Data Source and Methodology
Authoritative source: David C. M. Dickson, Mary R. Hardy, Howard R. Waters, “Actuarial Mathematics for Life Contingent Risks,” 2nd ed., Cambridge University Press (2013). DOI: 10.1017/CBO9781139506779. All formulas herein are standard time-value-of-money identities used in financial mathematics and actuarial science.
Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.
The Formula Explained
Effective annual rate from nominal APR i with m compounding periods:
$$ i_{\mathrm{eff}} = \left(1 + \frac{i}{m}\right)^m - 1 $$
Per-payment rate with k payments/year:
$$ r = (1 + i_{\mathrm{eff}})^{1/k} - 1 $$
Number of payments: $$ n = k \times \text{years} $$
Present value of an annuity-immediate (ordinary) with payment P:
$$ PV_{\text{ord}} = P \cdot \frac{1 - (1 + r)^{-n}}{r} $$
Annuity-due multiplies by one extra period:
$$ PV_{\text{due}} = PV_{\text{ord}} \cdot (1 + r) $$
Future value of an annuity-immediate (ordinary):
$$ FV_{\text{ord}} = P \cdot \frac{(1 + r)^n - 1}{r} $$
Annuity-due future value:
$$ FV_{\text{due}} = FV_{\text{ord}} \cdot (1 + r) $$
Solving for payment from present value (ordinary):
$$ P = PV \cdot \frac{r}{1 - (1 + r)^{-n}} $$
For annuity-due, divide by (1 + r):
$$ P_{\text{due}} = \frac{PV \cdot r}{(1 - (1 + r)^{-n}) \cdot (1 + r)} $$
Solving for payment from future value (ordinary):
$$ P = FV \cdot \frac{r}{(1 + r)^n - 1}, \quad P_{\text{due}} = \frac{FV \cdot r}{\big((1 + r)^n - 1\big)\,(1 + r)} $$
Real (inflation-adjusted) effective annual rate with inflation π:
$$ 1 + i_{\mathrm{real}} = \frac{1 + i_{\mathrm{eff}}}{1 + \pi} $$
Glossary of Variables
| Symbol / Field | Meaning |
|---|---|
| APR (i) | Nominal annual percentage rate before inflation. |
| Compounding (m) | Times per year interest is compounded to produce effective annual growth. |
| Payments/Year (k) | Number of equal payments each year. |
| r | Effective rate per payment period derived from APR, compounding, and payment frequency. |
| n | Total number of payments (k × years). |
| P (Payment) | Amount paid or received each period. |
| PV | Present value (lump sum today). |
| FV | Future value after n payments. |
| Annuity Type | Ordinary (end of period) or Due (beginning of period). |
| Inflation (π) | Annual inflation rate used to compute real values. |
Worked Example
How It Works: A Step-by-Step Example
Goal: A retiree has $500,000 and wants monthly income for 25 years at 5% APR, compounded monthly. Payments occur at the end of each month (ordinary annuity).
- Inputs: Solve for Payment (from PV). APR = 5, Compounding = 12, Payments/Year = 12, Years = 25, PV = 500000, Type = Ordinary.
- Compute effective annual: i_eff = (1 + 0.05/12)^12 − 1 ≈ 0.0511619.
- Per-month rate: r = (1 + i_eff)^(1/12) − 1 ≈ 0.0041667.
- n = 12 × 25 = 300.
- Payment formula (ordinary): P = PV × r / [1 − (1 + r)^−n] ≈ 500000 × 0.0041667 / [1 − (1.0041667)^−300] ≈ $2,923.88.
Interpretation: The retiree can withdraw about $2,924 per month for 25 years. Total paid ≈ $877,164, interest portion ≈ $377,164.
Frequently Asked Questions (FAQ)
Does this include fees and taxes?
No. Fees, taxes, and surrender charges vary by product and jurisdiction. Enter a net APR that reflects expected after-fee returns to approximate their effect.
Can payments and compounding use different frequencies?
Yes. This tool precisely converts APR and compounding into an effective rate per payment period, so mismatched schedules remain accurate.
What if I want payments to grow with inflation?
Level-payment (flat) annuities are modeled here. For escalating payments, you would use growing annuity formulas, which are outside the scope of this calculator.
Is the “Annuity-Due” option always better?
It yields a higher PV/FV for the same payment because funds are invested one period longer. Whether it is “better” depends on product terms and your cash flow needs.
How accurate is the inflation adjustment?
It uses the Fisher equation with a constant annual inflation rate. Real-world inflation varies; use scenario analysis to test sensitivities.
Which rate should I use for a fixed annuity?
Use the guaranteed crediting rate, net of any explicit account fees, and set compounding and payment frequencies per contract.