Tag: business mathematics and statistics

Questions Related to business mathematics and statistics

The dollar amount of mortgage loan multiplied monthly payment of mortgage loan per dollar is used to calculate

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

    1. semiannually mortgage payment

    2. daily mortgage payment

    3. monthly mortgage payment

    4. annually mortgage payment

    Show answer
    Correct Option: C
    Explanation:

    $\Rightarrow$  Dollar amount of mortgage loan multiplied monthly payment of mortgage loan per dollar is used to calculate $monthly\,mortgage\,payment.$

    $\Rightarrow$  The most common mortgage terms are 15 years and 30 years. Interest rate Annual fixed interest rate for this mortgage. Monthly payment (PI) Monthly principal and interest payment (PI). 
    $\Rightarrow$  Monthly payment (PITI) Monthly payment including principal, interest, homeowners insurance and property taxes.

    Find the present value and amount of an ordinary annuity of 8 quarterly payments of Rs 500 each, the rate of interest being 8% p.a. compounded quarterly.

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

    1. $4375$

    2. $4275$

    3. $4175$

    4. $4475$

    Show answer
    Correct Option: B
    Explanation:

    $\Rightarrow$  Here, $A=Rs.500,\, n=8$ and $r=\dfrac{8}{100}\times \dfrac{1}{4}=0.02$

    $\Rightarrow$  $V=\dfrac{A}{r}\times [1-(1+r)^{-n}]$
    $\Rightarrow$  $V=\dfrac{500}{0.02}\times [1-(1.02)^{-8}]$
    $\Rightarrow$  Now, let $x=(1.02)^{-8}$
    $\Rightarrow$  $log\,x=-8\,log\,1.02=-8(0.0086)$
    $\Rightarrow$  $log\,x=-0.0688$
    $\Rightarrow$  $x=0.8535$
    $\Rightarrow$  $V=\dfrac{500}{0.02} \times [1-0.8535]=Rs.3662.50$
    $\Rightarrow$  Now, $M=\dfrac{A}{r}\times [(1+r)^n -1]$
    $\Rightarrow$  $\dfrac{500}{0.02}\times [(1.02)^8-1]$
    $\Rightarrow$  Let $x=(1.02)^8$
    $\Rightarrow$  $log\, x =8\, log\, (1.02)=0.0688$
    $\Rightarrow$  $x=1.171$
    $\therefore$   $M=\dfrac{500}{0.02}\times [1.171-1]=Rs.4275$
    $\therefore$    The present value of annuity is $Rs.3662.50$ and amount is $Rs.4275$

    Payments if made at end of each period such as end of year is classified as __________________.

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

    1. ordinary annuity

    2. deferred annuity

    3. annuity due

    4. Both A and B

    Show answer
    Correct Option: D
    Explanation:

    Payments if made at end of each period such as end of year is classified as ordinary annuity and deferred annuity.

    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

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