Tag: financial mathematics

Questions Related to financial mathematics

Which of the following is  true about annuity?

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

  1. It is sequence of equal instalments.

  2. It is sequence of unequal instalments.

  3. It is paid at unequal interval of time.

  4. None of these

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  The true statement about annuity is $It\,\,is\,\,sequence\,\,of\,\,equal\,\,instalments.$

$\Rightarrow$  Series of payments at fixed intervals, guaranteed for a fixed number of years or the lifetime of one or more individuals.
$\Rightarrow$  Annuities are insurance products that provide long-term income through a stream of future payments. 

$\Rightarrow$  While investment annuities save money for retirement and beneficiaries, structured settlement annuities stem from personal-injury legal cases, wrongful-death claims or lottery payouts. When unexpected circumstances arise and require immediate funds, you can sell these payments for a lump sum of cash.

A man borrowed some money and returned it in 3 equal quarterly installments of Rs 4630.50 each. What sum did he borrow if the rate of interest was 20% p.a. compounded quarterly? 

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

  1. $12891.50$

  2. $12610$

  3. $13861.50$

  4. $13801.50$

Show answer
Correct Option: B
Explanation:

$\Rightarrow$   Here, $R=\left(20\times \dfrac{1}{4}\right)\%=5\%$


$\Rightarrow$   Sum borrowed = $\dfrac{4630.50}{\left(1+\dfrac{5}{100}\right)^1}+\dfrac{4630.50}{\left(1+\dfrac{5}{100}\right)^2}+\dfrac{4630.50}{\left(1+\dfrac{5}{100}\right)^3}$


$\Rightarrow$   Sum borrowed = $4630.50\times \left[\dfrac{100}{105}+\dfrac{(100)^2}{(105)^2}+\dfrac{(100)^3}{(105)^3}\right]$

$\Rightarrow$   Sum borrowed = $4630.50\times \dfrac{20}{21}\times \left[\dfrac{41}{21}+(\dfrac {20}{21})^2\right]$

$\Rightarrow$   Sum borrowed = $4410\times \dfrac{1}{21}\times \dfrac{1261}{21}=Rs.12610$

Three types of annuities are

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

  1. Annuity certain

  2. Annuity contingent

  3. Annuity perpetual

  4. All of the above

Show answer
Correct Option: D
Explanation:

$\Rightarrow$  Three types of annuities are : 

$(1)$ $Annuity\,\, certain$ - Annuity that, as a minimum, guarantees a fixed number of payments. It continues over the life of the annuitant, even if he or she lives beyond the number of payments specified in the annuity contract. In case the annuitant dies before exhausting the payments, a named beneficiary continues to receive the remaining number. Also called life annuity certain or life annuity certain and continuous.
$(2)$ $Annuity\,\, contingent$ - An annuity arrangement in which the beneficiary does not begin receiving payments until a specified event occurs. A contingent annuity may be set up to begin sending payments to a beneficiary upon the death of another individual who wishes to ensure financial stability for the beneficiary, or upon retirement or disablement of the beneficiary.
$(3)$ $Annuity\,\, perpetual$ - Annuity derived from an asset (such as an income generating security) where the life span of the annuitant (security holder or his or her beneficiary) is of no consequence.

A man borrowed some money and returned it in 3 equal quarterly installments of Rs 4630.50 each.  Find also the interest charged.

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

  1. $1281.50$

  2. $1291.50$

  3. $1181.50$

  4. $1381.50$

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  Here, we have $A=Rs.4630.50,\, n=3$ and $r=\dfrac{20}{100}\times \dfrac{1}{4}=0.05$

$\Rightarrow$  $V=\dfrac{A}{r}\times [1-(1+r)^{-n}]$

$\Rightarrow$  $V=\dfrac{4630.50}{0.05} \times [1-(1.05)^{-3}]$

$\Rightarrow$  $V=Rs.12610$
$\Rightarrow$  Now, Total money repaid = $3\times Rs.4630.50=Rs.13891.50$
$\therefore$   Interest paid = $Rs.13891.50-Rs.12610=Rs.1281.50$

An 8-year annuity due has a present value of $ $1,000$.  If the interest rate is $5$ percent,  the amount of each annuity payment is closest to which of the following? 

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

  1. $ $154.73$

  2. $ $147.36$

  3. $ $109.39$

  4. $ $104.72$

  5. $ $99.74$

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  We have, $V=\$1000,\, n=8$ and $r=5\%=0.05$.

$\Rightarrow$  We know, $V=\dfrac{A}{r}\times [1-(1+r)^{-n}]$
$\Rightarrow$  $A=\dfrac{V\times r}{[1-(1+r)^{-n}]}$

$\Rightarrow$  $A=\dfrac{1000\times 0.05}{[1-(1.05)^{-8}]}$

$\Rightarrow$  $A=\$154.73$

Susan purchased a new refrigerator priced at $ $675$. She made a down payment of $15$% of the price. Find the amount of the down payment.

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

  1. $\$100.25$

  2. $\$101.25$

  3. $\$101.75$

  4. $\$105$

Show answer
Correct Option: B
Explanation:

Susan purchased refrigerator for Rs.$675$.

Down payment made by her is $15\%$.
Therefore, $15$% of given number $675$ is $\dfrac{15}{100} \times 675 = $ 101.25$.
So, she paid Rs.$101.25$ as the down payment.

For son's education , a man sets aside Rs $4000$ at the end of every year for $8$ years . If the rate of interest is 15 % per annum C.I. , what is the value of his sinking fund.

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

  1. 54909.33

  2. 53909.33

  3. 52909.33

  4. 51909.33

Show answer
Correct Option: A
Explanation:

Using formula of sinking fund

$M=\cfrac{A}{r}[(1+r)^n-1]$
$\implies \cfrac{4000}{0.15}[(1+0.15)^8-1]=54907.2763\approx 54.909.83$

A man borrows Rs. $ 30000$ at $12 \%$ per annum compound interest from a bank and promises to pay off the loans in $20$ annual instalments beginning at the end of the first year . What is the annual payment necessary?

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

  1. $4016.76$

  2. $3013.54$

  3. $4065.24$

  4. $1034.54$

Show answer
Correct Option: A
Explanation:

Here, $V=Rs.30000,\,\,r=12\%=0.12$ and $n=20$.

We know $V=\dfrac{A}{r}[1-(1+r)^{-n}]$
Thus $30000=\dfrac{A}{0.12}[1-(1+0.12)^{-20}]$
$\Rightarrow$   $A=\dfrac{30000\times 0.12}{[1-(1+0.12)^{-20}]}$
$\Rightarrow$  $A=\dfrac{3600}{[1-(1.12)^{-20}]}$
$\Rightarrow$  $A=$ Rs. $4016.76$

A person borrowed some money and returned it in 3 equal quarterly instalments of Rs 4630.50 each. What sum (approximately) did he borrow if the rate of interest was 20 % per annum.compounded quarterly?

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

  1. 12613.48

  2. 10613.48

  3. 11613.48

  4. None of these

Show answer
Correct Option: D
Explanation:

$\Rightarrow$  We have $A=Rs.4630.50,\, n=3$ and rate of interest is $20\%$ which is compounded quarterly. So, $r=5\%$

$\Rightarrow$ We have to find sum borrow i.e. $V$
$\Rightarrow$  $V=\dfrac{A}{r}[1-(1+r)^{-n}]$

$\Rightarrow$  $V=\dfrac{4630.50}{0.05}[1-(1+0.05)^{-3}]$  

$\Rightarrow$  $V=\dfrac{463050}{5}[1-(1.05)^{-3}]$

$\Rightarrow$  $V=92610\times \dfrac{1261}{9261}$

$\Rightarrow$  $V=10\times 1261$

$\Rightarrow$  $V=Rs.12610$

A person takes a loan on compound interest and returns it in $2$ equal installments . If the rate of interest is $10$% per annum and the yearly installment is Rs $1682$. Find the interest charged with second installment.

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

  1. Rs.$613.2$

  2. Rs.$603.2$

  3. Rs.$513.2$

  4. Rs.$713.2$

Show answer
Correct Option: A
Explanation:

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

$\implies \cfrac{1682}{(1+\cfrac{10}{100})^1}$$+\cfrac{1682}{(1+\cfrac{10}{100})^2}$
$ \implies 1592.09+1390.08=2919.17\approx 2920$
$\implies A _2=2920(1+\cfrac{10}{100})^2=3533.2$
$CI=A _1-P=3533.2-2920=Rs.613.2$