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

Find the future value of an annuity of Rs. $500$ made annually for $7$ years at interest rate of $14\%$ compounded annually. Given that $(1.14)^7=2.5023$.

  1. $5,563.25$
  2. $5,365.35$
  3. $5,365.53$
  4. $5,356.35$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Annual payment A$=$Rs. $500$
$n=7$
$i=14\%=0.14$
$A(7, 0.14)=500\left[\displaystyle\frac{(1+1.014)^7-1}{0.14}\right]$
$=$Rs. $5365.35$

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

A limited company intends to create a depreciation fund to replace at the end of the 25th year assets costing Rs 100000.Calculate the amount (approximately) to be retained out of profits every year if the interest rate is 3%.

  1. 2755

  2. 3245

  3. 5431

  4. 1200
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This is a sinking fund calculation to reach a future value of 100,000 in 25 years at 3% interest. Using the formula PMT = FV * i / ((1+i)^n - 1), the result is approximately 2755.

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

Veena is allotted an LIG flat for which she has to make an immediate payment of $ $ 100,000$ and $10$ semi-annual payments of $ $50,000$ each, the first being made at the end of $3$ years. If money is worth $10\%$ per annum compounded half-yearly, find the cash price (in $) of the flat.

  1. $302,509$
  2. $400,509$
  3. $302,009$
  4. $402,509$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\Rightarrow$  Cash price = Down payment + Present value of annuity    --- ( 1 )

$\Rightarrow$  Here, down payment is $\$100,000$, while we have annuity 10 terms i.e. $n$, deferred for $2\dfrac{1}{2}$ years i.e. 5 terms.
$\Rightarrow$  Each installment, $A = \$50,000$
$\Rightarrow$  Rate of interest, $r=10\%\, p.a.$ compounded half-yearly = 0.05
$\Rightarrow$  $m=5$ and $m+n=15$
$\therefore$  Present value of annuity,
$\Rightarrow$  $V=\dfrac{A}{r}\times [\dfrac{1}{(1+r)^m}-\dfrac{1}{(1+r)^{m+n}}]$
$\Rightarrow$  $\dfrac{50,000}{0.05}\times [(1.05)^{-5}-(1.05)^{-15}]$
$\Rightarrow$  $\$ 302509.07$
Now, from ( 1 )
$\Rightarrow$  Cash price of the flat = $\$100,000+\$302,509=\$402,509$

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

An investor deposits Rs 1000 in a saving institution. Each payment is made at the end of year.If the payment deposited earns 12% interest compounded annually how much amount(approximately) will he receive at the end of 10 years.

  1. $1234$
  2. $2345$
  3. $17548$
  4. $4567$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$P=Rs.1000,r=12\%,t=10$

$A=\sum _{ n=1 }^{ 9 }{ P{ (1+\cfrac { r }{ 100 } ) }^{ n }+P } \ =\sum _{ n=1 }^{ 9 }{ 1000{ (1+\cfrac { 12 }{ 100 } ) }^{ n }+1000 } \ =3000\times 16.548\ =Rs.17548$

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

A machine costing Rs $2$ lacs has effective life of $7$ years and its scrap value is Rs $30000$. What amount (in Rs) should the company put into a sinking fund earning $5\%$ per annum so that it can replace the machine after its useful life? Assume that a new machine will cost Rs $3$ lacs after $7$ years.

  1. $30161.35$
  2. $33101.35$
  3. $33161.35$
  4. $33111.35$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\Rightarrow$  Cost of new machine is $Rs.3\,lacs$

$\Rightarrow$  Scrap value of old machine is $Rs.30000$
$\Rightarrow$  Hence, money required for new machine after 7 years  = $Rs.300000-Rs.30000=Rs.270000$.
$\Rightarrow$  If A is the annual deposit into sinking fund, then we have
$\Rightarrow$  Amount of annuity, $M=Rs.270000$
$\Rightarrow$  Number of periods = $7\, years$
$\Rightarrow$  $r=5\%=0.05$
$\therefore$   $M=\dfrac{A}{r}\times [(1+r)^n - 1]$
$\Rightarrow$  $270000=\dfrac{A}{0.05}\times[(1.05)^7-1]$

$\Rightarrow$  $A=\dfrac{270000\times 0.05}{(1.05)^7 - 1}$

$\therefore$   $A=33161.35$ Rs

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

A man decides to deposit Rs 3000 at the end of each year in a bank which pays 3% p.a compound interest . If the instalments are allowed to accumulate , what will be the total accumulation at the end of 15 years.

  1. Rs.$57450$
  2. Rs.$67780$
  3. Rs.$67050$
  4. Rs.$98450$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$P=Rs.3000,r=3\%,t=15$
$A=\sum _{ n=1 }^{ 15 }{ P{ (1+\cfrac { r }{ 100 } ) }^{ n } } \ =\sum _{ n=1 }^{ 15 }{ 3000{ (1+\cfrac { 3 }{ 100 } ) }^{ n } } \ =3000\times 19.15\ =Rs.57450$

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

Find the present value of a sequence of annual payments of Rs 25000 each , the first being made at the  end of 5th year and the last being paid at the end of 12th year, if money is worth 6%.

  1. $122875$
  2. $102875$
  3. $132875$

  4. None of the above
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

We know that formula of present value is 

$V=\cfrac{A}{r}[\cfrac{1}{(1+\cfrac{R}{100})^m}$$-\cfrac{1}{(1+\cfrac{R}{100})^{m+n}}]$
We have  annuity of $8$ terms $(n)$
For $4$ terms $(m)\implies m=4\ \implies m+n=12$
$V=\cfrac{25000}{0.06}[\cfrac{1}{(1+\cfrac{6}{100})^4}$$-\cfrac{1}{(1+\cfrac{6}{100})^{12}}]=Rs.122968.45$

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

A company borrows Rs 10000 on condition to repay it with compound interest at $5$% p.a . by annual instalments at Rs 1000 each. In how many years will the debt be paid off?

  1. 14.2

  2. 21.7

  3. 12.67

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$\Rightarrow$  Company borrow Rs.10000 i.e. $pv=Rs.10000$ and $I=5\%=0.05$. $A$ is also given which is $Rs.1000$
$\Rightarrow$  Present value of annuity regular
$\Rightarrow$  $pv=A\times [\dfrac{(1+I)^n-1}{I\times (1+I)^n}]$

$\Rightarrow$  $10000=1000\times [\dfrac{(1+0.05)^n-1}{0.05\times (1+0.05)^n}]$

$\Rightarrow$  $(1.05)^n-0.5\times (0.5)^n=1$
$\Rightarrow$  $(1.05)^n=2$
Taking log both sides
$\Rightarrow$  $n=\dfrac{log\,2}{log\, 1.05}$
$\therefore$   $n=14.2\, years$
Multiple choice business mathematics and statistics insurance and annuity amount of an annuity annuities financial mathematics

Mr Dev purchased a car paying Rs $90,000$ and promising to pay Rs 5000 every 3 months for the next 10 years. The interest is $6$% p.a. compounded quarterly. If at the end of 5th year , he wants to finish his liability by a single payment , how much should he pay?

  1. 90100

  2. 80100

  3. 34504

  4. 54345
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\Rightarrow$  We have $A=Rs.5000,\,I=\dfrac{6}{100}\times \dfrac{1}{4}=0.015$ and $n = 20$

$\Rightarrow$  If at the end of 5th year, i.e., at the time of 20th payment, he wants to finish off the liability, then lump sum payment required is,
$\Rightarrow$  $5000$ + Present value of the remaining 20 installments.
$\Rightarrow$  $5000+V$
$\Rightarrow$  $5000$ + $\dfrac{A}{I}[1-(1+I)^{-n}]$

$\Rightarrow$  $5000+\dfrac{5000}{0.015}[1-(0.015)^{-20}]$    ---- ( 1 )

$\Rightarrow$  Let $x=(1.015)^{-20}$
$\Rightarrow$  $log\,x=-20\,log\,(1.015)$
$\Rightarrow$  $log\,x=-20(0.0064)=-0.128=\bar{1}.8720$
$\Rightarrow$  $x=antilog\,(\bar{1}.8720)=0.7447$
Substitute value of $x$ in ( 1 ),
$\Rightarrow$  $5000+\dfrac{5000}{0.015}(1-0.7447)$

$\Rightarrow$  $5000+\dfrac{5000}{0.015}\times 0.2553$

$\Rightarrow$  $Rs.90100$

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

Amit buys a house for Rs 500000. The contract is that amit will pay Rs 200000 immediately and the balance in 15 equal instalments with 15 % p.a compound interest . How much has he to pay annually (approximately)?

  1. Rs$51,305$
  2. Rs$54,005$
  3. Rs$51,843$
  4. Rs$91,305$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Present value $=Rs.50,000Rs.20,000=Rs.30,000$

$P=\cfrac{A}{(1+\cfrac{R}{100})^n}$

$\implies 30,000=\cfrac{A}{1+(\cfrac{15}{100})}$$+\cfrac{A}{1+(\cfrac{15}{100})^2}+.....$4
$ \implies A\times 5,847 $

$\implies A=Rs51,305$