Corporate Finance

Lecture 4

Chapters 5 and 6 Textbook

The Time Value of Money and Valuation of Cash Flows

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The lecture material contains content owned by Kaplan Business School and other materials copyrighted by Parrino,R , Kidwell,D, Au

Yong,H, Dempsey,M , Morkel-Kingbury,N, Ekanayake,S, Kofoed,J and Murray,J (2014), Fundamentals of Corporate Finance, 2nd Edition,

John Wiley & Sons Australia, Ltd Do not remove this notice.

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3

Topic 3 Revision

1. Stand-alone risk (Ϭ) = __________(ß) + _______ risk

2. How do we quantify the total risk of asset returns? 3. What is market/systematic risk? Give one

example. 4. What is diversifiable/ unsystematic risk? Give one

example. 5. What is the CAPM?

Calculators • You can use any calculators that do not store text for

this course (i.e. Scientific, financial etc.) • You do not necessarily need a financial calculator

• You do not need to know how to use financial functions to do well in the course

• You may chose to use financial functions if you find it easier, however in tests and exams you need to show your working out

• Please do not bring a calculator that can store text into the exam (i.e. Graphics calculator)

1. Time value of money • How do managers determine the value of a

series of future cash flows? • What is the value of the stream of future cash

flows today?

MONEY HAS A TIME VALUE A dollar today is worth more than a dollar tomorrow

• We want to estimate what present values will be worth in the future and what future values are worth at the present

1- 5

1. Time value of money (cont.)

• Consuming today or tomorrow Would you prefer to get $100 today or tomorrow? • TVM is based on the belief that people prefer to consume

goods today rather than wait to consume similar goods tomorrow

• Positive time preference • Today’s dollar can be invested to earn interest or spent • Value of a dollar invested grows over time • Rate of interest determines trade-off between spending today

versus saving

1- 6

2. Interest Rate • Interest can be calculated as Simple Interest or Compound Interest

 Simple interest • Interest is only applied to (or earned on) the original amount

invested (the ‘principal’) S = Principal + Interest earned S = P + Pin S = P(1 + in)

where: P is the original amount invested and S = the total amount invested plus the return after n periods i = the interest rate for a given time period e.g. 1 year (but it

could be monthly or half-yearly or some other duration) n = the number of time periods e.g. 5 years

2. Interest Rate (cont.) Compound Interest

• Interest is paid on interest earned as well as on the original principal

S = P(1 + i)n

where: P is the original amount invested and S is the total amount invested plus the return after n periods i is the interest rate per period n is the number of periods

2. Interest Rate (cont.) Example: On 31 December 2010, John deposited $1,000 in a bank account that offered an interest rate of 10% per year. How much money ended up in John’s account after 3 years?

Simple interest (interest earned on original amount only)

S = P(1+in) = $1,000[1+(0.10x3)] = $1,300

Compound interest (interest earned on original amount plus interest already earned)

S = P(1+i)n = $1,000 x (1+0.10)3 = $1,331

Simple v compound interest • Let’s see step by step how compound interest is calculated

It may be easier to use formula: FV=PV(1+i)^n (where FV is “future value” and PV is “present value”)

 Why does compound interest give you a larger future value than simple interest?

Because interest is earned on top of interest already earned

Year A Starting Balance

B Interest (A*i)

C Starting Balance + Interest (A+B)

1 1000 1000*0.1=100 1100 2 1100 1100*0.1=110 1210 3 1210 1210*0.1=121 1331

11

3. Nominal and Effective Interest Rate

• Nominal Interest Rate (NIR) is a quoted interest rate.

• Effective Annual Interest Rate (EAR):

 the actual rate of interest to be earned or paid or

 the annual interest rate that reflects compounding within a year.

11 −  

  

+= m

m RateInterestQuoted

EAR

m= number of compounding periods per year

12

Nominal v Effective Rate • St George credit card rate = 13.24% • http://www.stgeorge.com.au/personal/credit-cards/credit-card-interest-rates

• Fine Print: Interest is compounded daily • What is the effective annual rate?

…Would you prefer a term deposit at 6% nominal interest rate that compounds interest monthly or daily?

11 −  

  

+= m

m RateInterestQuoted

EAR

%15.141 365 1324.0

1 365

=−  

  

+=EAR

4. Present Value VS Future Value • Financial decisions are evaluated either on a future value basis or

present value basis.

 Future value measures what one or more cash flows are worth at the end of a specified period.

 Present value is a measure of what one or more cash flows that are happen in the future will be worth today (at “time zero”)

Compounding is used to convert the cash flows to a future value

Discounting is the process of converting future cash flows to their present values

Timeline • A timeline shows the timing of cash flows. • In simple financial modeling, cash flows are assumed occur at the end of

periods.

-$1M +$0.5M +$0.7M+$0.4M

Time 0 Time 1 Time 2 Time 3

• Cash outflows = negative values • Cash inflows = positive values • Time 0 is today • Time 1 is the end of the first period ,or the

beginning of the second period.

e.g.

4. Cash Flow Patterns

5. Single amount cash flow 5.1 Present value Single

period 5.2 Present value Multiple

periods 5.3 Future value Single period 5.4 Future value Multiple

periods 6. Mixed cash flows

6.1 Present value 6.2 Future value

7. Annuity 6.1 Present value 6.2 Future value

8. Annuity Due 8.1 Present value 8.2 Future value

9. Perpetuity 10. Loan amortisation

You can use the formula, calculator functions or a PV/FV table to do the calculations. However, in this course we will be concentrating on using the formula.

5. Lump Sum Cash Flow 5.1 Present Value: Single-Period Investment

(Simple Interest)

Example: You want $1,000 after one year to buy a computer, so you deposit money in a bank today, earning interest at 10% per year. How much money should you invest now?

Formula: Present Value = Future Value

(1 + interest rate) PV = FV

(1 + i) Solution: PV = $1,000

(1+0.10) = $909.09

Info: FV = $1000 i = 10%

One year = single period PV = ?

Recall: S = P(1+in)

= P(1+i), n=1 (single period) P = S/(1+i), n=1 (single period)

Year 0 Year 1

1,000PV=?

Time line:

5.2 Present Value - Multiple Periods (Compounding Technique)

Example: You want $1,000 after 2 yrs to buy a computer so you deposit money in a bank today earning interest at 10% per year. How much money should you invest now?

Formula:

PV = FVn (1 + i)n

Solution: PV = $1,000 (1+0.10)2

= $826.45

Time line: 1 2

$1,000PV=?

Info: FV = $1000 i = 10% n = 2

years PV = ?

Recall: S = P(1+i)n P = S/(1+i)n

Use Formula

1,000

10

2

-826.45

2ndF C-CE Clearing Memories

Financial Calculator: Using the SHARP EL-735

Info: FV = $1,000 i = 10% n = 2 years PV = ?

PV = FVn (1 + i)n

COMP PV

n

i

FV

Use Calculator

Present Value Factor: 1/(1 + i)n (Appendix A-2: Present value factors, pp. 778-9 of Textbook)

Info: FV = $1,000 i = 10% n = 2 years

PV = ?

Interest rate per year Number of years 1% 5% 6% 7% 8% 9% 10%

1 0.990 0.952 0.943 0.935 0.926 0.917 0.909 2 0.980 0.907 0.890 0.873 0.857 0.842 0.826 3 0.971 0.864 0.840 0.816 0.794 0.772 0.751 4 0.961 0.823 0.792 0.763 0.735 0.708 0.683 5 0.951 0.784 0.747 0.713 0.681 0.650 0.621

PV = FVn (1 + i)n

= FVn x 1 . (1 + i)n

= 1,000 x 0.826 = 826.45

Use Table

5.3 Future Value: Single-Period Investment (Simple Interest)

Example: You place $1,000 in a bank savings account that pays interest at 10% per year for one year. How much money will you have in one year?

Formula: Future Value = Present Value x (1 + interest rate)

FV = PV x (1 + i) Solution:

FV = $1,000 x (1+0.10) = $1,100

Info: PV = $1000 i = 10%

one year = single period FV = ?

1,000 FV=?

Time 0 Year 1

5.4 Future Value - Multiple Periods (Compounding Technique)

Example: You place $1,000 in a bank savings account that pays 10% per year for two years. How much money will you have in two years?

Formula: FVn = PV x (1 + i)n

Solution: FV2 = $1,000 x (1+0.10)2

= $1,210 Time line:

1 2

$1,000 FV=?

Info: PV = $1000 i = 10% n = 2

years FV = ?

Use Formula

Future value when compounding more frequently

Compounding More Frequently Than Once a Year • The more frequently the interest payments are compounded, the larger the future value of $1 for a given time period.

FV n

= PV × (1+ i m) m × n

where: m = Number of compounding periods in a year n = Number of years i = Annual interest rate

Example: You deposit $5,000 in your bank which promises to pay an interest of 10% p.a. compounded quarterly. How much money will you have at the end of 2 years?

FV n

= PV × (1+ i m) m × n

FV2 = 5000 x (1+ (0.10/4))(4 x 2) = $6,092.01

5,000

2.5

8

6,092.01

2ndF C-CE Clearing Memories

COMP FV

n

i

PV+/-

Info: PV = $5,000 i = 10%

m = 4 n = 2 years FV = ?

10%/4

4 x 2

0

$1,000

1

$2,000

2 3

$3,000

i = 10%

Add for PV: $4,815.92

6. Mixed Cash Flows 6.1 Present Value of Multiple Cash Flows

( ) 09.909$

10.01 000,1$

11 =+ =PV

( ) 89.652,1$

10.01 000,2$

22 =+ =PV

Example: Suppose your friend needs cash and offers to pay you $1,000, $2,000, and $3,000 at the end of Y1, Y2 and Y3 respectively - if you will give him $6,000 cash today. You realize that because of the time value of money, the cash flows he has promised to pay are worth less than $6,000. If the interest rate on similar loans is 10%, how much should you pay for the cash flows your friend is offering

( ) 94.253,2$

10.01

000,3$ 33

= +

=PV

Time line

24

PV = FVn (1 + i)n

6.2 Future Value of Mixed Cash Flows

FVYear0 = $1,000(1+0.10)3 = $1,331

FVYear2 = $3,000(1+0.10)1 = $3,300

FVYear1 = $2,000(1+0.10)2 = $2,420

Add for Total Future Value: $7,051

( )tt rPVFV += 1 $1,000 $2,000 $3,000

Today Year 1 Year 2 Time line

Example: If you invest $1,000 today, $2,000 a year from now and $3,000 at the end of 2 years. How much money will you have at the end of 3 years, if the bank continues to pay 10% interest per year?

Year 3 i = 10%

7. Annuities An annuity is a series of cash flows of equal amount, equally spaced in time, for a certain period of time.

Annuities – Cash flows occur at the END of the period for a finite number of periods

$CF $CF $CF $CF $CF

0 1 2 3 4 5

0

$2,000

1

$2,000

2 3

$2,000

i = 8%

Add for PV: $5,154.19

7.1 Present Value for Annuities Cash Flows

( ) 85.851,1$

08.01 000,2$

11 =+ =PV

( ) 68.714,1$

08.01 000,2$

22 =+ =PV

Example: Suppose that a financial contract pays $2000 at the end of each year for 3 years and the appropriate discount rate is 8%. What is the most we should pay for this annuity? Using what we have learnt by now, we can do this in stages as follows:

( ) 66.587,1$

08.01 000,2$

33 =+ =PV

Time line

27

PV = FVn (1 + i)n

Info: CF= $2,000 i = 8%

n = 3 PVA = ?

Another, and quicker way to calculate, would be to use PVA (the Present Value of an annuity)

Use Formula

Info: CF= $2,000 i = 8%

n = 3 PVA = ?

( )

( ) 19.154,5$

08.01

1 1*

08.0 2000

annuity) (reg. 1

1 1*

3

=

  

   



+ −=

  

   



+ −=

PV

PV

ii CF

PV n

154,5$ 577.22000$

Pr1

= ×=

×=

  

   −

×=

FactorAnnuityPVCFPVA i

factorvalueesent CFPVA

n

n

Use Table

(Appendix A-4: PV Annuity Factor, pp.782-3 of Textbook)

Info: CF= $2,000 i = 8%

n = 3 PVA = ?

2,000

8

3

5,154.19

2ndF C-CE Clearing Memories

Financial Calculator: Using the SHARP EL-735

COMP PV

n

i

+/-

Use Calculator

Info: CF = $2,000 i = 8% n = 3 PVA = ?

 

  

 +

−×= nn ii CF

PVA )1(

1 1

PMT

7.2 Future Value for Annuities Cash Flows

FVYear1 = $1,000(1+0.08)3 = $1,259.71

FVYear3 = $1,000(1+0.08)1 = $1,080

FVYear2 = $1,000(1+0.08)2 = $1,166.40

Add for Total Future Value: $4,506.11

( )tt rPVFV += 1

$1,000 $1,000 $1,000

Today Year 1 Year 2 Time line

Example: Suppose that you plan to save $1000 at the end of every year for 4 years with the goal of buying a racing bicycle. The bike costs $4500. If your bank pays 8% p.a. will you have enough money to buy the bike at the end of 4 years?

Year 3 i = 8%

Year 4

$1,000

Info: CF= $1,000 i = 8%

n = 4 FVA = ?

FVA = Future value of an annuity

Use Formula

Info: CF= $1,000 i = 8%

n = 4 FVA = ?

( )( ) ( )( ) 11.506,4$FVA

108.01* 08.0

1000 FVA

11*FVA

n

4 n

n

=

−+=

−+= ni i

CF

8. Annuities Due

An annuity is a series of cash flows of equal amount, equally spaced in time, for a certain period of time.

Annuities Due – Cash flows occur at the BEGINNING of the period for a finite number of periods (Prepay)

$CF $CF $CF $CF $CF

0 1 2 3 4 5

$CF

What is the difference between an ordinary annuity and an annuity due?

Ordinary Annuity

PMT PMTPMT

0 1 2 3 i%

PMT PMT

0 1 2 3 i%

PMT

Annuity Due

Relationship between ordinary annuity and annuity due

• Each period’s cash flow thus earns extra period of interest compared to ordinary annuity

• Present or future value of annuity due is always higher than that of ordinary annuity

Annuity due = Ordinary annuity value × (1+i)

)1( )1(

1 1

)1(

i ii

C PV

iAnnuityOrdinaryPVDueAnnuityPV

n +× 

  

 +

−×=

+×= 8.1 Present Value for Annuities Due Cash Flows

Ordinary Annuity versus Annuity Due

PV of the ordinary annuity =3,312 PV of the annuity due = 3577  Calculated as 3,312 x 1.08

8.2 Future Value for Annuities Due Cash Flows Example: You plan to save $2,000 every year for 30 years to a retirement account paying 8%. If you retire at the end of 30 years, how much will you have?

$2,000 $2,000 $2,000 $2,000 (last deposit)

Start year 30

( )( ) ( )( ) ( )

71.691,244$

08.01108.01* 08.0

2000

)1(11*

)1(

30

=

+×−+=

+×−+=

+×=

ii i

CF FVA

ivaluefuturedueOrdinaryvaluefuturedueAnnuity n

n Info: CF = $2,000 i = 8% n = 30 FVA = ?

Time zero Maturity at end of year 30

9. Perpetuity A perpetuity is a series of cash flows of equal amount and equally spaced in time, which continues forever.

$CF $CF $CF $CF $CF

0 1 2 3 4 5

i CF

PV =

Example: An investment security promises to pay $1,000 per year in perpetuity. If the interest rate is 9% per year, what is the value of this investment?

11.111,11$ 09.0 000,1$

=

=

= i C

PV

Quiz 4 1. Compounding accelerates the growth of the total interest earned. a. True b. False 2. The more frequently the interest payments are compounded, the larger the future value of $1 for a given time period. a. True b. False 3. The present value of an investment of $1,000 to be received in three years at a discount rate of 10 percent is $751.31. a. True b. False

Quiz 4 4. Future value: Ning Gao is planning to buy a house in five years. She is looking to invest $25,000 today in an index managed fund that will provide her a return of 12 percent annually. How much will she have at the end of five years? (Round to the nearest dollar.) a. $45,000 b. $39,338 c. $44,059 d. $40,000

5. Multiple compounding periods (FV): Your brother has asked you to help him with choosing an investment. He has $5,000 to invest today for a period of two years. You identify a bank term deposit that pays an interest rate of 4.25 percent with the interest being paid quarterly. What will be the value of the investment in two years? a. $5,434 b. $5,441 c. $5,107 d. $5,216

Quiz 4 6. When you pay the same amount every month on your car loan for a period of three years, the stream of cash flows is called an annuity. a. True b. False 7. In ordinary annuities, cash flows occur at the beginning of each period. a. True False 8. The effective annual interest rate (EAR) is defined as the annual growth rate that takes compounding into account. a. True b. False

1. Explain the time value of money concept 2. Differentiate between simple and compound interest 3. Calculate effective interest rates and apply to problems 4. Calculate lump-sum Present value and Future value 5. Perform single amount cash flow – PV and FV

calculations 6. Perform mixed cash flows – PV and FV calculations 7. Perform annuity – PV and FV calculations 8. Perform annuity due – PV and FV calculations 9. Perform perpetuity calculations 10. Appraise and analyse real-life financial situations using

financial mathematics

Learning Outcomes Topic 4