Time value of money
Presentation for the Corporate Finance
Ing. Tom Heryán, Ph.D.
Corporate Finance
Content of this lecture
• Time value of the money
• Basics of the interest rates
• Basics related to bonds
• Few combinations
Present vs. Future value
The time value of money is a key
concept in financial management,
respectively corporate finance.
It is important for company’s
investments, financing and dividend
decisions, which result in significant
cash flows over a variety of time
periods.
Imagine that somebody offers you 50 €
now or 50 € in one year. Would you
prefer money now or next year? You
would choose probably money now for
three main reasons:
The first one is time. You prefer money
now, because you can spend it. You can buy
whatever you want. You can invest it, if you
do not want to spend it now and profit from
it. In every case you take it now.
The second reason is influence of inflation.
If you take money now, you can buy some
certain amount of products, which you could
not buy in one year, because they will be
more expensive. Inflation causes increase of
their price.
And the third reason is risk. If you take
money now, nobody can take it back. It is
yours instead of the situation that somebody
promised you 50 € in one year and
something would happen and you would not
get them or at least not all of it.
Example FV
1. What amount will you be able to withdraw from your bank account after 9 years if
you deposit CZK 50,000 today and the deposit bears 2.5% p.a.?
• It is much better to use directly 1.025 ☺
• However, never forget the ZERO with % in its absolute value!
Example PV
2. What is the present value of the investment, which after 15 years will yield a return
of CZK 1 million? Alternative costs can be 8% p.a.
• Again, 1.08 instead (1+0.08)! ☺
Interest rate / average costs / yield ??
3. The entrepreneur expects this year's profit of 500 thousand CZK. For the year would
like to invest in new production technology 580 thousand CZK. At what yield can he
realize his plan if the profit invested?
• It is up to You which formulae You want to use to find out (i) ☺
• Please, be aware of a mistake with MINUS one!
Time value of the money
Time value
Time value
The most important question is:
What are we going to calculate ??
The future or the present value?
The answer is always clear when we understand
what we do know within an exercise.
The solution is always unknown then!
Example:
If we know future value we cannot calculate its
future value again!
Example Bonds
4. What is the present value of a bond with a nominal value of CZK 3,000, if you know
that the yield to maturity is 6.7% p.a.? The annual coupon payment is 5.5% and the
bond is payable in 5 years.
• Never forget on the final step – to discount a nominal value ☺
Explanation of the bond valuation
Time value: BONDS means PRESENT value!
BOND = Present value
In the case of the BOND, no matter what, its
market value, price to buy/sell, it is always the
PRESENT value!
All the time we would like to know it right now!
Furthermore, we do know its future value
because of we are going to pay the debt back for
its nominal value when its maturity expires.
Future or the Present? That’s the question
5. Compare the two following revenue if the alternative cost is 11% p.a .:
a) after 3 years you will receive CZK 10 million;
b) after 5 years you will receive CZK 20 million
• Of course, we are taking into account just the revenue of the
investment, not its liquidity nor its risk (Magic triangle).
Multi-period compounding
6. You want to deposit 50 thousand CZK to the bank for one year. One bank offers you
a Quarterly interest rate of 2.8% p.a. and the second bank a Continuous compounding
interest rate 2.7% p.a.
a) What is more advantageous for you?
b) What will be the situation in both banks after 2 and 3 years?
Interest vs. Discount rates
The preceding analysis is typical of
decision making in corporations
today, though real-world examples
are, of course, much more complex.
Unfortunately, any example with
risk poses a problem not faced in a
riskless example. Conceptually, the
correct discount rate for an
expected cash flow is the expected
return available in the market on
other investments of the same risk.
This is the appropriate discount rate to
apply because it represents an economic
opportunity cost to investors. It is the
expected return they will require before
committing funding to a project.
Because the choice of a discount rate is
so difficult, we merely wanted to broach
the subject here.
We must wait until the specific material
on risk and return is covered in later
lectures before a risk-adjusted analysis
can be presented.
Multi-period compounding with the rates
7. The Bank offers an interest rate on deposits of 3% p.a. Calculate the effective
average interest rate if the interest is on:
a) annually
b) semi-annually
c) quarterly
d) monthly
e) daily
Multiple-period compounding FV
8. What will be the value of 12 thousand CZK after 5 years for all those interest rates
from the previous example?
• Surely, we can compound each interest rate and
use that within the basic formulae. But this is a much
more precise way to count this example ☺
Compounding
Time value: Compounding
Time value: Compounding
As we see in the picture if we
compound more often (i.e. semiannual,
quarterly, monthly, daily, or
even continuously), it means we are
going to earn more money because
of the interests earned.
GOOD FOR US! ☺
Discounting
Time value: Discounting
Time value: Discounting
However, on the other hand,
when we are going to discount
the money, or pay even the
interests in the case of PV within
a loan, more often discounting
(i.e. continuous), it means higher
costs in form of interests paid.
DEFINITELY NOT GOOD
whether we are not a bank.
EAIR or p.a. compounding?
The distinction between the annual
percentage rate, APR or p.a., and the
effective annual interest rate (EAIR)
is frequently troubling to students.
We can reduce the confusion by
noting that the APR becomes
meaningful only if the compounding
interval is given. (e.g. 10Y)
In other words, we do not know
whether to compound semiannually,
quarterly, or over some other
interval. By contrast, the EAIR is
meaningful without a compounding
interval. (e.g. 10Y again)
There can be a big difference between an
APR and an EAIR when interest rates
are large. For example, consider
"payday loans".
Payday loans are short-term loans made
to consumers, often for less than two
weeks. They are offered by companies
such as Check Into Cash and AmeriCash
Advance. The loans work like this: You
write a check today that is postdated.
When the check date arrives, you go to
the store and pay the cash for the check,
or the company cashes the check. For
example, in one particular state, Check
Into Cash allows you to write a check
for $115 dated 14 days in the future, for
which they give you $100 today.
Combining current knowledges 1/5
9. What will be the value of your deposit at the bank if this deposit bears 1.7% p.a.
two years, and then another four years quarterly? The amount of the deposit is
90 thousand CZK.
• Always think carefully before You start compounding! ☺
Combining current knowledges 2/5
11. The CNB treasury bill of nominal value 100 thousand CZK with a maturity of one
year, the market price is CZK 89,000. Calculate the present value Factor and the
corresponding alternative cost.
• PV factor is just:
Combining current knowledges 3/5
12. Imagine you won 100,000 CZK today in a lottery. You are going to save this money
on your bank account and buy a used car in 5 years. You assume that you need CZK
161,050 at that time. What is the interest rate for your account? Is this rate a real one
nowadays?
• REAL ???
Nominal vs. Real GDP ???
Combining current knowledges 4/5
13. Mrs. ZHANG plans to buy lucrative land around the future highway. The land is
now for sale for CZK 8.5 million and experts expect the price to rise to around
CZK 9.1 million in one year. This would amount to a profit of CZK 600,000.
a) Is such an investment advantageous if she can invest money with a yield of 10% p.a.
b) What would the interest rate have to be in order for her to make a profit of
600 thousand CZK?
• Only in few cases it is up to You, whether You use the formulae or just the logic ☺
Combining current knowledges 5/5
10. What is the effective average interest rate if the account is
continuously compounded at 2.7%?
• It seems to be a simple one, doesn’t it? ☺
Summarizing
The first lecture of this course has
examined the concepts of future
value and present value.
Although these concepts allow us to
answer many problems concerning
the time value of money, the effort
involved can be excessive.
Consider a bank calculating the
present value of a 20-year monthly
mortgage—or annuity payments
of mortgage loans, stocks, etc.
Because many basic finance problems
are potentially time-consuming, we will
search for simplifications in a few future
lectures.
We will provide simplifying formulas
and examples for four classes of cash
flow streams:
▪ Perpetuity
▪ Growing perpetuity
▪ Annuity
▪ Growing annuity
Literature
1) Seminar 01 Examples
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.
• Please, use Your own calculator…not a smart phone or anything smart instead Your brain! ☺
Thank you for
your attention!
☺