Annuities: Ordinary and due

Annuities: Ordinary vs. Due

An annuity is a financial product that provides a series of equal payments made at regular intervals over a specified period of time. Annuities are commonly used in retirement plans, insurance products, and loan repayments.

Annuities are generally classified into two types based on when the payments are made:

1. Ordinary Annuity (also called Annuity in Arrears)
2. Annuity Due

Both types have important differences in terms of the timing of payments, and as a result, they are calculated differently.

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1. Ordinary Annuity

✅ Definition:

An ordinary annuity is a sequence of equal payments made at the end of each period. The most common example is a loan repayment schedule where you make payments at the end of each month or year.

🧮 Formula:

The formula for the Present Value (PV) of an ordinary annuity is:

PV = PMT × [ (1 - (1 + r)^(-n)) / r ]

Where:

• PMT = Payment amount per period
• r = Interest rate per period (as a decimal)
• n = Number of periods

The formula for the Future Value (FV) of an ordinary annuity is:

FV = PMT × [ ((1 + r)^n - 1) / r ]

Where:

• PMT = Payment amount per period
• r = Interest rate per period
• n = Number of periods

🔍 How it Works:

• Payments are made at the end of each period.
• For example, with a loan, the borrower typically makes a payment at the end of each month or year.

📋 Example:

You invest ₹1,000 at the end of each year for 5 years in an ordinary annuity at an interest rate of 6%. Let’s calculate the Future Value.

1. Future Value Formula:

FV = 1,000 × [ ((1 + 0.06)^5 - 1) / 0.06 ]
   = 1,000 × [ (1.338225 - 1) / 0.06 ]
   = 1,000 × 5.63708
   = ₹5,637.08

So, the Future Value of this ordinary annuity after 5 years would be ₹5,637.08.

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2. Annuity Due

✅ Definition:

An annuity due is a sequence of equal payments made at the beginning of each period. This means that the first payment is made immediately, at the start of the first period.

🧮 Formula:

The formula for the Present Value (PV) of an annuity due is:

PV = PMT × [ (1 - (1 + r)^(-n)) / r ] × (1 + r)

The formula for the Future Value (FV) of an annuity due is:

FV = PMT × [ ((1 + r)^n - 1) / r ] × (1 + r)

🔍 How it Works:

• Payments are made at the beginning of each period.
• For example, in a lease agreement, you may need to pay immediately at the start of each month.

📋 Example:

You invest ₹1,000 at the beginning of each year for 5 years in an annuity due at an interest rate of 6%. Let’s calculate the Future Value.

1. Future Value Formula:

FV = 1,000 × [ ((1 + 0.06)^5 - 1) / 0.06 ] × (1 + 0.06)
   = 1,000 × [ (1.338225 - 1) / 0.06 ] × 1.06
   = 1,000 × 5.63708 × 1.06
   = ₹5,974.15

So, the Future Value of this annuity due after 5 years would be ₹5,974.15.

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Key Differences Between Ordinary Annuity and Annuity Due

Feature	Ordinary Annuity	Annuity Due
Payment Timing	Payments are made at the end of each period	Payments are made at the beginning of each period
Formula Adjustments	No adjustment for time value of first payment	Payments are multiplied by (1 + r) to account for the earlier payment
Examples	Mortgage payments, bond coupon payments	Lease payments, insurance premiums, rent payments
Effect on Value	Lower future value compared to annuity due for the same terms	Higher future value because of earlier payments
Present Value Formula	$PV = PMT \times \left[\frac{(1 - (1 + r)^{-n})}{r} \right]$	$PV = PMT \times \left[\frac{(1 - (1 + r)^{-n})}{r} \right] \times (1 + r)$
Future Value Formula	$FV = PMT \times \left[\frac{((1 + r)^n - 1)}{r} \right]$	$FV = PMT \times \left[\frac{((1 + r)^n - 1)}{r} \right] \times (1 + r)$

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Why the Difference in Value?

Since payments in an annuity due are made at the beginning of each period, each payment has one extra period to earn interest or appreciate in value. This extra compounding period leads to a higher future value for annuity due compared to an ordinary annuity.

🧠 Key Takeaways:

• Ordinary Annuity: Payments are made at the end of each period. It is more common in loans and bonds.
• Annuity Due: Payments are made at the beginning of each period. It is commonly used in leases, insurance, and rent agreements.
• The future value of an annuity due is higher than that of an ordinary annuity due to the extra compounding period for each payment.

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