Present Value of Ordinary Annuity and Annuity Due: Differences and Applications

In finance, small differences in timing can create large differences in value.

The choice between an ordinary annuity and an annuity due—two structures with identical payments but different timing—can shift the present value of cash flows by thousands.

This article explores both structures in depth, walking through formulas, real-world uses, and examples that show how the time of payment impacts present value calculations.

What Is the Present Value of an Annuity?

The present value (PV) of an annuity is the current worth of a stream of future payments, discounted using a specified interest rate.

It reflects the idea that money received today is more valuable than the same amount received in the future, due to its earning potential.

This concept is central to loan structuring, investment analysis, and contract valuation.

Lump Sum vs. Annuity: A Quick Contrast

To understand annuities, it helps to start with the basics.

If someone offers you 1,000 today or 1,000 one year from now, you’d prefer the payment today—because you can invest it.

That’s time value of money (TVM).

But what if the offer is 1,000 every year for five years?

Now we’re in annuity territory—a series of regular payments.

And to value that stream, we discount each payment back to its present value, one period at a time.

Ordinary Annuity vs. Annuity Due

The difference is only in timing:

• Ordinary Annuity: Payments occur at the end of each period (e.g. mortgage payments).

• Annuity Due: Payments occur at the beginning of each period (e.g. rent, insurance premiums).

This small shift affects the present value, as payments received sooner are more valuable.

Present Value of an Ordinary Annuity

In an ordinary annuity, every payment is made at the end of the period.

Each future payment must be discounted one more period than in an annuity due.

Formula Breakdown

PV = P × [(1 – (1 + r)^–n) / r]

Where:

• P = payment per period

• r = interest rate per period

• n = number of periods

This is a finite geometric series, with the present value being the sum of each discounted payment.

Step-by-Step Example: Loan Repayment

You agree to repay a loan with monthly payments of 500 for 3 years, at a 6% annual interest rate (0.5% monthly).

• P = 500

• r = 0.005

• n = 36

Step 1: Plug into the formula

PV = 500 × [(1 – (1 + 0.005)^–36) / 0.005]PV = 500 × [(1 – 0.835645) / 0.005]PV = 500 × [0.164355 / 0.005]PV ≈ 500 × 32.871PV ≈ 16,435.50

This means the lender is effectively giving you 16,435.50 today in exchange for 36 monthly payments of 500.

Additional Example: Annual Pension Payments

Imagine receiving 10,000 per year for 10 years, with a 5% annual discount rate.

• P = 10,000

• r = 0.05

• n = 10

PV = 10,000 × [(1 – (1.05)^–10) / 0.05]PV ≈ 10,000 × 7.722PV ≈ 77,220

Present Value of an Annuity Due

In an annuity due, every payment is made one period earlier.

That means less discounting—each payment is worth slightly more.

Formula Breakdown

PV = P × [(1 – (1 + r)^–n) / r] × (1 + r)

This is the same formula as the ordinary annuity, but multiplied by (1 + r) to reflect the earlier timing.

Step-by-Step Example: Rent Payments

You pay 500 at the beginning of each month for 3 years, at a 6% annual rate (0.5% monthly).

We already found the ordinary annuity value to be 16,435.50.

Now apply the (1 + r) factor:

PV = 16,435.50 × (1 + 0.005)PV ≈ 16,435.50 × 1.005PV ≈ 16,517.67

The value is higher because each payment is received one month earlier.

Additional Example: Equipment Lease

You lease equipment for 2,000 per quarter, payable at the beginning of each quarter, for 5 years.

Assume an 8% annual rate, or 2% quarterly.

• P = 2,000

• r = 0.02

• n = 20

Ordinary PV = 2,000 × [(1 – (1.02)^–20) / 0.02] ≈ 2,000 × 16.351Ordinary PV ≈ 32,702

Annuity due PV = 32,702 × 1.02 ≈ 33,356

Why the Timing Difference Matters

Even small shifts in payment timing can impact valuation—especially in:

• Lease vs. purchase decisions

• Retirement or pension planning

• Pricing annuities or insurance contracts

The earlier the payment, the less it’s discounted, increasing the present value.

This becomes more significant as the interest rate or number of periods increases.

Key Differences and Practical Applications

Feature	Ordinary Annuity	Annuity Due
Timing of Payments	End of each period	Beginning of each period
Present Value	Lower	Higher
Discounting	One period more	One period less
Common Examples	Loans, bonds, mortgages	Rent, leases, insurance
Adjustment Needed	No	Multiply by (1 + r)

Use in Financial Modeling

In Excel or financial calculators, the PV function assumes ordinary annuity by default.

To calculate an annuity due, you must set the type argument (Excel: =PV(rate, nper, pmt, 0, 1)).

Failing to do so leads to undervalued or overpriced projections.

Sensitivity to Timing

Let’s compare two identical annuities:

• 1,000 per month, 10 years, 6% annual rate (0.5% monthly)

Type	PV
Ordinary Annuity	93,050
Annuity Due	93,050 × 1.005 = 93,515

A simple shift in timing adds nearly 500 in value—without changing any other input.

_____________

The difference between an ordinary annuity and an annuity due may seem minor, but it can affect present value by hundreds or thousands, depending on term and rate.

Recognizing the structure is essential when:

• Pricing contracts

• Modeling cash flows

• Comparing investment alternatives

Use the correct formula, apply the right timing adjustment, and your valuation will reflect the true economic value of the payment stream.