g = i) with Continuous Compounding	PV = PMT × (1 + i) / i

Notes for Clarity

• PV = Present Value
• FV = Future Value
• PMT = Payment per period
• i = interest rate per period (decimal form)
• n = number of periods
• m = compounding frequency per period
• t = total time (number of periods)
• g = growth rate per period (decimal form)
• e = Euler’s number ≈ 2.71828
• T = Type (0 = ordinary annuity, 1 = annuity due)