corporate finance 1

Corporate Finance Lecture 1

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Corporate Finance

Lecture 1 : Concept of Time Value of Money


Concept of Time Value of Money !   Why does $ has time value?

!   Money is capable of earning return !   Money has the same value at the same time reference

•  Nominal Cash Flows Vs Time Value of Nominal Cash Flows

!   A $ today is worth more than a $ tomorrow !   Pre-requisite information for time value of money applications

!   Direction (Inflows or Outflows) & Magnitude (How much?) of cash flows

!   Timing of cash flows !   Discount rate : Opportunity cost of capital or Required rate

of return


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Time Value Terminologies & Symbols Symbol Description DCF Discounted cash flow PMT Equal payments or receipts with annuities CFt Cash flow occurring at end of period t PV Present value FVn Future value at the end of period n PVAn Present value of an annuity with n equal payments

or receipts FVAn Future value of an annuity with n equal payments

or receipts


Time Value Terminologies & Symbols Symbol Description

i or I Interest rate (or “Discount rate”)

n or N Number of time periods

t Reference period (e.g., t = 1, t = 2, etc.)

FVIFi,n Future value interest factor

PVIFi,n Present value interest factor

FVIFAi,n Future value interest factor for an annuity

PVIFAi,n Present value interest factor for an annuity


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Time Value of Money Solution Methods

For the purpose of UOL, there are 2 possible methods to solve for Time of Money problems as financial calculator are not permitted for use in exam

I.  Numerical – using regular calculator without financial functions BUT must know how to solve for the power of n (positive & negative)

II.  Interest Tables – contained at Tables A-1, A-2, A-3 & A-4 provided as handouts


Time Line Conventions & Illustrations !   Each number represents the end of the respective periods !   Cash inflows are represented by positive numbers !   Cash outflows are represented by negative numbers

Time line for a lump sum of $100 to be received at the end of Year 2 :

0 i% 1 2 Year

100 Cash Flow


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Time Line Conventions & Illustrations Time line for a 3-year annuity of $100 :

Time line for an uneven cash flow stream of -$50 in Year 0, $100 in Year 1, $75 in Year 2, and $50 in Year 3 :

0 2 1

0 100 100 100

3 i%

100 75 50 -50

0 1 2 3 i%


Future Value (Compounding) Find the FV of $100 invested for 3 years in an account paying 10% p.a. interest :

FVn = PV(1+i)n = PV(FVIFi,n) = $100 (1.10)3 = $100 (FVIF10%,3) See Table A-1 = $100 (1.3310) = $133.10

0 1 2 3

(100) FV=? 10% p.a.


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Present Value (Discounting) Find the PV of $100 to be received in 3 years if the appropriate rate of return is 10% p.a.:

PV = FVn / (1+i)n = FVn (l+i)-n = FVn(PVIFi,n) = $100 (1.10)-3 = $100 (PVIF10%,3) See Table A-2 = $100(0.7513) = $75.13

0 1 2 3

PV = ? 100

10% p.a.


Solving for n in TVM Problems How long will it take a firm’s sales to double, if sales are growing at a 20% p.a.?

FVn = PV(1+i)n $2 = $1(1.20)n $2 = (1.20)n Look in Table A-1 for FVIF20%,n = 2 n ≈ 4 periods Note : The PV equation could be used to solve this problem too

0

(1)

N = ?

2


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Solving for i in TVM Problems What growth rate does a firm need to achieve if it is to triple its sales in 6 years?

FVn = PV(1+i)n $3 = $1(1+i)6 $3 = (1+i)6 Look in Table A-1 for FVIFi,6 = 3 i ≈ 20% p.a. Note : The PV equation could be used to solve this problem too

0

(1)

N = 6

3

i% p.a.?


Types of Multiple Cash Flows Even Cash Flows !   Annuities !   Perpetuities

Uneven Cash Flows !   Multiple Single Unequal Cash Flows !   Recognising Annuities / Perpetuities within a series of cash

flows


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Annuities !   Definition

Finite series of equal payments or receipts over regular intervals

!   Ordinary annuity Equal payments or receipts occur at the end of each period

0

PV

1 2 3

100 100 100 FV

i% p.a.


Annuities !   Annuity due

Equal payments or receipts occur at the beginning of each period

0

100

PV

2 1 3

100 100 FV

i% p.a.


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Future Value of an Ordinary Annuity What will an investment of $100 p.a. for 3 years accumulate to if the expected rate of return is 10% p.a.?

Time Line Approach :

0

100 100 110 121 $331

1 2 3 10% p.a. 100 x 1.1

x 1.12


Future Value of an Ordinary Annuity !   Additive Principle

!   Future Value of a Series of Cash Flows = Aggregate Future Values of each Cash Flow in the Series

!   If the series of cash flows is even then a simplified mathematical formula can be derived with the use of the concept of Arithmetic Progression series


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Future Value of an Ordinary Annuity Numerical Approach :


Future Value of an Ordinary Annuity Interest Table Approach : FVAn = PMT (FVIFAi,n)

= $100 (FVIFA10%,3) = $100 (3.3100) = $331 Table A-4 contains FVIFAi,n factors


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Present Value of an Ordinary Annuity What is the value of an investment that provides an income of $100 p.a. for 3 years if the required return is 10% p.a.?

Time Line Approach :

0 10% p.a. 1 2 3

100 100 100 $ 90.91 82.64 75.13 $248.68

1.1-2 1.1-1

1.1-3


Present Value of an Ordinary Annuity !   Additive Principle

!   Present Value of a Series of Cash Flows = Aggregate Present Values of each Cash Flow in the Series

!   If the series of cash flows is even then a simplified mathematical formula can be derived with the use of the concept of Arithmetic Progression series


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Present Value of an Ordinary Annuity

Numerical Approach


Present Value of an Ordinary Annuity Interest Table Approach : PVA = PMT (PVIFAi,n)

= $100 (PVIFA10%,3) = $100 (2.4869) = $248.69 Table A-3 contains PVIFAi,n factors


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Perpetuities !   Infinite series of equal payments or receipts over regular

intervals

!   Since perpetuities are infinite series, it is only possible to determine the PV

If i = 10% p.a., then PV of the perpetuity is

100 100 100

2 3 1 0


Perpetuities !   To derive the formula, let’s revisit the PV of an annuity

!   For very large n -> (1 + i)n becomes infinity as long as i > 0 &

!   tends to 0 & therefore


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Growing Perpetuities !   If the payments or receipts are not equal but they grow by a

constant rate g such that

!   PMT1 = PMT, PMT2 = PMT(1+g)1, PMT3 = PMT(1+g)2, …, PMTn = PMT(1+g)n-1, …

PMT PMT(1+g) PMT(1+g) 2

2 3 1 0


Uneven Cash Flow Stream 0 1 2 3 4

0 100 300 300 -50 $ 90.91

247.93 225.39

(34.15) $ 530.08

1.1-1

1.1-2

1.1-3

1.1-4

10% p.a.

0 1 2 3 4

0 100 300 300 -50 10% p.a.

$ 90.91 473.32

(34.15) $ 530.08

$300 (1.7355) $520.65

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