Question 15 Marks
(a) Anil bought a TV costing ₹ 13000 by making a down payment of ₹ 3000 and agreeing to make equal annual payment for three years. How much would be each payment if the interest on unpaid amount be 14% compounded annually ?
(b) Experts say that the baby boom generation (Indians born between 1946 and 1960) cannot count on a company pension or social security to provide comfortable retirement, as their parents did. It is recommended that they start to save early and regularly. Mahesh, a baby boomer, has deposit ₹ 200 each month for 20 years in an account that pays interest of 7.2% compounded monthly.
(i) How much will be in the account at the end of 20 years?
(ii) Mahesh believes he needs to accumulate ₹ 130,000 in the 20 years period to have enough for retirement. if he can not get higher rate to produce ₹ 130,000 in 20 years. To meet the goal, he must increase her monthly payment. What payment should he make each month? Given that (1.006) 240 = 4.2026
(b) Experts say that the baby boom generation (Indians born between 1946 and 1960) cannot count on a company pension or social security to provide comfortable retirement, as their parents did. It is recommended that they start to save early and regularly. Mahesh, a baby boomer, has deposit ₹ 200 each month for 20 years in an account that pays interest of 7.2% compounded monthly.
(i) How much will be in the account at the end of 20 years?
(ii) Mahesh believes he needs to accumulate ₹ 130,000 in the 20 years period to have enough for retirement. if he can not get higher rate to produce ₹ 130,000 in 20 years. To meet the goal, he must increase her monthly payment. What payment should he make each month? Given that (1.006) 240 = 4.2026
Answer
View full question & answer→(a) In the present case, we have present value of annuity ie., ₹ 10000 (13000-3000) and we have to calculate equal annual payment over the period of 3 years.
We know that,
$ \begin{array}{l} \text {Present value }=\text { C.F. } \times \frac{\left[(1+i)^n-1\right]}{i(1+i)^n} \\ \Rightarrow \quad \text { C.F. }=\frac{\text { Present value }}{\left[(1+i)^n-1\right]} \times i(1+i)^n \end{array} $
Here, $n=3, i=14 \%=\frac{14}{100}=0.14$
$ \begin{aligned} \therefore \quad \text { C.F. } & =\frac{10,000}{\left[(1+0.14)^3-1\right]} \times 0.14 \times(1+0.14)^3 \\ & =\frac{10,000}{\left[(1.14)^3-1\right]} \times 0.14 \times(1.14)^3 \end{aligned} $
$\begin{array}{l}=\frac{10,000}{(1.4815-1)} \times 0.14 \times 1.4815 \\ =\frac{10,000}{0.4815} \times 0.207416 \\ =\frac{2074.16}{0.4815}\end{array}$
=₹ 4307.71
Therefore each payment would be ₹ 4307.71.
(b) (i) The saving plan is an annuity with periodic payments PMT $=200, i=7.2 \%$ monthly $=$ $\frac{7.2}{100} \times \frac{1}{12}$ annually $=0.006$ and $n=12(20)=$ 240 The future value is $ \begin{aligned} FV & =\operatorname{PMT} \frac{(1+i)^n-1}{i} \\ \therefore \quad FV & =200 \times\left\{\frac{(1+0.006)^{220}-1}{0.006}\right\} \\ & =200 \times\left\{\frac{(1.006)^{240}-1}{0.006}\right\} \\ & =200 \times\left(\frac{4.2026-1}{0.006}\right) \end{aligned} $
$\begin{array}{l}=200 \times \frac{3.2026}{0.006} \\ =200 \times 533.7623\end{array}$
= ₹ 106, 752.468
He will accumulate ₹ 106, 252.47 in the account at the end of 20 years.
(ii) Mahesh's goal is to accumulate ₹ 130,000 in 20 years at 7.2% compounded monthly. Therefore the future value is F .V.= ₹ 130,000, the monthly interest rate is $i=\frac{0.072}{12}=0.006$ and the number of periods is n = 12(20) = 240 and the Using the sinking fund payment formula to find the payment PMT.
$\begin{aligned} \text { PMT } & =\frac{i \times F . V .}{(1+i)^n-1} \\ & =\frac{0.006 \times 130,000}{(1+0.006)^{240}-1} \\ & =\frac{780}{4.2026-1} \\ & =\frac{780}{3.2026}\end{aligned}$
= ₹ 243.5521
Mahesh will need payments of ₹ 243.55 each month for 20 years to accumulate at atleast ₹ 130,000.
We know that,
$ \begin{array}{l} \text {Present value }=\text { C.F. } \times \frac{\left[(1+i)^n-1\right]}{i(1+i)^n} \\ \Rightarrow \quad \text { C.F. }=\frac{\text { Present value }}{\left[(1+i)^n-1\right]} \times i(1+i)^n \end{array} $
Here, $n=3, i=14 \%=\frac{14}{100}=0.14$
$ \begin{aligned} \therefore \quad \text { C.F. } & =\frac{10,000}{\left[(1+0.14)^3-1\right]} \times 0.14 \times(1+0.14)^3 \\ & =\frac{10,000}{\left[(1.14)^3-1\right]} \times 0.14 \times(1.14)^3 \end{aligned} $
$\begin{array}{l}=\frac{10,000}{(1.4815-1)} \times 0.14 \times 1.4815 \\ =\frac{10,000}{0.4815} \times 0.207416 \\ =\frac{2074.16}{0.4815}\end{array}$
=₹ 4307.71
Therefore each payment would be ₹ 4307.71.
(b) (i) The saving plan is an annuity with periodic payments PMT $=200, i=7.2 \%$ monthly $=$ $\frac{7.2}{100} \times \frac{1}{12}$ annually $=0.006$ and $n=12(20)=$ 240 The future value is $ \begin{aligned} FV & =\operatorname{PMT} \frac{(1+i)^n-1}{i} \\ \therefore \quad FV & =200 \times\left\{\frac{(1+0.006)^{220}-1}{0.006}\right\} \\ & =200 \times\left\{\frac{(1.006)^{240}-1}{0.006}\right\} \\ & =200 \times\left(\frac{4.2026-1}{0.006}\right) \end{aligned} $
$\begin{array}{l}=200 \times \frac{3.2026}{0.006} \\ =200 \times 533.7623\end{array}$
= ₹ 106, 752.468
He will accumulate ₹ 106, 252.47 in the account at the end of 20 years.
(ii) Mahesh's goal is to accumulate ₹ 130,000 in 20 years at 7.2% compounded monthly. Therefore the future value is F .V.= ₹ 130,000, the monthly interest rate is $i=\frac{0.072}{12}=0.006$ and the number of periods is n = 12(20) = 240 and the Using the sinking fund payment formula to find the payment PMT.
$\begin{aligned} \text { PMT } & =\frac{i \times F . V .}{(1+i)^n-1} \\ & =\frac{0.006 \times 130,000}{(1+0.006)^{240}-1} \\ & =\frac{780}{4.2026-1} \\ & =\frac{780}{3.2026}\end{aligned}$
= ₹ 243.5521
Mahesh will need payments of ₹ 243.55 each month for 20 years to accumulate at atleast ₹ 130,000.