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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)”


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