mba/mfm 253 enhancing firm value

MBA/MFM 253 Enhancing Firm Value

The Big Picture

The Goal of Corporate Financial Management:

Maximizing the Value of the Firm

Measuring Firm Value

The firm has many stakeholders – we will focus on four: Shareholders, bondholders, financial markets, and society.Does an increase in stock price signal an increase in firm value?

What Determines Firm ValueValue?

Firm and Project RiskInput Costs

IndustryEconomic Environment

Financing mix (Debt vs Equity)Other?

How do you calculate value?

Goal of Financial Management:

Maximize the value of the firm as determined by: the present value of its expected cash flows, discounted back at a rate that reflects both the riskiness of the firms projects and the financing mix used to fund the projects.

Firm Value and Stock Prices

Is maximizing the value of the firm the same as maximizing the stock price?

Only if maximizing stock price does not have a negative impact on other

stakeholders in the firm.

The Classical Objective Function

STOCKHOLDERS

BONDHOLDERS

FINANCIAL MARKETS

SOCIETYManagers

Management and Stockholders

The Principal / Agent ProblemWhenever owners (principals) hire managers (agents) to operate the firm there is a potential conflict of interest. The managers have an incentive to act in their own best interest instead of the shareholders.

Management and StockholdersOther Problems

Lack of monitoring by shareholdersIndividual shareholders often due not take the time monitor the firm

Lack of independence and expertise on the board.Small ownership stake of directorsTake over defenses and acquisitions:

Greenmail, Golden Parachutes, and Poison Pills. Overvaluing synergy.

Reducing Agency Problems

One way to reduce agency problems is to make management think more like a stockholder.

Offer managers Options and WarrantsProblems – May increase incentive to mislead marketsMay increase incentive to take on extra risk

Reducing Agency Problems

More Effective Board of DirectorsBoards have become smallerFewer insiders on the boardIncreased compensation with optionsNominating committee instead of Chosen by CEOSarbanes Oxley and transparency

More active participation by large stockholders – institutional ownership

Empirical Evidence on Governance

Gompers, Ishii, and Metrick (2003)*Developed corporate governance index based on best practices.Buying stock in firms with high scores for governance and selling those with low scores resulted in large excess returns.

Disney Example

Reaction to decline in share price and captive board

Required executive sessions without CEONew definition of director independence that must be met by a majority of the boardReduction in committee size and rotation of committee chairsNew provisions for succession planningEducation and training for board members

Management and StockholdersBest Case

Best CaseManagers focus on stock price maximization

and therefore the shareholders best interest.

Shareholders are not powerless & do a good job of monitoring the firm. They make informed decisions about the board of

directors and exercise their voting powers. The board acts independent of the CEO.


STOCKHOLDERS

Maximizestockholder wealth

Monitor the firmHire & fire Managers / Board

BONDHOLDERS

FINANCIAL MARKETS

SOCIETYManagers

Conflicts Between Stockholders and Bondholders

Stock Price maximization may increase risk of default.

Risky projects that increase shareholder returns and increase chance of defaultFunding projects with increased debt increasing chance of default.Paying high dividend, decreasing cash available for interest payments

Bond Covenants and Other Solutions

Examples of CovenantsRestrictions on Investment policyRestrictions on Dividend PolicyRestrictions on Additional Leverage

ProblemsMay force firm to pass up profitable projects

Bond Innovations – Puttable bonds and convertible bonds

Conflicts Between Stockholders and Bondholders

Best CaseLenders are protected via covenants in the debt contracts and management considers both bond and stockholders

in decision making..

Lenders supply capital to the firm and receive a return based on risk


STOCKHOLDERS

Maximize stockholder wealth

Lend Money

Monitor the firmHire & fire Managers /

Board

BONDHOLDERS

FINANCIAL MARKETS

SOCIETYManagersBond Covenants

Managers and Financial Markets

The Information ProblemFirms may intentionally mislead financial markets. Both Public and Private information impact firm value

The Market ProblemEven if information is correct, the markets may not react properlyMarket overreactionInsider influenceAre Markets too focused on the short term?Markets and expectations

Improving Transparency

Increased information sharing by independent analysts Market Efficiencies

Low transaction costsFree and wide access to informationComplete markets (short selling, insider trading?)

Managers and Financial Markets

Best CaseManagement does not intentionally

mislead the Financial markets

The markets interpret information correctly


STOCKHOLDERS


Lend Money


Board

BONDHOLDERS

FINANCIAL MARKETS

SOCIETYManagers

Bond CovenantsProtect Lenders

Mangers do not use info to mislead markets

Fin Markets interpret info correctly

Firms and Society

Management decisions often have social costs (intentional and non intentional)

pollution, Johns Manville and Asbestos…

A problem exists if the firm is not accountable for the spillover costs that results from its operations.

Firms and Society

What responsibility do firms have in respect to the communities in which they operate and the well being of their customers?

One definition – Sustainability : meeting the needs of the present without compromising the ability of future generations to meet their own needsOthers?

Corporate Social Responsibility

Firms respond to financial incentivesPart of social responsibility depends on shareholders responding to poor decisions relating to social responsibility. (US Universities divesting in tobacco firms, customer boycotts etc.)Should the firm pursue “socially responsible” actions if it decreases shareholder returns (decreases the value of the firm)??

Social Welfare

Assuming that all shareholders are protected:

Does firm value maximization benefit society?The owners of the firms stock are society

Stock price maximization promotes efficiency in the allocation of resources

Promotes economic growth and employment

Firms and Society

Best CaseManagement decisions have little or no social

costs. Management acts in the best interest of society,

and attempts to be a good “corporate citizen”.

Any social costs can be traced back to the firm.


STOCKHOLDERS


Lend Money


Board

BONDHOLDERS

FINANCIAL MARKETS

SOCIETYManagers

Bond CovenantsProtect Lenders

Mangers do not use info to mislead markets

Fin Markets interpret info correctly

No Social Costs

Costs are traced to the firm

Sustainability

Brundtland Commission (United Nations 1987)

Sustainable Development is development that meets the needs of the present without compromising the ability of future generations to meet their own needs

Everything Constrained by Environment?

aa^ Scott Cato, M. (2009). Green Economics. London: Earthscan, pp. 36–37. ISBN 9781844075713.

http://en.wikipedia.org/wiki/Special:BookSources/9781844075713

http://upload.wikimedia.org/wikipedia/commons/1/15/Nested_sustainability-v2.gif

People, Places, and Profit

Adams, W.M. (2006). "The Future of Sustainability: Re-thinking Environment and Development in the Twenty-first Century." Report of the IUCN Renowned Thinkers Meeting, 29–31 January 2006. Retrieved on: 2011-06-

30

http://upload.wikimedia.org/wikipedia/commons/7/70/Sustainable_development.svg

Triple Bottom Line

SocialEnvironmentalFinancial

The current value of any financial action should reflect future costsCapacity to raise capital and repay providers of capital“Profit” incorporates social and environmental costs

Our Assumption

In class we will assume that management attempts to act in the best interest of all stakeholders. Therefore, stock price maximization and firm value maximization are basically the same thing.However, we know that in the “real world” there cases where stakeholders incur costs associated with share price maximization.

Other Systems

Germany and JapanIndustrial groups where businesses invest in each other, and make decisions in the best interest of the group.Potential Problems?

Less risk taking?Contagion effects within the groupConflicts of interest

Other Objectives?

Should firm value / stock maximization be replaced by other objectives?

Maximize Market ShareObservable – does not require efficient marketsBased on assumption that market share increases pricing power – and earnings (increasing firm value)

Profit MaximizationConsistent with Firm Value Max, creates problems with Accounting

Empire Building

Quick Outline of Class

Part 1 Review of basic tools and concepts Time Value of MoneyMeasuring Risk and Return

Part 2 Applying and extending the basic tools to financial decision making

Financial Decision Making

The Investment DecisionInvest in assets that earn a return greater than the minimum acceptable hurdle rate The Financial DecisionFind the right kind of debt for your firm and the right mix of debt and equityThe Dividend DecisionIf you cannot find investments that make your minimum acceptable rate, return cash to owners of your business

Quick Outline of Class - Part 2

Investment DecisionEstimating Hurdle Rate Chapter 3, 4Returns on projects Chapter 5

Financial Decision (Capital Structure)Does an optimal mix exist? Chapters 6, 7, 8Matching financing and projects Chapter 9

Dividend DecisionHow much cash is available? Chapter 10How do you return the cash? Chapter 11

Introduction to Valuation Chapter 12

A Simple Example

You deposit $100 today in an account that earns 5% interest annually for one year.

How much will you have in one year?Value in one year = Current value + interest earned

= $100 + 100(.05)= $100(1+.05) = $105

The $105 next year has a present value of $100 orThe $100 today has a future value of $105

Calculations

105 = 100(1.05)

or

FV = PV(1+r)

Rearranging

PV = FV/(1+r)

Present Value and Returns

The $105 is discounted to its current value using the present value interest factor 1/(1+r)The interest rate represents the return you receive from waiting for one period to receive the $105.The return also represents an amount of risk that is associated with the certainty of receiving $105 in the future.

Risk and Return

Assume that you have $100 to invest and there are two options

1. You can invest it in a savings account that pays 5% interest (the future return is known with certainty)

2. You can loan it to a friend starting a new business, if the business fails you get nothing, if the business succeeds you get $105Which option would you choose?

Risk and Return

Consider two other options1. You can invest it in a savings account

that pays 5% interest (the future return is known with certainty)

2. You can loan it to a friend starting a new business, if the business fails you get nothing, if the business succeeds you get $110Which option would you choose?

Rules of Thumb

Generally, accepting extra risk is compensated with a higher expected return.Most individuals (and financial managers) are risk averse: They avoid risk, choosing the least risky of two alternatives with an equal return. However they may be willing to accept extra risk if compensated by extra return.

Cost of Capital

The return represents the return the investor expects to earn in return for giving up the $100 today.The investor is choosing to forego other investments For the firm, this represents a cost, the cost of borrowing the $100 today and repaying an amount in the future.

Outline of Class - Part 2Applications of the Tools

The Investment Decision: Allocating scarce resources among possible projects under certainty and uncertainty. (estimating future cash flows and discounting them)The Financing Decision: What mix of Debt and Equity should be used? (the financing mix)The Dividend Decision: How much, if any should be returned to the shareholders?

The Investment Decision

The total value of the firm is an aggregate of the value of its individual projects. Choosing which projects to undertake will be based upon the concepts of present value.


Assume that you know that you can receive a 5% risk free return by investing in a security. Alternatively, you have a buyer willing to agree to pay you $105 at the end of a year for a product that you produce. To produce the product you need to invest $95 today. Would you be willing to pay $95 today to receive the $105?


The decision to invest depends upon the amount it would cost you to undertake the project and the opportunity cost of capital.Assume for now, that you are certain that the buyer will purchase the product, in other words the project is risk free.You can also receive a 5% return on a risk free security (5% is your opportunity cost of capital)

Accepting the project

It costs you $95 to undertake the project, if the project is undertaken, does firm value increase by $10 = $105 - $95?No, The present value of the project is only $100

Net Present Value

The Net Present Value represents the increase in present value.In this case the NPV is

The 5% return represents the opportunity cost of capital (the return forgone by investing in the project instead of the security)

05.1

105$95$5$NPV

The Investment Decision Again

Assume that you again know that you can receive a risk free 5% return. Would you be willing to pay $102 to produce the project today to receive $105 in one year?No, you just learned that given a 5% return, the PV of $105 is $100. The example above is asking you to pay $102 for an investment worth $100.

Net Present Value

The Net Present Value represents the increase in present value.In this case the NPV is

You would be better off investing in the security, with the same risk characteristics that pays a 5% return.

05.1

105$102$2$NPV

Net Present Value

In the first case you are paying $95 for an investment worth $100, you have increased value by $5.In the second case you are paying $102 for an investment that is worth $100, you have decreased value by $2.

Net Present Value Rule

Accept investments that have a positive net present value and reject projects that have a negative net present value.

Rate of Return Rule

The rate of return on the project is based upon the investment and the final payoff:

Accept projects with a Rate of Return greater than the opportunity cost of capital

%526.1010526.95$

95$105$

Cost Initial

Cost InitialC

Return

of Rate 1

Complications

Cash flows received from a project usually extend for more than one period.How do you measure risk and the appropriate level of return?Generally the future cash flows are not known with certainty.The return (and riskiness) depends upon the type of financing used by the firm.


Assume that still can receive a 5% risk free return by investing in a security. Alternatively, you can invest $100 to produce a product that will sell for $105 in one year if the economy grows at an average pace. If there is a recession you will only receive $100. If there is fast expansion you will generate $110.

Expected Return

The expected (or average) return from the project is $105 assuming each outcome is equally likely.The 5% return no longer represents the opportunity cost of capital. The 5% is a risk free return, whether you invest in the project should depend upon the initial cost and the opportunity cost of capital

The Opportunity Cost of Capital

Assume that you find a stock selling for $96.33 with the same outcomes (an expected price of $105 in normal conditions, $100 in a recession and $110 in a boom) The expected rate of return on the stock is:

This is also the Opportunity Cost of Capital

%909.33.96$

33.96$105$

Return

of Rate


To decide if you want to invest, you need to find the NPV of the project.

3.66972$96.33$100$1.09

$105100$NPV


Assume that the last problem still holds, but the risk free rate of interest is 3%. A banker approaches you and based upon your past history offers to loan you $100 at a 4% rate of interest to finance the project.The rate of interest is greater than the risk free rate (compensating for the risk) Should the project be undertaken?

Wrong Assumptions

Using the 4% as the cost of capital, the NPV of the project would be

Should the project be accepted?No – The opportunity cost of capital is 9%, you can accept the same risk and have an expected return of 9%

.961540$96154.100$100$1.04

$105100$NPV

What’s next?

More detailed review of time value of moneyMore detailed review of the relationship between risk and return

Time Value of Money

A dollar received (or paid) today is not worth the same amount as a dollar to be received (or paid) in the future WHY?

You can receive interest on the current dollar

A Simple Example Revisited

You deposit $100 today in an account that earns 5% interest annually for one year.How much will you have in one year?

Value in one year = Current value + interest earned= $100 + 100(.05)= $100(1+.05) = $105

The $105 next year has a present value of $100 orThe $100 today has a future value of $105

Using a Time Line

An easy way to represent this is on a time line

Time 0 1 year 5% $100 $105

Beginning ofFirst Year

End of First year

What would the $100 be worth in 2 years?

You would receive interest on the interest you received in the first year (the interest compounds)

Value in 2 years = Value in 1 year + interest = $105 + 105(.05)= $105(1+.05) = $110.25

Or substituting $100(1+.05) for $105 = [$100(1+.05)](1+.05) = $100(1+.05)2 =$110.25

On the time line

Time 0 1 2

Cash -$100 $105 110.25 Flow

Beginning of year 1

End of Year 1Beginning of

Year 2

End of Year 2

Generalizing the Formula

110.25 = (100)(1+.05)2

This can be written more generally:

Let t = The number of periods = 2 r = The interest rate per period =.05 PV = The Present Value = $100 FV = The Future Value = $110.25

FV = PV(1+r)t

($110.25) = ($100)(1 + 0.05)2

This works for any combination of t, r, and PV

Future Value Interest Factor

FV = PV(1+r)t (1+r)t is called the Future Value Interest Factor (FVIFr,t)

It can be found using the yx key on your calculator

OR (1+.05)2 = 1.1025 Either way original equation can be rewritten:

FV = PV(1+r)t = PV(FVIFr,t)FV=100(1.1025) = $110.25

Calculation MethodsFV = PV(1+r)t

Regular Calculator

Financial Calculator

Spreadsheet

Using a Regular Calculator

Calculate the FVIF using the yx key(1+.05)2=1.1025

Proceed as BeforePlugging it into our equation

FV = PV(FVIFrr,t)

FV = $100(1.1025) = $110.25


Financial Calculators have 5 TVM keysN = Number of Periods = 2

I = interest rate per period =5PV = Present Value = $100PMT = Payment per period = 0 FV = Future Value =?After entering the portions of the problem that you know, the calculator will provide the answer

Financial Calculator Example

On an HP-10B calculator you would enter:

2 N 5 I -100 PV 0 PMT FV

and the screen shows 110.25

Spreadsheet Example

Excel has a FV command =FV(rate,nper,pmt,pv,type) =FV(0.05,2,0,100,0) =110.25 note: Type refers to whether the

payment is at the beginning (type =1) or end (type=0) of the year

Calculating Present Value

We just showed that FV=PV(1+r)t

This can be rearranged to find PV given FV, r and t.Divide both sides by (1+r)t

which leaves PV = FV/(1+r)t

t

t

t r)(1

r)PV(1

r)(1

FV

Example

If you wanted to have $110.25 at the end of two years and could earn 5% interest on any deposits, how much would you need

to deposit today?

PV = FV/(1+r)t

PV = $110.25/(1+0.05)2 = $100.00

Present Value Interest Factor

PV = FV/(1+r)t 1/(1+r)t is called the Present Value Interest Factor (PVIFr,t) PVIF’s can be calculated with your

calculator

1/(1+.05)2 = 0.907029

The original equation can be rewritten:PV = FV/(1+r)t = FV(PVIFr,t)

PV = $110.25(.907029) = $100

Calculating PV of a Single Sum

Regular calculator -Calculate PVIFPVIF =1/ (1+r)t PV = 110.25(0.9070) =

100.00

Financial Calculator2 N 5 I - 110.25 FV 0 PMT PV =

100.00

SpreadsheetExcel command =PV(rate,nper,pmt,fv,type)Excel command =PV(.05,2,0,110.25,0)=100.00

Example

Assume you want to have $1,000,000 saved for retirement when you are 65 and you believe that you can earn 10% each year.

How much would you need in the bank today if you were 25?

PV = 1,000,000/(1+.10)40=$22,094.93

What if you are currently 35? Or 45?

If you are 35 you would needPV = $1,000,000/(1+.10)30 = $57,308.55

If you are 45 you would needPV = $1,000,000/(1+.10)20 = $148,643.63

This process is called discounting (it is the opposite of compounding)

Annuities

Annuity: A series of equal payments made over a fixed amount of time. An ordinary annuity makes a payment at the end of each period.Example A 4 year annuity that makes $100 payments at the end of each year.Time 0 1 2 3 4

CF’s 100 100 100 100

Future Value of an Annuity

The FV of the annuity is the sum of the FV of each of its payments. Assume 6% a year

Time 0 1 2 3 4 100 100 100 100 FV of

CF

100(1+.06)0=100.00100(1+.06)1=106.00100(1+.06)2=112.36100(1+.06)3=119.10

FV = 437.4616

FV of An Annuity

This could also be writtenFV=100(1+.06)0 +100(1+.06)1 +100(1+.06)2+

100(1+.06)3

FV=100[(1+.06)0 +(1+.06)1 +(1+.06)2+(1+.06)3]

or for any n, r, payment, and t

4

1t

t4.06)(1100FV

t

1j

jtr)(1PMTFV

FVIF of an Annuity (FVIFAr,t)

Just like for the FV of a single sum there is a future value interest factor of an annuity

This is the FVIFAr,t

t

1j

jtAnnuity r)(1PMTFV

r

1r)(1PMT)PMT(FVIFAFV

t

tr,Annuity

Calculation Methods

Regular calculator -Approximate FVIFAFVIFA = [(1+r)t-1]/r FV = 100(4.374616)

=437.4616 Financial Calculator

4 N 6 I 0 PV -100 PMT FV = 437.4616

SpreadsheetExcel command =FV(rate,nper,pmt,pv,type)Excel command =FV(.06,4,100,0,0)=437.4616

Present Value of an Annuity

The PV of the annuity is the sum of the PV of each of its payments

Time 0 1 2 3 4 100100 100 100

100/(1+.06)1=94.3396

100/(1+.06)2=88.9996

100/(1+.06)3=83.9619100/(1+.06)4=79.2094

PV = 346.5105

PV of An Annuity

This could also be written

PV=100/(1+.06)1+100/(1+.06)2+100/(1+.06)3+100/(1+.06)4

PV=100[1/(1+.06)1+1/(1+.06)2+1/(1+.06)3+1/(1+.06)4]

or for any r, payment, and t

t

1j

jAnnuity r)][1/(1PMTPV

4

1j

j.06)][1/(1100PV

PVIF of an Annuity PVIFAr,t

Just like for the PV of a single sum there is a future value interest factor of an annuity

t

1j

jAnnuity r)][1/(1CPV

This is the PVIFAr,t

r

)r1(

11

PMT)PMT(PVIFAPV tr,Annuity

t

Calculation Methods

Regular calculator -Approximate FVIFAPVIFA = (1-[1/(1+r)t])/r] FV = 100(3.465105) =346.5105


4 N 6 I 0 FV -100 PMT PV = 346.5105 Spreadsheet

Excel command =PV(rate,nper,pmt,fv,type)Excel command =PV(.06,4,100,0,0)=346.5105

Annuity Due

The payment comes at the beginning of the period instead of the end of the period.

Time 0 1 2 3 4

CF’s Annuity 100 100 100 100

CF’s Annuity Due 100 100 100 100

How does this change the calculation methods?

FV an PV of Annuity Due

FVAnnuity Due There is one more period of compounding for each payment, Therefore:

FVAnnuity Due = FVAnnuity(1+r)

PVAnnuity Due There is one less period of discounting for each payment, ThereforePVAnnuity Due = PVAnnuity(1+r)

Uneven Cash Flow Streams

What if you receive a stream of payments that are not constant? For example:

Time 0 1 2 3 4 100 100 200 200 FV of CF

200(1+.06)0=200.00 200(1+.06)1=212.00100(1+.06)2=112.36100(1+.06)3=119.10

FV = 643.4616

FV of An Uneven CF Stream

The FV is calculated the same way as we did for an annuity, however we cannot factor out the payment since it differs for each period.

t

1j

j-tjsCF'Uneven r)(1CFV

PV of an Uneven CF Streams

Similar to the FV of a series of uneven cash flows, the PV is the sum of the PV of each cash flow. Again this is the same as the first step in calculating the PV of an annuity the final formula is therefore:

t

1j

jjsCF'Uneven r)][1/(1CPV

Quick Review

FV of a Single Sum FV = PV(1+r)t

PV of a Single Sum PV = FV/(1+r)t

FV and PV of annuities and uneven cash flows are just repeated applications of the above two equations

t

1j

j-tjsCF'Uneven )(1CFV r

t

1j

jjsCF'Uneven r)][1/(1CPV

t

1j

jAnnuity r)][1/(1CPV

t

1j

j-tAnnuity r)(1CFV

Perpetuity

Cash flows continue forever instead of over a finite period of time.

1j

jPerpetuity r)][1/(1CPV

1j

jr)][1/(1r1

rCPVP erpetuity

Growing Perpetuity

What if the cash flows are not constant, but instead grow at a constant rate?The PV would first apply the PV of an uneven cash flow stream:

n

1t

ttsCF'Uneven r)][1/(1CFPV

Growing Perpetuity

However, in this case the cash flows grow at a constant rate which implies

CF1 = CF0(1+g)

CF2 = CF1(1+g) = [CF0(1+g)](1+g)

CF3 =CF2(1+g) = CF0(1+g)3

CFt = CF0(1+g)t

Growing Perpetuity

The PV is then Given as:

1j

jj0Perpetuity Growing r)/(1g)(1CFPV

1j

j

j

01j

jj0 r)(1

g)(1CFr)/(1g)(1CF

1j0j

j

0 gr

g)(1CF

r)(1

g)(1CF

Semiannual Compounding

Often interest compounds at a different rate than the periodic rate. For example:

6% yearly compounded semiannualThis implies that you receive 3% interest each six months

This increases the FV compared to just 6% yearly

Semiannual CompoundingAn Example

You deposit $100 in an account that pays a 6% annual rate (the periodic rate) and interest compounds semiannually

Time 0 1/2 1 3% 3%

-100 106.09 FV=100(1+.03)(1+.03)=100(1.03)2=106.09

Effective Annual Rate

The effective Annual Rate is the annual rate that would provide the same annual return as the more often compounding

EAR = (1+rnom/m)m-1 m= # of times compounding per period Our example EAR = (1+.06/2)2-1=1.032-1=.0609

Real and Nominal Rates of Interest

The real rate of interest represents the change in purchasing power. It is equal to the nominal rate of interest adjusted for inflation.

1+rnomial=(1+rreal)(1+inflation)

In-Class Practice Problems 1

1. Assume you are currently 30 and you want to retire at age 65. If you need $1,500,000 saved for retirement, How much would you need save today to fund your retirement assuming you can earn 6% each year?

2. Instead of 1) How much should you deposit at the end of each of the next 35 years assuming your deposits earn 6% each year?

In-Class Practice Problems 2

1. Your sister recently had a new baby daughter and has asked you to help plan for her college education. She estimates that the cost of tuition will be $50,000 a year. If the first payment for her new daughter’s college education will be 18 years from today and she earns 8% on any deposits, how much would she need in the bank today to pay for 4 years of education?

2. If instead she makes a payment at the end of each of the next 18 years, how much should each payment be?

mba/mfm 253 enhancing firm value

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