Growing annuity &
Perpetuity
Lecture for Corporate Finance
Ing. Tomáš Heryán, Ph.D.
Corporate Finance
FIU/BAFIK
Outline of the lecture
• PV and FV of growing annuity
• PV of perpetuity
• Selected interesting combinations
Growing annuity
The present value of a growing
annuity formula calculates the
present day value of a series of
future periodic payments that grow
at a proportionate rate. A growing
annuity may sometimes be referred
to as an increasing annuity.
For a growing annuity, each cash
flow increases at a certain rate. This
formula is the general formula for
summing the discounted future cash
flows along with using 1 + g to
factor in that each future cash flow
will increase at a specific rate.
Like all financial formulas that involve a
rate, it is important to correlate the rate
per period to the number of periods in
the present value of a growing annuity
formula. If the payments are monthly,
then the rate would need to be the
monthly rate (=EAIR).
The future value of a growing annuity
can be calculated by working out each
individual cash flow by (a) growing the
initial cash flow at g; (b) finding future
value of each cash flow at the interest
rate i and (c) then summing up all the
component future values. The future
value of a growing annuity can also be
calculated by growing the present value
of the growing annuity at i for n periods.
Growing vs. Ordinary Annuity
Growing vs. Ordinary Annuity
Growing vs. Ordinary Annuity
In the picture we see that we can really save
more money using the growing annuity than just
ordinary annuity.
When the growth will be approx. as the inflation,
then we are able to save the purchasing power of
our money.
Fight against inflation
1. You are going to start saving for your children's future studies. This year you will be
able to save 5 thousand CZK. This amount will increase by 4% each year according to
your calculations. The alternative cost is 8%. How much will you save in 15 years?
• Each other year You will save 4% more than the previous year.
Present value of growing annuity
2. What is the present value of your inheritance if your trustee is obliged to pay you a
yearly amount starting this year for 10 payments, while this year you will be paid 2,000
USD (right now)? We assume alternative costs of 11%.
• It is still important remember what You have already learned! ☺
PV of growing annuity
PV of growing annuity
PV of growing annuity
The inflation can have really destructive impact
on the purchasing power of our money. However,
on the other hand, if we will withdraw more
money each other year (growing A), we will have
smaller amount than using the ordinary annuity
from the point of view of present value.
Future value of growing annuity
3. You started saving for retirement this year. You are going to deposit 2% of your
annual income every year, which is 150 thousand. CZK and will grow by 4% each year
for the duration of your employment. You anticipate 8% savings per year. What amount
will you have if you retire in 40 years?
• Please, be careful of the amount of annuity payment.
Two periods
4. What amount of funds will you dispose of in 2034, if you save 40 thousand CZK and
next year by 4% more? While you will stop saving in 2032, the money will be left in
your account. Your account bears an interest rate of 6% p.a.
• Combining of two periods is common ☺ CONTROLLING is necessary.
Be careful of a solution
5. The couple saved 1.5 million CZK. They plan to use this money annually during
their retirement over 20 years. The interest rate on such savings is 10% p.a. How much
can they spend the first time if they want to increase the amount by 5% p.a.?
• A little more complicated one? ☺
Outline of the lecture
6. What is the present value of your heritage? Since this year, you will obtain a certain
amount each year for 10 payments. This year it is 25,000, - CZK and every next year
by 10% more. The alternative cost is 12%.
• However, this is a usual example ☺
Bond yields (JUNK vs. INVEST. GRADE)
Comparison of bond yields
Comparison of bond yields
Junk bonds are high-paying bonds with a lower credit rating than investmentgrade
corporate bonds, Treasury bonds, and municipal bonds. Junk bonds are
typically rated 'BB' or lower by Standard & Poor's and 'Ba' or lower by Moody's.
Despite their name, junk bonds can be valuable investments for informed
investors, but their potential high returns come with the potential for high risk.
PV of perpetuity
A perpetuity is a security that pays
for an infinite amount of time. In
finance, perpetuity is a constant
stream of identical cash flows with
no end.
Specifically, the perpetuity formula
determines the amount of cash flows
in the terminal year of operation. In
valuation, a company is said to be a
going concern, meaning that it goes
on forever.
An annuity is a stream of cash flows. A
perpetuity is a type of annuity that lasts
forever, into perpetuity. The stream of
cash flows continues for an infinite
amount of time.
In finance, a person uses the perpetuity
calculation in valuation methodologies
to find the present value of a company's
cash flows when discounted back at a
certain rate.
An example of a financial instrument
with perpetual cash flows is the Britishissued
bonds known as consols. By
purchasing a consol from the British
government, the bondholder is entitled to
receive annual interest payments forever.
Perpetuity means infinity
Cash flow diagram of perpetuity
Perpetuity means infinity
Because of the never ending period, FV of
perpetuity does not exist! Therefore, we should
calculate only PV.
Examples PERPETUITY
7. Calculate the market price of the share if the alternative cost is 5% and this year's
dividend is100 CZK per share. The company's dividend policy is stable.
• The question is just whether the dividend was not already paid.
Examples of shares 1/4
8. What is the market price of a share if DPS a) will be 100 CZK; b) was 100 CZK this
year. We do expect DPS to increase by 10% each year and alternative costs of 15%?
• If the dividend was already paid out we should count with the increased one.
Examples of shares 2/4
9. What is the market price of a share if the dividend per share (DPS) was CZK 100 per
share last year? We do expect DPS to increase by 10% each year and alternative costs
of 15% this year and beyond?
• You should calculate present value, so for the current year.
Examples of shares 3/4
10. The company decided to pay a dividend of CZK 400 per share next year, with the
dividend increasing by 4% per year. What is the current value of dividend income if the
interest rate is 14%.
• This kind of decision is made during board meeting when dividends
have been already paid.
Examples of shares 4/4
11. What should be the market value of one share of the company, if the expected
profit this year is 5 million CZK, the company has 10 thousand of outstanding shares
and shareholders will pay 50% of earnings in the form of dividends? The expected
annual increase in profit is 5%. The alternative cost is 12%.
• As the first You should calculate DPS 250,- CZK ☺
Perpetuity means forever
12. You own the land you rent for an indefinite period (forever). The land yield will
increase annually by 5%. If the cash flow in the first year is CZK 10,000 and the
interest rate is 10%, what is the value of the land?
• You should receive the rent right now and calculate with future cash flows.
Harder combinations 1/3
13. SINCE next year, you intend to deposit a certain amount each year up to 2030
(including this year as well as 2030). Then you want to withdraw CZK 12,000 each
year from your account during the following years (2031 - 2047). If your account bears
2% p.a., how much do you have to deposit?
• You have to have the money saved to withdraw 12,000 (PV).
However, as the first think You have to save this money as FV from
this certain amount (AFV)
Harder combinations 2/3
14. SINCE next year, you intend to deposit a certain amount each year up to 2030
(including this year as well as 2030). The first year it will be 10,000, - CZK and every
next year by 2% more. During the following years (2031 - 2036), how much will you
be able to withdraw from your account each year? The account bears 3% p.a.
• As the first, You will save this 10,000 into the future (FV). Then You
will withdraw certain amount from the money You have already saved
(APV)
Harder combinations 3/3
15. You have won CZK 1 million in the lottery and you will save your winnings to the
bank today. You will add CZK 100,000 annually to the same account for 10 years
between 2025 and 2034. How much will you have in your account at the beginning of
2055? The account bears 3% p.a. at a monthly basis.
• Three steps should be done ☺
Stock prices’ development
S&P 500
S&P 500
The S&P 500 index consists of most but not all of the largest companies in the United States. The S&P market cap is 70 to 80% of
the total US stock market capitalization. It is a commonly used benchmark for stock portfolio performance in America and abroad.
Beating the performance of the S&P with less risk is the goal of nearly every portfolio manager, hedge fund and private investor.
Literature:
1) Seminar 05, Corporate Finance.
2) ROSS, S. A., R. W. WESTERFIELD, J. JAFFE & B. D. JORDAN, 2019. Valuation
and Capital Budgeting. In: Corporate Finance, PART II, pp. 85-298. ISBN 978-1-
260-09187-8.
3) BERK, J. & P. DeMARZO, 2017. The Time Value of Money. In: Corporate
Finance, Chap. 4, pp. 130-174. ISBN 978-1-292-16016-0.
Thank you for
your attention!
☺