Calculating Present and Future Value of
Annuities
value is useful in determining the total cost of
the loan.
KEY TAKEAWAYS
for example, a series of five $1,000 payments
made at regular intervals.
•
•
Recurring payments, such as the rent on
an apartment or interest on a bond, are
sometimes referred to as "annuities."
In ordinary annuities, payments are made at
the end of each period. With annuities due,
they're
•
made at the beginning of
•
The future value of an annuity is the
total value of payments at a specific
point in time.
•
The present value is how much money
would be required now to produce those
future payments.
the period.
Two Types of Annuities
Because of the time value of money—the
concept that any given sum is worth more now
than it will be in the future because it can be
invested in the meantime—the first $1,000
payment is worth more than the second, and so
on. So, let's assume that you invest $1,000 every
year for the next five years, at 5% interest. Below
is how much you would have at the end of the
five-year period.
Annuities, in this sense of the word, break down
into two basic types: ordinary annuities and
annuities due.
Ordinary annuities: An ordinary annuity makes
(or requires) payments at the end of each period.
For example, bonds generally pay interest at the
end of every six months.
Annuities due: With an annuity due, by contrast,
payments come at the beginning of each period.
Rent, which landlords typically require at the
beginning of each month, is a common example.
Calculating the Future Value of an Ordinary
Annuity
Future value (FV) is a measure of how much a
series of regular
payments will be worth at
some point in the future, given a
specified interest rate. So, for example, if you
plan to invest a certain amount each month or
year, it will tell you how much you'll have
accumulated as of a future date. If you are
making regular payments on a loan, the future
Rather than calculating each payment
individually and then adding them all up,
however, you can use the following formula,
which will tell you how much money you'd have
in the end:
Calculating the Future Value of an Annuity Due
An annuity due, you may recall, differs from an
ordinary annuity in that the annuity due's
payments are made at the beginning, rather
than the end, of each period.
Calculating the Present Value of an Ordinary
Annuity
In contrast to the future value calculation, a
present value (PV) calculation tells you how
much money would be required now to produce
a series of payments in the future, again
assuming a set interest rate.
Using the same example of five $1,000 payments
made over a period of five years, here is how a
present value calculation would look. It shows
that $4,329.58, invested at 5% interest, would be
sufficient to produce those five $1,000
payments.
To account for payments occurring at the
beginning of each period, it requires a slight
modification to the formula used to calculate the
future value of an ordinary annuity and results in
higher values, as shown below.
The reason the values are higher is that
payments made at the beginning of the period
have more time to earn interest. For example, if
the $1,000 was invested on January 1 rather
than January 31 it would have an additional
month to grow.
The formula for the future value of an annuity
due is as follows:
Again, please note that the one-cent difference
in these results, $5,801.92 vs. $5,801.91, is due
to rounding in the first calculation.
Calculating the Present Value of an Annuity Due
Similarly, the formula for calculating the present
value of an annuity due takes into account the
fact that payments are made at the beginning
rather than the end of each period.
For example, you could use this formula to
calculate the present value of your future rent
payments as specified in your lease. Let's say you
pay $1,000 a month in rent. Below, we can see
what the next five months would cost you, in
terms of present value, assuming you kept your
money in an account earning 5% interest.
What is Annuity Formula?
The annuity formula helps in determining
the values for annuity payment and
annuity due based on the present value of
an annuity due, effective interest rate, and
several periods. Hence, the formula is
based on an ordinary annuity that
is calculated based on the present value of
an ordinary annuity, effective interest rate,
and several periods.
What Does Present Value Mean in the Annuity
Formula?
The word present value in the annuity formula
refers to the amount of money needed today to
fund a series of future annuity payments. The
value of money over time is worth more as the
sum of money received today has greater value
than the sum of money received in the future.
Examples Using Annuity Formula
P = $20,000 × 0.011...
Example 1: Dan was getting $100 for 5 years
every year at an interest rate of 5%. Find the
future value of this annuity at the end of 5 years?
Calculate it by using the annuity formula.
P = $220
Solution:
Jane won a lottery worth $20,000,000 and has
opted for an annuity payment at the end of each
year for the next 10 years as a payout option.
Determine the amount that Jane will be paid as
annuity payment if the constant rate of interest
in the market is 5%.
The future value
Given: r = 0.05, 5 years = 5 yearly payments, so
n = 5, and P = $100
FV = P×((1+r)n−1) / r
FV = $100 × ((1+0.05)5−1) / 0.05
FV = 100 × 55.256
Therefore, the value of each payment is $220.
Example 3:
Solution:
Given:
FV = $552.56
PVA (ordinary) = $20,000,000 (since the annuity
to be paid at the end of each year)
Therefore, the future value of annuity after the
end of 5 years is $552.56.
r = 5%
n = 10 years
Example 2:
If the present value of the annuity is
$20,000. Assuming a monthly interest rate of
0.5%, find the value of each payment after every
month for 10 years. Calculate it by using the
annuity formula.
Solution:
Given:
r = 0.5 % = 0.005
n = 10 years x 12 months = 120 , and PV =
$20,000
Using formula for present v alue
PV = P×(1−(1+r)-n) / r
Or, P = PV × ( r / (1−(1+r)−n))
P = $20,000 × (0.005 / (1−(1.005)−120))
P = $20,000 × (0.005/ (1−0.54963))
Using the Annuity Formula,
Annuity = r * PVA Ordinary / [1 – (1 + r)-n]
Annuity = 5% × 20000000 / [1 - (1 + 0.05)-10
Annuity = $2,564,102.56
Therefore, Jane will pay an annuity amount of
$2,564,102.56
Present Value of an
Annuity Formula (PV)
The formula for calculating the present
value (PV) of an annuity is equal to the sum
How an Annuity Works (Step-by-Step)
of all future annuity payments – which are
An annuity provides periodic payments for a
specific number of years until reaching
maturity.
divided by one plus the yield to maturity
Unique to annuities, there is no final lump
sum payment (i.e. the principal) paid back at
the end of the borrowing term, as with zerocoupon bonds.
of periods.
Unlike a perpetuity, an annuity also comes with
a pre-determined maturity date, which marks
the date when the final interest payment is
received.
Since there is no final principal repayment, each
payment is equal in value. However, payments
received earlier are more valuable due to the
“time value of money.”
Earlier cash flows can be reinvested earlier and
for a longer duration, so these cash flows carry
the highest value (and vice versa for cash flows
received later).
Typically, the most common types of issuers of
annuities are the following:
•
Insurance Companies (e.g. Retirement
Planning)
•
Mutual Funds
•
Brokerage Firms
•
Mortgage and Auto Financing
(YTM) and raised to the power of the number
PV = Σ A / (1 + r) ^ t
Where:
•
PV = Present Value
•
A = Annuity Payment Per Period ($)
•
t = Number of Periods
•
r = Yield to Maturity (YTM)
Alternatively, a simpler approach consists of two
steps:
1. First, the annuity payment is divided by
the yield to maturity (YTM), denoted as
“r” in the formula.
2. Next, the result from the previous step is
multiplied by one minus [one divided by
(one + r) raised to the power of the
number of periods].
Present Value (PV) = (A / r) (1 – (1 / (1 + r) ^ t))
Ordinary Annuity vs. Annuity Due: What is the
Difference?
When calculating the present value (PV) of an
annuity, one factor to consider is the timing of
the payment.
•
Ordinary
Annuity → Cash
Received at End of Period
•
Annuity Due → Cash Flows Received at
Beginning of Period
Flows
Present Value of an Annuity Due Table (PV)
The term “annuity due” means receiving the
payment at the beginning of each period (e.g.
monthly rent).
On the other hand, an “ordinary annuity” is more
so for long-term retirement planning, as a fixed
(or variable) payment is received at the end of
each month (e.g. an annuity contract with an
insurance company).
•
•
Fixed Annuity: The insurance company
provides interest in the form of periodic
payments that must meet the minimum
rate of return threshold.
Variable Annuity: The insurance
company allows owners to allocate their
annuity payments into certain low-risk
investment vehicles, such as mutual
funds.
The trade-off with fixed annuities is that an
owner could miss out on any changes in market
conditions that could have been favorable in
terms of returns, but fixed annuities do offer
more predictability.
Present Value of an Ordinary Annuity Table (PV)
Present Value of Annuity Calculation Example
(PV)
In our illustrative example, we’ll calculate an
annuity’s present value (PV) under two different
scenarios.
1. Ordinary Annuity
2. Annuity Due
The assumptions listed below are to be used for
the entirety of the exercise.
•
Annuity Payment: $1,000
•
Yield (r): 5%
•
Periods (t): 20 Years
First, we will calculate the present value (PV) of
the annuity given the assumptions regarding the
bond.
The “PV” Excel function can be used here, as
shown below.
•
Present Value (PV) = PV (r, Periods, –
Annuity Payment, 0, “0” or “1”)
•
Present Value (PV) = PV (5%, 20, –
$1,000, 0, IF (Annuity Type Cell
=“Ordinary”,0,1))
Note: Since we have two scenarios, we’ll create a
toggle to alternate between the two options –
which is the “IF(Annuity Type Cell =“Ordinary,”
0,1)”.
The two present value (PV) amounts calculated
on the annuity bond are the following:
•
Ordinary Annuity: $12,462
•
Annuity Due: $13,085
From there, we can also calculate the future
value (FV) using the formula below:
•
Future Value (FV) = – FV (r, t, Annuity
Payment, 0, “0” or “1”)
•
Future Value (FV) = – FV (5%, 20, $1,000,
0, IF (E5 = “Ordinary”, 0, 1))
The two future value (FV) amounts calculated on
the annuity bond are the following:
•
Ordinary Annuity: $33,066
•
Annuity Due: $34,719
We’ll calculate the yield to maturity (YTM) using
the “RATE” Excel function in the final step.
•
Yield to Maturity (YTM) = RATE (t,
Annuity Payment, 0, – FV, “0” or “1”)
•
Yield to Maturity (YTM) = RATE (20,
$1,000, 0, – FV, IF (E5 = “Ordinary”, 0,
1))”
The annuity bond has a yield of 5% under either
scenario.
What is Perpetuity?
A Perpetuity refers to a constant stream of cash
flows payments anticipated to continue
indefinitely.
In the prior example, the size of the cash flow
(i.e. the $1,000 annual payment) is kept constant
throughout the entire duration of the
perpetuity.
How to Calculate PV of Perpetuity (Step-by-Step)
In a perpetuity, the series of cash flows received
by the investor is expected to be received
forever (i.e. a never-ending stream of cash
flows).
For instance, if an investment comes with terms
stating that a $1,000 payment will be paid out at
the end of each year with an indefinite end, this
represents an example of a zero-growth
perpetuity (i.e. the annual payout remains the
same through the life of the investment).
Despite the cash flows theoretically lasting
“forever,” the present value (PV) – i.e. the
approximate valuation of the total potential
stream of cash flows as of the current date – can
still be calculated.
The “time value of money,” a fundamental
concept in corporate finance, states that the
further away from the date of when a cash flow
payment is received, the greater the reduction in
its value today.
As a result, the present value (PV) of the future
cash flows of a perpetuity eventually reaches a
point where the cash flow payments in the far
future have a present value of zero.
Perpetuity vs. Annuity: What is the Difference?
•
Perpetuity: To reiterate, perpetuities are
cash flows are expected to continue
forever with no ending date.
•
Annuity: In contrast, annuities comes
with a pre-determined maturity date,
which is when the final cash flow
payment is received.
Growing Perpetuity vs. Zero-Growth Perpetuity
However, for growing perpetuities, there is a
perpetual
(or
“continuous”) growth
rate attached to the series of cash flows.
If we assume equal initial payment amounts,
a growing perpetuity will thus be valued higher
than one with zero-growth, all else being equal.
For example, if the investment stated that
$1,000 would be issued in the following year but
at a 2% growth rate, then the annual cash flows
would increase 2% year-over-year (YoY).
Since the cash flows increase each year, the
growth rate helps offset the discount rate used
to calculate the present value (PV).
Perpetuity Formula
In order to calculate the present value (PV) of a
perpetuity with zero growth, the cash flow
amount is divided by the discount rate.
Present Value of Zero-Growth Perpetuity (PV)
= Cash Flow ÷ Discount Rate
The discount rate is a function of the opportunity
cost of capital – i.e. the rate of return that could
be obtained from other investments with a
similar risk profile.
For a growing perpetuity, on the other hand, the
formula consists of dividing the cash flow
amount expected to be received in the next year
by the discount rate minus the constant growth
rate.
Present Value of Growing Perpetuity (PV) = Year
1 Cash Flow ÷ (Discount Rate – Growth Rate)
What is a Growing Perpetuity?
A Growing Perpetuity is a series of future cash
flows expected to grow indefinitely at a
constant rate.
How to Calculate Present Value of Growing
Perpetuity (Step-by-Step)
A growing perpetuity is defined as a stream of
payments anticipated to grow at a constant rate
for an infinite number of periods.
Perpetuities are unique in that their cash flows
continue indefinitely with no ending date,
whereas annuities are a stream of cash flows
with a stated maturity, i.e. there is a predefined
date on which the final payment is received.
Since the periodic cash flows increase at a
fixed growth rate, each payment exceeds the
gross amount paid in the prior period.
The reason we can estimate the valuation of a
stream of perpetual cash flows is because of the
“time value of money” concept, which states
that a dollar today is worth more than a dollar
received in the future.
All future cash flows must thereby be discounted
to
the
present
date
using
an
appropriate discount rate that reflects the
riskiness of the cash flows (and the expected
return).
The more time between the present date and
the date on which a payment is expected to be
received, the more pronounced the effects of
discounting become, i.e. a greater discount is
applied.
That said, the present value (PV) of a growing
perpetuity gradually declines in value until
eventually reaching a point at which the present
value of the future cash flows drops to zero.
The process of calculating the present value (PV)
of a growing perpetuity consists of three steps:
•
Step 1. Determine the Cash Flow in the
Next Period (t=1)
•
Step 2. Subtract the Discount Rate (r) by
the Constant Growth Rate (g)
•
Step 3. Divide the Cash Flow (t=1) by (r –
g)
Note that the discount rate must be greater than
the growth rate assumption, or else the present
value of the growing perpetuity never reaches
zero (and thus, its present value would be an
infinite value).
Present Value of Growing Perpetuity Formula
The formula to calculate the present value of a
growing perpetuity is as follows.
Present Value of Growing Perpetuity = CF
t=1 ÷ (r – g)
Where:
•
CF t=1 → Periodic Cash Flow in Year 1
•
r → Discount Rate (Cost of Capital)
•
g → Constant Growth Rate
Growing Perpetuities
Perpetuities
vs.
Zero
Growth
The distinction between growing perpetuities
and zero growth perpetuities is the periodic cash
flows do not remain constant in the case of a
growing perpetuity.
If we are given two identical perpetuities—
where the only difference is the growth rate—
the present value of a growing perpetuity will be
greater than that of a zero-growth perpetuity.
For growing perpetuities, there is a constant
growth rate attached to the series of cash flows,
which partially offsets the effects of the
opportunity cost of capital.
Therefore, the perpetuity with growth continues
to retain value for a longer period into the future
compared to a perpetuity with no growth. But in
either case, the present value of the cash flows
in the far future eventually reaches zero.
Present Value of Growing Perpetuity Calculation
Example
Suppose you’re presented with the following
two options to pick from:
•
Option 1. $15,000 in Cash Today
•
Option 2. Perpetual Interest Payments
of $1,000 Continuously Growing at 3%
per year
Furthermore, we’ll assume that if Option 1 is
chosen, the rate of return that you could earn on
the $15k in cash is 10%.
Solved Examples Using Ordinary Annuity
Formula
•
In order to determine which investment is more
profitable, we’ll need to calculate the present
value of the growing perpetuity.
•
Year 0 Payment = $1,000
•
Discount Rate = 10%
•
Growth Rate = 3%
The first step is to increase the initial interest
payment by the 2% growth rate assumption to
arrive at the next period payment amount.
•
Year 1 Payment = $1,000 * (1 + 3%) =
$1,030
From there, we’ll subtract the growth rate from
our cost of capital assumption.
•
Present Value (PV) = $1,030 ÷ (10% – 3%)
= $14,714
If the only criteria for the investment decision
were picking the option that is of greater value,
Option 1 would be the right choice ($15k vs.
$14.7k).
Example 1: Alan was getting $100 for 5
years every year at an interest rate of
5%. Find the future value using the
ordinary annuity formula at the end of 5
years?
Solution:
The future value
Given: r = 0.05, 5 years = 5 yearly payments, so
n = 5, and P = $100
FV = P×((1+r)n−1) / r
FV = $100 × ((1+0.05)5−1) / 0.06
FV = 100 × 55.256
FV
=
$552.56
Answer: The longer-term value of annuity after
the end of 5 years is $552.56.
•
Example 2: If the present value of the
annuity is $20,000. Assuming a monthly
interest rate of 0.5%, find the value of
each payment after every month for
10 years.
Solution
To find: The value of each payment
Given:
r = 0.5% = 0.005
n = 10 years x 12 months = 120, and PV = $20,000
Using formula for present value
-n
PV = P×(1−(1+r) ) / r
Or, P = PV × ( r / (1−(1+r)−n))
P = $20,000 × (0.005 / (1−(1.005)−120))
P = $20,000 × (0.005/ (1−0.54963))
P = $20,000 × 0.011...
P = $220
Answer: The value of each payment is $220.
How to Calculate Annuity Payment? (Step by
Step)
The calculation of annuity payment can be
derived by using the PV of ordinary annuity in the
following steps:
1. Firstly, determine the PV of the annuity
and confirm that the payment will be
made at the end of each period. It is
denoted by PVA Ordinary.
2. Next, determine the interest rate based
on the current market return. Then, the
effective rate of interest is computed by
dividing the annualized interest rate by
the number of periodic payments in a
year, and it is denoted by r. r =
Annualized interest rate / Number
regular payments in a year
3. Next, determine the number of periods
by multiplying the number of periodic
payments in a year and the number of
years, and it is denoted by n. n = number
of regular payments in a year * Number
of years
4. Finally, the annuity payment based on
PV of an ordinary annuity is calculated
based on PV of an ordinary annuity (step
1), effective interest rate (step 2), and
some periods (step 3), as shown above.
The calculation of annuity payment can also be
derived by using the PV of an annuity due in the
following
steps:
Step 1: Firstly, determine the PV of the annuity
and confirm that the payment will be made at
the beginning of each period. It is denoted by
PVA Due.
Step 2: Next, determine the interest rate based
on the current market return. Then, the effective
rate of interest is computed by dividing the
annualized interest rate by the number of
periodic payments in a year, and it is denoted by
r. r = Annualized interest rate / Number regular
payments
in
a
year
Step 3: Next, determine the number of periods
by multiplying the number of periodic payments
in a year and the number of years, and it is
denoted by n. n = number of regular payments in
a
year
*
Number
of
years
Step 4: Finally, the annuity payment based on PV
of an annuity due is calculated based on PV of an
annuity due (step 1), effective interest rate (step
2), and several periods (step 3), as shown above.
Example #1
Let us take the example of David, who won a
lottery worth $10,000,000. He has opted for an
annuity payment at the end of each year for the
next 20 years as a payout option. Determine the
amount that David will be paid as annuity
payment if the constant rate of interest in the
market is 5%.
Given below is the data used for the calculation
of annuity payments.
We will use the same data as the above example
for the calculation of Annuity payments.
PVA Ordinary = $10,000,000 (since the annuity to
be paid at the end of each year)
Therefore, the calculation of annuity payment
can be done as follows –
•
Annuity = 5% * $10,000,000 / [1 – (1 +
5%)-20]
Calculation of Annuity Payment will be –
Therefore, the calculation of annuity payment
can be done as follows –
•
Annuity = r * PVA Due / [{1 – (1 + r)-n} * (1
+ r)]
•
Annuity = 5% * $10,000,000 / [{1 – (1 +
5%)-20} * (1 + 5%)]
Calculation of Annuity Payment will be –
•
Annuity = $802,425.87 ~ $802,426
Therefore, David will pay annuity payments of
$802,426 for the next 20 years in case
of ordinary annuity.
Example #2
Let us take the above example of David and
determine the annuity payment if paid at the
beginning of each year with all other conditions
the same.
•
Annuity = $764,215.12 ~ $764,215
Therefore, David will pay annuity payments of
$764,215 for the next 20 years in case of an
annuity due.
Problem 1: Present value of annuity
Problem 5: Present value of ordinary annuity
You are making car payments of $315/month for
the next 3 years, you know that your car loan has
an interest rate of 12.4%, discounted monthly,
what was the initial price of the car?
Mr. Mohammad Ali has received a job offer from
a large investment bank as an accountant. His
base salary will be $35,000 constant to date of
retirement. He will receive his first annual salary
payment one year from the day he begins to
work. In addition, he will get an immediate
$10,000 bonus for joining the company. Mr. Ali
is expected to work for 25 years. What is the
present value of the offer if the discount rate is
12 percent?
Answer:
=$9,429.53
Problem 2: Present value of annuity table
Solution:
Mr. Naeem has won a scholarship which pays
him $5,000 per year for 3 years beginning a year
from today. He wants to know the present value
of the scholarship using a discount rate of 7%.
Solve by Factor Formula?
PVA25 = 274,509.87
Solution:
PV2 = 9,245.76 / (1 + 0.08) 2
Answer: $7,926.75
Problem 4: PV of annuity using intra-year
discounting
Find the present value of an annuity with
periodic payments of $2,000, for a period of 10
years at an interest rate of 6%, discounted
semiannually by factor formula?
Solution:
PVA10 = 2,000 (PVIFA 6%/2, 10*2)
PVA10 = 2,000 (2.1065)
Answer: $4,213
Bonus = 10,000
Answer: $284,509.87
Problem 6: Present value of annuity due
PVA6 = $17,022.53
Mr. Khaild will receive $8,500 a year for the next
15 years from her trust. If a 7 percent interest
rate is applied, what is the current value of the
future payments if first receipt occurs today?
PV4 = 17,022.53/ (1 + 0.08/4) 4*4
Solution:
Answer: $ 12,400
Problem 9: Present value of an ordinary annuity
table
Find the present value of due annuity with periodic
payments of $2,000, for a period of 10 years at an
interest rate of 6%, discounted semiannually by
factor formula and table?
Solution:
Answer: $82,836.48
Problem 7: Present value of an annuity due
What is the present value of an annuity due that
makes 5 annual payments of $200 each if the
discount rate is 12% by general formula constant
rate and general floating formula?
Solution:
Answer: $807.47
Answer: $807.48
Problem 8: Present value of an ordinary annuity
A 10-year annuity pays $900 four times in year. The
first $900 will be paid five years from now. If the
stated interest rate is eight percent, discounted
quarterly, what is the present value of this annuity?
Solution:
What Is the Present Value of an Annuity?
The present value of an annuity is the current value
of future payments from an annuity, given a
specified rate of return, or discount rate. The higher
the discount rate, the lower the present value of the
annuity.
Present value (PV) is an important calculation that
relies on the concept of the time value of money,
whereby a dollar today is relatively more "valuable"
in terms of its purchasing power than a dollar in the
future.
It's important to note that the discount rate used in
the present value calculation is not the same as the
interest rate that may be applied to the payments in
the annuity. The discount rate reflects the time value
of money, while the interest rate applied to the
annuity payments reflects the cost of borrowing or
the return earned on the investment.
Formula and Calculation of the Present Value of an
Annuity
The formula for the present value of an ordinary
annuity, is below. An ordinary annuity pays interest
at the end of a particular period, rather than at the
beginning:
Example of the Present Value of an Annuity
Assume a person has the opportunity to receive an
ordinary annuity that pays $50,000 per year for the
next 25 years, with a 6% discount rate, or take a
$650,000 lump-sum payment. Which is the better
option? Using the above formula, the present value
of the annuity is:
Present value=$50,000×1−(1(1+0.06)25)0.06=$639,1
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