fn3092: corporate finance [“cf”] introductionsim.ascentiz.com/cf/notes/1s-cf-c1.pdf · fn3092:...

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©Jonathan Peh. Not for resale/transfer 1

FN3092:

CORPORATE FINANCE [“CF”]

INTRODUCTION

By Jonathan Peh, CFA, MSc

1

IMPORTANT:

• Your course material is the subject guide from UOL

and the essential reading texts cited therein.

• Other materials such as notes, assignments and

working solutions provided by the lecturer are for the

purpose of facilitating his teaching. By accessing

these materials, you agree to use them only for

your personal reference and not to share them

with others. You are not allowed to reproduced in

part or in whole the materials without the written

consent of the lecturer.

• Tape and/or video recording of any session is not

permitted.

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IMPORTANT:

• The lecturer reserves the right to deny your access to

all materials if:

• you are repeatedly late or absent from lecture

without valid reasons and/or

• fail to submit the required written assignments.

• Do not attend the class that you are not registered for.

Anyone caught “gate-crashing” will be reported to the

school.

• Do not engage in private conversation or talk on the

phone during the lecture.

About this course

• This course is aimed at students interested in understanding asset pricing and corporate finance. It provides a theoretical framework used to address issues in project appraisal and financing, the pricing of risk, securities valuation, market efficiency, capital structure and mergers and acquisitions. It provides students with the tools required for further studies in financial intermediation and investments.

• If you are taking this course as part of a BSc degree, courses which must be passed before this course may be attempted are 2 Introduction to economics and 5A Mathematics 1 or 5B Mathematics 2 or 174 Calculus.

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Learning Outcomes • At the end of this course, and having completed the Essential reading

and activities, you should be able to:

• explain how to value projects, and use the key capital budgeting techniques (NPV

and IRR)

• understand the mathematics of portfolios and how risk affects the value of the asset in equilibrium under the fundaments asset pricing paradigms (CAPM and APT)

• know how to use recent extensions of the CAPM, such as the Fama and French three-factor model, to calculate expected returns on risky securities

• explain the characteristics of derivative assets (forwards, futures and options), and how to use the main pricing techniques (binomial methods in derivatives pricing and the Black–Scholes analysis)

• discuss the theoretical framework of informational efficiency in financial markets and evaluate the related empirical evidence

• understand the trade-off firms face between tax advantages of debt and various costs of debt

• understand and explain the capital structure theory, and how information asymmetries affect it

• understand and explain the relevance, facts and role of the dividend policy

• understand how corporate governance can contribute to firm value

• discuss why merger and acquisition activities exist, and calculate the related gains and losses.

5

Subject Guide Structure • Chapter 1: here we focus on the evaluation of real investment projects using the

net present value technique and provide a comparison of NPV with alternative forms of project evaluation.

• Chapter 2: we look at the basics of risk and return of primitive financial assets and mean–variance optimisation. We go on to derive and discuss the capital asset pricing model (CAPM).

• Chapter 3: we present the arbitrage pricing theory, proposed as an alternative to the CAPM and discuss multifactor models. We study several recent multifactor models, such as the Fama and French three factor model, and observe that they can explain a large fraction of the variation in risky returns.

• Chapter 4: here we look at derivative assets. We begin with the nature of forward, future, option and swap contracts, then move on to pricing derivative assets via absence-of-arbitrage arguments. We also include a description of binomial option pricing models and end with the Black–Scholes analysis.

• Chapter 5: in this chapter, we examine the efficiency of financial markets. We present the concepts underlying market efficiency and discuss the empirical evidence on efficient markets. We also note that returns may be predictable even in efficient markets if risk is also predictable and discuss evidence in support of predictability of long horizon returns.

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Subject Guide Structure • Chapter 6: here we turn to corporate finance issues, treating the decision over a

corporation’s capital structure. The essential issue is what levels of debt and equity finance should be chosen in order to maximise firm value.

• Chapter 7: this chapter is complementary to Chapter 6, however, rather than looking at values, as in Chapter 6, this chapter analyses discount rates. We learn that if there are no taxes, while the return on equity gets riskier as the level of debt increases, the average rate the firm pays to raise money is unchanged. In the presence of taxes, as debt increases, the average rate the firm pays to raise money decreases due to tax shields.

• Chapter 8: we look at more advanced issues in capital structure theory and focus on the use of capital structure to mitigate governance problems known as agency costs and how capital structure and financial decisions are affected by asymmetric information.

• Chapter 9: here we examine dividend policy. What is the empirical evidence on the dividend payout behaviour of firms, and theoretically, how can we understand the empirical facts?

• Chapter 10: we look at mergers and acquisitions, and ask what motivates firms to merge or acquire, what are the potential gains from this activity, and how can this be theoretically treated? We also explore how hostile acquisitions may serve as a discipline device to mitigate governance problems.

7

Essential and Further reading

• Essential reading

• Hillier, D., M. Grinblatt and S. Titman Financial Markets

and Corporate Strategy. (Boston, Mass.; London: McGraw-Hill, 2008) European edition [ISBN 978007119027].

• Further reading

• Brealey, R., S. Myers and F. Allen Principles of Corporate Finance. (Boston, Mass., London: McGraw-Hill, 2008) ninth international edition [ISBN 9780071266758].

• Copeland, T., J. Weston and K. Shastri Financial Theory and Corporate Policy. (Reading, Mass.; Wokingham: Addison-Wesley, 2005) fourth edition [ISBN 9780321223531].

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Online study resources

• You can access the VLE, the Online Library and your

University of London email account via the Student Portal

at:

http://my.londoninternational.ac.uk

• You should receive your login details in your study pack. If

you have not, or you have forgotten your login details,

please email [email protected] quoting your

student number.

9

Examination Structure

• 3-hour paper consists of two sections. Each section

comprises of four questions that contain both numerical

and discursive elements. Students are to attempt four

questions with at least one question from each section.

• A list of formulas is provided in the exam paper (refer next

slide). Extracts from Present Value and Annuity Discount

tables, the area under a Normal Curve and log tables are

previously given at the end of exam paper.

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http://my.londoninternational.ac.uk/




mailto:[email protected]

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List of formulas

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PV table from 2011 paper

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Annuity table from 2011 paper

13

Areas under standard normal distribution

from 2010 paper

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FN3092-CORPORATE FINANCE CHAPTER 1: PRESENT VALUE

CALCULATIONS AND THE VALUATION OF

PHYSICAL INVESTMENT

By Jonathan Peh, CFA, MSc.

15

Learning outcomes

At the end of this chapter, and having completed the Essential reading and activities, you should be able to:

• analyse optimal physical and financial investment in perfect capital markets setting and derive the Fisher separation result (defer)

• justify the use of the NPV rules via Fisher separation (defer)

• compute present and future values of cash-flow streams and appraise projects using the NPV rule

• evaluate the NPV rule in relation to other commonly used evaluation criteria

• value stocks and bonds via NPV.

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Essential reading

• Hillier, D., M. Grinblatt and S. Titman Financial Markets

and Corporate Strategy.(Boston, Mass.; London:

McGraw-Hill, 2008)

• Chapter 9 (Discounting and Valuation),

• Chapter 10 (Investing in Risk-Free Projects),

• Chapter 11 (Investing in Risky Projects)

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Time Value of Money

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Time value of money – Future Value • A dollar today is worth more than a dollar in one year or at any time in the future.

The reason for this is that the dollar today can be invested, and start to earn interest immediately.

• Future value (“FV”) of a dollar today earning a rate of return, r per year, for t years will have a compounded value of:

𝑭𝑽 = $𝟏 × (𝟏 + 𝒓)𝒕

• Compounding accounts for interest on interest. For example, if you reinvest your money (i.e. the principal and interest paid by end of year 1) for another year, you will earn interest on the interest for the second year.

year FV of $1 @ 10% per year FV of $1 @ 20% per year

1 1.100

1.200

2 1.210

1.440

3 1.331

1.728

4 1.464

2.074

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Activity

1. Assume a compound borrowing and lending rate of 10% annually. What is the future value of $6,000 at end of year 4?

2. If interest is compound semi-annually (i.e. interest is computed every 6 monthly) at 10% per year, what is the future value of $6,000 at end of year 4?

3. For Q1 above (i.e. interest compound annually), what if the rates are different in each year, such as 8% per year for first year, 9% per year for second year, 11% per year for third year and 12% per year for fourth year? How does it compare to the answer in Q1?

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Present Value • Present value (“PV”) also term as discount value represents the value today of a

cash flow received in t years’ time. It is the opposite perspective of the future value concept.

• For example, if you expected to be repaid $1 in year t, how much are you willing to invest or lend today (i.e. what is the PV?) if the interest rate in the market is r per year

𝑃𝑉 =$1

(1 + 𝑟)𝑡

• The above discounting process would also account for interest on interest.

year PV of $1 @ 10% per year PV of $1 @ 20% per year

1

0.909

0.833

2

0.826

0.694

3

0.751

0.579

4

0.683

0.482

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Activity

1. Assume a compound borrowing and lending rate of 10% annually. What is the present value of $2,000 to be received in three years time?

2. If interest is compound semi-annually at 10% per year, what is the present value of $2,000 to be received in three years time?

3. For Q1 above (i.e. interest compound annually), what if the rates are different in each year, such as 9% per year for first year, 10% per year for second year and 11% per year for third year? How does it compare to the answer in Q1?

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Additivity Property

𝑃𝑉 𝐶𝑡 + 𝐶𝑛 = 𝑃𝑉 𝐶𝑡 + 𝑃𝑉 𝐶𝑛

• Which means the present value of two separate cash flows, one at year t and another at year n are simply the sum of individual present value of the cash flows.

• If there are two similar risk investments, A and B. A pays $100 in one year time while B pays $100 in two years time. How much are you willing to pay for these investment separately as well as in combination if the interest rate for similar risk investment is 10% per year?

• Based on PV, you are willing to pay $90.90 𝑖. 𝑒. 100

(1.1)1 for A and $82.64

𝑖. 𝑒. 100

(1.1)2 for B.

• In total, you would be willing to pay $173.54 (i.e. $90.90+$82.64) for both A and B.

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Timeline

t 0 n

Ct Cn

Evaluating investment with multi-period

cash flows • If there exists another Investment C that pays $100 each

in year 1 and year 2 where it is regard as having the same

risk as A and B above, the investment must be worth

$173.54. Otherwise, there will be arbitraging opportunity.

• Thus, value of investment with multi-period cash flows is

the sum of the present value of all the future cash flows

𝑃𝑉 =𝐶1(1 + 𝑟)1

+𝐶2(1 + 𝑟)2

+⋯+𝐶𝑛1 + 𝑟 𝑛

𝑤ℎ𝑒𝑟𝑒 𝐶𝑛 𝑟𝑒𝑓𝑒𝑟𝑠 𝑡𝑜 𝑐𝑎𝑠ℎ 𝑓𝑙𝑜𝑤 𝑎𝑡 𝑦𝑒𝑎𝑟 𝑛

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Activity

• Assume current interest rate is 8% per year. What is the

value of an investment that pays $100 each in year 3, 5

and 7?

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Project Appraisal Techniques

• Net Present Value (“NPV”)

• Internal Rate of Return (“IRR”)

• Payback

• Multiples method

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NPV Rules

• The NPV of the project is the sum of the present values of cash receipts, less the sum of the present values of the cash payments:

𝑁𝑃𝑉 = 𝐶𝑖(1 + 𝑟)𝑖

− 𝐼0

𝐾

𝑖=1

Where

"𝐼0" is immediate cost of project,

“r” is rate of return or opportunity cost of capital (i.e. Weighted Average Cost of Capital or WACC will be covered in Chapter 7)

“𝐶𝑖” is the net cash flow (i.e. cash receipts less cash payments) in year i

• Projects with positive NPV will be accepted while those with negative NPV are rejected. Firm is indifferent to project with zero NPV.

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Example

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Activity

• Assume an interest rate of 5 per cent. Compute the NPV

of Project D, and state whether the project should be

accepted or not.

• Project D costs £1,500 immediately. In year 1 it generates

£1,000. In year 2 there is a further cost of £2,000. In years

3, 4 and 5 the project generates revenues of £1,500 per

annum.

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Mutually exclusive projects

• Mutually exclusive projects are a set of projects which

only one can be chosen at a given time. (e.g. manual

versus computer controlled machine)

• For mutually exclusive projects, pick the project that has

the greatest positive NPV

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Example – FE 2005ZA Q1a

𝑁𝑃𝑉𝐵 =600

1.11+500

1.12 + 100

1.13 - 1500 = - $466.19

𝑁𝑃𝑉𝑐 =200

1.12 + 1400

1.13 - 1500 = - $282.87

Which project would you accept?

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Example – FE 2006ZAQ4d&e

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NPV of A = - 388 NPV of B = - 392. Reject both??

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FE 2006 ZA Q4d & e

• NPV of A = -388

• NPV of B = - 392

• Choose project with highest NPV, project A

• As Project A’s costs are mostly incurred in earlier periods,

while B’s in later periods, an increase in interest rate (i.e.

the discount rate) would act to reduce B’s cost more than

A and thus, would make B more attractive to A at some

higher level of rates.

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FE 2006 ZA Q4d & e (alternative solution)

• Consider choosing Machine A instead of B, the incremental cash flows are:

• If interest rate increases, this will cause NPV to decline. If the rate is sufficient high, it may cause the NPV to be negative which then favour B over A.

Year

0

Year 1 Year 2 Year 3 Year 4

Cash

flows of A

minus B

- 80 30 55 5 5

𝑁𝑃𝑉 =30

1.071 + 55

1.072 + 5

1.073 + 5

1.074 - 80 = 3.97 > 0 (choose A)

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Investment appraisal issues

• Capital rationing (i.e. Limited funds)

• Project infinitely divisible?

• Replacement decision

• Machine with different life span?

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Capital Rationing (i.e. Limited Funds)

• It may be the case that we are faced with a pool of projects, all of which have positive NPVs, but we only have access to an amount of money that is less than the total investment cost of the entire project pool.

• Given present values obey additivity principle, it follows that the NPV also possesses the additivity property.

𝑁𝑃𝑉 𝐴 + 𝐵 = 𝑁𝑃𝑉 𝐴 +𝑁𝑃𝑉 𝐵

• We should calculate all project combinations that are feasible (i.e. the total investment in these projects can be financed with our current funds). Then calculate the NPV of each combination by summing the NPVs of its constituents, and finally choose the combination that yields the greatest total NPV.

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Example

• A company has limited fund of $200,000 with following

pool of projects that cost in total $290,000. Which

projects should it accept assuming they are (i) Infinitely

divisible and (ii) NOT infinitely divisible?

Project Initial

investment

NPV

A $120,000 $80,000

B $80,000 $70,000

C $50,000 $50,000

D $40,000 $31,000

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Solution (Infinitely Divisible Projects)

• Compute the profitability index (“PI”) [i.e. NPV/ Investment] of

each project and rank them from highest to lowest. Invest in

project with highest PI to lowest PI till capital is fully utilised.

For the example, invest fully in Project C, B, D and remaining

$30,000 [i.e. $200,000 - $170,000] in Project A.

Project Initial

investment,

I0

NPV Profitability

Index

(i.e. NPV / I0)

Ranking Cumulative

outlay

C $50,000 $50,000 1.000 1 $50,000

B $80,000 $70,000 0.875 2 $130,000

D $40,000 $31,000 0.775 3 $170,000

A $120,000 $80,000 0.667 4 $290,000

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Solution (Projects Not Infinitely Divisible)

• The following are feasible combinations of projects where total initial investment is less than or equal to the limited fund of $200,000.

• The company should choose projects B, C and D as they yield the greatest NPV.

Projects Total initial cost Total NPV

A&B $200,000 $150,000

A&C $170,000 $130,000

A&D $160,000 $111,000

B&C&D $170,000 $151,000

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Question 1:

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[FE 2011 ZA Q5a]

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Replacement decision

• When considering a scenario where we have to select mutually

exclusive projects with different life spans and where each project can

be replicated in exact cash flow patterns, the simple NPV rule might

not necessarily give the correct advice. Consider following example:

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NPV evaluation

• Suppose Lion plc has a cost of capital of 10% per annum.

The NPV of running these two machines can be

calculated as follows:

• For machine A:

• For machine B:

𝑁𝑃𝑉 = −10,000

1.11−10,000

1.12−10,000

1.13−10,000

1.14− 100,000 = −131,699

𝑁𝑃𝑉 = −15,000

1.11−15,000

1.12−15,000

1.13− 75,000 = −112,303

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Annual Equivalent Value (AEV)

• On the basis of the NPV calculations, it seems to cost the company less to run Machine B ($112,290 compared to $131,690). However, if the operation is a going concern and we have to replace the machine once it has expired, how do we know if Machine B still gives the best value to the company?

• To answer this question, we need to find a way to compare the two machines’ cash flows in a consistent manner. This can be done by converting a project’s NPV into its annual equivalent value (AEV). That is, an annuity that lead to same NPV of project. Using Machine A as an example, the annual equivalent value, x, is:

−𝑥

1.11−𝑥

1.12−𝑥

1.13−𝑥

1.14= −131,699

𝑥 = 131,699 ÷1

1.11+1

1.12+1

1.13+1

1.14= 131.699 ÷ 3.17 = 41,545

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Computing AEV

• AEV of 41,545 for Machine A is thus computed using:

𝐴𝐸𝑉 =𝑁𝑃𝑉

𝐴10%,4𝑦𝑒𝑎𝑟𝑠

• Where A10%, 4 years is the annuity factor at 10% for 4 years=1

1.11+1

1.12+

1

1.13+1

1.14= 3.17

• For Machine B, 𝐴𝐸𝑉 =𝑁𝑃𝑉

𝐴10%, 3𝑦𝑒𝑎𝑟𝑠=112,303

2.487= 45,156

• Where A10%, 3 years = 1

1.11+1

1.12+1

1.13 =2.487

• Since we are comparing cost and AEV of A < AEV of B, machine A is selected

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Conditions for using AEV (for info)

• The comparison of two projects’ AEV is only valid

provided that:

• Projects can be replicated in exactly the same cash flow patterns

whenever they expire.

• Projects have similar risk to the company.

• Technological changes are unlikely to affect the efficiency of either

project.

• The expiration of the project will be many years hence (in theory,

infinitely).

• If these conditions are not met, then AEV would not be a

sensible method to determine the replacement policy

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Question 2:

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[FE 2011 ZB Q5a]

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Advantage and Disadvantage of NPV

Advantage

• lead to maximisation of shareholders’ wealth

• It takes into consideration all cash flows and time value of money

• It can be applied to deal with mutually exclusive projects

• It can deal with non-conventional cash flows

• It has realistic assumptions about how the capital markets work in

real life

• Disadvantage

• NPV is depending on the accuracy of cash flows projection and

determination of cost of capital

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Other project appraisal techniques

• Internal rate of return rule

• Payback rule

• Multiples method

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Internal rate of return (“IRR”) rule

• The internal rate of return on an investment or project is the annualised effective compounded return rate of the project.

• The IRR is determined by using the NPV calculation where it refers to the discount rate that makes the net present value (NPV) of all cash flows (both positive and negative) generated from a particular investment equal to zero.

0 = 𝑐𝑖

(1 + 𝐼𝑅𝑅)𝑖− 𝐼0

𝑁

𝑖=1

• The decision rule is to accept a project or investment if its IRR is higher than pre-determined required rate of return (known as hurdle rate)

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Solving the IRR using numerical method

(i.e. trial and error method) • Calculation of the IRR need not be straightforward.

Rearranging the above equation shows us that the IRR is

a solution to a nth order polynomial in r.

• In general, the solution must be found by some iterative

process, for example, a (progressively finer) grid search

method.

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Example

𝑁𝑃𝑉𝐴 =400

(1 + 𝑟)1+700

(1 + 𝑟)2+200

(1 + 𝑟)3− 1000

• Substituting r = 10%, 𝑁𝑃𝑉𝐴 =92.41

• Substituting r = 20%, 𝑁𝑃𝑉𝐴 = - 64.80

Year 0 1 2 3

Project Y -1000 400 700 200

51

Relationship of NPV and discount rate for

project with convention cash flows

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Approximating IRR

• IRR is between 10% and 20%, using linear interpolation

𝐼𝑅𝑅 = 𝑟1 +(𝑟2−𝑟1)

(𝑁𝑃𝑉1−𝑁𝑃𝑉2)× 𝑁𝑃𝑉1

Where

𝑟1 is the lower discount rate (i.e. 10%)

𝑟2 is the higher discount rate (i.e. 20%)

𝑁𝑃𝑉1 is the NPV using 𝑟1 (i.e. $92.41)

𝑁𝑃𝑉2 is the NPV using 𝑟2 (i.e. - $64.80)

𝐼𝑅𝑅 = 10% +(20% − 10%)

92.41 − (−64.80)× 92.41 = 15.878%

If hurdle rate set by company is 10%, accept project since IRR > hurdle rate

53

Question 3:

• 7(b) Suppose you were able to negotiate a discount and

the license only cost you $1 million. What is this project's

IRR? (7 marks)

• To facilitates your calculation of the project IRR, assume

following project’s cash flows (in million):

Year 0 Year 1 Year 2 Year 3 Year 4

-1001 230 245.6 261.98 879.18

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[FE 2012 ZA Q7b]

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Advantage and Disadvantage of IRR

method Advantages:

• It uses all relevant cash flows, not accounting profits, arising from a project

• It takes into account the time value of money.

• The difference between the IRR and the cost of capital can be seen as a margin of safety.

Disadvantages: • The main limitations of using IRR in investment appraisals are that it

may not give the correct decision (i.e. may not lead to maximisation of shareholders’ wealth) in the following scenarios:

• when comparing mutually exclusive projects

• when projects have non-conventional cash flows

• when the cost of capital varies over time.

• It discounts all flows at the IRR rate not the cost of capital rate.

55

Multiple IRRs • As IRR is a solution to a nth order polynomial in r, the solution may not

be unique. That is, there may exist more than one rate that lead to zero NPV or there could be no solution.

Conventional cash flows project Non - conventional cash flows

project

NPV NPV

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Mutually exclusive projects

• Consider a firm that is faced with a choice between two mutually exclusive investment projects (A and B). The locus of NPV-rate of return pairings for each of these projects is given on Figure 1.7 (refer next slide).

• In the evaluation of mutually exclusive projects, use of the IRR rule may lead to choices that do not maximise expected shareholder wealth.

• The first thing to note from the figure is that the IRR of project A exceeds that of B. Also, both IRRs exceed the hurdle rate, r*. Hence, both projects are acceptable but, using the IRR rule, one would choose project A as its IRR is greatest.

• However, if we assume that the hurdle rate is the true opportunity cost of capital (which should be employed in an NPV calculation), then Figure 1.7 indicates that the NPV of project B exceeds that of project A.

57

IRR may not lead to maximisation of

shareholders’ wealth

• IRR Rule : As IRR of A > IRR of B > Hurdle rate, r* (Choose A)

• NPV Rule: If r* is the true opportunity cost of capital, then NPV of B > NPV A > 0 (Choose B)

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Examples: NPV versus IRR (Mutually Exclusive

projects)

59

Time varying cost of capital

• If the cost of capital changes over time, NPV can easily

accommodate this. Suppose the cost of capital is rt for the tth

year. The NPV of a project with different cost of capital over its

lifetime can be given in the following equation:

• NPV assumes that cash flows received during the life of the

project can be reinvested in the capital market at the cost of

capital whereas IRR assumes they can be reinvested at the

IRR which is not a realistic assumption in the real world.

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Question 4: Past exam questions

• 3c. Explain why Net Present Value is a better investment

appraisal technique than Internal Rate of Return. (6

marks) [FE 2011 ZA Q3c]

• 4(b) Describe the NPV and IRR approaches to project

evaluation. When do they agree? Give an example of

when they disagree. (8 marks) [FE2013 ZB Q4b]

61

Payback Rule

• Payback period measures the shortest time to recover the

initial investment outlay from the cash flows generated

from the investment. A company will accept an investment

if the payback period is less than or equal to a target

period set by the company, e.g. three year.

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Example Year 0 1 2 3 4 5

Project A -1000 400 700 200 100 100

Cumulative cash flow -1000 -600 100 300 400 500

Project B -1000 200 300 400 2000 3000

Cumulative cash flow -1000 -800 -500 -100 1900 4900

If target payback period set by company is 3 years,

Payback period of A = 2 years < 3 years (Accept)

Payback period of B = 4 years > 3 years (Reject)

If we can assume that 700 for A in year 2 and 1900 for B in year 4 are received

evenly throughout the year,

Payback period of A = 1 year + 600/700 year = 1.857 years < 3 years (Accept)

Payback period of B = 3 years + 100/2000 year = 3.05 years > 3 years (Reject)

Do you really want to reject B?

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Advantage and Disadvantage of Payback

Rule Advantages:

• It is computationally straightforward.

• It considers the actual cash flows, not profits, arising from a project.

Disadvantages:

• It ignores cash flows beyond the payback period and hence it does

not provide a full picture of a project.

• It does not consider the time value of money (even though the

discounted payback period takes care of that).

• The target payback period is somehow arbitrary

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Discounted Payback Rule

• This is similar to payback period except that the cash

flows from the investment are first discounted to time 0

and the shortest time to recover the initial investment

outlay will then be measured.

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Example

Year 0 1 2 3 4 5

Project A -1000 400 700 200 100 100

Discount Factor 0.833 0.694 0.579 0.482 0.402

Present value 333.20 485.80 115.80 48.20 40.20

Cumulative

discounted cash flow

-1000 - 666.80 -181.00 - 65.20 -17.00 23.20

Assume that the discount rate for project A is 20% per annum

The discounted payback period of project A is 5 years.

While discounted payback period consider the time value of money, it

suffer the same problem of payback period in which cash flows beyond

the discounted payback period are ignored and hence it does not

provide a full picture of a project.

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Multiples method

• The multiples method assesses the firm’s value based on the value of a comparable publically traded firm.

• The multiples method is not an exact science but rather a convenient way to incorporate market beliefs. It should always be used in conjunction with another method, such as NPV.

• Common multiples to use are • market value to earnings,

• market value to EBITDA,

• market value to cash flow, and

• market value to book value.

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Example

• Consider a public traded company’s (let say Firm A) market value to earnings ratio (such as P/E ratio computed as price per share over earning per share). The ratio is publicly available and it tells us how much a dollar of earnings contributes to the present value according to the market’s consensus view.

• Now consider Firm B which not public listed and hence, its market value is not available but we are likely to know its earning. If we assume B is similar to A in all aspects (e.g. capital structure, business, earning growth etc.) and should have similar P/E as A, we can value share of Firm B taking the publicly available ratio and multiply by Firm B’s earning per share.

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Multiple methods

• Some firms, especially younger firms, have no earnings or even negative earnings. In this case it may be better to value the firm as of some future date in which the firm’s cash flows have stabilised, and then to discount to today’s value.

• An alternative is to use more creative multiples, for example • price to patent ratio,

• price to subscriber ratio, or

• price to Ph.D. ratio.

• It is often better to take an average over several comparable firms to calculate the multiple.

• If you believe the firm being valued is better or worse than the comparable firms, you can shade the multiple down or up, as in the example below.

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Subject Guide Example

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Valuing conglomerate using multiples

method • The multiples method allows us to check whether the value of a

conglomerate is equal to the sum of its parts.

• To estimate the value of each business division of a conglomerate we can calculate each division’s earnings and multiply it by the average value to earnings multiple of standalone firms in the same sector. Adding up the value of all divisions gives us an estimated value for the conglomerate, this estimate is on average 12% greater than the traded value of the conglomerate. This is called the conglomerate discount.

• The reasons for the conglomerate discount are not fully understood. It is possible that conglomerates are a less efficient form of organisation due to inefficient capital markets. It is also possible that the multiples method is inappropriate here because single segment firms are too different from divisions of a conglomerate operating in the same industry.

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Advantage and Disadvantage of Multiple

Method • Advantages

• the multiples approach incorporates a lot of information in a simple way.

• It does not require assumptions on the discount rate and growth rate (as is necessary with the NPV approach) but just uses the consensus estimates from the market.

• Disadvantages: • It relies on the assumption that the comparable companies are truly

similar to the company one is trying to value; there is no simple way of incorporating company specific information.

• By using market information, we are assuming that the market is always correct. This approach would lead to the biggest mistakes in times of biggest money making opportunities: when the market is overvalued or undervalued.

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Using PV techniques to value stocks and

bonds

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Valuation of financial assets (e.g. bond,

stock)

• Value of any financial asset is the sum of the

present value of all future cash flows from the

asset:

𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝐴𝑠𝑠𝑒𝑡 =𝐶1(1 + 𝑟)1

+𝐶2(1 + 𝑟)2

+⋯+𝐶𝑛(1 + 𝑟)𝑛

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Type of Bonds

• Zero coupon bond

• A discount or zero coupon bond is an instrument that promises to

pay the bearer a given sum (known as the principal) at the end of

the instrument’s lifetime. For example, a simple five-year discount

bond might pay the bearer $1,000 after five years have elapsed.

• Coupon bond

• coupon bonds not only repay the principal at the end of the term

but in the interim entitle the bearer to coupon payments that are a

specified percentage of the principal. Assuming annual coupon

payments, a three-year bond with principal of £100 and coupon

rate of 8 per cent will give annual payments of £8, £8 and £108 in

years 1, 2 and 3.

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Valuation of annual coupon bond

• In more general terms, assuming the annual coupon rate is c (e.g. 12% per annum), the principal is P (e.g. 100), the time to maturity is K (e.g. 5 years) and the required annual rate of return on this type of bond is rb (also known as yield to maturity, e.g. 10%), the price of the bond can be written as:

𝑃𝑟𝑖𝑐𝑒 = 𝑐𝑃

(1 + 𝑟𝑏)𝑖

𝐾−1

𝑖=1

+𝑃(1 + 𝑐)

(1 + 𝑟𝑏)𝑘

𝑃𝑟𝑖𝑐𝑒 = 0.12(100)

(1 + 0.1)𝑖

5−1

𝑖=1

+100(1 + 0.12)

(1 + 0.1)5

𝑃𝑟𝑖𝑐𝑒 =12

1.11+12

1.12+12

1.13+12

1.14+112

1.15= 107.58

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Using PV annuity factor

• From above, we can calculate the PV of all coupons (which is a set of annuity) and the PV of principal separately


1.11+12

1.12+12

1.13+12

1.14+12

1.15+100

1.15

= 12 ×1

1.11+1

1.12+1

1.13+1

1.14+1

1.15+100

1.15

Where following can be calculated using PV annuity factor

1

(1+𝑟)1+1

(1+𝑟)2+1

(1+𝑟)3+⋯+

1

(1+𝑟)𝑛=1

𝑟−

1

𝑟(1+𝑟)𝑛

Thus,

𝑃𝑟𝑖𝑐𝑒 = 12 ×1

0.1−

1

0.1(1 + 0.1)5+100

1.15= 107.58

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Valuation of semi-annual coupon bond

• If the above bond is semi-annual coupon bond, the price is

𝑃𝑟𝑖𝑐𝑒 =

𝑐2𝑃

(1 +𝑟𝑏2)𝑖

2𝐾−1

𝑖=1

+𝑃(1 +

𝑐2)

(1 +𝑟𝑏2)2𝑘

𝑃𝑟𝑖𝑐𝑒 = 0.06(100)

(1 + 0.05)𝑖

10−1

𝑖=1

+100(1 + 0.06)

(1 + 0.05)10

𝑃𝑟𝑖𝑐𝑒 = 6 ×1

0.05−

1

0.05(1 + 0.05)10+100

1.0510= 107.72

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Valuation of zero coupon bond

• If above bond does not coupon, the price is

𝑃𝑟𝑖𝑐𝑒 =𝑃

(1 + 𝑟𝑏)𝑘


(1 + 0.1)5= 62.09

• Price of zero coupon bond will always sell at a discount to

its principal (also known as face value or par value of

bond)

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Activity

• Using the previous formula, value a seven-year bond with

principal $1,000, annual coupon rate of 5 per cent and

required annual rate of return of 12 per cent. (Hint: the

use of a set of annuity tables might help.)

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Valuing common equity using dividend

discount model [“DDM”] • Aside from issues such as voting rights, the share simply delivers a

stream of future dividends to the holder.

• Assume that we are currently at time t, that the corporation is infinitely long-lived (such that the stream of dividends goes on forever) and that we denote the dividend to be paid at time t+i by Dt+i.

• Also assume that dividends are paid annually with first dividend received at time t+1 and not current time, t. Denoting the required annual rate of return on this equity share to be re, then a present value argument would dictate that the share price (P) should be defined by the following formula:

𝑃 = 𝐷𝑡+𝑖(1 + 𝑟𝑒)

𝑖

∞

𝑖=1

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Gordon growth model of equity valuation

• A simplification of the preceding formula is available when we assume that the dividend paid grows at constant percentage rate g per annum.

• Then, assuming that a dividend of D0 has just been paid, the future stream of dividends will be D0(1+g), D0(1+g)2, D0(1+g)3 and so on.

• This type of cash-flow stream is known as a perpetuity with growth, and its present value can be calculated very simply. In this setting the price of the equity share is:

𝑃 =𝐷0(1 + 𝑔)

𝑟𝑒 − 𝑔

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Required rate of return on the stock

• Re-arrange the above Gordon growth model

𝑟𝑒 =𝐷0(1 + 𝑔)

𝑃+ 𝑔

• The first term is the expected dividend yield on the stock

and the second is expected dividend growth

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Question 5:

• Derive the Gordon Growth Model, and explain how you

can use this model to work out the required rate of return

for an investment project in terms of the dividend yield of

the project. (5 marks) [2008ZAQ2(a)]

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Question 6:

• Suppose that starting next year you will receive a

cash flow D, which will grow at a rate g per year.

The appropriate discount rate is r. Derive the

Gordon Growth Model for the present value of this

cash flow. (9 marks) [2013ZA Q2(c)]

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Activity

Attempt the following questions:

1. An investor is considering buying a certain equity share. The stock has just paid a dividend of £0.50, and both the investor and the market expect the future dividend to be precisely at this level forever. The required rate of return on similar equities is 8 per cent. What price should the investor be prepared to pay for a single equity share?

2. A stock has just paid a dividend of $0.25. Dividends are expected to grow at a constant annual rate of 5 per cent. The required rate of return on the share is 10 per cent. Calculate the price of the stock.

3. A single share of XYZ Corporation is priced at $25. Dividends are expected to grow at a rate of 8 per cent, and the dividend just paid was $0.50. What is the required rate of return on the stock?

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