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Insurance and annuity - class-XI

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Annuity, where the payments start after specified no. of periods, is known as

  1. Immediate Annuity

  2. Deferred annuity

  3. Contingent annuity

  4. Perpetual annuity


Correct Option: B
Explanation:

An annuity which begins payments only after a period is a deferred annuity
Annuity, where the payments start after specified no. of periods, is known as Deferred annuity.

Which of the following is an example of annuity contingent ?

  1. Car Loan

  2. House Loan

  3. Daughter's Marriage

  4. All of the above


Correct Option: C
Explanation:
$\Rightarrow$  $Daughter's\,\,Marriage$ is an example of annuity contingent. 
$\Rightarrow$  Annuity contingent is an annuity arrangement in which the beneficiary does not begin receiving payments until a specified event occurs.
 $\Rightarrow$  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.
$\Rightarrow$  Car loan and House loan is not an example of annuity contingent, it's an example of annuity certain.

Security payment type in which payments are made at equal intervals of time and every payment amount is same is classified as _______________.

  1. fixed interval investment

  2. fixed payment investment

  3. annuity

  4. lump sum amount


Correct Option: C
Explanation:

An annuity is a series of equal payments in equal time periods. Usually, the time period is $1$ year, which is why it is called an annuity, but the time period can be shorter, or even longer. These equal payments are called the periodic rent. The amount of the annuity is the sum of all payments.
such security payment type in which payments are made at equal intervals of time and every payment amount is same is classified as annuity.

If payment of security is paid as $ $100$ at end of year for three years, it is an example of

  1. fixed payment investment

  2. lump sum amount

  3. fixed interval investment

  4. annuity


Correct Option: D
Explanation:

The amount of the annuity is the sum of all payments.
such security payment type in which payments are made at equal intervals of time and every payment amount is same is classified as annuity.
If payment of security is paid as $100 at end of year for three years, it is an example of annuity.

What is true about Annuity Due ?

  1. It is an annuity in which payments are made at the end of each payment period.

  2. It is an annuity in which payments are made at the beginning of each payment period.

  3. It is an annuity in which payments are made in the middle of each payment period.

  4. None of the above


Correct Option: B
Explanation:

$\Rightarrow$   True statement about Annuity Due is,

$-It\,is\,an\,annuity\,in\,which\,payments\,are\,made\,at\,the\,beginning\,of\,each\,payment\,period.$
$\Rightarrow$  Annuity due is an annuity whose payment is to be made immediately at the beginning of each period. 
$\Rightarrow$  A common example of an annuity due payment is rent, as the payment is often required upon the start of a new month as opposed to being collected after the benefit of rent has been received for an entire month.
$\Rightarrow$  All payments are in the same amount.
$\Rightarrow$  All payments are made at the same intervals of time

Which of the following is true about Annuity Contingent ?

  1. It is made till the happening of an event.

  2. It is made for fixed number of intervals of time.

  3. Loans for home comes under it

  4. All of the above


Correct Option: A
Explanation:

$\Rightarrow$  True statement about Annuity contingent is $It\,is\,made\,till\,the\,happening\,of\,an\,event.$

$\Rightarrow$  An annuity arrangement in which the beneficiary does not begin receiving payments until a specified event occurs. 
$\Rightarrow$  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.

Which of the following is not an example of annuity contingent ?

  1. Daughter's Marriage Loan

  2. Life Insurance Plans

  3. Mortgage

  4. All of the above


Correct Option: C
Explanation:
$\Rightarrow$  $Mortgage$ is not an example of annuity contingent.
$\Rightarrow$   Annuity contingent is an annuity arrangement in which the beneficiary does not begin receiving payments until a specified event occurs.
 $\Rightarrow$  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.
$\Rightarrow$  Daughter's Marriage loan and Life insurance plans are examples of annuity contingent.

The Future amount of annuity, $(M)$, can be found by

  1. $ M=\dfrac{A}{r} \times \left[(1+r)^{n}+1\right] $

  2. $ M=\dfrac{r}{A} \times \left[(1+r)^{n}-1\right] $

  3. $ M=\dfrac{A}{r} \times \left[(1+r)^{n}-1\right] $

  4. $ M=\dfrac{r}{A} \times \left[(1+r)^{n}+1\right] $


Correct Option: C
Explanation:

The future amount of annuity (M) can be founded by the formula

$M=\dfrac { A }{ r } \times \left[ { \left( 1+r \right)  }^{ n }-1 \right] $
Where $A$ is amount and $r$ is rate and $n$ is duration

An annuity left unpaid for a certain number of years is called ________ for that number of years.

  1. Deferred Annuity

  2. Uniform Annuity

  3. Forborne Annuity

  4. Immediate Annuity


Correct Option: C
Explanation:

An annuity left unpaid for a certain number of years is called forborne annuity for that number of years.

If the periodic payments are made at the end of each period, the annuity is called:

  1. Annuity due

  2. An immediate annuity

  3. Ordinary annuity

  4. (B) or (C)


Correct Option: D
Explanation:

If the periodic payments are made at the end of each period, the annuity is called an immediate annuity or ordinary annuity.

If the period payments start only after a certain specified period it is called.

  1. Contingent Annuity

  2. Deferred Annuity

  3. Perpetual Annuity

  4. Annuity certain


Correct Option: B
Explanation:

If the period payments start only after a certain specified period it is called deferred annuity.

An annuity whose payments continue till the happening of an event, the date of which cannot be foretold is called.

  1. Contingent Annuity

  2. Deferred Annuity

  3. Perpetual Annuity

  4. Annuity certain


Correct Option: A
Explanation:

An annuity whose payments continue till the happening of an event, the date of which cannot be foretold is called contingent annuity.

Find the amount of an annuity of Rs. 400 per quarter payable for 6 years at 8% p.a.
[Given : $(1.02)^{24} = 1.608$]-

  1. Rs. 11,260

  2. Rs. 12,160

  3. Rs. 13,200

  4. None.


Correct Option: B
Explanation:

Formula for calculating the amount of an annuity,


$F=R \dfrac{\left ( 1+\dfrac{r}{m} \right )^{m \times n} -1}{\dfrac{r}{m}}$

$F=400 \dfrac{\left ( 1+\dfrac{8/100}{4} \right )^{4 \times 6} -1}{\dfrac{8/100}{4}}$

$F=400 \dfrac{\left ( 1+\dfrac{8}{100} \times \dfrac{1}{4} \right )^{24} -1}{\dfrac{8}{100} \times \dfrac{1}{4} }$

$F=400 \dfrac{(1.02)^{24} -1}{\dfrac{1}{50} }$

$F = 12,160$


Find the least number of years for which an annuity of Rs. 1,000 must run in order that its amount exceed Rs. 16,000 at 5% p.a. compounded monthly.
[Given : Log 18 = 1.2553, log 105 = 2.8212]

  1. 12 years

  2. 11 years

  3. 13 years

  4. None.


Correct Option: C
Explanation:
$F=R \dfrac{\left ( 1+\dfrac{r}{m} \right )^{m \times n} -1}{\dfrac{r}{m}}$

$16000=1000 \dfrac{\left ( 1+\dfrac{5}{100} \right )^n -1}{\dfrac{5}{100}}$

$16=1 \dfrac{\left ( 1+\dfrac{5}{100}  \right )^n -1}{\dfrac{5}{100}  }$

$0.8=(1.05)^n-1$

$1.8=(1.05)^n$

Applying log on both sides, we get,

$\log 1.8 = n \log 1.05$

$\Rightarrow n=13$

Least number of years = $13$ years

If the periodic payments are all equal, the annuity is called level of.

  1. Deferred Annuity

  2. Uniform Annuity

  3. Forborne Annuity

  4. Immediate Annuity


Correct Option: B
Explanation:

If the periodic payments are all equal, the annuity is called level of uniform annuity.

Annuity payable for a fixed number of intervals is called.

  1. Contingent Annuity

  2. Deferred Annuity

  3. Perpetual Annuity

  4. Annuity certain


Correct Option: D
Explanation:

Annuity payable for a fixed number of intervals is called annuity certain.

An annuity which continues forever(infinite number of years) is called.

  1. Contingent Annuity

  2. Deferred Annuity

  3. Perpetual Annuity

  4. Annuity certain


Correct Option: C
Explanation:

Perpetual annuity is an annuity which continues forever i.e. infinite number of year.

Present value of annuity, $(V)$, can be found by

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

  2. $ V=\dfrac{A}{r} \times \left[1-(1+r)^{(-n)}\right] $

  3. $ V=\dfrac{A}{r} \times \left[1-(1+r)^{(n)}\right] $

  4. $ V=\dfrac{r}{A} \times \left[1-(1+r)^{(n)}\right] $


Correct Option: B
Explanation:

The present value annuity factor is used for simplifying the process of calculating the present value of an annuity. A table is used to find the present value per dollar of cash flows based on the number of periods and rate per period. Once the value per dollar of cash flows is found, the actual periodic cash flows can be multiplied by the per dollar amount to find the present value of the annuity.
$v= \frac{A}{r} \times [1-(1+r)^{(-n)}]$
where , A =annuity , r =rate per period , n= number of periods

Annuity where payments are made at the end of each payment period, i.e. 1st payment is made at the end of the 1st payment interval, and so on, is known as 

  1. Perpetual annuity

  2. Contingent annuity

  3. Ordinary annuity

  4. Immediate annuity


Correct Option: C,D
Explanation:

Answer is Ordinary or Immediate Annuity.

  1. An ordinary annuity or immediate annuity is where payments are made at the end of each payment period, i.e. 1st payment is made at the end of the 1st payment interval, and so on. Examples are repayment of car loans, house mortgage etc.
  2. A contingent annuity is one where the term depends upon some event whose occurrence is not fixed. An example is periodic payments of life insurance premiums which stop when the person dies.
  3. A perpetual annuity is an annuity whose term does not end, i.e. it extends till infinity. Thus there is no last payment; they go on forever. An example is freehold property, where you can earn rent in perpetuity.

A house is sold for $ Rs \ 30,000$ cash or $ Rs\ 17, 500$ cash down payment and instalments of $ Rs \ 1, 600$ per month for eight months. Determine the approximate rate of interest for instalment.

  1. $6.5 \%$

  2. $6 .8 \%$

  3. $ 6. 2 \%$

  4. None of these

  5. $6.3 \%$


Correct Option: A
Explanation:

$\Rightarrow$  Cash price = $Rs.30,000$

$\Rightarrow$  Cash down payment = $Rs. 17500$
$\Rightarrow$  Total amount paid in 8 monthly installments = $Rs.(1600\times 8)=Rs.12800$
$\Rightarrow$  Total amount paid under installment paln = $Rs.17500+Rs.12800=Rs.30300$
$\Rightarrow$  Interest charged = $Rs.30300-Rs.30000=Rs.300$
$\Rightarrow$  Principal for 1st month = $Rs.30000-Rs.17500=Rs.12500$
$\Rightarrow$  Principal for 2nd month = $Rs.12500-Rs.1600=Rs.10900$
$\Rightarrow$  Principal for 3rd month = $Rs.10900-Rs.1600=Rs.9300$
$\Rightarrow$  Principal for 4th month = $Rs.9300-Rs.1600=Rs.7700$
$\Rightarrow$  Principal for 5th month = $Rs.7700-Rs.1600=Rs.6100$
$\Rightarrow$  Principal for 6th month = $Rs.6100-Rs.1600=Rs.4500$
$\Rightarrow$  Principal for 7th month = $Rs.4500-Rs.1600=Rs.2900$
$\Rightarrow$  Principal for 8th month = $Rs.2900-Rs.1600=Rs.1300$
$\Rightarrow$  Total principal = $Rs.55200$
$\Rightarrow$  The last installment of Rs.1600 includes Rs.1300 plus Rs.300 interest.
$\Rightarrow$  Time = 1 month = $\dfrac{1}{12}$year, Interest = Rs.$300$
$\Rightarrow$  Interest = $\dfrac{P\times T\times R}{100}$
$\Rightarrow$  $R=\dfrac{I\times 100}{P\times T}$
$\Rightarrow$  $R=\dfrac{300\times 100}{55200\times \dfrac{1}{12}}=\dfrac{300\times 100\times 12}{55200}=\dfrac{150}{23}=6.5$
$\therefore$   $Rate =6.5\%$

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

  1. Rs. $3660.20$

  2. Rs. $3662.50$

  3. Rs. $4275$

  4. Rs. $3660$


Correct Option: B
Explanation:

Given, $A=$ Rs $500$, $n= 8$

Also, $ r= \dfrac{8}{100} \times \dfrac{1}{4} =0.02 $

$ \therefore V= \dfrac{A}{r} \times \left[1-(1+r)^{(-n)}\right]=\dfrac{500}{0.02} \times \left[1-(1.02)^{(-8)}\right] $

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 $

Thus, the present value of annuity is Rs. $3662.50$.

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?

  1. Rs. $12000$

  2. Rs. $12100$

  3. Rs. $12160$

  4. Rs. $13000$


Correct Option: C
Explanation:

Here, we have to find present value $(V)$ of an ordinary annuity certain.

Given, $A=$ Rs. $4630.50$, $n= 3$

Also $ r= \dfrac{20}{100} \times \dfrac{1}{4}=0.05 $

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

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

$=$ Rs. $12610 $

Thus, the sum borrowed was Rs. $12160$.

Find the Amount of an ordinary annuity of $8$ quarterly payments of Rs. $500$ each, the rate of interest being $8\%$ p.a. compounded quarterly.

  1. Rs. $3660.20$

  2. Rs. $3662.50$

  3. Rs. $4275$

  4. Rs. $3670$


Correct Option: C
Explanation:

Here, $A=$ Rs. $500$, $n= 8$

Also $ r= \dfrac{8}{100} \times \dfrac{1}{4} =0.02 $

$ M=\dfrac{A}{r} \times [(1+r)^{(n)}-1]=\dfrac{500}{0.02} \times [(1.02)^{8}-1] $

Let $ x= (1.02)^(8) $

$\Rightarrow \log{x}=8\log{1.02}=0.0688 $

$ \Rightarrow x= 1.171 $

$ \Rightarrow  M= \dfrac{500}{0.02} \times [1.171-1] =$ Rs $4275 $

Thus, the amount is Rs. $4275$.

The present value of an amount ____ its future value.

  1. Greater than

  2. Less than

  3. Equal to

  4. Not equal to


Correct Option: B

A man borrows Rs $37500$ and agrees to repay in semi-annual installments of Rs $2250$ each, the first due in $6$ months. How many payments must he make if rate of interest is $6\%$ compounded semi-annually?

  1. $23$

  2. $24$

  3. $25$

  4. $22$


Correct Option: B
Explanation:

$\Rightarrow$  Here, we have to find the number of payments, $n$.

$\Rightarrow$  $V=Rs.37500$  and $A=Rs.2250$
$\Rightarrow$  Rate of interest compounded semi-annually = $\dfrac{1}{2}\times 6\% = \dfrac{1}{2}\times \dfrac{6}{100}=0.03$
$\Rightarrow$  $V=\dfrac{A}{r}\times [1-(1+r)^{-n}]$

$\Rightarrow$  $37500=\dfrac{2250}{0.03}\times [1-(1.03)^{(-n)}]$

$\Rightarrow$  $1-(1.03)^{-n}=\dfrac{37500\times 0.03}{2250}$

$\Rightarrow$  $(1.03)^{(-n)}=0.5$

$\Rightarrow$  $-n\, log(1.03)=log(0.5)$

$\Rightarrow$  $-n(0.0128)=-0.3010$

$\Rightarrow$  $n=\dfrac{-0.3010}{-0.0128}$

$\therefore$    $n=23.51 \approx 24$

Find the Present value of an annuity due of Rs $500$ per quarter for $8$ years and $9$ months at $6\%$ compounded quarterly.

  1. Rs $27032.30$

  2. Rs $23137.98$

  3. Rs $13740.86$

  4. Rs $24017.25$


Correct Option: C
Explanation:

Here, rate of interest, r =$1.5$% per interest period =$0.015$
Number of interest periods, $n = 4 \times 8 +3 = 35$
Each installment, $A=Rs $ $500$
Present value of annuity due,
$v = \dfrac{A}{r} \times (1+r) \times [1-(1+r)^{-n}]$

= $\dfrac{500}{0.015} \times 1.015 \times [1-(1.015)^{-35}]$
= $13740.86$

A man borrowed some money and returned it in $3$ equal quarterly installments of Rs $4630.50$ each. Find the interest charged (in Rs) on the sum he borrowed, if the rate of interest was $20\%$ p.a. compounded quarterly?

  1. $1731.50$

  2. $1200$

  3. $1300$

  4. $1251.80$


Correct Option: A
Explanation:

Here, we have to find present value $(V)$ of an ordinary annuity certain.

Given, $A=$ Rs $4630.50$, $n= 3$

Also $ r= \dfrac{20}{100} \times \dfrac{1}{4}=0.05 $

$ \therefore V=\dfrac{A}{r} \times [1-(1+r)^{(-n)}] =\dfrac{4630.50}{0.05} \times [1-(1.05)^{(-3)}]=$ Rs. $ \: 12610 $

Thus, the sum borrowed was Rs. $12160$

Now, total money repaid $ = 3 \times 4630.50 =$ Rs. $ \: 13891.50 $

Therefore, interest paid $ =$ Rs, $ \: 13891.50$ $-$  Rs. $ \: 12160 =$ Rs. $ \: 1731.50 $.

Find the amount of an annuity due of Rs $500$ per quarter for $8$ years and $9$ months at $6\%$ compounded quarterly.

  1. Rs $27452.30$

  2. Rs $23137.98$

  3. Rs $13740.86$

  4. Rs $24671.30$


Correct Option: B
Explanation:

Number of interest periods, $n = 4 \times 8 +3 = 35$
Each installment, A=Rs $500$
Present value of annuity due,
$v = \frac{A}{r} \times (1+r) \times [1-(1+r)^{-n}]$
= $\frac{500}{0.015} \times 1.015 \times [1-(1.015)^{-35}]$
= $13740.86$
$v = \frac{A}{r} \times (1+r) \times [1-(1+r)^{-n}-1]$
= $\frac{500}{0.015} \times 1.015 \times [1-(1.015)^{-35}-1]$
= $23137.98$

Three equal instalments each of $Rs 200$ were paid at the end of the year for the sum borrowed at $20 \%$ interest compounded annually. Find the sum. 

  1. $ 600$

  2. $421.3$

  3. $ 400$

  4. $ 431.1$


Correct Option: B
Explanation:
Each Installment = $\dfrac{P{\cdot}r}{100[1-{\{ \dfrac{100}{100+r} \}}^n]}$
Here Installement$ = 200$
Rate of interest $(r) = 20%$
Number of years$ (n) = 3$
Solving the above equation gives $P = 421.3$

Process of loan repayment by installment payments is classified as

  1. appreciation of loan

  2. amortizing a loan

  3. depreciation a loan

  4. appreciation of investment


Correct Option: B
Explanation:

$\Rightarrow$   Process of loan repayment by installment payment is classified as $amortizing\,a\,loan.$

$\Rightarrow$   All repayments of interest-bearing debts by a series of payments, usually in size, made at equal intervals of time is called an amortization. Mortgages and many consumer loans are repaid by this method.
$\Rightarrow$  An amortized loan is a loan with scheduled periodic payments that consist of both principal and interest. An amortized loan payment pays the relevant interest expense for the period before any principal is paid and reduced. 
$\Rightarrow$   This is opposed to loans with interest-only payment features, balloon payment features and even negatively amortizing payment features.

Find the present value (in Rs) of a sequence of annual payments of Rs $10000$ each, the first being made at the end of $5^{th}$ year and the last being made at the end of $12^{th}$ year, if money is worth $6\%$.

  1. $40187.38$

  2. $49087.38$

  3. $40107.38$

  4. $49187.38$


Correct Option: D
Explanation:

Here, we have a deferred annuity of $8$ terms(n), deferred for $4$ terms.Each installment, A=Rs $10000$
Rate of interest, $r=$ $6\%$= $0.06$
m= $4$, $m + n=$ $12$
Using the formula, present value
$V = \dfrac{A}{r}$ $\times \dfrac{1}{(1+r)^m}$ - $\frac {1}{(1+r)^{m+n}}$
$V = \dfrac{10000}{0.06} \times [(1.06)^{-4}-(1.06)^{-12}]$$ =49187.38$

Find SI if, Amount =  $Rs \ 1120$ , Rate = $2\dfrac{2}{5}\%$ per year , Time = $5$ years 
  1. $1200$

  2. $201$

  3. $120$

  4. $134.4$


Correct Option: D
Explanation:

$ SI= prt$

$p$ is the principal amount on which interest is to be calculated
$r$ is the rate of interest at which the loan is taken
$t$ is the time period for which the simple interest is to be calculated
From the question we know that,
$p=Rs 1120, $ $r =2.4$ percent and $t=5$ years
$\Rightarrow SI = (1120)(2.4)(0.01)(5)= 134.4$

A sum of Rs $2500$ is invested at a rate of $5 \%$ per annum for a term of $5$ years. Find the simple interest received at the end of the term.

  1. $1250$

  2. $625$

  3. $1200$

  4. $1500$


Correct Option: B
Explanation:

$\Rightarrow$  Here, $P=Rs.2500,\, R=5\%,\, T=5\,years$.

$\Rightarrow$  $Simple\, Interest=\dfrac{P\times R\times T}{100}$
$\Rightarrow$  $\dfrac{2500 \times 5\times 5}{100}$
$\Rightarrow$  $25\times 25$
$\therefore$  $Simple\, Interest\, =Rs.625$

A man borrows $Rs \ 1820$ and undertakes to payback with a compound interest of $20 \%$ per annum in a $3$  equal yearly installments at the end of first year, second year and third year. Find the amount of each installments 

  1. $864$

  2. $800$

  3. $900$

  4. $820$


Correct Option: A
Explanation:

Each Installment = $\dfrac{P{\cdot}r}{100[1-{{ \frac{100}{100+r} }}^n]}$

here $P = Rs1820$
$r=0.2$
$n=3$
solving gives $Installment = 864Rs$

Mr. Gupta has been accumulating a fund at $8\%$ effective, which will provide him with an annual income of Rs $30000$ for $15$ years, the first payment being paid on his $60^{th}$ birthday. If he wishes to reduce the number of payments to $10$, find how much annual income (in Rs) will he receive?

  1. $38268$

  2. $30268$

  3. $38208$

  4. $38068$


Correct Option: A
Explanation:

In first case, we have annuity due of $15$ terms.
Its present value (as on Mr. Gupta's $60^{th}$ birthday),
$v = \frac{A}{r} \times (1+r) \times [1-(1+r)^{-n}]$
= $\frac{3000}{0.08} \times 1.08 \times [1-(1.08)^{-15}]$
Now if only $10$ payments are to be received, we have annuity due of $10$ terms. If A is the amount of each annual installment,
$\frac{A}{0.08} \times 1.08 \times [1-(1.08)^{-10}]$
Thus,
$\frac{A}{0.08} \times 1.08 \times [1-(1.08)^{-10}]$ = $\frac{3000}{0.08} \times 1.08 \times [1-(1.08)^{-15}]$
A = $30000 \times \frac{[1-(1.08)^{-15}]}{[1-(1.08)^{-10}]}$
A = $38268$

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.

  1. $4375$

  2. $4275$

  3. $4175$

  4. $4475$


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 __________________.

  1. ordinary annuity

  2. deferred annuity

  3. annuity due

  4. Both A and B


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?

  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


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? 

  1. $12891.50$

  2. $12610$

  3. $13861.50$

  4. $13801.50$


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

  1. Annuity certain

  2. Annuity contingent

  3. Annuity perpetual

  4. All of the above


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.

  1. $1281.50$

  2. $1291.50$

  3. $1181.50$

  4. $1381.50$


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? 

  1. $ $154.73$

  2. $ $147.36$

  3. $ $109.39$

  4. $ $104.72$

  5. $ $99.74$


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.

  1. $\$100.25$

  2. $\$101.25$

  3. $\$101.75$

  4. $\$105$


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.

  1. 54909.33

  2. 53909.33

  3. 52909.33

  4. 51909.33


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?

  1. $4016.76$

  2. $3013.54$

  3. $4065.24$

  4. $1034.54$


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?

  1. 12613.48

  2. 10613.48

  3. 11613.48

  4. None of these


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.

  1. Rs.$613.2$

  2. Rs.$603.2$

  3. Rs.$513.2$

  4. Rs.$713.2$


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

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 principal (approximately).

  1. $2920$

  2. $3450$

  3. $2346$

  4. $2275$


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$
$\implies A=Rs.2920$

A sinking fund is created for the redemption of debentures of Rs 10,000 at the end of 25 years. How much money should be provided out of profits each year for the sinking fund if the investment can earn interest at the rate 4% per annum?

  1. 2408.19

  2. 1408.19

  3. 3408.19

  4. 5408.19


Correct Option: A
Explanation:

Using formula of sinking fund

$M=\cfrac{A}{r}[(1+r)^n-1]$
$\implies 1000= \cfrac{A}{0.04}[(1+0.04)^{25}-1]\implies A=2408.19$

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 (approximately) with first installment.

  1. Rs. $202$

  2. Rs. $192$

  3. Rs. $92$

  4. Rs. $292$


Correct Option: D
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$
$\implies A _1=2920(1+\cfrac{10}{100})=3212$
$CI=A _1-P=3212-2920=Rs.292$

Which of the following is not an example of annuity certain ?

  1. Car Loan

  2. Daughter's Marriage

  3. House Loan

  4. All of above


Correct Option: B
Explanation:
$\Rightarrow$  $Daughter's\,\,Marriage$ is not an example of annuity certain.
$\Rightarrow$  Annuity certain is 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. 
$\Rightarrow$ 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.
$\Rightarrow$  House loan and Car loan are examples of annuity certain.

A man borrowed some money and paid back in 3 equal instalments of Rs 2160 each. What sum did he borrow if the rate of interest charged was 20 % p.a .compounded annually?

  1. 4551.12

  2. 4334.24

  3. 4768.97

  4. None of these


Correct Option: A
Explanation:

We know that,

$A=P(1+\cfrac{r}{100})^n$
$\implies 2160=P(1+\cfrac{20}{100})^2\ \implies P=Rs.4551.12$

A man borrowed some money and paid back in $3$ equal instalments of Rs. $2160$ each. The rate of interest charged was $20\%$ p.a .compounded annually. Find the total interest charged in Rs.(approximately).
  1. $1928$

  2. $1980$

  3. $1930$

  4. $1954$


Correct Option: C
Explanation:

$\Rightarrow$  We have, $A=Rs.2,160,\, n=3\, and\, r=20\%$

$\Rightarrow$  First we have to find Sum borrowed i.e. $V$.
$\Rightarrow$  Using, $V=\dfrac{A}{r}[1-(1+r)^{-n}]$


$\Rightarrow$  $V=\dfrac{2160}{0.2}[1-(1.2)^{-3}]$

$\Rightarrow$  $V=10800[1- \dfrac{1000}{(12)^3}]$

$\Rightarrow$  $V=10800[\dfrac{(1728-1000)}{1728}]$

$\Rightarrow$  $V=\dfrac{10800\times 728}{1728}$

$\Rightarrow$  $V=Rs.4550$.
$\Rightarrow$  Total interest = $(A\times n)-V=2160\times 3-4550=Rs. 1930$

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. What is the cash value (approximately) of the car ?

  1. 238467

  2. 235467

  3. 228467

  4. None of these


Correct Option: A
Explanation:

$\Rightarrow$  We have, $A=Rs.5000,\, n=40$ and $I=\dfrac{6}{100}\times \dfrac{1}{4}=0.015$

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

$\Rightarrow$  $V=\dfrac{5000}{0.015}[1-(1+0.015)^{-40}]$      ---   ( 1 )

$\Rightarrow$  Let $x=(1.015)^{-40}$
$\Rightarrow$  $log\, x=-40(0.0064)$
$\Rightarrow$  $log\, x=-0.256=(\bar{1}.7440)$
$\Rightarrow$   $x=antilog (\bar{1}.7440)$
$\Rightarrow$   $x=0.5546$
     Substitute this value in ( 1 ),
$\Rightarrow$  $V=\dfrac{5000}{0.015}(1-0.5546)$

$\Rightarrow$  $V=\dfrac{5000}{0.015}\times 0.4454=Rs.148466.67$

$\therefore$   Total cash value of the car = $Rs.90000+Rs.148466.67$
$\Rightarrow$  Total cash value of car = $Rs.2,38,467$ approx.

Find the amount(approximately) of an annuity immediate of Rs 2000 per annum for 20 years. The rate of interest is $\dfrac{27}{2}$ % per annum.

  1. 171703

  2. 19000

  3. 21345

  4. 43251


Correct Option: A
Explanation:
$n=20,r=(27/2)\%,P=Rs.2000$
$A=P(1+\cfrac{r}{100})^n$
$\implies A=2000[(1+\cfrac{(27/2)}{100})^{20}$$+(1+\cfrac{(27/2)}{100})^{19}......$$+(1+\cfrac{(27/2)}{100})^{1}]\\A=2000\times 85.85\\ A=Rs.171700\approx 171703$

Mr Assem decides to deposit Rs 5000 at the end of year in a bank which pays compound interest the rate of 5% per annum. What will be his total accumulation (approximately) at the end of 15 years?

  1. Rs.$140092$

  2. Rs.$907892$

  3. Rs.$100892$

  4. Rs.$107892$


Correct Option: D
Explanation:
$A=P(1+\cfrac{r}{100})^n$
$\implies A=5000[(1+\cfrac{5}{100})$$+(1+\cfrac{5}{100})^{14}......$$+(1+\cfrac{5}{100})^{1}+5000]\\=5000\times 21.5718\\ Rs.107892$

Mr Dev purchased a car paying Rs $90,000$ and promised to pay Rs 5000 every 3 months for the next 10 years. The interest is $6$% p.a. compounded quarterly. If Mr Dev misses first 6 payments , how much should he pay at the time of 7th payment (approximately) to bring himself up to date.

  1. 36333

  2. 46532

  3. 56734

  4. 60322


Correct Option: A
Explanation:

$\Rightarrow$  We have $A=Rs.5000,\, n=7$  $I=\dfrac{6}{100}\times \dfrac{1}{4}=0.015$

$\Rightarrow$  At the time of 7th payment, equivalent amount of first 6 missed payment has also to be paid. Thus total payment to be made at the end of 7th period is the amount of annuity of 7 terms. Hence, amount required required to be paid is,
$\Rightarrow$  $M=\dfrac{A}{I}[(1+I)^{n}-1]$


$\Rightarrow$  $M=\dfrac{5000}{0.015}[(1.015)^{7}-1]$           -----  ( 1 )
$\Rightarrow$   Let $x=(1.015)^{7}$
$\Rightarrow$   $log\,x=7\,log\,(1.015)$
$\Rightarrow$   $log\,x=7\,(0.0064)=0.0448$
$\Rightarrow$   $x=antilog\,(0.0448)$
$\Rightarrow$   $x=1.109$
Substitute value of $x$ in ( 1 ) we get,
$\Rightarrow$  $M=\dfrac{5000}{0.015}(1.109-1)$

$\Rightarrow$  $M=\dfrac{5000}{0.015}\times 0.109$

$\Rightarrow$  $M=\dfrac{5000\times 1000}{15}\times \dfrac{109}{1000}$

$\Rightarrow$  $M=Rs.36333.33\approx Rs.36333$

A bank pays interest at the rate of 8 % per annum compounded half yearly. Find how much should be deposited in the bank (approximately) at the beginning of each half year in order to accumulate Rs 8000 for 3 years.

  1. 994

  2. 1161

  3. 4532

  4. 2341


Correct Option: B
Explanation:
$A=P(1+\cfrac{r}{100})^n$
$r=8\%/2=4\%, n=2\times 3=6$
$\implies 8000=P[(1+\cfrac{4}{100})^6$$+(1+\cfrac{4}{100})^{5}......$$+(1+\cfrac{4}{100})^{1}]\\ \implies 8000=P\times 21.5718\\ \implies P=Rs.1159\approx Rs.1161$

Find the future value of an ordinary amount of Rs 4000 each six months for 10 years at 8% per annum , compounded semi-annually.

  1. $123877$

  2. $175640$

  3. $213457$

  4. $324156$


Correct Option: A
Explanation:

$P=Rs.4000$

Amount payable in half yearly so $,r=8\%/2=4\%,t=10\times 2=20$
$A=\sum _{ n=1 }^{ 20 }{ P{ (1+\cfrac { r }{ 100 } ) }^{ n } } \ =\sum _{ n=1 }^{ 20 }{ 4000{ (1+\cfrac { 4 }{ 100 } ) }^{ n } } \ =4000\times 30.969\ =Rs.123877$

Find the least number of years for which an annuity Rs 1000 must run in order that its amount just exceeds Rs 16000 at 5% pa. compounded annually.

  1. $12$

  2. $9$

  3. $2$

  4. $25$


Correct Option: A
Explanation:

$P=4000, r=5\%$

$A=\sum _{ n=1 }^{ n _0 }{ P{ (1+\cfrac { r }{ 100 } ) }^{ n } } \ =\sum _{ n=1 }^{ n _0 }{ 1000{ (1+\cfrac { 5 }{ 100 } ) }^{ n }\  } >1600\ =\sum _{ n=1 }^{ n _0 }{ { (1+\cfrac { 5 }{ 100 } ) }^{ n }\  } >16$
Since it is a sum of a infinite GP. We can write it as a
$\cfrac  {(1.05^{n _0}-1)}{(1.05-1)}\times 1.05>16$
$\implies (1.05)^{n _0}>1.7619$
Taking log on both sides
$\implies \log{(1.05)^{n _0}}>\log 1.7619$
$\implies n _0>\cfrac{\log (1.7619)}{\log (1.05)}$
$\implies n _0>11.6$
$\implies n _0=12$
Hence, least no. of years$=12$.

Find the amount of the annuity of Rs 150 payable in half yearly instalments  for 15 years at 4% per annum interest also payable half yearly

  1. Rs.$9903.45$

  2. Rs.$3003.45$

  3. Rs.$8103.45$

  4. Rs.$3103.45$


Correct Option: D
Explanation:

$P=Rs.150$

Amount payable in half yearly so $,r=4\%/2=2\%,t=15\times 2=30$
$A=\cfrac{1}{2}\sum _{ n=1 }^{ 30 }{ P{ (1+\cfrac { r }{ 100 } ) }^{ n } } \ =\cfrac{1}{2}\sum _{ n=1 }^{ 30 }{ 150{ (1+\cfrac { 2 }{ 100 } ) }^{ n } } \ =75\times 41.38\ =Rs.3103.45$

A sum of Rs 2522 is borrowed from a money lender at 5% per annum compounded annually.If this amount is to be paid back in 3 equal instalments , find the annual instalments (approximately).

  1. 925

  2. 800

  3. 875

  4. 567


Correct Option: A
Explanation:

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

$\implies 2522= \cfrac{A}{(1+\cfrac{5}{100})^1}$$+\cfrac{A}{(1+\cfrac{5}{100})^2}+$$\cfrac{A}{(1+\cfrac{5}{100})^3}$
$ \implies A=926\approx 925$

What is the amount of annuity(approximately) due of Rs 100 yearly payable half yearly for 15 years at 10 % compound interest per annum half yearly.

  1. 3491

  2. 1456

  3. 5434

  4. 2341


Correct Option: A
Explanation:

$r=\cfrac{10\%}{2}=5\%,n=2\times 15=30,A=Rs.100$

Amount of annuity due payable half yearly
$=(1/2)\cfrac{A}{r}\times (1+r)\times[(1+r)^n-1]$
$=(1/2)\cfrac{100}{0.05}\times (1+0.05)\times[(1+0.05)^{30}-1]\=Rs.3488\approx  Rs.3491$

Which of the following comes under Annuity due?

  1. Life insurance Premium

  2. Recurring Deposit Payments

  3. Advance Payment of monthly house rent

  4. All of the above


Correct Option: D
Explanation:

An annuity is a contract aimed at generating steady income during retirement, where in lump sum payment is made by an individual to obtain certain amounts immediately or at some point of future
all of above comes under annuity.
It includes Life insurance Premium, Recurring Deposit Payments, Advance Payment of monthly house rent.

Suppose you deposit $ \$900$ per month into an account that pays $4.8 \%$ interest, compounded monthly. How much money will you have after $9$ months? 

  1. $ $8432.97$

  2. $ $1372.44$

  3. $ $9812.97$

  4. None of these


Correct Option: D
Explanation:

Here, $P=\$900,\,R=4.8\%$ and $T=9$ months $=\dfrac{3}{4}$ year

$\Rightarrow$ $A=P\left (1+\dfrac{R}{100}\right)^{4T}$
$\Rightarrow$ $A=900\times \left (1+\dfrac{4.8}{100}\right)^{4\times \frac{3}{4}}$
$\Rightarrow$ $A=900\times \left (\dfrac{131}{125}\right)^3=900\times 1.1510$
$\Rightarrow$ $A=\$1035.9$

Identify the correct term for the following definition: It is any asset donated to and for the perpetual benefit of a non-profit institution. The donation is usually made with the requirement that the principal remain intact and money earned from investing the principal be used for a specific purpose.

  1. Scholarship fund

  2. Sinking fund

  3. Endowment

  4. None of these


Correct Option: C
Explanation:

"It is any asset donated to and fro for the perpetual benefit of a non-profit institution. The donation is usually made with the requirement that the principal remain intact and money earned from investing the principal be used for a specific purpose".

Correct term for he above defination is the Endowment.

What is true about deferred annuity ?

  1. It is an annuity in which the first payment is postponed for period of times.

  2. It is annuity when payments are made at the end of each payment.

  3. It is annuity when payments are made at the beginning of each payment.

  4. None of the above


Correct Option: A
Explanation:

Deferred payment annuities typically offer tax-deferred growth at a fixed or variable rate of return, just like regular annuities. Often deferred payment annuities are purchased for under-age children, with the benefit payments postponed until they reach a certain age. Deferred payment annuities can be helpful in retirement planning.
Option (A) is correct

What is true about deferred annuity ?

  1. It is an annuity when the payments are made at the end of payment period.

  2. It is an annuity when the payments are made at the beginning of payment period.

  3. It is an annuity when the payments are made at the middle of payment period.

  4. None of the above


Correct Option: A
Explanation:

$\Rightarrow$  True statement about deferred annuity is,

$-\,It\,is\,an\,annuity\,when\,the\,payment\,are\,made\,at\,the\,end\,of\,payment\,period.$
$\Rightarrow$  A deferred annuity is an insurance contract designed for long-term savings. 
$\Rightarrow$  Unlike an immediate annuity, which starts annual or monthly payments almost immediately, investors can delay payments from a deferred annuity indefinitely. During that time, any earnings in the account are tax-deferred.

Which of the following is correct regarding endowment?

  1. Endowments are given to non-profit organizations with the intention that they be used to advance the mission of the organization for the long term.

  2. Endowments of large institutions, sometimes become significant players in the financial world due to the significant amount of money that the endowment is investing.

  3. Both are correct

  4. None of the above


Correct Option: C
Explanation:

$A$ and $B$ both are correct statements regarding endowment which are:

Endowments are given to non-profit organizations with the intention that they be used to advance the mission of the organization for the long term.
Endowments of large institutions, sometimes become significant players in the financial world due to the significant amount of money that the endowment is investing.

What amount should be set aside at the end of each year to amount Rs 10 lakhs at the end of 15 years at 6% per annum compound interest?

  1. $5298994$

  2. $3297000$

  3. $4297994$

  4. None of the above


Correct Option: D
Explanation:

$A=10,00,000, t=15,r=6\%$

$A=\sum _{ n=1 }^{ 14 }{ P{ (1+\cfrac { r }{ 100 } ) }^{ n }+P } $
$\implies 10,00,000=P[\sum _{ n=1 }^{ 14 }{ { (1+\cfrac { 6 }{ 100 } ) }^{ n }+ 1} ]$
$\implies 10,00,000\times (22.276+1)\ \implies P=Rs.42963$

Find the amount of annuity of Rs. $4000$ p.a. for $10$ years reckoning compound interest at $10\%$ p.a.

  1. Rs. $63,706$

  2. Rs. $63,670$

  3. Rs. $67,360$

  4. Rs. $63,760$


Correct Option: D
Explanation:
$Payment  \space  'P'= 4000$
$N=10$
$R=10%=10/100=0.1$
$A=P(\dfrac{(1+r)^n - 1)}{r}) $
$=4000(\dfrac{(1.1)^{10} -1}{0.1})$
$=63760$

Find the amount of annuity of Rs. 4,000 per annum for 10 years reckoning interest at 10% p.a.
[Given : $(1.1)^{10} = 2.594$]

  1. Rs. 63,760

  2. Rs. 63,670

  3. Rs. 63,205

  4. None.


Correct Option: A
Explanation:

$P=Rs.4000$

$n=10$
$x=10$%
    $=\dfrac { 10 }{ 100 } =0.1$
$M=\dfrac { P }{ r } \left( { \left( 1+r \right)  }^{ n }-1 \right) $
     $=\dfrac { 4000 }{ 0.1 } \left( { \left( 1.1 \right)  }^{ 10 }-1 \right) $
     $=40000(2.594-1)$
     $=63760$.
$\therefore $  Amount of annuity $=63760$

The present value of an annuity of Rs. $3,000$ for $15$ years at $4.5\%$ p.a. CI is?

  1. Rs. $23,809.41$

  2. Rs. $32,218.63$

  3. Rs. $32,908.41$

  4. None of the above


Correct Option: B,D
Explanation:

Payment $P=3000$

$n=15$ years
rate=$4.5%$
$\therefore r=\cfrac { 4.5 }{ 100 } \ =0.045\ PV=P\left( \cfrac { 1-(1+r)^{ -n } }{ r }  \right) \ =3000(\cfrac { 1-(1+(1+0.045)^{ -15 } }{ 0.045 } )\ =3000(\cfrac { 1-(1.045)^{ -15 } }{ 0.045 } )\ =32218.63$
 $\therefore PV$ of the annuity is 
 $32218.63.$

Rs. $200$ is invested at the end of each month in an account paying interest $6\%$ per year compounded monthly. What is the future value of this annuity after $10$th payment? Given that $(1.005)^{10}=1.0511$.

  1. $2,044$

  2. $2,404$

  3. $2,440$

  4. $2,004$


Correct Option: A
Explanation:

A$=$Rs. $200$
$n=10$
i$=6\%$ p.a. $=6/12\%$ per month $=0.005$
Future value of annuity after $10$ months is given by
A(n, i)$=A\left[\displaystyle\frac{(1+i)^n-1}{i}\right]$
$A(10, 0.005)=200\left[\displaystyle\frac{(1+0.005)^{10}-1}{0.005}\right]$
$=$Rs. $2,044$.

Belose Infrastructures  just issued 10 million Rs100-par bonds payable carrying 8% coupon rate and maturing in 5 years. The bond indenture requires GI to set up a sinking up to pay off the bond at the maturity date. Semi-annual payments are to be made to the fund which is expected to earn 10% per annum. Find the amount of required periodic contributions.

  1. 7905155

  2. 7950515

  3. 8950515

  4. 6950515


Correct Option: B

An annuity is?

  1. A fixed sum

  2. Paid at regular intervals

  3. Under certain stated conditions

  4. All of the above


Correct Option: D
Explanation:

An annuity is 

$(1)$A fixed sum
$(2)$ Paid at regular intervals
$(3)$Under certain stated conditions.

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. $5,356.35$


Correct Option: B
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. $5365.35$

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


Correct Option: A

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. $402,509$


Correct Option: D
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 = $\$100,000+\$302,509=\$402,509$

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. $4567$


Correct Option: C
Explanation:

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

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

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. $33111.35$


Correct Option: C
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

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.$98450$


Correct Option: A
Explanation:

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

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


Correct Option: D
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$

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


Correct Option: A
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
$\Rightarrow$  $n=\dfrac{log\,2}{log\, 1.05}$
$\therefore$   $n=14.2\, years$

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


Correct Option: A
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 ),
$\Rightarrow$  $5000+\dfrac{5000}{0.015}(1-0.7447)$

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

$\Rightarrow$  $Rs.90100$

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$91,305$


Correct Option: A
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 $

$\implies A=Rs51,305$

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 $(1+0.08)^{10}=2.15892500$.

  1. $1,44,865.625$

  2. $1,56,454.875$

  3. $1,54,654.875$

  4. $1,44,568.625$


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 ?

  1. Rs. $6,000$

  2. Rs. $5,968$

  3. Rs. $4,787$

  4. Rs. $4,763$


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.
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