Skip to main content

What is an Annuity? How does it calculate?

Annuity: It is a fixed series of payment in fixed interval in the specified period. For example monthly pension payments, loan payments etc. Annuity payments are made monthly, annually, half yearly and quarterly. Payment frequency and interest compounded must be equal . It is basically two types:
Ø  Ordinary annuity
Ø  Annuity due

Ordinary Annuity: It is a fixed series of payment or amount receives at the end of each period at fixed interval to the certain specified period. It is beneficial for the loan payment because according to the time value of money we have to pay less in future in comparison to present.

Annuity Due: it is a fixed series of payment or amount receives at the beginning of each period at fixed interval to the certain specified period. For example rent payments. It is beneficial for the investment purpose because we will get long time to earn interest on it.

Formula
Present Value

Ordinary Annuity
Annuity Due
PV
A*[(1-(1+i) –n / i] or 
A*[(1-(1/(1+i) n ))/ i]
A * [1-(1+i) –n / i] * (1+i) or A*[(1-(1/(1+i) n ))/ i]
N
In [((1-PV*i) -n /A]/ In (1+i)
[-In(1+i(1-PV/A)/In (1+i)] +1
A
PV/[(1-(1+i)-n/i]
PV/[1-(1+i) –n/i]*(1+i)

Formula
Future value

Ordinary Annuity
Annuity Due
FV
A* [(1+i) n-1/i]
A* [(1+i) n -1 / i] * (1+i)
N
In (1+FV*i/A) / In (1+i)
In[(1+(FV/A(1+i))*i)/In(1+i)]+1
A
FV/((1+i)n/i)-1
FV/((1+i)n-1/i)*(1+i)

Where,
FV= Future Value
PV= Present Value
A= Monthly payment
i= rate of interest
n= number of period

Example 1: Calculate future value of an annuity of Rs.2000 paid at the end of every 6 month for 5 years @ 7% compounded half yearly.

FVOA=   A* [(1+i) n-1/i]
=2000*[(1+0.035) 10-1/0.035]
=23462.79

In case monthly payment is unknown :
A = FV/ ((1+i) n/i)-1
= 23462.79/ [(1+0.035) 10-1/0.035]
=23462.79/11.731
=2000

Example 2:Mr. Chauhan wants to know present value of ordinary annuity if he paid Rs 10,000 per month and assume interest 6% compounded monthly for 4 years?

PVOA= A [1-(1+i) –n/ i]
= 10, 000[1-(1+0.005) -48/ 0.005]
=425803.18

In case monthly payment is unknown:
APV A = PV/ [(1-(1+i)-n/i]
= 425803.18/[1-(1+0.005)-48/0.005]
=10000

Example 3:Mr. Sohan has taken a house on rent and he paid Rs. 5000 rent at the beginning of each quarter @ 7.5% compounded quarterly for 8 years. Find out the future value?

FVAD= A * [(1+i) n-1/ i] * (1+i)
=5000* [(1+0.01875) 32-1/0.0 1875] * (1+0.01875)
=216539.678*1.01875
=220599.797

In case monthly payment is unknown:
A FV AD = FV/ ((1+i) n-1/i)*(1+i)
=220599.79/ [(1+0.01875) 32-1/0.01875]*(1+0.01875)
=5000

Example4:  Calculate present value if you made payment at the beginning of each year of Rs.2 0,000 @ 5 % compounded annually for 4 years.

PVAD= A * [1-(1+i) –n / i] * (1+i)
= 20,000*[1-(1+0.05)-4/ 0.05]* (1+0.05)
=74464.9626

In case monthly payment is unknown:
APV AD = PV/ [1-(1+i) –n/i]*(1+i)
=74464.96/[1-(1+0.05) -4/0.05]*(1+0.05)
=20,000

Example 5: Mr. X invests Rs.2000 in ordinary annuity per month in his saving a/c from his salary and earns 7% interest compounded monthly .So in how many years he will earn Rs.500, 000?

NFV A= In (1+FV*i/A) / In (1+i)
=In (1+ (500,000*0.00583/2000))/In (1+0.00583)
=154.67
= 13 years

In case of Future Value of Annuity Due "n" is :
NFV AD = In [(1+ (FV/A (1+i))*i)/In(1+i)]+1
= In [(1+ (500000/2000(1+0.00583))*0.00583)/In (1+0.00583)] +1
= 154.08                      
=13 years

Example 6: In how many number of payments Rs. 10,000 invested in ordinary annuity@ 4% compounded annually and the present value is Rs.2, 00,000?

NPV A = In [(1-PV*i) -n /A]/ In (1+i)
= [In (1-(2, 00,000*0.04) -1/10000))]/ In (1+0.04)]
=41.03 years

In case of Present value of Annuity Due "n" is :
NPV AD = [-In (1+i(1-PV/A)/In (1+i)] +1
= [-In (1+0.04 (1-2, 00, 000/10000))/In (1+0.04)] +1
= [-In (0.24)/In (1.04)] +1
=[1.4271164/0.03922071] +1

=37 years

 Calculations of Annuity in excel:

 Future Value:
A
B
1
Monthly Payment (pmt)
2000
2
Number of Periods (nper)
4
3
Rate
7%
4
Future Value
INR 11,70,000
=FV(rate,nper,pmt,[pv])
B4: “=FV (B3,B2,B1,0)”

Monthly payment:
A
B
1
Present Value
65000
2
Number of Periods (nper)
4
3
Rate
7%
4
pmt
INR 455,111.11
=pmt(rate,nper,pv,[fv])
B4: “=PMT(B3,B2,B1)”

Present Value:
A
B
1
Pmt
20000
2
Number of Periods (nper)
4
3
Rate
7%
4
PV
INR 2856.45
=pv(rate,nper,pmt,[fv])
B4: “=PV(B3,B2,B1)”


Comments

Post a Comment

Popular posts from this blog

How to calculate Cost of Preference Share Capital?

Cost of Preference Share Capital:  An amount paid by company as dividend to preference shareholder is known as Cost of Preference Share Capital. Preference share is a small unit of a company’s capital which bears fixed rate of dividend and holder of it gets dividend when company earn profit. Dividend payable is not a tax deductible amount. So, there is no tax adjustments required for comparing with cost of debt. Formula for Cost of Preference Share: Irredeemable Preference Share Redeemable Preference Share K p  = Dp/NP K p  = D p +((RV-NP)/n )/ (RV+NP)/2 Where, K p  = Cost of Preference Share D p  = Dividend on preference share NP = Net proceeds from issue of preference share (Issue price – Flotation cost) RV = Redemption Value N = Period of preference share Example:  A company issues 20,000 irredeemable preference share at 8% whose face value is Rs.50 each at 4% discount. Find out the Cost of ...
90 comments
Read more

What is the difference between Cheque book and Pass book?

 Cheque book is issued by bank in customers / account holder request. With the help of this book account holder can withdraw cash from his/her account. Bank does not charge any fee to issued cheque book to its customer. But afterward bank charges some amount for using bank facility like cheque book, Debit card etc.So, Automatic some definite amount deducted from customer bank account. Pass book is  also issued by bank to its customer. It helps to record all the bank related activity according to date that is withdrawal and deposit. It is recorded by bank but the book is kept by customer to know the current balance of  his /her account.  Point of difference Pass book Cheque book What is the meaning of pass book and cheque book? Passbook is a book in which all withdrawal and deposit against customer account is recorded.   Cheque book is a book of cheques which are used to withdrawal the money to b...
Post a Comment
Read more

Numericals with solutions of Net income Approach

Net income approach questions and answers:   Questions:  Find out the value of the firm with the help of given information: Particulars Amount Earnings before interest and tax 3, 50, 000 Cost of equity 10% Cost of debt 7.2% Debenture 1,00,000 Find out the overall cost of capital with the help of net income approach. (Assume tax rate-10%) Solution: Particulars Amount Earnings before interest and tax 3, 50, 000 Less: Interest @7.2% 7, 200 Earnings before tax 3, 42, 800 Less: Tax@10% 34, 280 Net income 3, 08, 520 Cost of equity 10% Market value of equity (S =net income/ cost of equity) 30, 85, 200 Market value of debt (B) 1, 00, 000 Value of the firm (S+B) 31, 85, 200 Questions:  Find out the overall cost of capital if the equity capitalisation rate is 12...
Post a Comment
Read more