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**MBA/MFM 253 Enhancing Firm Value**

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**Maximizing the Value of the Firm**

The Big Picture The Goal of Corporate Financial Management: Maximizing the Value of the Firm

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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?

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**What Determines Firm Value?**

Firm and Project Risk Input Costs Industry Economic Environment Financing mix (Debt vs Equity) Other? How do you calculate value?

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**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.

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**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.

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**The Classical Objective Function**

STOCKHOLDERS BONDHOLDERS Managers SOCIETY This is the utopian world. None of the assumptions are really defensible as written, and skepticism is clearly justified: Why do we need these assumptions? Since, in many large firms, there is a separation of ownership from management, managers have to be fearful of losing their jobs and go out and maximize stockholder wealth. If they do not have this fear, they will focus on their own interests. If bondholders are not protected, stockholders can steal from them and make themselves better off, even as they make the firm less valuable. If markets are not efficient, maximizing stock prices may not have anything to do with maximizing stockholder wealth or firm value. If substantial social costs are created, maximizing stock prices may create large side costs for society (of which stockholders are members). Note that corporate finance, done right, is not about stealing from other groups (bondholders, other stockholders or society) but about making the firm more productive and valuable. FINANCIAL MARKETS

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**Management and Stockholders**

The Principal / Agent Problem Whenever 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.

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**Management and Stockholders Other Problems**

Lack of monitoring by shareholders Individual shareholders often due not take the time monitor the firm Lack of independence and expertise on the board. Small ownership stake of directors Take over defenses and acquisitions: Greenmail, Golden Parachutes, and Poison Pills. Overvaluing synergy.

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**Reducing Agency Problems**

One way to reduce agency problems is to make management think more like a stockholder. Offer managers Options and Warrants Problems – May increase incentive to mislead markets May increase incentive to take on extra risk

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**Reducing Agency Problems**

More Effective Board of Directors Boards have become smaller Fewer insiders on the board Increased compensation with options Nominating committee instead of Chosen by CEO Sarbanes Oxley and transparency More active participation by large stockholders – institutional ownership

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**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.

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**Disney Example Reaction to decline in share price and captive board**

Required executive sessions without CEO New definition of director independence that must be met by a majority of the board Reduction in committee size and rotation of committee chairs New provisions for succession planning Education and training for board members

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**Management and Stockholders Best Case**

Managers 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.

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**The Classical Objective Function**

STOCKHOLDERS Monitor the firm Hire & fire Managers / Board Maximize stockholder wealth BONDHOLDERS Managers SOCIETY This is the utopian world. None of the assumptions are really defensible as written, and skepticism is clearly justified: Why do we need these assumptions? Since, in many large firms, there is a separation of ownership from management, managers have to be fearful of losing their jobs and go out and maximize stockholder wealth. If they do not have this fear, they will focus on their own interests. If bondholders are not protected, stockholders can steal from them and make themselves better off, even as they make the firm less valuable. If markets are not efficient, maximizing stock prices may not have anything to do with maximizing stockholder wealth or firm value. If substantial social costs are created, maximizing stock prices may create large side costs for society (of which stockholders are members). Note that corporate finance, done right, is not about stealing from other groups (bondholders, other stockholders or society) but about making the firm more productive and valuable. FINANCIAL MARKETS

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**Conflicts Between Stockholders and Bondholders**

Stock Price maximization may increase risk of default. Risky projects that increase shareholder returns and increase chance of default Funding projects with increased debt increasing chance of default. Paying high dividend, decreasing cash available for interest payments

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**Bond Covenants and Other Solutions**

Examples of Covenants Restrictions on Investment policy Restrictions on Dividend Policy Restrictions on Additional Leverage Problems May force firm to pass up profitable projects Bond Innovations – Puttable bonds and convertible bonds

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**Conflicts Between Stockholders and Bondholders**

Best Case Lenders 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

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**The Classical Objective Function**

STOCKHOLDERS Monitor the firm Hire & fire Managers / Board Maximize stockholder wealth Bond Covenants BONDHOLDERS Managers SOCIETY Lend Money This is the utopian world. None of the assumptions are really defensible as written, and skepticism is clearly justified: Why do we need these assumptions? Since, in many large firms, there is a separation of ownership from management, managers have to be fearful of losing their jobs and go out and maximize stockholder wealth. If they do not have this fear, they will focus on their own interests. If bondholders are not protected, stockholders can steal from them and make themselves better off, even as they make the firm less valuable. If markets are not efficient, maximizing stock prices may not have anything to do with maximizing stockholder wealth or firm value. If substantial social costs are created, maximizing stock prices may create large side costs for society (of which stockholders are members). Note that corporate finance, done right, is not about stealing from other groups (bondholders, other stockholders or society) but about making the firm more productive and valuable. FINANCIAL MARKETS

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**Managers and Financial Markets**

The Information Problem Firms may intentionally mislead financial markets. Both Public and Private information impact firm value The Market Problem Even if information is correct, the markets may not react properly Market overreaction Insider influence Are Markets too focused on the short term? Markets and expectations

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**Improving Transparency**

Increased information sharing by independent analysts Market Efficiencies Low transaction costs Free and wide access to information Complete markets (short selling, insider trading?)

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**Managers and Financial Markets**

Best Case Management does not intentionally mislead the Financial markets The markets interpret information correctly

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**The Classical Objective Function**

STOCKHOLDERS Monitor the firm Hire & fire Managers / Board Maximize stockholder wealth Bond Covenants Protect Lenders BONDHOLDERS Managers SOCIETY Lend Money This is the utopian world. None of the assumptions are really defensible as written, and skepticism is clearly justified: Why do we need these assumptions? Since, in many large firms, there is a separation of ownership from management, managers have to be fearful of losing their jobs and go out and maximize stockholder wealth. If they do not have this fear, they will focus on their own interests. If bondholders are not protected, stockholders can steal from them and make themselves better off, even as they make the firm less valuable. If markets are not efficient, maximizing stock prices may not have anything to do with maximizing stockholder wealth or firm value. If substantial social costs are created, maximizing stock prices may create large side costs for society (of which stockholders are members). Note that corporate finance, done right, is not about stealing from other groups (bondholders, other stockholders or society) but about making the firm more productive and valuable. Mangers do not use info to mislead markets Fin Markets interpret info correctly FINANCIAL MARKETS

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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.

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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 needs Others?

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**Corporate Social Responsibility**

Firms respond to financial incentives Part 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)??

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**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

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**Firms and Society Best Case**

Management 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.

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**The Classical Objective Function**

STOCKHOLDERS Monitor the firm Hire & fire Managers / Board Maximize stockholder wealth Bond Covenants Protect Lenders Costs are traced to the firm BONDHOLDERS Managers SOCIETY Lend Money No Social Costs This is the utopian world. None of the assumptions are really defensible as written, and skepticism is clearly justified: Why do we need these assumptions? Since, in many large firms, there is a separation of ownership from management, managers have to be fearful of losing their jobs and go out and maximize stockholder wealth. If they do not have this fear, they will focus on their own interests. If bondholders are not protected, stockholders can steal from them and make themselves better off, even as they make the firm less valuable. If markets are not efficient, maximizing stock prices may not have anything to do with maximizing stockholder wealth or firm value. If substantial social costs are created, maximizing stock prices may create large side costs for society (of which stockholders are members). Note that corporate finance, done right, is not about stealing from other groups (bondholders, other stockholders or society) but about making the firm more productive and valuable. Mangers do not use info to mislead markets Fin Markets interpret info correctly FINANCIAL MARKETS

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**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

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**Everything Constrained by Environment?**

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

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**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 Retrieved on:

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**Triple Bottom Line Social Environmental Financial**

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

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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.

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**Other Systems Germany and Japan**

Industrial 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 group Conflicts of interest

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Other Objectives? Should firm value / stock maximization be replaced by other objectives? Maximize Market Share Observable – does not require efficient markets Based on assumption that market share increases pricing power – and earnings (increasing firm value) Profit Maximization Consistent with Firm Value Max, creates problems with Accounting Empire Building

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**Quick Outline of Class Part 1 Review of basic tools and concepts**

Time Value of Money Measuring Risk and Return Part 2 Applying and extending the basic tools to financial decision making

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**Financial Decision Making**

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

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**Quick Outline of Class - Part 2**

Investment Decision Estimating Hurdle Rate Chapter 3, 4 Returns on projects Chapter 5 Financial Decision (Capital Structure) Does an optimal mix exist? Chapters 6, 7, 8 Matching financing and projects Chapter 9 Dividend Decision How much cash is available? Chapter 10 How do you return the cash? Chapter 11 Introduction to Valuation Chapter 12

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**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.

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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 = $ (.05) = $100(1+.05) = $105 The $105 next year has a present value of $100 or The $100 today has a future value of $105

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Calculations 105 = 100(1.05) or FV = PV(1+r) Rearranging PV = FV/(1+r)

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**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.

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**Which option would you choose?**

Risk and Return Assume that you have $100 to invest and there are two options You can invest it in a savings account that pays 5% interest (the future return is known with certainty) You can loan it to a friend starting a new business, if the business fails you get nothing, if the business succeeds you get $105 Which option would you choose?

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**Which option would you choose?**

Risk and Return Consider two other options You can invest it in a savings account that pays 5% interest (the future return is known with certainty) You can loan it to a friend starting a new business, if the business fails you get nothing, if the business succeeds you get $110 Which option would you choose?

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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.

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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.

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**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.

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**Outline of Class - Part 2 Applications 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?

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**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.

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**The Investment Decision**

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?

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**The Investment Decision**

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)

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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

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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)

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**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.

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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.

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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.

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Net Present Value Rule Accept investments that have a positive net present value and reject projects that have a negative net present value.

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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

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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.

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**The Investment Decision**

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.

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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

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**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

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**The Investment Decision**

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

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**The Investment Decision**

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?

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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%

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**What’s next? More detailed review of time value of money**

More detailed review of the relationship between risk and return

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**Time Value of Money You can receive interest on the current dollar**

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

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**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 = $ (.05) = $100(1+.05) = $105 The $105 next year has a present value of $100 or The $100 today has a future value of $105

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**Using a Time Line An easy way to represent this is on a time line**

Time year 5% $100 $105 Beginning of First Year End of First year

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

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**On the time line Time 0 1 2 Cash -$100 $105 110.25 Flow Beginning**

of year 1 End of Year 1 Beginning of Year 2 End of Year 2

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**Generalizing the Formula**

= (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)( )2 This works for any combination of t, r, and PV

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**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 = Either way original equation can be rewritten: FV = PV(1+r)t = PV(FVIFr,t) FV=100(1.1025) = $110.25

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**Calculation Methods FV = PV(1+r)t**

Regular Calculator Financial Calculator Spreadsheet

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**Using a Regular Calculator**

Calculate the FVIF using the yx key (1+.05)2=1.1025 Proceed as Before Plugging it into our equation FV = PV(FVIFrr,t) FV = $100(1.1025) = $110.25

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**Financial Calculator Financial Calculators have 5 TVM keys**

N = Number of Periods = 2 I = interest rate per period =5 PV = Present Value = $100 PMT = Payment per period = 0 FV = Future Value =? After entering the portions of the problem that you know, the calculator will provide the answer

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**Financial Calculator Example**

On an HP-10B calculator you would enter: 2 N 5 I PV 0 PMT FV and the screen shows

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**Spreadsheet Example Excel has a FV command =FV(rate,nper,pmt,pv,type)**

=110.25 note: Type refers to whether the payment is at the beginning (type =1) or end (type=0) of the year

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**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

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Example If you wanted to have $ 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

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**Present Value Interest Factor**

PV = FV/(1+r)t /(1+r)t is called the Present Value Interest Factor (PVIFr,t) PVIF’s can be calculated with your calculator 1/(1+.05)2 = The original equation can be rewritten: PV = FV/(1+r)t = FV(PVIFr,t) PV = $110.25( ) = $100

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**Calculating PV of a Single Sum**

Regular calculator -Calculate PVIF PVIF =1/ (1+r)t PV = (0.9070) = Financial Calculator 2 N 5 I FV 0 PMT PV = Spreadsheet Excel command =PV(rate,nper,pmt,fv,type) Excel command =PV(.05,2,0,110.25,0)=100.00

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

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

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**What if you are currently 35? Or 45?**

If you are 35 you would need PV = $1,000,000/(1+.10)30 = $57,308.55 If you are 45 you would need PV = $1,000,000/(1+.10)20 = $148,643.63 This process is called discounting (it is the opposite of compounding)

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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 CF’s

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**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 FV of CF 100(1+.06)0=100.00 100(1+.06)1=106.00 100(1+.06)2=112.36 100(1+.06)3=119.10 FV =

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**FV of An Annuity FV=100(1+.06)0 +100(1+.06)1 +100(1+.06)2+ 100(1+.06)3**

This could also be written FV=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

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**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

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**Calculation Methods Regular calculator -Approximate FVIFA**

FVIFA = [(1+r)t-1]/r FV = 100( ) = Financial Calculator 4 N I 0 PV PMT FV = Spreadsheet Excel command =FV(rate,nper,pmt,pv,type) Excel command =FV(.06,4,100,0,0)=

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**Present Value of an Annuity**

The PV of the annuity is the sum of the PV of each of its payments Time 100/(1+.06)1= 100/(1+.06)2= 100/(1+.06)3= 100/(1+.06)4= PV =

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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

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**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 This is the PVIFAr,t

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**Calculation Methods Regular calculator -Approximate FVIFA**

PVIFA = (1-[1/(1+r)t])/r] FV = 100( ) = Financial Calculator 4 N 6 I 0 FV PMT PV = Spreadsheet Excel command =PV(rate,nper,pmt,fv,type) Excel command =PV(.06,4,100,0,0)=

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Annuity Due The payment comes at the beginning of the period instead of the end of the period. Time CF’s Annuity CF’s Annuity Due How does this change the calculation methods?

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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, Therefore PVAnnuity Due = PVAnnuity(1+r)

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**Uneven Cash Flow Streams**

What if you receive a stream of payments that are not constant? For example: Time FV of CF 200(1+.06)0=200.00 200(1+.06)1=212.00 100(1+.06)2=112.36 100(1+.06)3=119.10 FV =

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**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.

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**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:

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**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

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

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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:

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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

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Growing Perpetuity The PV is then Given as:

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**Semiannual Compounding**

Often interest compounds at a different rate than the periodic rate. For example: 6% yearly compounded semiannual This implies that you receive 3% interest each six months This increases the FV compared to just 6% yearly

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**Semiannual Compounding An Example**

You deposit $100 in an account that pays a 6% annual rate (the periodic rate) and interest compounds semiannually Time 0 1/ % 3% FV=100(1+.03)(1+.03)=100(1.03)2=106.09

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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= =.0609

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**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)

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**In-Class Practice Problems 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? 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?

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**In-Class Practice Problems 2**

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

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