This Present Value of Annuity Calculator is based on the standard monthly compounding formula used for discounting future cash flows. It is essential for accurate valuation of pensions, lottery winnings, and settlement payments.
Welcome to the **Present Value of Annuity Calculator**. An annuity represents a series of equal payments received at regular intervals. This tool helps you determine the current worth (Present Value) of that future income stream. By providing any three of the four inputs—Present Value (PV), Periodic Payment (PMT), Annual Rate (r), or Time in Years (t)—you can solve for the missing variable to make informed financial decisions.
Present Value of Annuity Calculator
Present Value of Annuity Formula
The core formula for Present Value of an Ordinary Annuity (PV) is:
PV = PMT × [ 1 – (1 + i)-n / i ]
Where i is the monthly rate (r/12) and n is the total number of periods (t × 12).
1. Solve for Present Value (PV):
PV = PMT × [ ( 1 – (1 + i)-n ) / i ]
2. Solve for Periodic Payment (PMT):
PMT = PV × [ i / ( 1 – (1 + i)-n ) ]
3. Solve for Time in Years (t):
t = -ln( 1 – (PV × i / PMT) ) / (12 × ln(1 + i))
4. Solve for Annual Rate (r):
r = Iterative Numerical Approximation (e.g., Bisection Method)
Formula Source: Investopedia – Present Value of Annuity
Variables Explained
- PV – Present Value: The current lump-sum value of the future stream of payments (annuity).
- PMT – Periodic Payment: The fixed amount of money received at the end of each period (assumed monthly).
- r – Annual Discount Rate: The annual nominal interest rate used to discount the future payments to their present value.
- t – Time in Years: The total length of time the payments will be received, in years.
- i – Periodic Rate: The monthly interest rate (r / 12).
- n – Number of Periods: The total number of payments (t × 12).
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What is Present Value of Annuity?
The Present Value of an Annuity (PVA) is the value today of a series of future payments or receipts, assuming a specific rate of return (or discount rate). This rate is used to “discount” the future payments, as money received today is worth more than money received tomorrow due to its earning potential.
PVA is a core concept in finance used to value long-term financial commitments. For instance, if you win a lottery prize that pays $5,000 per month for 20 years, the PVA calculator tells you what that entire stream of payments is worth in a lump sum today, based on current interest rates.
Understanding PVA is crucial for investors making decisions about asset valuation, bond pricing, and comparing different investment opportunities with varying cash flow schedules.
How to Calculate Present Value (Example)
Let’s calculate the Present Value (PV) of receiving $500 per month (PMT=$500) for 15 years (t=15) at a 4% annual discount rate (r=0.04):
- Determine Periods (n) and Monthly Rate (i): $n = 15 \times 12 = 180$ periods. $i = 0.04 / 12 \approx 0.003333$.
- Calculate the Discount Factor: $(1 + i)^{-n} = (1.003333)^{-180} \approx 0.54964$.
- Calculate the PVA Factor: $\frac{1 – (1 + i)^{-n}}{i} = \frac{1 – 0.54964}{0.003333} \approx 135.108$.
- Determine the Present Value (PV): $PV = PMT \times \text{Factor} = \$500 \times 135.108 \approx \$67,554.00$.
Frequently Asked Questions (FAQ)
PVA calculates the present value of a series of *multiple* future payments. The PV of a Lump Sum calculates the present value of a *single* payment received in the future.
Why is the discount rate called ‘r’ and not ‘i’ in the inputs?‘r’ represents the Annual Discount Rate, which is the figure typically quoted. ‘i’ is the calculated periodic (monthly) rate used inside the formula ($i = r/12$).
Is the PVA formula the same as the Loan Amortization formula for Principal?Yes. The principal loan amount (P) is the Present Value of the stream of fixed monthly payments (PMT) that will be made over the loan term. The formulas are mathematically identical.
Why does the calculated Present Value increase if the discount rate decreases?A lower discount rate means future money is worth relatively more today, as the opportunity cost (or risk) of waiting for the money is lower. Therefore, the PV increases.