UNISEMINAR
Seminar
Extras
Exams
Practice
Theory
Introduction
Finance Academic Year 2012/2013, Block 5
Introduction
Uniseminar – Finance
Uniseminar – Finance
Welcome to Uniseminar!
Introduction
I n t r o d u c t i o n
Uniseminar offers E x a m P r e p a r a t i o n S e m i n a r s , S u m m a r y S c r i p t s a n d L e a r n i n g C a r d s for students of the Maastricht University. It is our goal to op-‐
timally prepare you for your exams and to make your own exam preparation as
efficient as possible. In order to achieve this goal, we have developed a system of
seminars in combination with an extensive summary script, which is proven for several years by now.
In university it is often the case that there is a lot of material available for a course and that the importance of this material is hard to evaluate. Since we, as
students, have made this experience as well, you are provided with a Uniseminar Summary Script of the corresponding course. This folder contains all exam-‐ relevant material and it gives you a good summary of all course topics. The con-‐
tent of the folder is created by experienced Master or PhD students, who have taught this course already several times. As a consequence, it is possible for you to concentrate on the actual exam preparation, rather than spending hours searching and printing the right material.
At the end of week 6 of your block, normally during the weekend, our E x a m
P r e p a r a t i o n S e m i n a r s take place. These seminars are taught by above-‐
average students, who have already mastered their studies at the Maastricht University and have a great deal of experience in tutoring. Since they have stud-‐
ied and taught at the Maastricht University they know exactly where potential problems may lie and are therefore able to optimally teach you the whole theory
of the course and practice perfectly tailored examples with you. Furthermore you can bring in your own questions during the seminar and discuss individual prob-‐ lems during the breaks.
You are able to pick up your S u m m a r y S c r i p t a n d L e a r n i n g C a r d s in ad-‐
vance of the Seminar in order to already start preparing so that you can discover
your own difficulties early enough. Later in the Seminar you will then know what
Introduction
Uniseminar – Finance
your weaknesses are and be able to pay special attention to these sections or ask
questions about it. Our Summary Script and Learning Cards are updated every
year according to the current course’s content and we are always trying to opti-‐
mize the folder as much as possible.
A b o u t U U s
Uniseminar was founded 5 years ago by two students at the University of St. Gallen in order to make Exam Preparation more efficient and coherent. Since
2005 we have expanded our vision and are now offering seminars and material for an efficient exam preparation in Switzerland, the Netherlands, Italy and Ger-‐ many.
Thanks to this longstanding experience, we were able to build up a team of highly
qualified tutors and editors and are therefore able to guarantee high quality of exam preparation.
The team of Uniseminar is grown strongly over the years and comprehends sev-‐
eral mathematicians, statisticians and economists, who all bring a great didactical experience. All tutors of Uniseminar have been teaching their field for years and
know exactly what is important in order to optimally prepare and pass the exam.
Uniseminar – Finance
S u m m a r y SS c r i p t
Introduction
This aim of this folder is to support you with your exam preparation for ‘Finance’
as much as possible. Usually it consists of five different sections. As follows, a short overview of the content of this folder:
1 . T h e o r y : The theory script summarizes the whole theory of the course in
a simple and understandable way. Concepts are explained with the help of demonstrative examples. It is structured according to the seven weeks of the course and is one of the most important parts of your exam prepara-‐ tion.
2. P r a c t i c e : The practice part contains practice exercises to each week and therefore to each chapter of the theory script. By this, you can deepen your
theoretical knowledge with practical exercises and you can go through the exercises of these topics again, which you have not understood so well un-‐
til now.
3. E x a m s : In this part you will find old exams of the Maastricht University, as well as one practice exam constructed by Uniseminar. During the semi-‐ nar you will then receive a further practice exam.
4. E x t r a s : In the Extras part, you will find a formula sheet as well as an ex-‐ tensive glossary, containing all important definitions.
5. S e m i n a r : In this part, we have provided you with some notepaper so that you can take notes during the seminar. Furthermore you will receive a fur-‐
ther practice exam during the seminar, which you can file in here. In case
you have not subscribed for the Finance seminar yet, you can do so on our
website -‐ www.uniseminar.nl -‐ at any time.
Seminar
Extras
Exams
Practice
Theory
T
Theory
Finance Academic Year 2012/2013, Block 5
Theory
T h e o r y
Uniseminar – Finance
The theory script summarizes the whole theory of the course in a simple and
understandable way. Concepts are explained with the help of demonstrative examples. It is structured according to the seven weeks of the course and is one of the most important parts of your exam preparation. Although practice is very
important, it is even more crucial to understand the basic concepts of the course
in order to be able to calculate and understand all different kinds of exercises and exam questions.
T a b l e o o f C C o n t e n t s 0 Finance
1 Institutional Frameworks
2 Mergers & Takeovers
1
1
7
3 Financial Decision Making & Interest Rates
12
5 Valuing Bonds
30
7 Weighted Average Cost of Capital
51
4 Investment Decisions
6 Stocks and Portfolio Valuation
8 Investor Behavior & Capital Market Efficiency 9 Financial Options
10 Financial Statements
22
38 55 60
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0
F i n a n c e
Theory
This first course in finance gives an introduction into financial markets and the
firms that are interacting within them. Moreover, it explains which advantages
and disadvantages certain financial decisions have. Finally, the course also has an
analytical focus and thereby teaches the necessary quantitative skills to value
different types of investments or financial products.
As a summary this manual is structured very similarly to the course. The first
section will lay out a conceptual framework of companies and financial markets, while the second section focuses on the mathematic tools that will be important.
The sections thereafter will then use these tools to make investment decision or to analyze and price financial products.
1
I n s t i t u t i o n a l F F r a m e w o r k s
Before we can jump into the important issues like making investment decisions, bond pricing or stock valuation we need to clarify some general terms that are of
great importance in the field of finance. Therefore this section deals with the
institutional set-‐up of financial markets and the companies that participate in these markets. The main purpose thereby is to give you a broad picture and leaving out the analytical details. In the next sections the focus then shifts and the topics are analyzed in a more mathematical and technical way.
1 . 1 S t a k e h o l d e r s o o f aa ff i r m
In general everybody who is in some sort connected to a company is called a s t a k e h o l d e r of that company. However, obviously there are lots of different
groups who are connected to a certain company and the interests of these groups
are often not in line with each other. The next section will further elaborate on problems that arise out of a clash of interests.
1
Theory
Uniseminar – Finance
The owners of a firm are called the s t o c k h o l d e r s or s h a r e h o l d e r s . This expression is, however, only common for companies that are listed on an
exchange and thereby have a publicly traded amount of shares. Their owners do generally not manage such corporations as ownership changes whenever the
shares of the company are sold on a stock exchange. Therefore another group of stakeholders are the m m a n a g e r s of the firm. Obviously this group coincides with
employments in general who also have a concern about the performance of the
company. Finally, companies commonly borrow money either from a bank or by
issuing bonds in the financial markets. Anyone who lends money to a company
becomes a cc r e d i t o r of that company. Moreover, someone who bought a bond of b o n d h o l d e r . a certain company in the financial markets is referred to as a b
Whenever a company goes bankrupt its bondholders have priority over the
stockholders. That is creditors of a company get paid back before the owners of the company receive payments. On the contrary, bondholders only borrow
money to a company and therefore have no impact on the decisions taken by the company. Shareholders own the company and can influence decision taken by the company because they can elect the managers that run the company.
1 . 2 L e g a l tt y p e s o o f cc o m p a n i e s
Before one can analyze the behavior of a firm in the financial markets one needs to think about the characteristics of the firm. In general we can distinguish
between four different l e g a l f o r m s . However, as regulations differ across
countries these types will not be exactly the same for all countries, even though the general scheme is quite similar. The following table summarizes the different types of firms as well as some advantages and disadvantages for the USA.
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Uniseminar – Finance
T y p e D e s c r i p t i o n S o l e Most common p r o p r i e t o r s h i p type of company. The company is managed by its single owner. P a r t n e r s h i p
L i m i t e d L i a b i l i t y C o m p a n y ( L L C )
C o r p o r a t i o n
Theory
P r o Ownership and control are exercised by the same person
C o n t r a Owner of the company is liable for the entire company losses. Company dissolves itself if the owner dies. Similar to sole Ownership and General partners are proprietorship, control are not liable for the entire with the separated and company losses. difference that ownership is not The death of a the firm can have limited to one general partner multiple owners person. dissolves the called partners. Limited partners company. Can also have are only liable to Limited partners are limited partners the extent of their not involved in who bear no investment. management company risk. decisions. Partnership with Limited partners Agency problems can only limited are now allowed arise if limited partners. to run the partners do not Inspired by the business. chose to manage the German GmbH Limited partners company. and established are only liable to in the USA in the extent of their 1977. investment. Company is Shareholders are Agency problems owned by liable to the extent arise because shareholders and of their management and the shares are investment in ownership are publicly traded. shares. clearly separated. Thereby the owners do not run the company.
Finally, one needs to distinguish between two different types of corporations.
The criterion that distinguishes between these two types is tax regulation. In an 3
Theory
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Uniseminar – Finance
F i n a n c i a l D D e c i s i o n M M a k i n g & & II n t e r e s t R R a t e s
If you borrow money from a bank you have to pay i n t e r e s t on it. If you save
money you will earn interest. This section will first outline why financial markets exist and how the interest rate is determined in the process of balancing savings
and investments. This is the so-‐called model of intertemporal choice. Thereafter
the main part of the section will show how a known interest rate can be used to
shift different streams of payments through time. Shifting payments forward in time is referred to as compounding, while discounting means shifting payments
backwards in time. Furthermore, this section discusses some special topics like arbitrage and the separation principle
3 . 1 I n t e r t e m p o r a l cc o n s u m p t i o n aa n d ff i n a n c i a l m m a r k e t s
The reason why financial markets exist is that people want to shift income between different points in time. This means that you can either increase your
future income by saving today or you can increase your current income through borrowing. Obviously saving reduces your current income, while borrowing decreases your future income.
Whether a certain person borrows or saves depends on the set of preferences as
well as the stream of current and future incomes. The first means that some people value present consumption much higher as future consumption and hence prefer to spend everything as soon as possible. Such a person will rather
use the financial markets to borrow and thereby increase his current
consumption. The second means that – given someone’s income is very low at present but will be very high in the future, this person is likely to shift some future income to the present through borrowing in the financial markets.
12
Uniseminar  –  Finance  Â
Â
Theory Â
Â
The  above  graph  represents  exactly  the  different  options  outlined  before.  First  of Â
all,  imagine  a  certain  consumer  that  has  an  income  of  đ?‘Œđ?‘Œ  in  period  one  and  an Â
income  of  đ?‘Œđ?‘Œ  in  period  two.  If  there  are  no  financial  markets  the  consumer  has  no Â
chance  of  shifting  income  between  periods  and  consumes  exactly  đ?‘Œđ?‘Œ  in  period  one Â
and  đ?‘Œđ?‘Œ  in  period  two.  However,  if  financial  markets  exist  it  is  possible  to  shift Â
income.  Thereby  it  is  easiest  to  think  about  the  extreme  scenarios.  If  that  consumer  decides  to  spend  nothing  in  period  one  but  everything  in  period  two Â
he  can  spend  at  most  đ?‘Œđ?‘Œ 1 + đ?‘&#x;đ?‘&#x; + đ?‘Œđ?‘Œ .  This  comes  from  the  fact  that  he  can  save  đ?‘Œđ?‘Œ Â
in  the  financial  markets  and  earns  an  interest  of  đ?‘&#x;đ?‘&#x;đ?‘Œđ?‘Œ  on  it.  The  other  extreme  is  that  the  consumer  would  like  to  spend  everything  right  now  and  nothing  in  period  two.  That  means  that  he  can  spend  a  maximum  of  đ?‘Œđ?‘Œ +
 now.  The  logic Â
behind  this  is  that  he  must  be  able  to  pay  back  the  money  that  he  borrows  now. Â
To  be  able  to  do  so  he  cannot  borrow  more  than  the  discounted  value  of  his  future  income.  All  other  positions  on  the  straight  ff i n a n c i a l  m a r k e t s  l i n e  can Â
be  reached  through  either  borrowing  or  saving.  Whenever  period-Ââ€?one  13 Â
Uniseminar – Finance
5
V a l u i n g B B o n d s
Theory
The previous sections introduced interest rates and how they can be used for
investment decisions. In real life, however, the story may be more difficult. As soon as money is stored other than in a bank account one does not simply receive a fixed interest. In the financial markets we hardly find any products without risk
that pay a fixed interest rate. What we do find are bonds, stocks and derivatives. This section will deal with bonds, which is the instrument that comes closest to
putting money in a bank account and receiving interest payments. However, as this section will show, there can also be major differences.
5 . 1 B o n d R R a t i n g s
The first difference to our previous calculations is that bonds carry some risk. As the payoffs of a bond are fixed, the main concern is that the issuer of the bond
becomes insolvent and cannot pay. This is referred to as default risk. The d e f a u l t r i s k of a bond depends on the issuer of the bond and its
creditworthiness. In general one has to distinguish between two groups of
issuers: governments and corporations. On average the default risk of a government bond will be lower than that of a corporate bond.
The higher the default risk of a certain bond is the lower will be the price of that bond. This should make sense as you are not willing to buy a bond which carries
a higher risk that you lose your investment at the same price than a bond which carries a lower such risk. As it is very difficult for an individual investor to assess the default risk of a certain bond there are rr a t i n g aa g e n c i e s that rate bonds. The
biggest three rating agencies are Standard & Poor, Fitch and Moody’s. The following table summarizes the different rating grades.
31
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Uniseminar – Finance
5 . 2 P r i n c i p a l , C C o u p o n aa n d M M a t u r i t y
Every bond can be defined by three characteristics, namely the principal, the
coupon and the maturity. The c o u p o n is a payment that the holder of a bond m a t u r i t y defines the time span of the contract. Hence receives every year. The m
at maturity the bond expires and the bondholder receives the last coupon
payment and the p r i n c i p a l v a l u e (also called f a c e v a l u e ) of the bond.
Sometimes the principal of a bond is also referred to as the face value.
Depending on these characteristics it is possible to price a bond if one knows the interest rate. This will be done in the next section. What we can already say now
is that the higher the coupon payments are the higher will be the price of the bond. Moreover, if the price of a bond is higher than the principal value we say that a bond sells aa t aa p p r e m i u m . If the opposite is true and the price of the bond
is lower than the principal we say that the bond sells a t aa d d i s c o u n t .
32
Theory   Â
7 Â
Â
W e i g h t e d   A A v e r a g e   C C o s t   o o f   C C a p i t a l Â
Uniseminar  –  Finance Â
Whenever  a  company  wants  to  decide  whether  or  not  to  do  a  certain  investment Â
it  needs  to  know  the  interest  rate  to  be  able  to  discount  the  free  cash  flows.  To  discount  the  cash  flows  our  models  in  section  three  simply  assumed  that  the Â
interest  rate  was  known  and  equal  to  a  certain  value.  In  real  life,  however,  the  interest  rate  at  which  a  certain  company  needs  to  discount  its  cash  flows  depends  on  how  the  company  finances  itself  in  the  financial  markets.  This  decision  is  also  called  financing  structure.  Therefore  the  interest  rate  is  different  for  each Â
company  and  is  referred  to  as  the  w e i g h t e d  a v e r a g e  c o s t  o f  c a p i t a l  ( W A C C )  of  a  certain  company.  Moreover,  the  last  sections  have  shown  us  that Â
there  are  basically  two  ways  for  a  company  to  finance  capital.  The  first  is  issuing Â
bonds  and  thereby  taking  on  debt  and  the  second  is  to  issue  stock  and  thereby Â
increase  the  equity.  The  remainder  of  the  section  explains  what  the  costs  of  capital  are  for  both  these  options  and  how  they  can  be  combined  to  find  the  WACC.  Â
7 . 1  T h e   cc o s t   o o f   d d e b t Â
The  cc o s t   o o f   d d e b t  for  a  certain  company  is  nothing  else  than  the  return  that  an Â
investor  can  earn  on  the  bond  of  that  company.  This  is  because  the  return  that Â
investor  earns  must  be  paid  by  the  company.  Hence,  when  we  need  to  find  the  cost  of  debt  of  a  company  we  need  to  employ  the  tools  of  section  four  and  calculate  the  yield  to  maturity  that  can  be  earned  on  the  bonds  which  were Â
issued  by  that  particular  company.  Recap  that  there  are  two  kind  of  bonds  for  which  the  yield  to  maturity  can  be  easily  computed.  The  first  is  a  zero  coupon Â
bond  of  any  maturity,  the  second  is  a  coupon  paying  bond  with  a  maturity  of  at  most  two  years.  The  formulas  are  given  again  below.  Zero  coupon  bond  with  N  years  to  maturity:  đ?‘&#x;đ?‘&#x; =
50 Â
− 1 Â
Uniseminar  –  Finance  Â
Coupon  bond  with  one  year  to  maturity: Â
đ?‘&#x;đ?‘&#x; =
Â
− 1 Â
Coupon  bond  with  two  years  to  maturity:   đ?‘&#x;đ?‘&#x; = 0.5
Theory Â
+
0.5
+
− 1 Â
As  soon  as  you  managed  to  compute  y  with  the  help  of  those  formulas  you  got  the  cost  of  debt  of  a  particular  firm.  This  is  then  the  first  part  of  the  total  WACC  of  that  firm.  Â
Example Â
A  company  issues  debt  in  form  of  a  4  year  zero  coupon  bond  with  a  face  value  of  1,000â‚Ź  which  can  be  bought  at  a  price  of  754â‚Ź.  What  is  the  companies  cost  of  debt? Â
1000 đ?‘&#x;đ?‘&#x; = 754
Â
− 1 = 0.0731 Â
7 . 2  T h e   cc o s t   o o f   ee q u i t y Â
The  second  part  of  the  WACC  of  any  firm  is  the  cc o s t  o f  e q u i t y .  Similar  to  the  previous  example,  the  cost  of  equity  is  simply  equal  to  the  return  that  an  investor  can  earn  by  holding  the  stock  of  a  company.  Therefore  we  have  to  use  the  CAPM Â
formula  to  derive  the  return  on  the  stock  of  the  company.  To  do  so  we  need  to Â
know  the  expected  return  of  the  market  portfolio,  the  risk  free  interest  rate  as Â
well  as  the  beta  of  the  company.  Most  of  the  time  beta  will  just  be  given  together  with  the  other  two  variables.  However,  sometimes  beta  is  not  known  directly  and  needs  to  be  computed  as  the  average  of  other  companies  that  operate  in  the  same  industry.  In  that  case  it  will  be  equal  to  the  following.  �� =
â‹Ż
Â
51 Â
Theory
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Uniseminar – Finance
I n v e s t o r B B e h a v i o r & & C C a p i t a l M M a r k e t E E f f i c i e n c y
The previous discussions always built on the assumption of efficient markets and that investors should always invest in a combination of the market portfolio and
a risk-‐free bond. In reality, however, one can easily see that this does not take place but that most market participants try to beat the market. This section will discuss deviations from the security market line in the form of alpha stocks, various forms of trading biases, trading strategies which try to exploit these inefficiencies, and introduce a multifactor model of risk as alternative to CAPM.
8 . 1 A l p h a ss t o c k s
The first anomaly we will discuss is a stock’s a l p h a , which is simply the
difference between the expected return of a stock and its return according to the security market line. This occurrence is pictured in the figure below.
α
By definition this can only occur when the market portfolio is not efficient, as otherwise all expected returns would lie on the security market line and could be
perfectly estimated using beta and the CAPM formula. There are investors which specialize in finding alpha stocks (e.g. special hedge funds) and through buying 54
Uniseminar – Finance
Theory
positive alpha stocks and short selling negative alpha stocks bring the market back to equilibrium very quickly. Thus, although it contradicts CAPM’s
assumption of an always-‐efficient market, savvy investors should keep the
market close to efficient most of the time. Thus, these specialized investors serve two purposes. First, they gather profits through being faster as their competitors. And second, they strongly contribute to keep the financial markets efficient.
This “normalization” of stock prices can even take place without any actual trade, just because competition between investors is so though that no trade becomes possible before prices have gone back to the SML. This is called n o -‐-‐tt r a d e
t h e o r e m . In fact, although it might seem smart for individual investors to search
for alpha stocks this is most of the time not advisable simply because of the information disadvantage compared to professional traders and institutional
investors. Thus, most individual investors are best of just investing in a combination of the market portfolio and a risk-‐free bond. Thereby they can
guarantee themselves the return of an average investor.
8 . 2 T r a d i n g b b i a s e s
Trading biases can be separated into individual biases and systematic biases.
Most of them have a background in psychology and particularly systematic biases have significant consequences for the validity of the efficient market hypothesis.
Individual investors often do not hold an efficient portfolio. Instead they tend to diversify too little and trade too much, thereby receiving suboptimal returns for
the amount of risk they take. Four of the most common psychological
explanations will be discussed in turn. First, investors may suffer from
f a m i l i a r i t y b b i a s , which means that they favor to invest in companies they are
familiar with. Second, investors often rank their relative performance against a personal peer group (i.e. family, friends & colleagues) above absolute
performance. This phenomenon is called r e l a t i v e w e a l t h c o n c e r n s and may
lead them to build undiversified portfolios that more or less mirror those of their 55
Theory
1 0 F i n a n c i a l SS t a t e m e n t s
Uniseminar – Finance
Financial Accounting provides essential information for everybody working in Finance. The information is conveyed in the form of f i n a n c i a l s t a t e m e n t s ,
which are business documents that companies use to represent their finances to the public. There are four important financial statement documents that will be addressed in the following: 1. Income Statement 2. Balance Sheet
3. Cash-‐Flow Statement
4. Statement of Retained Earnings
These statements are the effective means, with which information is communicated to stakeholders. Furthermore, international standardized rules such as IFRS make it possible to draw international comparisons between different companies based on these documents.
1 0 . 1 I n c o m e ss t a t e m e n t (( s t a t e m e n t o o f o o p e r a t i o n s )
The i n c o m e s t a t e m e n t reports revenues or expenses for the period and
measures the operating performance of a business. The bottom line is the net profit/loss or "Net Income". It tells how well a company performed during the
period, that is, whether its operations have been profitable or not.
N e t II n c o m e = Total Revenues and Gains – Total Expenses and Losses
Before arriving at net income there are several other important profit figures often used by analysts for different purposes: G r o s s p p r o f i t = Total Sales – Cost of Sales
O p e r a t i n g I n c o m e = Gross profit – SGA Expenses – R&D Expenses – Depreciation & Amortization
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Uniseminar – Finance
Theory
E a r n i n g s b e f o r e i n t e r e s t a n d t a x e s ( E B I T ) = Operating income – Other income
P r e t a x ii n c o m e = EBIT – Interest income + Interest expense
Investors and analysts can use the balance sheet for further evaluation purposes by building different ratios. Some of the most important are: P r o f i t a b i l i t y R R a t i o s : Operating Margin = =
Net Profit Margin = =
A c c o u n t s R R e c e i v a b l e D D a y s =
I n v e s t m e n t R R e t u r n s :
Return on Equity (ROE) = Return on Assets (ROA) =
R a t i o = P r i c e -‐-‐ee a r n i n g s (( P / E ) R
=
1 0 . 2 B a l a n c e SS h e e t (( S t a t e m e n t o o f ff i n a n c i a l p p o s i t i o n )
The b b a l a n c e ss h e e t shows the financial position of a company at a specific point
of time. It reports 3 items: assets, liabilities and stockholders’ equity (or simply
equity). Assets are recorded at the left hand side of the balance sheet (or at the
top) and Liabilities plus Shareholder´s equity are reported on the right hand side
(or at the bottom). The balance sheet gives investors an idea of how much has
been invested in the company, what the company owns and what it owes. As the
name already implies, the two sides of the balance sheet have to balance out, which is referred to as the b a l a n c e s h e e t i d e n t i t y ( o r a c c o u n t i n g
e q u a t i o n ) . The rationale behind this is that the company has to pay for all assets it owns by either borrowing money from creditors (liabilities) or to raise money from shareholders (equity).
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Seminar
Extras
Exams
Practice
P
Practice Exercises
Finance Academic Year 2012/2013, Block 5
Practice
P r a c t i c e E E x e r c i s e s
Uniseminar – Finance
This part contains practice exercises to each week and therefore to each chapter of the theory script. By this, you can deepen your theoretical knowledge with
practical exercises and you can go through the exercises of these topics again,
which you have not understood so well until now. Although you may think that you already have done enough exercises during the weeks, these exercises are
tailored specifically to your needs and try to teach you the most important topics of the exam in a practical manner.
T a b l e o o f C C o n t e n t s
F i n a n c e
Exercises
Solutions
1
23
Practice
3
Uniseminar – Finance
F i n a n c i a l D D e c i s i o n M M a k i n g & & II n t e r e s t R R a t e s
E x e r c i s e 3 . 1 – – A A r b i t r a g e & & F F i n a n c i a l D D e c i s i o n -‐-‐m m a k i n g a ) What is the NPV decision rule?
b ) What is an arbitrage opportunity and why should it not be possible according to the law of one price?
c ) What is the separation principle?
d ) Suppose you can choose between receiving 100,000€ today or receiving a payment of 200,000€ in 10 years. The risk-‐free interest rate is 5%. Using the NPV decision rule, which option should you choose?
e ) You are a decision maker at Silverman Investments Inc. and have the opportunity to make a 2,000,000€ investment into an American investment
opportunity, which will yield a cash flow of $3,000,000 in two years. The risk
free rate in Europe is 5%, while it is 3% in the US and the exchange rate over the whole period is 1€ : $1.45. Should you use this investment opportunity?
f ) You have the opportunity to make a risk-‐free investment of 50,000€ today,
which will yield 55,000€ in one year. How big can the interest rate for the project be so that it is still attractive?
g ) You have four different investment opportunities. The table below summarizes their individual cash flows: Project A B C D
Cash flow today (€) Cash flow in two years (€) -‐50,000 70,000 -‐20,000 30,000 -‐150,000 170,000 -‐100,000 110,000
All cash flows are certain and the risk-‐free interest rate is 8%. 1. Calculate the NPV of each project
4
2. You can choose to invest in two projects. Which should you take?
Uniseminar – Finance
E x e r c i s e 3 . 2 – – T T h e tt i m e v v a l u e o o f m m o n e y
Practice
a ) There is the possibility to invest in a zero-‐coupon bond with a principal
amount of 50,000€, which will be paid in 5 years. The competitive market interest rate is 7%. How much is this bond worth today?
b ) You just secured a job at an auditing firm and you plan to save 5,000€ a year over a time span of 3 years. The fixed interest rate for your savings plan is
6% and the first payment is done today. What is the present value of this savings plan?
c ) What is the future value in 4 years of a 10,000€ today? Assume an interest rate of 5%.
d ) You have the option to receive 10,000€ today or 15,000 in 5 years. Which is the best alternative assuming an interest rate of: 1. 0% 2. 5%
3. 10% 4. 20%
e ) As manager for ABC Investments Inc. you have discovered an interesting
investment opportunity. It would cost 50,000€ today and return 25,000€ one year from now, 20,000€ two years from now and 15,000 in three years. 1. What is the NPV for an interest rate of 6%?
2. What is the NPV for an interest rate of 12%?
f ) What is the price of a consol bond by the Dutch government which pays 500€ yearly? Assume an interest rate of 5% and you buy the bond immediately before the yearly payment.
g ) What would be the price of the consol above if the Dutch government would increase payments by 2% every year to account for inflation?
h ) You plan to invest into a retirement plan which requires yearly payments of 2,500€ over the next 40 years. Assuming a constant interest rate of 6%, what is the present value of this retirement portfolio?
5
Uniseminar – Finance
5
V a l u i n g B B o n d s
Practice
E x e r c i s e 5 . 1 – – V V a l u i n g b b o n d s
a ) What is the difference between yield to maturity (YTM) and coupon rate?
b ) What price could a 1,000€ bond have which is: 1. At a premium 2. At par
3. At a discount
c ) Why would anybody buy a zero-‐coupon bond although it does not pay any interest?
d ) Where is the threshold between investment grade bonds and junk bonds?
e ) Consider a 10-‐year bond with a face value of 1,000€ and a coupon rate of 4.0%.
1. What is the coupon payment (CPN) if interest is paid annually? 2. What is the CPN if interest is paid monthly?
f ) Consider three different 1,000€ zero-‐coupon bonds which are assumed to be without any default risk. What is their individual yield to maturity (YTM)? 1. Price: 950€ -‐ Maturity: 2 years 2. Price: 900€ -‐ Maturity: 5 years
3. Price: 750€ -‐ Maturity: 10 years
g ) What is the price of a 1,000€ bond with a maturity of 4 years, an annual coupon rate of 5.0% and a yield to maturity of 4.5%?
h ) Unilever issued a 1,000€ bond with 10 years until maturity, an annual coupon of 6% and a yield to maturity of 8%.
1. What is the price on the issue date?
2. What is the price immediately before the first coupon payment?
3. What is the price immediately after the first coupon payment?
11
Practice
6
S t o c k s & & P P o r t f o l i o V V a l u a t i o n
Uniseminar – Finance
E x e r c i s e 6 . 1 – – V V a l u i n g ii n d i v i d u a l ss t o c k s
a) What does the total return of an investment in a stock encompass?
b) In a dividend discount-‐model, do you use the risk-‐free interest rate as discount rate?
c) Please explain the tradeoff of dividend payout from the perspective of a shareholder.
d) How does stock valuation with multiples (comparables) work?
e) Why is the efficient market hypothesis of essential importance for most valuation techniques?
f) The current stock price of ABC Cars Inc. is 100€ and it is expected that it will pay out a 5€ dividend in one year. The expected price of ABC Cars’ stock after the dividend payment in one year is 120€. Please calculate: 1. Expected dividend yield
2. Expected capital gain rate 3. Expected total return
4. Implied equity cost of capital
g) DEF Cars Inc. currently pays a dividend of 8€ per share per year. It has 10,000 shares with a current price of 250€outstanding. If this dividend is
assumed to be constant over time, dividend is paid out at the end of each year, and its cost of equity is 12%, what is its firm value at the beginning of
this year?
h) Assume the same details as in the previous task; only that dividends are expected to grow by 4% each year. What is the firm value in this case?
i) Assume ABC Cars Inc. has a stock price of 100€ and EPS of 8€. In comparison, DEF Cars Inc. has EPS of 15€. Using comparables, what should be the stock price of DEF Cars Inc.?
12
Uniseminar – Finance
7
W e i g h t e d A A v e r a g e C C o s t o o f C C a p i t a l (( W A C C )
Practice
E x e r c i s e 7 . 1
a) What is the interest tax shield?
b) What is the effect of the interest tax shield on WACC?
c) ABC Cars Inc. has 10 million shares outstanding with a current market value of 35€ per share. Furthermore, it has 200 million € debt outstanding. Assume
an equity cost of capital of 14%, debt cost of capital of 9% and a corporate tax rate of 30%.
1. What is ABC Cars’ unlevered cost of capital?
2. What is ABC Cars’ after-‐tax debt cost of capital? 3. What is ABC Cars’ WACC?
d) The CFO of DEF Cars Ltd. asks you as new intern to calculate the WACC of the company. Based on its industry beta you could determine the unlevered cost
of capital to be 10%. Furthermore, you know that the company is 40% debt financed and 60% equity financed and the cost of equity is 12%. Assuming a corporate tax rate of 30%, what should be the WACC of the company?
e) GMAT Enterprises Ltd. has outstanding debt of 100,000€ and 10,000 shares
outstanding worth 22€ each. If we assume a beta of 0 for debt and a beta of 1.2 for equity, what is the company’s beta?
f) Longman Bows Inc. is financed by both equity and debt. It has 50,000 zero coupon bonds with a time to maturity of 10 years and a face value of 1,000€ outstanding. They currently trade at 487€. Furthermore, it has 250,000
shares outstanding with a current share price of 350€. The current market return on risk-‐free t-‐bills is 3%, while the market portfolio yields a return of 12%. Although you do not know the firm’s beta, you were able to gather the
betas of Longman’s 4 top competitors. They are 0.43, 0.89, 1.22 and 0.93 respectively.
1. Please calculate Longman’s cost of debt 17
Uniseminar – Finance
4
I n v e s t m e n t d d e c i s i o n s
S o l u t i o n tt o E E x e r c i s e
Practice
4 . 1
a ) The most popular alternatives to NPV are:
1. P a y b a c k i n v e s t m e n t r u l e : The time it takes to pay back the initial investment (ignoring discounting)
2. I n t e r n a l r a t e o f r e t u r n : The discount rate that sets the NPV to zero
3. E c o n o m i c p r o f i t / E V A : The difference between cash flows generated by a project and its costs of capital
b ) The IRR investment rule cannot be used for comparing investment opportunities with d i f f e r e n t s c a l e s . One of the investment opportunities may well have the highest IRR but be so small compared to the other
investment opportunities that the NPV is a lot smaller. Furthermore, the IRR decision rule does not account for tt i m i n g aa n d rr i s k o o f cc a s h ff l o w s .
c ) The p r o f i t a b i l i t y i n d e x is simply calculated as NPV divided by the “bottleneck” resource consumed. It is a measure of “bang for the buck” and can help to discover which projects provide the highest NPV per unit of
resource consumed. Thereby they help in deciding on an appropriate investment portfolio.
d ) NPV = -‐100,000 + 125,000/1.085 = -‐14,927€ Do NOT take the investment opportunity!
Maximum deviation IRR:
IRR = (125,000/100,000)1/5 – 1 = 4.56%
Maximum deviation = 8.00% -‐ 4.56% = 3.44%
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Practice
Uniseminar – Finance
If the cost of capital decreases by MORE THAN 3.44%, we should change our decision and invest in this project!
e ) Payback period = 10,000/800 = 12.5 months
f ) Solution:
1. NPV:
Initial investment = -‐10,000,000
Annuity of CFs = ((2,000,000-‐1,200,000)/0.06)*(1-‐1/1.0630) = 11,011,865€
Recycling costs = 500,000/1.0630 = 87,055€
NPV = -‐10,000,000 + 11,011,865 – 87,055 = 924,810€
2. Payback period = 10,000,000/(2,000,000-‐1,200,000) = 12.5 years
34
Uniseminar – Finance
6
S t o c k s & & P P o r t f o l i o V V a l u a t i o n
S o l u t i o n tt o E E x e r c i s e
Practice
6 . 1
a) The total return of stock encompasses the d d i v i d e n d y i e l d (expected annual
dividend divided by current stock price) and the cc a p i t a l g a i n r a t e (capital
gain expected divided by current stock price).
b) No, you cannot use the risk-‐free interest rate because the cash flows are not riskless. Instead you need to use the ee q u i t y c o s t o f c a p i t a l , which is equal
to the expected return of other available investments with the same risk profile.
c) Although it is nice for the shareholder to receive dividend payouts this is not
always the best option for him. Because it pulls money out of the corporation it potentially decreases growth which may result in capital gains.
d) An alternative to the dividend-‐discount model is the m e t h o d o f
c o m p a r a b l e s . Here we use multiples and firm value of comparable firms to
predict the firm value of our firm of investigation. Popular multiples are P/E-‐
ratio, Price to book ratio or the ratio of firm value to Sales/EBIT/EBITDA.
e) The e f f i c i e n t m a r k e t h y p o t h e s i s states that all securities are fairly priced according to their expected future cash flows. This makes the existence of positive-‐NPV trading opportunities impossible and allows us to
infer the market price of a security out of its expected cash flows (e.g. for bond or stock valuation).
f) Solution:
1. Expected dividend yield = 5/100 = 5% 39
Practice
Uniseminar – Finance
2. Expected capital gain rate = (120-‐100)/100 = 20% 3. Expected total return = 5% + 20% = 25%
4. Implied equity cost of capital = Expected total return = 25%
g) Share price = Perpetuity = 8/0.12 = 66.67€
Firm value = 66.67€ * 10,000 shares = 666.667€
h) Share price = Perpetuity = 8/(0.12-‐0.04) = 100€ Firm value = 100€ * 10,000 shares = 1,000,000€
i) ABC’s P/E Ratio = 100/8 = 12.50
DEF’s stock price = 15 * 12.50 = 187.50€
j) Stock price = 0.7*(50/1.08) + 0.3*(100/1.08) = 60.19€
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Seminar
Extras
Exams
E
Exams
Finance Academic Year 2012/2013, Block 5
Exams
E x a m s
Uniseminar – Finance
You should start early with the calculation of exams, because you need to get a general feeling of how the exams are built up. You will soon discover how the
exams are constructed and that there are general tendencies, which repeat from exam to exam. In this part you will find old exams of the Maastricht University, as
well as one practice exams constructed by Uniseminar. During the seminar you
will then receive a further practice exam.
T a b l e o o f C C o n t e n t s
P r a c t i c e E E x a m 1 1 (( i n c l . ss o l u t i o n s )
1
O l d E E x a m s (( i n c l . ss o l u t i o n s )
1 7
Practice Exam 2010
Exam 2008
Exam 2007
Uniseminar – Finance
U n i s e m i n a r P P r a c t i c e E E x a m 1 1
1 T/?/F
2 T/?/F 3 T/?/F 4 T/?/F 5 T/?/F 6 T/?/F 7 T/?/F 8 T/?/F
9 T/?/F
10 T/?/F
11 T/?/F
12 T/?/F 13 T/?/F 14 T/?/F 15 T/?/F
Practice Exam
There are clear limitations on who is allowed to own the stock of a corporation.
A partnership becomes a sole proprietorship if one of the two general partners dies.
If a corporation becomes bankrupt, then shareholder are compensated before bondholders. The Sarbanes-‐Oxley Act has been introduced to decrease agency costs.
The agency problem is completely solved due to the fact that there exists a labor market for talented CEOs.
The capital budgeting decision is the choice of the firm which securities it should issue.
In a limited liability company the owners of the firm are only liable for at most 50% of the losses of the company.
The stock market is an auction market.
Firms who wish to raise funds by selling equity on a stock market for the first time involve in an IPO. Managers have a lower tolerance to risk than shareholders. The mangers of a firm are also called principals.
Financial markets exist because people wish to shift consumption from one point in time to another.
Whether a person saves or not is not determined by the interest rate but solely by the tastes of that person.
The model of intertemporal choice predicts that the financial markets can decrease the level of utility of some people. In the model of intertemporal choice the interest rate equalizes the amount of people that wish to save with those that wish to borrow. 45
Practice Exam
16 T/?/F 17 T/?/F 18 T/?/F
19 T/?/F 20 T/?/F
21 T/?/F 22 T/?/F 23 T/?/F 24 T/?/F 25 T/?/F 26 T/?/F 27 T/?/F 28 T/?/F 29 T/?/F
46
Uniseminar – Finance
If the current risk free interest rate is 5% and the average risk premium of stocks is 4%, then you should be indifferent between 100€ today and 109€ in one year. If the annual rate is equal to 5% and you receive interest every quarter, then the effective rate will be equal to 5.0945%. The APR will never be higher than the EAR.
The EAR tends to increase with the amount of compounding periods per year.
If you give the amount of 1000€ to a bank and the interest rate is equal to 10%, then this theoretically allows you to get an amount of 100€ each year until forever.
Paying back a mortgage of 200,000€ within 8 years requires a fixed monthly payment of 2749.23 if the interest rate is equal to 6%.
The present value of an infinite stream of payments of which the first payment is 120€ and that grows at a rate of 5% is 240,000 if the interest rate is equal to 5.05%. A company should never engage in negative net present value projects. If the payback period of a project is shorter than 2 years one should always do that project.
The incremental IRR gives the value of the interest rate that makes one indifferent between two different projects. The NPV of a project that requires an initial investment of 2000€ and pays 220€ from year two until eternity will be equal to -‐ 307.69€ if the interest rate is equal to 13%.
The IRR of a project that requires an initial investment of 144€ and pays back 88€ in year one and year two will be equal to 13.77%. It is possible that a certain project has multiple internal rates of return.
The payback rule is superior to the NPV rule because it gives the same result and is simpler to apply.
Practice Exam
U n i s e m i n a r P P r a c t i c e E E x a m 1 1 – – SS o l u t i o n s
Uniseminar – Finance
1. F Everybody is allowed to buy the stocks of a corporation, this is one of the unique features.
2. F A partnership dissolves itself if one of the general partners dies.
3. F If a corporation becomes bankrupt bondholders are compensated first. In some cases shareholders can even lose control of the firm, which is then passed over to bondholders.
4. T The Sarbanes-‐Oxley Act has been introduced to improve the monitoring of managers. Thereby it decreases agency costs.
5. F The agency problem becomes less severe. It cannot be entirely solved as long as the incentives of the managers differ from those of the owners. 6. F The financing decision determines which securities the firm should issue. The capital decision determines in which asset the firm should invest. 7. F In a limited liability the owners of the firm a liable only to the amount of their invest
8. T The stock market is an auction market. Only the primary stock market is a brokered market as it deals with the introduction of new stock to the public. 9. T Initial public offering (IPO) says it all.
10. T Shareholders can diversify contrary to managers and hence prefer a larger amount of risk. 11. F The owners of a firm are called principals. The managers are called agents.
12. T Through saving people can shift consumption forward in time and through borrowing they can shift it backwards. This is the prime economic reason for the existence of financial markets.
13. F The interest rate, the stream of earnings and tastes have an impact on the decision to save or borrow. The interest rate defines the slope of the line, the stream of payments the point in which there is no saving and or borrowing and the tastes the shape of the indifference curves. 52
Seminar
Extras
E
Extras
Finance Academic Year 2012/2013, Block 5
Extras
E x t r a s
Uniseminar – Finance
In this part you find several extras that will be very helpful for your exam
preparation. In this course you will find a formula sheet as well as a glossary that contains all the relevant definitions.
T a b l e o o f C C o n t e n t s
F o r m u l a SS h e e t G l o s s a r y
Formula Sheet
Finance Academic Year 2012/2013, Block 5
Glossary
Uniseminar – Finance
“ C ” c o r p o r a t i o n s -‐-‐ Corporations that have no restrictions on who owns their “ shares or the number of shareholders, and therefore cannot qualify for subchapter S treatment and are subject to direct taxation.
1 0 -‐-‐K K -‐-‐ The annual form that U.S. companies use to file their financial statements with the U.S. Securities and Exchange Commission (SEC).
1 0 -‐-‐Q Q -‐-‐ The quarterly reporting form that U.S. companies use to file their financial statements with the U.S. Securities and Exchange Commission (SEC).
A c c o u n t s p a y a b l e -‐-‐ The amounts owed to creditors for products or services
purchased with credit.
A c c o u n t s r e c e i v a b l e -‐-‐ Amounts owed to a firm by customers who have purchased goods or services on credit.
A c c o u n t s r e c e i v a b l e d a y s -‐-‐ An expression of a firm’s accounts receivable in terms of the number of days’ worth of sales that the accounts receivable represents. A d j u s t e d b e t a s -‐-‐ A beta that has been adjusted towards 1 to account for estimation error. A f t e r -‐-‐tt a x i n t e r e s t r a t e -‐-‐ Reflects the amount of interest an investor can keep after taxes have been deducted.
A l p h a -‐-‐ The difference between a stock’s expected return and its required return according to the security market line.
o p t i o n s -‐-‐ The most common kind of option, they allow their holders A m e r i c a n o to exercise the option on any date up to, and including, the expiration date. A m o r t i z a t i o n -‐-‐ A charge that captures the change in value of acquired assets. Like depreciation, amortisation is not an actual cash expense.
A m o r t i z i n g l o a n -‐-‐ A loan on which the borrower makes monthly payments that include interest on the loan plus some part of the loan balance. 96
Uniseminar – Finance
Glossary
A n n u a l p e r c e n t a g e r a t e ( A P R ) -‐-‐ Indicates the amount of interest earned in one year without the effect of compounding.
A n n u a l rr e p o r t -‐-‐ The yearly summary of business sent by U.S. public companies to their shareholders that accompanies or includes the financial statement.
A n n u i t y -‐-‐ A stream of equal periodic cash flows over a specified time period. These cash flows can be inflows of returns earned on investments or outflows of funds invested to earn future returns.
A n n u i t y s p r e a d s h e e t -‐-‐ An Excel spreadsheet that can compute any one of the five variables of NPER, RATE, PV, PMT and FV. Given any four input variables the spreadsheet computes the fifth.
A r b i t r a g e -‐-‐ The practice of buying and selling equivalent goods or portfolios to take advantage of a price difference. A r b i t r a g e o p p o r t u n i t y -‐-‐ Any situation in which it is possible to make a profit without taking any risk or making any investment.
A r b i t r a g e P r i c i n g T h e o r y ( A P T ) -‐-‐ A model that uses more than one portfolio to capture systematic risk. The portfolios themselves can be thought of as either the risk factor itself or a portfolio of stocks correlated with an unobservable risk factor. Also referred to as multifactor model.
A s k p r i c e -‐-‐ The price at which a market maker or a specialist is willing to sell a security.
A s s e t s -‐-‐ The cash, inventory, property, plant and equipment, and other investments a company had made.
A t -‐-‐tt h e -‐-‐m m o n e y -‐-‐ Describes options whose exercise prices are equal to the current stock price. A u d i t o r -‐-‐ A neutral third party that corporations are required to hire that checks the annual financial statements to ensure they are prepared according to GAAP, and to verify that the information is reliable. 97
Formula Sheet
Finance Academic Year 2011/2012, Block 5
Formula Sheet
I n s t i t u t i o n a l F F r a m e w o r k s
Uniseminar – Finance
T a x a t i o n o o f d d i f f e r e n t cc o r p o r a t i o n s : “C”-‐corporation:
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑡𝑡𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
= 𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 ∗ 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑡𝑡𝑡𝑡𝑡𝑡 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ∗ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡𝑡 𝑟𝑟.
“S”-‐corporation:
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑡𝑡𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
= 𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 ∗ 𝑛𝑛𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡𝑡 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟
Effective tax rate of taxation: 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑡𝑡𝑡𝑡𝑡𝑡 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 =
𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 − 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑃𝑃𝑃𝑃𝑒𝑒 𝑡𝑡𝑡𝑡𝑡𝑡 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎
R a t e s I n t e r e s t R
P r e s e n t aa n d F F u t u r e v v a l u e cc a l c u l a t i o n w w i t h ss i n g l e p p a y m e n t s : 𝐹𝐹𝐹𝐹 = 𝐾𝐾 ∗ 1 + 𝑟𝑟
𝑃𝑃𝑃𝑃 =
𝐾𝐾 1 + 𝑟𝑟
A n n u a l p p e r c e n t a g e rr a t e aa n d ee f f e c t i v e aa n n u a l rr a t e : 𝐴𝐴𝐴𝐴𝐴𝐴 𝐸𝐸𝐸𝐸𝐸𝐸 = 1 + 𝑛𝑛
− 1
A n n u i t i e s : 𝑃𝑃𝑃𝑃 = 2
𝐶𝐶 𝐶𝐶 + 1 + 𝑟𝑟 1 + 𝑟𝑟
+
𝐶𝐶 1 + 𝑟𝑟
+ ⋯+
𝐶𝐶 1 + 𝑟𝑟
=
𝐶𝐶 1 1− 1 + 𝑟𝑟 𝑟𝑟