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

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] $
Reveal answer Fill a bubble to check yourself
B Correct answer
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

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

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
Reveal answer Fill a bubble to check yourself
C,D Correct answer
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.

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

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. $6.3 \%$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 = $\dfrac{P\times T\times R}{100}$
$\Rightarrow$  $R=\dfrac{I\times 100}{P\times T}$
$\Rightarrow$  $R=\dfrac{300\times 100}{55200\times \dfrac{1}{12}}=\dfrac{300\times 100\times 12}{55200}=\dfrac{150}{23}=6.5$
$\therefore$   $Rate =6.5\%$

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

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. $3660$
Reveal answer Fill a bubble to check yourself
B Correct answer
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. $3662.50$.

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

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. $13000$
Reveal answer Fill a bubble to check yourself
C Correct answer
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. $12160$.

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

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. $3670$
Reveal answer Fill a bubble to check yourself
C Correct answer
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. $4275$.

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

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. $22$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 = $\dfrac{1}{2}\times 6\% = \dfrac{1}{2}\times \dfrac{6}{100}=0.03$
$\Rightarrow$  $V=\dfrac{A}{r}\times [1-(1+r)^{-n}]$

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

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

$\Rightarrow$  $(1.03)^{(-n)}=0.5$

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

$\Rightarrow$  $-n(0.0128)=-0.3010$

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

$\therefore$    $n=23.51 \approx 24$

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

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 $24017.25$
Reveal answer Fill a bubble to check yourself
C Correct answer
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,
$v = \dfrac{A}{r} \times (1+r) \times [1-(1+r)^{-n}]$

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