Annuities and Financial Mathematics - Class XI

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.

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.

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.

$\Rightarrow$   True statement about Annuity Due is,

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

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

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 an immediate annuity or ordinary annuity.

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 contingent annuity.

Formula for calculating the amount of an annuity,

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 annuity certain.

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

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

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.

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

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. $3662.50$.

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. $12160$.

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. $4275$.

Due to the time value of money, a dollar today is worth more than a dollar in the future because it can be invested to earn interest.

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

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,

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

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 $.

Number of interest periods, $n = 4 \times 8 +3 = 35$

Each installment, A=Rs $500$

Present value of annuity due,

$v = \frac{A}{r} \times (1+r) \times [1-(1+r)^{-n}]$

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

= $13740.86$

$v = \frac{A}{r} \times (1+r) \times [1-(1+r)^{-n}-1]$

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

= $23137.98$

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

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$

$ SI= prt$

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

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

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 = $38268$

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

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

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

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

$\Rightarrow$  Three types of annuities are : 

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

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

Susan purchased refrigerator for Rs.$675$.

Using formula of sinking fund

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

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

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

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

Using formula of sinking fund

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

We know that,

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

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

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

$P=Rs.4000$

$P=4000, r=5%$

$P=Rs.150$

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

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

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.

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

"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".

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

$\Rightarrow$  True statement about deferred annuity is,

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

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

$P=Rs.4000$

Payment $P=3000$

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. $2,044$.

This is a sinking fund problem. The periodic payment is calculated using the formula for an annuity: PMT = FV * i / ((1+i)^n - 1). With semi-annual periods, n=10 and i=0.05.

An annuity is 

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. $5365.35$

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.

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

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

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

We know that formula of present value is 

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

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

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 $(1+i)$
$=1,44,865.625\times (1+0.08)$
$=1,56454.875$.