Questions & Answers
Question
AnswersFind the present value of an annuity of Rs.$3000$ for $15$ years at $4.5\% $ p.a. CI-
A) Rs.$23809.41$
B) Rs.$32218.63$
C) Rs.$32908.41$
D) None of these
Answer
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Hint: Use the formula for present value (PV) =${\text{P}} \times \dfrac{{\left( {1 - {{\left( {1 + {\text{r}}} \right)}^{{\text{ - n}}}}} \right)}}{{\text{r}}}$ where P is payment, r is rate and n is time. Put the given values to get the answer.
Complete step-by-step answer:
Given, Payment P=Rs. $3000$of an annuity, rate%=$4.5\% $ and time n=$15$years
Here rate=$\dfrac{{4.5}}{{100}} = 0.045$
Now we know that formula to find annuity is-
$ \Rightarrow {\text{P = PV}} \times \dfrac{{\text{r}}}{{\left( {1 - {{\left( {1 + {\text{r}}} \right)}^{{\text{ - n}}}}} \right)}}$
Where PV = present value of an annuity, P= payment of an annuity, r=rate and n=time
We can write the formula as-
$ \Rightarrow $ Present value (PV) =${\text{P}} \times \dfrac{{\left( {1 - {{\left( {1 + {\text{r}}} \right)}^{{\text{ - n}}}}} \right)}}{{\text{r}}}$
On putting the values in the formula, we get-
$ \Rightarrow {\text{PV = }}\dfrac{{3000 \times \left( {1 - {{\left( {1 + 0.045} \right)}^{{\text{ - 15}}}}} \right)}}{{0.045}}$
On simplifying, we get
$ \Rightarrow {\text{PV = 3000}} \times \dfrac{{\left( {1 - {{\left( {1.045} \right)}^{{\text{ - 15}}}}} \right)}}{{0.045}} = 600 \times \dfrac{{\left( {1 - 0.5167204} \right)}}{{0.009}}$
On further solving we get,
$ \Rightarrow {\text{PV = 200}} \times \dfrac{{0.4832796}}{{0.003}} = 200 \times 161.0932$
$ \Rightarrow {\text{PV = 32218}}{\text{.64}}$
Hence, the correct answer is ‘B’.
Additional Information: Annuity is the term for the equal payments made at equal intervals of time with compound interest on the payment made. The present value of an annuity is the sum of all the payments made in installments. It is given by the formula-
$ \Rightarrow $ (PV) =${\text{P}} \times \dfrac{{\left( {1 - {{\left( {1 + {\text{r}}} \right)}^{{\text{ - n}}}}} \right)}}{{\text{r}}}$ Where PV = present value of an annuity, P= payment of an annuity, r=rate and n=time.
Note: In this question, the student may go wrong when they put the value of rate% directly in the formula because in the formula we have to put the value of rate not rate%.
Complete step-by-step answer:
Given, Payment P=Rs. $3000$of an annuity, rate%=$4.5\% $ and time n=$15$years
Here rate=$\dfrac{{4.5}}{{100}} = 0.045$
Now we know that formula to find annuity is-
$ \Rightarrow {\text{P = PV}} \times \dfrac{{\text{r}}}{{\left( {1 - {{\left( {1 + {\text{r}}} \right)}^{{\text{ - n}}}}} \right)}}$
Where PV = present value of an annuity, P= payment of an annuity, r=rate and n=time
We can write the formula as-
$ \Rightarrow $ Present value (PV) =${\text{P}} \times \dfrac{{\left( {1 - {{\left( {1 + {\text{r}}} \right)}^{{\text{ - n}}}}} \right)}}{{\text{r}}}$
On putting the values in the formula, we get-
$ \Rightarrow {\text{PV = }}\dfrac{{3000 \times \left( {1 - {{\left( {1 + 0.045} \right)}^{{\text{ - 15}}}}} \right)}}{{0.045}}$
On simplifying, we get
$ \Rightarrow {\text{PV = 3000}} \times \dfrac{{\left( {1 - {{\left( {1.045} \right)}^{{\text{ - 15}}}}} \right)}}{{0.045}} = 600 \times \dfrac{{\left( {1 - 0.5167204} \right)}}{{0.009}}$
On further solving we get,
$ \Rightarrow {\text{PV = 200}} \times \dfrac{{0.4832796}}{{0.003}} = 200 \times 161.0932$
$ \Rightarrow {\text{PV = 32218}}{\text{.64}}$
Hence, the correct answer is ‘B’.
Additional Information: Annuity is the term for the equal payments made at equal intervals of time with compound interest on the payment made. The present value of an annuity is the sum of all the payments made in installments. It is given by the formula-
$ \Rightarrow $ (PV) =${\text{P}} \times \dfrac{{\left( {1 - {{\left( {1 + {\text{r}}} \right)}^{{\text{ - n}}}}} \right)}}{{\text{r}}}$ Where PV = present value of an annuity, P= payment of an annuity, r=rate and n=time.
Note: In this question, the student may go wrong when they put the value of rate% directly in the formula because in the formula we have to put the value of rate not rate%.