Computational Finance
Annuity
An annuity is a series of equal payments made at regular intervals over
time. These payments can be monthly, quarterly, yearly, etc.
Examples of Annuities:
Loan Payments: Paying your car loan or home loan in fixed monthly amounts.
Investments: Depositing a fixed amount every year into a retirement fund.
Insurance Plans: Receiving a fixed pension every month after retirement.
Based on
Time of Payment
The term of Annuity
The timing of corresponding process
Annuity
The term of Annuity
Annuity Certain
Contingent
Perpetuity
Time of Payment
Ordinary Annuity: Payments are made at the end of each period (e.g., home
loan EMI).
Annuity Due: Payments are made at the beginning of each period (e.g., rent
payments
Deffered Annuity:
Ordinary annuity
An ordinary annuity is a series of equal payments made regularly at the end of
each period (like a month or year). For example:
When you pay your rent at the end of every month.
When you pay your car loan or home loan at the end of each month.
The main thing to remember is that payments happen after the period is over.
That's why it’s called “ordinary.”
Eg . you take a loan of ₹1,00,000, and you agree to pay it back in equal monthly
installments of ₹10,000 over 12 months. Each ₹10,000 payment is made at the end
of the month.
This is an ordinary annuity because:
The payments are equal (₹10,000 each).
Payments happen at the end of each month (not at the start).
You invest ₹10,000 every year into a savings plan for 5 years. The interest rate is 5% per
year, and payments are made at the end of each year.
Question:
What is the total value of the annuity after 5 years (future value)?
Formula for Future Value (FV):
FV=PMT×(1 + r)^n - 1/r
Where:
PMT = ₹10,000 (annual payment)
r = 5% or 0.05 (annual interest rate)
n = 5 years (number of periods)
Calculation:
FV=10,000×(1+0.05) ^5−1/0.05
FV=10,000×5.5256=₹55,256
PMT= FV×r/(1+r)^n −1
PV=PMT×r1−(1+r)^−n /r
Situation: You want to save ₹10,00,000 for retirement in 20 years. You plan to
deposit money at the end of each year into an account earning 8% annually.
How much should you deposit per year?
Given Data:
FV (Target Amount) = ₹10,00,000
Annual Interest Rate = 8% (r = 0.08)
Years (n) = 20
Using the future value annuity formula:
Situation: You take a loan of ₹5,00,000 at an annual interest rate of 12%, to
be repaid over 5 years with monthly payments.
Given Data:
PV (Loan Amount) = ₹5,00,000
Annual Interest Rate = 12% (or 1% per month, r = 0.01)
Number of months (n) = 5 × 12 = 60 months
Using the ordinary annuity payment formula:
Ordinary Annuity
•Loan Amount (Principal, P): ₹1,00,000
•Annual Interest Rate (r): 10%
•Loan Duration: 5 years
•Payment Timing: End of each year
•PMT= PV x r /1−(1+r)^−n
•pay ₹26,379 per year for 5 years.
Car Loan (Quarterly Payments)
Situation: You buy a car for ₹3,00,000, taking a loan at 10% annual interest,
to be repaid in 4 years with quarterly payments.
Given Data:
PV (Loan Amount) = ₹3,00,000
Annual Interest Rate = 10% (or 2.5% per quarter, r = 0.025)
Number of quarters (n) = 4 × 4 = 16
Using the PMT(loan payment) formula:
You take a home loan of ₹50,00,000 at 9% annual interest, to be repaid
over 15 years with monthly payments. However, at the end of the 15th year,
you will make a balloon payment of ₹10,00,000 to clear the remaining
balance.
A person deposits ₹10,000 at the end of each year into a savings account
that earns 8% annual interest. They continue this for 5 years. How much will
they have at the end of 5 years?
A company wants to determine how much they should invest today to
withdraw ₹50,000 per year for the next 6 years, assuming an interest rate of
9% per year.
A person plans to retire in 20 years and wants to save enough money to
accumulate ₹1 crore by retirement. They invest a fixed amount at the end of
each month in a fund that earns 10% annual interest, compounded
monthly.
To accumulate ₹1 crore in 20 years, the person must invest ₹14,547 per
month in a 10% annual return investment.
Annuity due
An annuity due is a type of annuity where payments are made at the beginning of each period, rather than
at the end. This contrasts with an ordinary annuity, where payments are made at the end of each period.
Key Features of Annuity Due:
Payments occur at the start of each period.
The value of an annuity due is usually higher than that of an ordinary annuity because each payment is
invested for an extra period.
Common examples include rent payments, insurance premiums, and lease agreements, where payments
are made in advance.
Formula for the Future Value (FV) of an Annuity Due:
FV=P×(1+r)^n−1/r×(1+r)
where:
P = Payment per period
r = Interest rate per period
n = Number of periods
Formula for the Present Value (PV) of an Annuity Due:
PV=P× [1−(1+r)^−n/r] ×(1+r)
Imagine you are renting an apartment and need to pay ₹10,000 at the beginning
of each month for 3 months. The landlord requires advance payments, making this
an annuity due. Assume an interest rate of 2% per month.
Step 1: Future Value Calculation
Using the Future Value formula:
FV=P×(1+r)^n−1/r×(1+r)
FV=10,000 × (1.02)^3-1/0.02×(1.02)
=30,600
at the end of 3 months, the future value of payments would be ₹30,600.
Year
Deposit (₹)
Interest Earned (₹)
1
10,000
500
10,500
2
10,000
1,025
21,525
3
10,000
1,995
31,520
Future Value at End of
Year (₹)
You need to pay rent of ₹12,000 per year for 5 years, paid at the beginning
of each year. The discount rate (interest rate) is 6% per year.
You save ₹15,000 per year at the beginning of each year for your child's
education for 4 years. The account earns 5% annual interest.
You are repaying a loan with ₹20,000 annual payments made at the
beginning of each year for 3 years. The bank charges 7% interest per year.
ou deposit ₹25,000 per year into a retirement account at the beginning of
each year for 6 years. The interest rate is 8% per year.
•Annual Payments: ₹50,000
•Payment Type: At the beginning of each year (Annuity Due)
•Number of Payments: 6 years
•Interest Rate:
•1st year: 8%
•2nd year: 7%
•3rd year: 6%
•4th year: 5%
•5th year: 4%
•6th year: 3%
•Total Present Value (PV): ₹1,089,093
•Total Future Value (FV): ₹886,785
Using the Present Value formula:
PV=10,000×2.9=29,000
3 advance payments is ₹29,000.
Deferred Annuity
A Deferred Annuity is an annuity where payments start after a waiting period, instead of starting immediately.
This allows the invested money to grow before withdrawals begin.
How It Works:
Accumulation Phase – You invest money (lump sum or installments).
Growth Period – The money earns interest or investment returns over time.
Payout Phase – After a set period (e.g., after 10 years or retirement), you start receiving regular payments.
Example:
Suppose you invest ₹5 lakhs in a deferred annuity plan today. The policy states that payments will begin after
10 years. During these 10 years, your investment grows with interest. After 10 years, you start receiving
monthly payouts for a fixed duration or for life.
Types of Deferred Annuities:
Fixed Deferred Annuity – Earns a guaranteed interest rate.
Variable Deferred Annuity – Growth depends on market investments (stocks, bonds).
Indexed Deferred Annuity – Returns are linked to a market index (e.g., Nifty 50).
Example of Deferred Annuity
Scenario:
Ramesh, aged 40, plans for his retirement at 60.
He invests ₹50,000 per year in a Fixed Deferred Annuity for 20 years.
The annuity offers 7% annual interest.
The payouts will start at age 60 and continue for 15 years as ₹X per year.
Calculate the Future Value (FV) at Year 20
Since Ramesh is investing ₹50,000 per year, we use the FV of an Ordinary Annuity
formula:
FV=PMT×(1+r)n−1/r
Where:
PMT = ₹50,000 (Annual Investment)
r = 7% = 0.07 (Annual Interest Rate)
n = 20 years
V=50,000× (1.07)^20−1 /0.07
=₹27,64,000
At age 60, Ramesh will have ₹27.64 lakhs accumulate
Now, Ramesh will receive equal yearly payments for 15 years.
We use the Present Value of an Annuity formula:
PMT=PV/[1−(1+r)^−n/r ]
Where:
PV = ₹27,64,000 (Total fund at age 60)
r = 7% = 0.07
n = 15 years
PMT=27,64,000/1−(1.07)^−15 /0.07
PMT = =₹3,03,602 per year
Ramesh will receive ₹3,03,602 per year for 15 years after retirement.
Year
Investment (₹)
Interest Earned (7%)
Total at Year-End (₹)
1
50,000
0
50,000
2
50,000
3,500 (7% of 50,000)
1,03,500
3
50,000
7,245 (7% of 1,03,500)
1,60,745
4
50,000
11,252 (7% of 1,60,745)
2,21,997
5
50,000
15,540 (7% of 2,21,997)
2,87,537
10
50,000
41,073 (7% of 5,86,749)
6,77,822
15
50,000
91,399 (7% of
13,05,708)
14,47,107
20
50,000
1,80,480 (7% of
25,83,520)
₹27,64,000
Year
Start Balance
(₹)
Interest
Earned (7%)
(₹)
Withdrawal
(₹)
End Balance
(₹)
21 (Age 60)
27,64,000
1,93,480
3,03,602
26,53,878
22
26,53,878
1,85,772
3,03,602
25,35,048
23
25,35,048
1,77,453
3,03,602
24,08,899
24
24,08,899
1,68,623
3,03,602
22,73,920
25
22,73,920
1,59,174
3,03,602
21,29,492
30
14,12,018
98,841
3,03,602
12,07,257
35 (Age 75)
3,16,251
22,138
3,03,602
0 (Fully
Withdrawn)
Example 1: Future Value (FV) of a Deferred Annuity
Scenario:
Rahul invests ₹20,000 per year in a fixed deferred annuity with an interest
rate of 6% per year. The annuity is deferred for 5 years and he continues
investing for 10 years. What will be the Future Value (FV) after 10 years?
Formula for Future Value of an Ordinary Annuity:
FV= PMT×(1+r) ^n −1 / r
PMT = ₹20,000 (Annual Investment)
r = 6% = 0.06 (Annual Interest Rate)
n = 10 years (Investment Period)
Deferred Annuity with a 10-Year Waiting Period
Scenario:
Investor: Priya (Age 35, plans retirement at 55).
Investment: ₹1,00,000 per year for 10 years.
Interest Rate: 8% per year.
Deferment: No withdrawals for 10 years (age 35 to 45).
Payout Period: 15 years (age 55 to 70).
₹31,25,502 3,18,355
Scenario:
Investor: Raj (Age 50, plans to start withdrawals at 55).
Investment: ₹75,000 per year for 5 years.
Interest Rate: 6% per year.
Payout: ₹X per month for 20 years (Age 55-75).
Scenario:
Investor: Amit (Age 30) plans retirement at Age 55.
Investment:
₹50,000 per year for 5 years (Age 30-35).
₹75,000 per year for 5 years (Age 36-40).
₹1,00,000 per year for 5 years (Age 41-45).
Interest Rate:
6% for first 5 years.
7% for the next 5 years.
8% for the last 5 years.
Deferment: No withdrawals for 10 more years (Age 45-55).
Answer
Payout Period: 20 years (Age 55-75)
Compute Future Value (FV) for Each Investment Phase
Total FV at Age 55
Convert to 20-Year Payouts
Scenario:
Investor: Sunita (Age 35) plans retirement at Age 60.
Investment: ₹10,000 per month for 15 years.
Interest Rate: 8% annually (0.67% per month).
Deferment: No withdrawals for 10 years (Age 50-60).
Payouts: Inflation-adjusted withdrawals (3% increase each year) for 25 years
(Age 60-85).
Q1.A parent wants to save for their child’s higher education, which starts in
15 years. They estimate that they will need ₹12,00,000 to cover expenses
over 4 years, withdrawing ₹3,00,000 per year. The education fund will earn
7% annually during the withdrawal phase. Before withdrawals begin, they
invest money in a savings account at 9% annual interest.
A business signs a 10-year lease for office space with an annual rent of
₹5,00,000 in the first year. The rent increases by 5% per year. The business
wants to calculate the total cost of the lease.
An entrepreneur takes a business loan of ₹30,00,000 at 10% annual
interest, to be repaid in 8 years with equal annual payments. The business is
expected to generate ₹6,00,000 per year in profits, growing at 3% annually.
The entrepreneur wants to check if the business can sustain loan payments.
Perpetuity
A perpetuity is a type of annuity that pays a constant amount forever,
without an end date. Unlike regular annuities, which have a fixed number of
payments, a perpetuity continues indefinitely.
Key Features of Perpetuity:
Constant Payments: The same amount is paid at regular intervals (usually
annually).
No End Date: Payments continue forever.
Fixed Interest Rate: The payment amount is typically determined based on a
fixed interest rate
•Government Bonds: Some government bonds pay a fixed amount indefinitely (e.g., a bond paying interest without a maturity date).
•Charitable Donations: Charities sometimes set up perpetuity funds to ensure continuous funding.
Present Value of Perpetuity (PV):
The Present Value (PV) of a perpetuity is calculated using the formula:
PV=C / r
If you receive ₹20,000 every year forever, and the annual interest rate is 5%,
the Present Value (PV) of this perpetuity would be:
PV=20,000 /0.05=₹400,000
This means that the value of receiving ₹20,000 every year forever is ₹400,000
today if the interest rate is 5%.
You inherit a fixed annual payment of ₹25,000 forever, and the interest rate
is 6% per year. What is the present value of this perpetuity?
In some cases, perpetuities grow over time. This is called an increasing
perpetuity, where the payment increases by a fixed percentage each period.
Scenario:
You are promised ₹50,000 in the first year, but this amount increases by 5%
annually, and the interest rate is 7%. What is the present value of this
increasing perpetuity?
Formula for Present Value of Increasing Perpetuity:
The formula for the present value of an increasing perpetuity is:
PV=C/ r−g
C=₹50,000 (first-year payment)
𝑟=7%=0.07 (interest rate)
𝑔=5%=0.05 (growth rate)
A university sets up an endowment fund to provide ₹5,00,000 per year for
scholarships forever. The fund earns an annual return of 9%, but due to
inflation and administrative costs, only 7% of the total fund can be withdrawn
each year.
Question:
How much money should be initially deposited in the fund to ensure it provides
₹5,00,000 annually forever?
A retired individual wants to set up an increasing annuity trust that will
provide ₹2,00,000 in the first year and increase by 5% per year forever to
adjust for inflation. The trust’s investments yield an annual return of 8%.
Question:
How much money should be initially deposited to ensure the growing payments
last forever?
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