Tag: amount of an annuity

Questions Related to amount of an annuity

Z invests Rs. $10,000$ every year starting for today for next $10$ years. Suppose interest rate is $8\%$ per annum compounded annually. Calculate future value of the annuity. Given that $(1+0.08)^{10}=2.15892500$.

business mathematics and statistics insurance and annuity amount of an annuity annuities financial mathematics

  1. $1,44,865.625$

  2. $1,56,454.875$

  3. $1,54,654.875$

  4. $1,44,568.625$

Show answer
Correct Option: B
Explanation:

Step-$1$: Calculate future value as though it is an ordinary annuity
Future value of the annuity as if it is an ordinary annuity
$=10,000\left[\displaystyle\frac{(1+0.08)^{10}-1}{0.08}\right]$
$=10,000\times 14.4865625$
$=Rs. 1,44,865.625$
Step-$2$: Multiply the result by $(1+i)$
$=1,44,865.625\times (1+0.08)$
$=1,56454.875$.

A machine costs Rs. $98,000$ and its effective life is estimated at $12$ years. If the scrap value is Rs. $3, 000$, what should be cut out of the profit at the end of each year to accumulate at compound rate of $5\%$ per annum so that a new machine can be purchased after $12$ years ?

business mathematics and statistics insurance and annuity amount of an annuity annuities financial mathematics

  1. Rs. $6,000$

  2. Rs. $5,968$

  3. Rs. $4,787$

  4. Rs. $4,763$

Show answer
Correct Option: B
Explanation:
Effective cost of the machine is $98000–3000 = 95000$.
We know that Future value of annuity (FV) $=$ annuity $\times$ Compount Value factor of Annuity (CVAF)
That is $\text{FV}= \text{annuity} \times \text{CVAF} _{(5\%, 12)}$
$\text{FV}= \text{annuity} \times \left(\dfrac{(1+r)^n-1}{r}\right)$, where $r=0.05, n=12$
$\Rightarrow 95000 = \text{annuity} \times \dfrac{(1+0.05)^{12}-1}{0.05} $
$\Rightarrow 95000 = \text{annuity} \times \dfrac{0.795856}{0.05} $
$\Rightarrow 95000=\text{annuity} \times 15.917$
$ \therefore \text{annuity} = \dfrac{95000}{15.917} =5968$
Therefore, Rs. $5,968$ should be cut out of the profit at the end of each year to accumulate at compound rate of $5\%$ per annum, so that a new machine can be purchased after $12$ years.