Present Value Of Annuity Derivation

Understanding the Present Value of an Annuity: A Comprehensive Derivation

The present value of an annuity is a fundamental concept in finance, crucial for making informed decisions about investments, loans, and retirement planning. An annuity is a series of equal payments made at fixed intervals over a specified period. Understanding how to calculate its present value – the current worth of those future payments – is essential. This article will delve into the detailed mathematical derivation of the present value of an ordinary annuity and an annuity due, explaining the underlying principles and providing a clear, step-by-step approach.

Introduction: What is Present Value and Why is it Important?

Before diving into the derivation, let's establish a clear understanding of present value (PV). Simply put, the present value is the current worth of a future sum of money or stream of cash flows, given a specified rate of return. This rate of return, often represented as the discount rate, accounts for the time value of money – the principle that money available today is worth more than the same amount in the future due to its potential earning capacity.

The importance of present value calculations cannot be overstated. In personal finance, it helps in comparing different investment options, assessing the affordability of a loan, and planning for retirement. In corporate finance, it’s vital for evaluating projects, making capital budgeting decisions, and determining the value of businesses.

Deriving the Present Value of an Ordinary Annuity

An ordinary annuity involves a series of equal payments made at the end of each period. Let's derive the formula for its present value:

1. Defining Variables:

• PV: Present Value of the annuity
• PMT: Periodic payment (the equal amount paid each period)
• r: Discount rate (interest rate per period)
• n: Number of periods

2. Breaking Down the Annuity:

Imagine an annuity with payments made at the end of each of the ‘n’ periods. To find the present value, we need to discount each individual payment back to the present time (time zero).

• Payment 1 at the end of period 1 has a present value of PMT/(1+r)^1
• Payment 2 at the end of period 2 has a present value of PMT/(1+r)^2
• Payment 3 at the end of period 3 has a present value of PMT/(1+r)^3
• ...and so on, until
• Payment n at the end of period n has a present value of PMT/(1+r)^n

3. Summing the Present Values:

The present value of the entire annuity is the sum of the present values of all individual payments:

PV = PMT/(1+r)^1 + PMT/(1+r)^2 + PMT/(1+r)^3 + ... + PMT/(1+r)^n

This is a geometric series with ‘n’ terms, a first term of PMT/(1+r), and a common ratio of 1/(1+r).

4. Using the Geometric Series Formula:

The sum of a geometric series is given by:

Sum = a(1 - r^n) / (1 - r)

where ‘a’ is the first term and ‘r’ is the common ratio. Applying this to our annuity:

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

5. Simplifying the Formula:

By multiplying the numerator and denominator by (1+r), we can simplify the equation to the commonly used formula:

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

This formula allows us to directly calculate the present value of an ordinary annuity, given the periodic payment, discount rate, and number of periods.

Deriving the Present Value of an Annuity Due

An annuity due differs from an ordinary annuity in that payments are made at the beginning of each period. The derivation follows a similar approach but with a key difference in the timing of the payments.

1. Defining Variables: The variables remain the same as in the ordinary annuity derivation (PV, PMT, r, n).

2. Breaking Down the Annuity Due:

The present value of an annuity due considers that each payment is received one period earlier than in an ordinary annuity.

• Payment 1 at the beginning of period 1 has a present value of PMT
• Payment 2 at the beginning of period 2 has a present value of PMT/(1+r)^1
• Payment 3 at the beginning of period 3 has a present value of PMT/(1+r)^2
• ...and so on, until
• Payment n at the beginning of period n has a present value of PMT/(1+r)^(n-1)

3. Summing the Present Values:

PV = PMT + PMT/(1+r)^1 + PMT/(1+r)^2 + ... + PMT/(1+r)^(n-1)

Again, this is a geometric series, but with a slightly different structure.

4. Using the Geometric Series Formula (modified):

We can apply the geometric series formula, but we need to account for the first payment (PMT) which is not discounted. Alternatively, a simpler approach is to recognize that an annuity due is simply an ordinary annuity with one extra payment at the beginning. Therefore:

PV (Annuity Due) = PV (Ordinary Annuity) + PMT

Substituting the formula for the present value of an ordinary annuity:

PV (Annuity Due) = PMT * [(1 - (1 + r)^-n) / r] + PMT

5. Simplifying the Formula:

Factoring out PMT, we arrive at the formula for the present value of an annuity due:

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

6. Alternative Derivation:

Another approach is to directly apply the geometric series formula: The series is:

PV = PMT [1 + (1+r)^-1 + (1+r)^-2 + ... + (1+r)^-(n-1)]

Using the geometric series formula, we get:

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

Simplifying this expression, it will lead to the same formula as above.

Illustrative Example: Comparing Ordinary and Annuity Due

Let's consider a scenario to illustrate the difference between the present values of an ordinary annuity and an annuity due. Suppose you are promised $1,000 per year for 5 years. The discount rate is 5%.

Ordinary Annuity:

Using the formula: PV = PMT * [(1 - (1 + r)^-n) / r]

PV = $1000 * [(1 - (1 + 0.05)^-5) / 0.05] ≈ $4329.48

Annuity Due:

Using the formula: PV = PMT * [1 + (1 - (1 + r)^-n) / r]

PV = $1000 * [1 + (1 - (1 + 0.05)^-5) / 0.05] ≈ $4543.94

Notice that the present value of the annuity due is higher because the payments are received earlier. This highlights the importance of understanding the timing of cash flows when calculating present values.

Understanding the Underlying Principles: Time Value of Money

The derivations above rely on the fundamental principle of the time value of money. Money received today is worth more than the same amount received in the future because:

• Investment potential: You can invest today's money and earn interest or returns.
• Inflation: The purchasing power of money erodes over time due to inflation.
• Risk: There's always an element of risk associated with receiving money in the future. There's a chance that the payment may not be made.

The discount rate used in the present value calculations reflects these factors. A higher discount rate indicates a higher level of risk or a greater opportunity cost of foregoing immediate consumption.

Frequently Asked Questions (FAQ)

Q1: What if the payments are not equal?

A: If the payments are unequal, you cannot use the annuity formulas. You'll need to discount each payment individually and sum the present values.

Q2: What if the interest rate changes over time?

A: The formulas presented assume a constant interest rate throughout the annuity's lifespan. If the rate changes, you need to discount each payment using the appropriate rate for that period. This becomes more complex and often requires spreadsheet software or financial calculators.

Q3: Can I use these formulas for perpetuities?

A: A perpetuity is an annuity that continues forever. As ‘n’ approaches infinity, the term (1 + r)^-n approaches zero. Therefore, the present value formula for an ordinary perpetuity simplifies to:

PV = PMT / r

The present value of a perpetuity due is simply:

PV = PMT / r + PMT

Q4: What are some real-world applications of these calculations?

A: These calculations are crucial for:

• Loan amortization: Determining monthly payments and the total interest paid on a loan.
• Bond valuation: Assessing the fair value of a bond based on its coupon payments and maturity value.
• Retirement planning: Estimating the amount needed to save today to achieve a desired retirement income.
• Lease valuation: Determining the present value of lease payments.
• Capital budgeting: Evaluating the profitability of long-term investment projects.

Conclusion

The derivation of the present value of an annuity, both ordinary and due, provides a powerful tool for financial decision-making. Understanding the underlying principles and the mathematical steps involved allows for a more profound grasp of the time value of money and its implications across various financial contexts. While the formulas themselves might seem complex at first glance, with practice and a solid understanding of the underlying concepts, they become invaluable assets for anyone dealing with financial planning and investment analysis. Remember to always choose the correct formula (ordinary annuity or annuity due) based on the timing of the payments. The accuracy of your calculations directly impacts the soundness of your financial decisions.