Chapter 5. Investment and Analysis methods Summary (2016)
Resumen InglésUniversidad  Universidad de Barcelona (UB) 
Grado  Empresa internacional  3º curso 
Asignatura  Finances II 
Año del apunte  2016 
Páginas  11 
Fecha de subida  26/03/2016 
Descargas  13 
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FINANCES II: Investment and analysis
methods (II)
1. The Internal Rate of Return (IRR)
IRR is the discount rate (r) that makes the NPV be = 0.
Rules:
Companies are interested only in those investments that have r > opp.cost of k.
Companies choose the investment with the higher r.
Distinguishing r and k


Both r and opp. cost of k appear as a discount rate in NPV formula.
The r is a relative profitability measure that depends only on the amount and timing
of the project cash flows. “Relative” means that regardless of the amount of the
principal, this will be my investment cost.
The opp. cost of k is established in the capital market, and represents the expected
rate of return (ROR) offered by other assets with the same risk.
Different types of IRR
Constant cash flows (C1 = C2 = C3 =…= Cn)
Constant cash flows + Nº of years unlimited
Methodology to calculate IRR
*It is quite difficult to calculate IRR for long term projects, since we have to solve an equation of
ndegrees. Anyway, currently we use special financial software to solve it.
Example 1
Example 2
*Note that the higher the r is, the lower the NPV.
The hypothesis of reinvestment of net cash flows in “r”
If reinvestment rate “ r’ “ is equal to r (r’ = r) we can verify:
*It is very unrealistic though.
IRR’s inconsistency
*We will work only with Simple Investment.
Example 1’: Multiple IRRs, cash flows in billions of m.u.
*There can be as many internal rates of return (IRR) as changes in the sign of the cash flows!
Example 2’: there is no valid solution for IRR
There is no internal rate of return (r or IRR), which means that this project will have a positive
NPV at all discount rates.
The Effective Annual Interest Rate or the Annual Equivalent Rate (AER)
The “r” can be used in terms of months, quarters, years…, but normally an AER is often used a
reference to “r” but always in a yearly basis.
So we need to distinguish between quoted (nominal) interest rate and the effective annual
interest rate (or AER)
An Annual Equivalent Rate (AER) is an indicative reference of a cost of effective yield of a
financial product. It includes a nominal interest rate, expenses, bank charges and a term of
transaction.
Calculates an overall/global profitability on an annual basis
Applies to the savings products as well as to the loans
Its homogeneous character facilitates easier comparative analysis between various
financial products.
Example
Suppose a bank offers you an automobile loan at an annual percentage rate of 12% with
interest to be paid monthly. What is the effective annual rate(EAR)?
The EAR is needed to use comparable “r” for the investment projects with the different timing
of cash flows.
Example 1: AER
A bank gives a 2year loan of 10.000€ to one of its clients. The annual interest to be charged is
7%. On maturity date of a loan, the client has to return the loan (a principal value) and pay the
interests corresponding to a period.
1. What is the IRR that a bank obtains in this operation?
2. What is the AER, assuming that an interest has to be paid each semester?
(2 year loan with a 7%
interest annually)
(2 year loan, with a 7% interest annually, so a 3,5% interest semianually or semestrally)
(TAE=AER)
Example 2: AER
A bank gives a 2year loan of 20.000€ to one of its best clients. The annual interest to be
charged is 7%. So, each year a client has to pay the interests (annually interests). On maturity
date of a loan, a client must return a principal value of a loan and pay the interests
corresponding to a period.
1. What is the IRR that a bank obtains in this operation?
2. What is the AER, assuming that an interests has to be paid each quarter?
(2 year loan with a 7% annual
interest)
(2 year loan with a 7% annual interest, so a 1,75% quarter interest. TAE=AER)
Similarities and differences between NPV and IRR
For simple investment projects, IRR is a profitability limit which determines the viability of the
project, which depends on the opp. cost of capital (k).
The Fisher rate
The Fisher rate (f) is the interest rate that equates the NPV of 2 investment projects.
IRR and NPV give the same ranking to the projects, only if
fisher rate doesn’t exist in the first quadrant. In case of
existence of intersection, the ranking between IRR and NPV
doesn’t match.
*What does it mean that “it gives the same ranking”?
Example: Fisher rate
Example 1: Fisher Rate
The company you work for decided to install a new equipment for which there are 2
alternative simple investment projects with the following financial flows:
In order to make a final decision you have to present the following information:
a) To calculate IRR for each project
b) To check whether the Fisher rate exists or not in the first quadrant
c) To make a graphical analysis and financial interpretation
Example 2: IRR vs NPV choice if exist different reinvestment ratios):
A company ACEROX S.A. is considering to buy a new playback machine that produces video
faster than it did the previous one.
After running a market’s study it was found that there were two possibilities such as Acex and
Laxus machines for which a financial department made an estimation of the expected cash
flows for 2 years.
In case of choosing Laxus, the company would have 100.000 m.u available to make a
temporary financial investment.
Knowing that the financing cost (k) was 7%, financial director asked the project’s director to
make a complete analysis of these 3 investments.
The selection and ranking of the projects taking into consideration homogeneous initial
payments compare an option of purchase of Acex with the set of the other 2 possibilities of
investment.
FINAL SOLUTION
FINAL SOLUTION II
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