Question

Find the present value of an ordinary annuity of 8 quarterly payments of Rs.500 each, the rate of interest being 8% per annum compounded quarterly.
(A) Rs.3660.20
(B) Rs 3662.50
(C) Rs 4275
(D) Rs 3660

Answer 1

Hint: We start solving this problem by considering the formula for the present value of annuity, $V=\dfrac{A}{r}\times \left[ 1-{{\left( 1+r \right)}^{-n}} \right]$. Then we find the rate per quarter from the given rate per annum by dividing with four. Then we substitute the values given in the formula and simplify it to find the present value.

Complete step by step answer:
We are given that amount of each installment is Rs 500 and we are given that number of installments are 8 quarterly payments.
The rate of interest is 8% per annum compounded quarterly.
Now let us consider the formula for the present value of annuity,
$V=\dfrac{A}{r}\times \left[ 1-{{\left( 1+r \right)}^{-n}} \right]$
V = Present Value of annuity
A = Amount of each instalment
r = Rate of interest
n = number of instalments
As rate of interest is given per annum but the interest is compounded quarterly,
\[Rate\text{ }=\dfrac{8\%}{4}=2\%\]
So, comparing to the given information we get that,
A = 500, r = 2%, n = 8
Using these values, we get the present value as,
\[\begin{align}
  & \Rightarrow V=\dfrac{500}{\dfrac{2}{100}}\times \left[ 1-{{\left( 1+\dfrac{2}{100} \right)}^{-8}} \right] \\
 & \Rightarrow V=\dfrac{50000}{2}\times \left[ 1-{{\left( \dfrac{102}{100} \right)}^{-8}} \right] \\
 & \Rightarrow V=\dfrac{50000}{2}\times \left[ 1-{{\left( \dfrac{100}{102} \right)}^{8}} \right] \\
 & \Rightarrow V=\dfrac{50000}{2}\times \left[ 1-{{\left( 0.98039 \right)}^{8}} \right] \\
\end{align}\]
Calculating the value of \[{{\left( 0.98039 \right)}^{8}}\] and substituting we get,
\[\begin{align}
  & \Rightarrow V=\dfrac{50000}{2}\times \left[ 1-0.8535 \right] \\
 & \Rightarrow V=\dfrac{50000}{2}\times \left[ 0.1465 \right] \\
 & \Rightarrow V=3662.50 \\
\end{align}\]
So, we get the present value as Rs. 3662.50.
Hence answer is Option B.

Note:
 The common mistake one makes while solving this problem is one might take the formula for the present value for annuity as, $V=\dfrac{A}{r}\times \left[ {{\left( 1+r \right)}^{n}}-1 \right]$. But that formula is for the amount or final value, that is $M=\dfrac{A}{r}\times \left[ {{\left( 1+r \right)}^{n}}-1 \right]$.