200244_1468273595_FINC19011BUSINESSFINANCE-Wk02-.pptx

8/10/2019 200244_1468273595_FINC19011BUSINESSFINANCE-Wk02-.pptx

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Chapter 6Discounted cash flows

and valuation

Prepared by

Alex Proimos & Chee Jin Yap


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Basics


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Multiple cash flows

Future value of multiple cash flows

Solving future value problems with multiple cash flows:

1. Draw timeline to ascertain each cash flow is placed

in correct time period

2. Calculate future value of each cash flow for its time

period

3. Add up the future values


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Multiple cash flows

Present value of multiple cash flows

First, prepare timeline to identify magnitude and timing

of cash flows

Next, calculate present value of each cash flow

Finally, add up all present values.

Sum of present values of stream of future cash flows is

their current market price, or value


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Present value of three cash flows


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Another MathYour grandfather has agreed to deposit a certainamount of money each year into an account paying7.25 per cent annually to help you go to university.Starting next year, and for the following 4 years, heplans to deposit $2250, $8150, $7675, $6125, and$12 345 into the account. How much will you have at

the end of the 5 years?Solution:


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Annuity, Perpetuity


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Level cash flows: annuities and

perpetuities

Present value of an annuity Many situations exist where businesses and individuals

would face either receiving or paying constant amount

for length of period Annuitystream of cash flows when company faces stream of

constant payments on a bank loan for a period of time

Individual investors may make constant payments on home or

car loans, or invest fixed amount year after year saving forretirement


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perpetuities

Annuities and perpetuities

Annuity: any financial contract calling for equally spaced

level cash flows over finite number of periods

Ordinary annuity: constant cash flows occurring at end of eachperiod

Annuity due: constant cash flows occurring at beginning ofeach period

Perpetuity: contract calling for cash flow payments tocontinue forever


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Annuity Present Value of Ordinary Annuity

CF is the series of equal cash flows per year

i is the rate of interest per annum

m is the number of compounding per year

n is the number of years

x

11

(1 )( / )

m n

n

i / mPVA CF m

i / m


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A Math on PV of an Ordinary AnnuityDynamo Ltd is expecting annual payments of $34225for the next 7 years from a customer. What is the

present value of this annuity if the discount rate is 8.5per cent?

Solution:

m=1


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A Math on PV of an Ordinary AnnuityDynamo Ltd is expecting annual payments of $34225for the next 7 years from a customer. What is the

present value of this annuity if the discount rate is 8.5per cent?

Solution (Same Math: without using Formula)

PV of the 1stcash flow =$34225

(1 + .085)= $31543.78

PV of the 2ndcash flow = $34225(1 + .085)

= $29072.61

PV of the 3rdcash flow =$34225

(1 + .085)= $26795.03

PV of the 4thcash flow =$34225

(1 + .085)= $24695.88


(1 + .085)= $22761.18


(1 + .085)= $20798.05


(1 + .085)= $19334.60

Total PV = $175181.13

Same result using

Formula

Yet, formulaIs better particularlysince (m X n) can belarge


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Extra Math Involving Ordinary Annuity

You have borrowed $10,000, repayable by equal half-

yearly instalments for five years. If interest is 5% p.a.compounded half-yearly, what will be your half-yearlyrepayments?

Solution:

x

2 x 5

11(1 )

( / )

$10,000; ?; 2; 5% 0.05; 5

11(1 )

10,000 ( / 2)

10,000 x 4.376032

CF=$2285.18

m n

n

n

i / mPVA CF m

i / m

PVA CF m i n

0.05 / 2CF

0.05 / 2

CF

This CF=$2285.18 is annual.

So, half-yearly repayment is

CF/2 = $2285.18/2 = $1142.5


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Annuity Future Value of Ordinary Annuity

CF is the series of equal cash flows per year

i is the rate of interest per annum

m is the number of compounding per year

n is the number of years

x(1 ) 1( / )

m n

n

i / mFVA CF m

(i / m)


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A Math on FV of an Ordinary AnnuityMike White is planning to save up for a trip to Europe in 3 years.He will need $10000 when he is ready to make the trip. He plans

to invest the same amount at the end of each of the next 3 yearsin an account paying 6 per cent. What is the amount he will

have to save every year to reach his goal of $10000 in 3 years?

Solution:

On Next Slide


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A Math on FV of an Ordinary Annuity

m=1


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Annuity Due

Annuities due

Annuity is called an annuity due when there is an

annuity with payment being incurred at beginning of

each period rather than at end

Rent or lease payments typically made at beginning of

each period rather than at end


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Ordinary annuity vs annuity due


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Annuity Due

Annuities due

Annuity transformation method shows relationship

between ordinary annuity and annuity due

Each periods cash flow thus earns extra period of

interest compared to ordinary annuity

present or future value of annuity due is always higher thanthat of ordinary annuity

Annuity due = Ordinary annuity value (1+i)


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O di it


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Slide 25

e.g., Ordinary annuityversus annuity due

12750 1

1 1 1.080.08 (1.08)

750 7.5361 1.08 5,652.06 1.08

$6,104.22

Due Ord

Due

Due

PVA PVA i

PVA

PVA

An investment opportunity requires a payment of $750 for 12 years, starting ayear from today. If your required rate of return is 8 per cent, what is the valueof the investment today? Now, assume this is an annuity due with paymentsstarting today. Recalculate the PV of this investment.

11

1Ord n

CFPVAi i

$5,652.06

PV f di


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e.g. PV of an ordinaryannuity

Slide 26

12

11

1

750 11 750 7.53610.08 (1.08)

n n

CFPVA

i i

$5, 652.06

Present value of an ordinary annuity:An investmentopportunity requires a payment of $750 for 12 years,starting a year from today. If your required rate of return is8 per cent, what is the value of the investment today?


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Perpetuities

A perpetuity is constant stream of cash flows that goeson for infinite period

In share markets, preference shares issues are

considered to be perpetuities, with issuer paying aconstant dividend to holders

Equation for present value of a perpetuity can be

derived from present value of an annuity equation with

n tending to infinity:


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i

ii

i

CF

CFCF

CFPVA

)01()1(

1

1

annuityforfactorvaluePresent

Perpetuities


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Cash flows that grow at a

constant rate In addition to constant cash flow streams, one may have

to deal with cash flows that grow at constant rate overtime

These cash-flow streams called growing annuities orgrowing perpetuities


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h fl h


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constant rate

Growing annuity

Use this equation to value the present value of growing

annuity when the growth rate is less than discount rate

h fl h


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PVA

=CF

1

(i- g)


constant rate

Growing perpetuity

When cash flow stream features constant growing

annuity forever


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Slide 36

Growing annuity:Gull Petroleum Ltd owns several service stations. Management islooking to open a new station in the northern suburbs of Perth. One possibility theyare evaluating is to take over a station located at a site that has been leased from thestate government. The lease, originally for 99 years, currently has 73 years beforeexpiration. The service station generated a net cash flow of $92,500 last year, andthe current owners expect an annual growth rate of 6.3 per cent. If Gull uses a

discount rate of 14.5 per cent to evaluate such businesses, what is the present valueof this growing annuity?

e.g. Growing annuity

73

1

73

1 92,500 1.063 1.0631 1

( ) 1 (0.145 0.063) 1.145

98,327.501 0.928384279

0.082

1199115.854 0.995593154

n

Growing

CF gPVA

i g i

$1,193,831.54


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e.g. Growing perpetuity Growing perpetuity:You are evaluating a growing perpetuity

product from a large financial services company. The productpromises an initial payment of $20,000 at the end of this year andsubsequent payments that will thereafter grow at a rate of 3.4 percent annually. If you use a 9 per cent discount rate for investment

products, what is the present value of this growing perpetuity?

Slide 37

1 20,000

( ) (0.09 0.034)

CFPVA

i g

$357,142.86


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Term Loans

L l h fl iti d


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perpetuities

Preparing a loan amortisation schedule

Amortisation: the way the borrowed amount (principal)

is paid down over life of loan

Monthly loan payment is structured so each monthportion of principal is paid off; at time loan matures, it is

entirely paid off

L l h fl iti d


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perpetuities


Amortised loan: each loan payment contains some

payment of principal and an interest payment

Loan amortisation schedule is a table showing: loan balance at beginning and end of each period

payment made during that period

how much of payment represents interest

how much represents repayment of principal

L l h fl iti d


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perpetuities


With amortised loan, larger proportion of each monthspayment goes towards interest in early periods

as loan is paid down, greater proportion of each payment isused to pay down principal

Amortisation schedules are best done on a spreadsheet


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NOTE THE MATH ON LOANAMORTIZATION ON PAGE 194

(FIG. 6.5)


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The effective annual interest rate

Interest rates can be quoted in financial markets invariety of ways

Most common quote, especially for a loan, is annualpercentage rate (APR)

APR represents simple interest accrued on loan or

investment in a single period; annualised over a year by

multiplying it by appropriate number of periods in a year


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Calculating the effective annual rate (EAR)

Correct way to calculate annualised rate is to reflect

compounding that occurs; involves calculating effective

annual rate (EAR) Effective annual interest rate (EAR) defined as annual

growth rate that takes compounding into account


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Calculating the effective annual rate (EAR)

EAR = (1 + Quoted rate/m)m1

- mis the # of compounding periods during a year EAR conversion formula accounts for number of

compounding periods, thus effectively adjusts

annualised interest rate for time value of money

EAR is the true cost borrowing and lending.


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Few More Maths on

Annuity

C l l ti it


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Slide 48

Calculating annuity

PMT Brian is a Year 9 student. He plans to buy a car in 4 years.

He estimates that the car will cost him $22,000 in 4 years.How much money should Brian save each year if he wants

to buy the car? Assume his savings account earns 5.65per cent annually.

4

1

(1 ) 1 1.0565 1

22,000 0.0565

22,00022,000 4.351949362 $5,055.21

4.351949362

n

n

m

i

FVA CF CFi

CF CF

e g Calculating annuity


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Slide 49

e.g. Calculating annuityn

Christine wants to accumulate $247,609.95 so that she can purchase an apartment.She plans to invest $25,000 at the end of each year. If Christine's savings earn11.4% p.a., how long will it take for her to reach her goal?

1

(1 ) 1 1.114 1 247,609.95 25,000

0.114

247,609.95 1.114 1 1.114 1 9.904398

25,000 0.114 0.114

9.904398 0.114 1.114 1 1.129101372 1 1

n n

n

n n

n

m

iFVA CF

i

.114

2.129101372 1.114 log2.129101372 log1.114

0.328196339

0.328196339 0.04688519 70.04688519

n

nn

n n years


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WORKSHOP & OTHERACTIVITIES

Workshop Questions &


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Chapter 6 - Parrino et. al. 2014

Critical Thinking Questions

6.3, 6.4, 6.8, 6.9

Numerical Problems:

6.1, 6.2, 6.4, 6.5, 6.7, 6.9, 6.11, 6.12, 6.13, 6.14,6.15, 6.16, 6.18, 6.19, 6.21, 6.22, 6.23, 6.25, 6.27,6.29, 6.31, 6.35

Workshop Questions &Problems

Documents

200244_1468273595_FINC19011BUSINESSFINANCE-Wk02-.pptx