last study topics present value (pv) – a dollar today is worth more than a dollar tomorrow net...

Last Study Topics• Present Value (PV)– A dollar today is worth more than a dollar

tomorrow• Net Present Value (NPV)– NPV = PV – INV

• NPV Rule– Accept the project that makes a net contribution

to value.• Rate Of Return Rule– Rate of Return is > Cost of Capital

11/16/2014 Instructor: Mr. Wajid Shakeel Ahmed

Topics covered

• Opportunity Cost of Capital

• Investment vs. Consumption

• Managers and the Interests of Shareholders


Opportunity Cost of Capital

• The opportunity cost of capital is such an important concept.

• Theory of valuation endorses that rate of return of the project must always be greater to the rate of return of an alternative opportunity.

• This alternative rate becomes an opportunity cost of capital.



ExampleYou may invest $100,000 today. Depending on the state of the economy, you may get one of three possible cash payoffs:

140,000110,000$80,000Payoff

BoomNormalSlumpEconomy

000,110$3

000,140000,100000,80C payoff Expected 1


Continue

• But what’s the right discount rate?• You search for a common stock with the same

risk as the investment. • Stock X turns out to be a perfect match i.e., X’s

price next year, given a normal economy, is forecasted at $110.

• The stock price will be higher in a boom and lower in a slump, but to the same degrees as your investment ($140 in a boom and $80 in a slump).


Continue

Example - continuedThe stock is trading for $95.65. Depending on the state of the economy, the value of the stock at the end of the year is one of three possibilities:

140110$80eStock Pric

BoomNormalSlumpEconomy


Continue

Example - continuedThe stocks expected payoff leads to an expected return.

15%or 15.65.95

65.95110profit expectedreturn Expected

110$3

14011080C payoff Expected 1

investment


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• The amount it would cost investors in the stock market to buy an expected cash flow is $110,000. (They could do so by buying 1,000 shares of stock X @ $110 each)

• It is, therefore, also the sum that investors would be prepared to pay you for your project.



Example - continuedDiscounting the expected payoff at the expected return leads to the PV of the project.

650,95$1.15

110,000PV


Continue

• To calculate net present value, deduct the initial investment:

NPV = 95,650 - 100,000 = -$4,350

• The project is worth $4,350 less than it costs. • It is not worth undertaking.


Alternative

• Compare the expected project return with the cost of capital:

• The 10 % return on project < 15 % return on stock of equal risk.

• The project is not worthwhile.


10%or 10.100

100110profit expectedreturn Expected

investment

Discount rate equals the opportunity cost of capital

• The opportunity cost of capital is the return earned by investing in the best alternative investment.

• This return will not be realized if the investment under consideration is undertaken.

• Thus, the two investments must earn at least the same return.

• This return rate is the discount rate used in the net present value calculation.


A source of Confusion

• Bank A will lend you the $100,000 that you need for the project at 8 percent. Does that mean that the cost of capital for the project is 8 %? Or does not?

• Does Not - First, the interest rate on the loan has nothing to do with the risk of the project.

• Second, whether you take the loan or not, you still face the choice between the projects.


Case: Norman

• Norman Gerrymander has just received a $2 million bequest. How should he invest it?

• There are four immediate alternatives. – Which of these investments have positive NPVs?– Which would you advise Norman to take?

Lets have a look each of them!


Alternative A• A - Investment in one-year U.S. government

securities yielding 5 percent.• Step 1: calculate the payoff after 1 year.

= 2.0M X 5% = 2.1 M.

• Step 2: Discounting the expected payoff at the expected return leads to the PV of the project.


Continue

• Step 3: To calculate net present value, deduct the initial investment:

NPV = $2.0M - $2.0M = $0

• The ‘A’ investment has not contributed to the net value.

M0.2$1.05

M 2.1PV


Alternative B• B - A loan to Norman’s nephew Gerald, who

has for years aspired to open a big Cajun restaurant in Duluth.

• Gerald had arranged a one-year bank loan for $900,000, at 10 percent, but asks for a loan from Norman at 7 percent.

• Step 1: calculate the payoff after 1 year.

= 0.9 M X 7% = 0.96 M.


Continue


M88.0$1.10

M 0.96PV


Continue• Step 3: To calculate net present value, deduct

the initial investment:NPV = $0.88M - $.9M = -$0.02M

• The project is worth $0.02M less than it costs. • It is not worth undertaking.• The ‘B’ investment has brought decline to the

net value.


Alternative C• C - Investment in the stock market. The

expected rate of return is 12 percent.


= 2.0 M X 12% = 2.24 M.


Continue


M0.2$1.12

M 2.26PV



the initial investment:NPV = $2.0M - $2.0M = $0

• The ‘C’ investment has not contributed to the net value nor a decline.


Alternative D• B - Investment in local real estate, which

Norman judges is about as risky as the stock market.

• The opportunity at hand would cost $1 million and is forecasted to be worth $1.1 million after one year.


= $ 1.1 M.


Continue


M98.0$1.12

M 1.1PV



the initial investment:NPV = $0.98M - $1.0M = -$0.02M

• The project is worth $0.02M less than it costs. • It is not worth undertaking.• The ‘D’ investment has brought decline to the

net value.


Continue• Results from four Alternatives;– The ‘A’ investment has not contributed to the net value

nor a decline. – The ‘B’ investment has brought decline to the net value. – The ‘C’ investment has not contributed to the net value

nor a decline. – The ‘D’ investment has brought decline to the net

value.

• Decision;

– Norman should invest in either the risk-free government securities or the risky stock market, depending on his tolerance for risk.


Investment vs. Consumption• Some people prefer to consume now. Some

prefer to invest now and consume later.

• Borrowing and lending allows us to reconcile these opposing desires which may exist within the firm’s shareholders.

• Could a positive-NPV project for Ms. X be a negative-NPV proposition for Mr. Y? Could they find it impossible to agree on the objective of maximizing the market value of the firm?


Continue

• This could be answer;• Well functioned capital markets allows

investors with different time patterns of income and desired consumption to agree on whether investment projects should be undertaken.

• Lets consider an example !


Ant vs Grasshopper

• A is an ant, who wishes to save for the future; G is a grasshopper, who would prefer to spend all his wealth.

• Suppose both of them can earn a return of about 14 percent on every $100 investment. The interest rate is 7 percent.


Investment vs. ConsumptionThe grasshopper (G) wants to consume now. The ant (A) wants to wait. But each is happy to invest. A prefers to invest 14%, moving up the red arrow, rather than at the 7% interest rate. G invests and then borrows at 7%, thereby transforming $100 into $106.54 of immediate consumption. Because of the investment, G has $114 next year to pay off the loan. The investment’s NPV is $106.54-100 = +6.54


Investment vs. Consumption• The grasshopper (G) wants to consume now.

The ant (A) wants to wait. But each is happy to invest. A prefers to invest 14%, moving up the red arrow, rather than at the 7% interest rate. G invests and then borrows at 7%, thereby transforming $100 into $106.54 of immediate consumption. Because of the investment, G has $114 next year to pay off the loan. The investment’s NPV is $106.54-100 = +6.54

100 106.54 Dollars Now

Dollars Later

114

107

A invests $100 now and consumes $114 next year

G invests $100 now, borrows $106.54 and consumes now.


Calculation• Return of A: INV = C0; C1 = Payoff; – r = C1 -C0 / C0 * 100 =

= 14%

• NPV for G: INV = C0; C1 = Payoff; INT = r;– PV = C1/ (1+r)t = = $106.54

NPV = PV – C0 =


Explanation• In our example the ant and the grasshopper

placed an identical value on same project and were happy to share in its construction.

• They agreed because they faced identical borrowing and lending opportunities.

• Whenever firms discount cash flows at capital market rates, they are implicitly assuming that their shareholders have free and equal access to competitive capital markets.


Continue

• Despite their different tastes, both A and G are better off by investing in the project and then using the capital markets to achieve the desired balance between consumption today and consumption at the end of the year.

• If A and G were shareholders in the same enterprise, there would be no simple way for the manager to reconcile their different objectives.


Summary

• Opportunity Cost of Capital– Rate of an Alternative investment opportunity

having a similar risk.

• Investment vs. Consumption– Manager finds it difficult to reconcile the different

objectives of the shareholders.


Calculation of NPV and ROR• The opportunity cost of capital is 20 percent for all four

investments.Initial Cash Cash Flow

Investment Flow, C0 in Year 1, C11 10,000 18,0002 5,000 9,0003 5,000 5,7004 2,000 4,000

a. Which investment is most valuable?b. Suppose each investment would require use of the same parcel of land. Therefore you can take only one. Which one?


Continue


Continue

• Answer to question ‘a’; – Investment 1, since this investment with respect

to others has generate highest NPV with Max rate of return.

• Answer to question ‘b’; – Investment 1, since this investment with respect

to others has highest contributed net value.


Decision Rules

• Here then we have two equivalent decision rules for capital investment;

– Net present value rule; Accept investments that have positive net present values.

– Rate-of-return rule; Accept investments that offer rates of return in excess of their opportunity costs of capital.


Fundamental Result• Our justification of the present value rule was

restricted to two periods and to a certain cash flow.

• However, the rule also makes sense for uncertain cash flows that extend far into the future. The argument goes like this:

• 1- A financial manager should act in the interests of the firm’s owners, its stockholders.


Continue• 2- Stockholders do not need the financial

manager’s help to achieve the best time pattern of consumption.

• 3- How then can the financial manager help the firm’s stockholders? – There is only one way: by increasing the market

value of each stockholder’s stake in the firm.


Continue• Despite the fact that shareholders have

different preferences, they are unanimous in the amount that they want to invest in real assets.

• This means that they can cooperate in the same enterprise and can safely delegate operation of that enterprise to professional managers.


Continue• These managers do not need to know

anything about the tastes of their shareholders and should not consult their own tastes.

• Their task is to maximize net present value. If they succeed, they can rest assured that they have acted in the best interest of their shareholders.


Managers and Shareholder Interests

• Do managers really looking after the interests of shareholders?

• This takes us back to the principal–agent problem.

– Several institutional arrangements that help to ensure that the shareholders’ pockets are close to the managers’ heart.


Continue• Utilizing their Voting power;– If shareholders believe that the corporation is

underperforming and that the board of directors is not sufficiently aggressive in holding the managers to task, they can try to replace the board in the next election.

• E.g; chief executives of Eastman Kodak, General Motors, Xerox, Lucent, Ford Motor, etc were all forced to step aside.


Continue• As a result the stock price tumbles. – This damages top management’s reputation and

compensation.• Part of the top managers’ paychecks comes

from bonuses tied to the company’s earnings or from stock options, which pay off if the stock price rises but are worthless if the price falls below a stated threshold. – This should motivate managers to increase

earnings and the stock price.


Continue• If managers and directors do not maximize

value, there is always the threat of a hostile takeover.

• The further a company’s stock price falls, due to lax management or wrong-headed policies, the easier it is for another company or group of investors to buy up a majority of the shares.– The old management team is then likely to find

themselves out on the street and their place is taken by a fresh team.


Continue

• Should managers look after the interests of shareholders?

• For this question we need to understand that In most instances there is little conflict between doing well (maximizing value) and doing good.– Profitable firms are those with satisfied customers

and loyal employees and vice versa;


Continue

• Ethical issues do arise in business as in other walks of life.– when we say that the objective of the firm is to

maximize shareholder wealth, we do not mean that anything goes.

• In business and finance, as in other day-to-day affairs, there are unwritten, implicit rules of behavior.

• To work efficiently together, we need to trust each other.


Summary

• Opportunity Cost of Capital– Rate of an Alternative investment opportunity

having a similar risk.• Investment vs. Consumption– Manager finds it difficult to reconcile the different

objectives of the shareholders.• Managers and the Interests of Shareholders– Are managers taking care of the interests of the

Shareholders or should they?


How to Calculate Present Values

Principles of Corporate FinanceBrealey and Myers Sixth Edition

Chapter 3

©The McGraw-Hill Companies, Inc., 2000Irwin/McGraw Hill11/16/2014 Instructor: Mr. Wajid Shakeel Ahmed

Topics Covered

• Valuing Long-Lived Assets• PV Calculation Short Cuts• Compound Interest• Interest Rates and Inflation• Example: Present Values and Bonds


Present Values

Discount Factor = DF = PV of $1

Discount Factors can be used to compute the present value of any cash flow.


Present Values

Discount Factor = DF = PV of $1


DFr t

1

1( )


Present Values


DFr t

1

1( )

1

11 1 r

CCDFPV


Present Values

Replacing “1” with “t” allows the formula to be used for cash flows that exist at any point in time.

t

tt r

CCDFPV

1


Present ValuesExample

Suppose you will receive a certain cash inflowof $100 next year (C1 = $100) and the rate of interest on

one-year U.S. Treasury notes is 7 percent (r1 = 0.07). What would be the present value of Cash inflow ?

94$)07.1(100

)1(1 r

CPV



Suppose you will receive a certain cash inflowof $100 in two year (C2 = $100) and the rate of interest on

two-year U.S. Treasury notes is 7.7 percent (r1 = 0.077). What would be the present value of year 2 Cash inflow ?

21.86$)077.1(100

)1(1 r

CPV



You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years?



You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years?

PV 30001 08 2 572 02

( . )$2, .


Present Values PVs can be added together to evaluate

multiple cash flows.

PV C

r

C

r

1

12

21 1( ) ( )....


Present Values PVs can be added together to evaluate

multiple cash flows, and called as discounted cash flow or (DCF) formula;

tt

t

rCPV

)1(


Present Values• If a dollar tomorrow is worth less than a

dollar today, one might suspect that a dollar the day after tomorrow should be worth even less.

• But, lets assume - given two dollars, one received a year from now and the other two years from now, the value of each is commonly called the Discount Factor. Assume r1 = 20% and r2 = 7%.


Present Values• Given two dollars, one received a year from now

and the other two years from now, the value of each is commonly called the Discount Factor. Assume r1 = 20% and r2 = 7%.

87$.

83$.

2

1

)07.1(00.1

2

)20.1(00.1

1

DF

DF


Continue• If First we lend $1,000 for one year at 20

percent, we have;– FV = PV (1+rt)t

=

• Go to the bank and borrow the present value of this $1,200 at 7 percent interest, we have;– PV = FV / (1+rt)t

=


Continue

• If we going to find out the net present value of our investment we get,

– NPV = PV – Investment =

“Just imagine in this game of lending and borrowingHow much money you can earn without taking risks”


Continue

• Of course this story is completely fanciful.– “There is no such thing as a money machine.”

• In well-functioning capital markets, any potential money machine will be eliminated almost instantaneously by investors who try to take advantage of it.



Assume that the cash flows from the construction and sale of an office building is as follows. Given a 7% required rate of return, create a present value worksheet and show the net present value.

000,300000,100000,150

2Year 1Year 0Year


Present ValuesExample - continued

Assume that the cash flows from the construction and sale of an office building is as follows. Given a 7% required rate of return, create a present value worksheet and show the net present value.

400,18$

900,261000,300873.2

500,93000,100935.1

000,150000,1500.10Value

Present

Flow

Cash

Factor

DiscountPeriod

207.11

07.11

TotalNPV


NPV FormulaMathematically; PV @ 7%

∑PV = PV of C1+ PV of C2

= -$93,500 + $261,900

= $168,400

NPV = ∑ PV - INV

= $168,400 - $150,000

= $18,400


Short Cuts

• Sometimes there are shortcuts that make it very easy to calculate the present value of an asset that pays off in different periods.

• These tolls allow us to cut through the calculations quickly.


Short CutsPerpetuity - Financial concept in which a cash flow is theoretically received forever.

PV

Cr

cashflow

luepresent va

Return


Short CutsPerpetuity - Financial concept in which a cash

flow is theoretically received forever.

r

CPV 1

ratediscount

flow cash FlowCash of PV


Case : Investment A• An investment costs $1,548 and pays $138 in

perpetuity. If the interest rate is 9 percent, what is the NPV?– PV = C / r

= $1533.33

– NPV = PV - INV = $1533.33 - $1548

= -$ 14.67


Short CutsAnnuity - An asset that pays a fixed sum each

year for a specified number of years.

trrrC

1

11annuity of PV


Case: leasing a carExample

You agree to lease a car for 4 years at $300 per month. You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease?


ContinueExample - continued

You agree to lease a car for 4 years at $300 per month. You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease?

10.774,12$

005.1005.

1

005.

1300Cost Lease 48

Cost


Case: Endowment FundsExample - Suppose, for example, that we begins to wonders what it would cost to contribute in a endowment fund with an amount of $100,000 a year for only 20 years?

400,851$

10.110.

1

10.

1000,100Cost Lease 20

Cost


Alternatively• We can simply look up the answer in the annuity table given on the next slide. • This table gives the present value of a dollar to be received in each of t periods. • In our example t = 20 and the interest rate r = .10, and therefore;• We look at the twentieth number from the top in the 10 percent column. It is 8.514. Multiply 8.514 by $100,000, and we have our answer,$851,400.


Compound Interest

• There is an important distinction between compound interest and simple interest.– When money is invested at compound interest,

each interest payment is reinvested to earn more interest in subsequent periods.

– In contrast, the opportunity to earn interest on interest is not provided by an investment that pays only simple interest.


Compound Interest

02468

1012141618

Number of Years

FV

of

$1

10% Simple

10% Compound


Compounding InterestSuppose that the bank starts with $10 million of automobile loans outstanding. This investment grows to ;

$10 * 1.005 = $10.05 million after month 1, $10 * 1.0052 = $10.10025 million after month 2, and $10 1.00512 $10.61678 million after 12 months.

Thus the bank is quoting a 6 percent APR but actually earns 6.1678 percent if interest payments are made monthly.

In general, an investment of $1 at a rate of r per annum compounded m times a year amounts by the end of the year to [1 + (r/m)]m, and the equivalent annually compounded rate of interest is [1 + (r/m)]m - 1.


Compound Interest i ii iii iv vPeriods Interest Value Annuallyper per APR after compoundedyear period (i x ii) one year interest rate

1 6% 6% 1.06 6.000%

2 3 6 1.032 = 1.0609 6.090

4 1.5 6 1.0154 = 1.06136 6.136

12 .5 6 1.00512 = 1.06168 6.168

52 .1154 6 1.00115452 = 1.06180 6.180

365 .0164 6 1.000164365 = 1.06183 6.183


Explanation• 1$ at the end of the year can be calculated as;– [1 + (r/m)]m ;

• Since, we have r = 6%; periods = m = 12, 52;– = ; =– = $1.06168 = $1.06180

• Annually compounded rate;– [1 + (r/m)]m – 1– = ; =


Continuous Compounding


• Eventually one can quote a continuously compounded rate, so that payments were assumed to be spread evenly and continuously throughout the year.

• In terms of our formula, this is equivalent to letting m approach infinity.

As m approaches infinity [1 + (r/m)]m approaches (2.718)r or er = (2.718)r

•By the end of t years ert = (2.718)rt

Example 1


• Suppose you invest $1 at a continuously compounded rate of 11 percent (r=.11) for one year (t =1). The end-year value is e =.11, which can be calculated as;

ert = (2.718).11*1 = $1.116

•In other words, investing at 11 percent a year continuously compounded is exactly the same as investing at 11.6 percent a year annually compounded.

Example 2• Suppose you invest $1 at a continuously

compounded rate of 11 percent (r=.11) for two year (t =2). The end-year value is e =.11, which you can be calculated as;

ert = (2.718).11*2 = $1.246

• In other words, investing at 11 percent for 2 years continuously compounded is exactly the same as investing at 24.6 percent a year annually compounded.


Example 3• Suppose that we start thinking more seriously

and decided to found a home for us, which will cost $100,000 a year, starting and spread evenly over 20 years. Annual compounding rate is 10%. What sum should we set aside?

• Here we going to apply Annuity formula– Step 1: calculate continuously compounded rate– r = 9.53% or ( e .0953 = 1.10)


Continue

• Step2: calculate the PV of the annuity;– PV = C [1/r – (1/r*1/ert)]

=

= $893,200


Alternatively• we could have cut these calculations short by

using PV of annuity Table, given on the next slide.

• This shows that, if the annually compounded return is 10 percent, then $1 a year spread over 20 years is worth $8.932.


Inflation

Inflation - Rate at which prices as a whole are increasing.

Nominal Interest Rate - Rate at which money invested grows.

Real Interest Rate - Rate at which the purchasing power of an investment increases.


Inflation

1 real interest rate = 1+nominal interest rate1+inflation rate


Inflation

1 real interest rate = 1+nominal interest rate1+inflation rate

approximation formula

Real int. rate nominal int. rate - inflation rate


InflationExample

If the interest rate on one year govt. bonds is 5.9% and the inflation rate is 3.3%, what is the real interest rate?

Savings

Bond


InflationExample


1

1

+

+

real interest rate =

real interest rate = 1.025

real interest rate = .025 or 2.5%

1+.0591+.033 Savings

Bond


InflationExample


1

1

+

+

real interest rate =

real interest rate = 1.025

real interest rate = .025 or 2.5%

Approximation =.059-.033 =.026 or 2.6%

1+.0591+.033 Savings

Bond


Valuing a BondExample

If today is October 2000, what is the value of the following bond?

An IBM Bond pays $115 every Sept for 5 years. In Sept 2005 it pays an additional $1000 and retires the bond.

The bond is rated AAA (WSJ AAA YTM is 7.5%).

Cash Flows

Sept 0102 03 04 05

115 115 115 115 111511/16/2014 Instructor: Mr. Wajid Shakeel Ahmed

Valuing a BondExample continued

If today is October 2000, what is the value of the following bond? An IBM Bond pays $115 every Sept for 5 years. In Sept 2005 it pays an

additional $1000 and retires the bond. The bond is rated AAA (WSJ AAA YTM is 7.5%).

84.161,1$

075.1

115,1

075.1

115

075.1

115

075.1

115

075.1

1155432

PV


Bond Prices and Yields

0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10 12 14

5 Year 9% Bond 1 Year 9% Bond

Yield

Pri

ce


last study topics present value (pv) – a dollar today is worth more than a dollar tomorrow net...

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