Question

If an amount $R$ is paid at the end of every year for '$n$' years, then the net present value of the annuity at an interest rate of '$i$' is

• A. \(R[\{(1 + i)^n - 1\}/i]\)
• B. \(R[\{(1 + i)^n - 1\}/\{i(1 + i)^n\}]\)
• C. \(R(1 + i)^n\)
• D. \(R/(1 + i)^n\)

Hint:

Remember the timing of the payments for ordinary annuities (end of the period) versus annuities due (beginning of the period). The present value of an annuity due is the present value of an ordinary annuity multiplied by \((1 + i)\).

Answer 1

The Correct Option is B

Step 1: Understand the concept of the present value of an ordinary annuity.
An ordinary annuity is a series of equal payments made at the end of each period for a specified number of periods. The present value of an ordinary annuity is the current worth of these future payments, discounted at a given interest rate. 
Step 2: Derive the formula for the present value of an ordinary annuity.
Let \(PV\) be the present value of the annuity, \(R\) be the periodic payment, \(i\) be the interest rate per period, and \(n\) be the number of periods. The present value of each individual payment can be calculated as follows:
Present value of the payment at the end of year 1: \(R(1 + i)^{-1}\)
Present value of the payment at the end of year 2: \(R(1 + i)^{-2}\)
...
Present value of the payment at the end of year \(n\): \(R(1 + i)^{-n}\)
The present value of the entire annuity is the sum of the present values of all these individual payments:
$$PV = R(1 + i)^{-1} + R(1 + i)^{-2} + \cdots + R(1 + i)^{-n}$$This is a geometric series with the first term \(a = R(1 + i)^{-1}\), the common ratio \(r = (1 + i)^{-1}\), and \(n\) terms. The sum of a geometric series is given by:$$S_n = a \frac{1 - r^n}{1 - r}$$Substituting the values for our annuity:$$PV = R(1 + i)^{-1} \frac{1 - (1 + i)^{-n}}{1 - (1 + i)^{-1}}$$$$PV = \frac{R}{1 + i} \frac{1 - \frac{1}{(1 + i)^n}}{1 - \frac{1}{1 + i}}$$$$PV = \frac{R}{1 + i} \frac{\frac{(1 + i)^n - 1}{(1 + i)^n}}{\frac{(1 + i) - 1}{1 + i}}$$$$PV = \frac{R}{1 + i} \frac{(1 + i)^n - 1}{(1 + i)^n} \frac{1 + i}{i}$$The \((1 + i)\) terms cancel out:$$PV = R \frac{(1 + i)^n - 1}{i(1 + i)^n}$$ 
Step 3: Compare the derived formula with the given options.
The derived formula for the present value of an ordinary annuity, \(PV = R[\{(1 + i)^n - 1\}/\{i(1 + i)^n\}]\), matches option (2). 
Step 4: Analyze the other options to understand why they are incorrect.
(1) \(R[\{(1 + i)^n - 1\}/i]\): This formula represents the future value of an ordinary annuity, not the present value. It calculates the accumulated amount at the end of \(n\) periods if \(R\) is invested at the end of each period.
(3) \(R(1 + i)^n\): This formula calculates the future value of a single sum \(R\) invested for \(n\) periods at an interest rate \(i\).
(4) \(R/(1 + i)^n\): This formula calculates the present value of a single payment \(R\) received at the end of \(n\) periods, discounted at an interest rate \(i\).