Lectures on Corporate Finance

If you want to save yourself some time, you can download all of the notes in two lecture note packets: Corporate finance class syllabus (MBA) and project.
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Alexa Actionable Analytics for the Web. The dividend is a function of the profitability of the corporation. Since this profitability is quite variable for most companies, the cashflows from a stock will be risky. Historically, though, at least in the U. S, the return from holding equity has been on average much higher than the return from holding government debt.

Corporate Finance MBA Class (Spring 2017)

Some of the largest financial markets are markets in derivatives, securities whose payoff depend on the price of some other security, or even on the prices of real as opposed to financial goods. Futures markets are markets where one can fix a price today for a future delivery of some good. Options markets are markets where one can fix a price today for a future contingent delivery of some good. The primary market is at the issue of a security. Treasury securities are often issued to the general public by an auction where anybody can send in bids.

This is then the primary market for treasury securities. When a corporation issues equities for the first time, the Initial Public Offering, this is the primary market for equities. When securities are traded after they have been issued in the primary market , they are said to be traded in the secondary market. In terms of volume and value, the secondary market dwarfs the primary market. Many do not understand why secondary markets are important, because they only seem to be "zero sum games. These people miss some important services that financial markets provide: Relative to the amounts being bought and sold in financial markets, the costs of transacting are small.

Hey, all you are doing is shuffling paper around. The costs are not zero How do you think stockbrokers survive? It is this that makes it justifiable for us to make an assumption of perfect capital markets, markets where it is costless to transact. Their existence would in practice lead to extreme investment strategies, like: Example The current price of a contract for one dollar to be delivered in 12 months is 0. Even if a corporation sells one million units of this contract, it does not afFect its price, the company will get 0. No more, no less. One to deliver one dollar in 6 months when half of the coupon is paid , and another to deliver one dollar in 12 months.

The basket contains 40 units of the first contract, and 1, units of the second contract. Do you know any supermarket that violates the rule of value additivity when ringing up a customer? Financial super markets don't either! This is also called the "no arbitrage" assumption. The no arbitrage assumption is intimately connected to the value additivity assumption, since a violation of value additivity will also be a violation of the no free lunch assumption. They are still different principles. One says how to compute portfolio value; the other prescribes how to price a particular portfolio.

An efficient market is really never surprised. Bad news happens with a certain frequency, and the market knows this. The market just does not know how the occurrence of bad news is distributed over time. Of course, the precision of the market's beliefs depends on the available information, but it is never systematically biased. Any systematic biases will be used to make profits above those justified by the risk of a strategy. Chapter 3 On Value Additivity Contents 3.

The price of a basket of goods apples, pears, The Value of the Firm A corporation holds a certain number of assets. Applying value additivity, the value of those assets is simply the sum of the values of the components. That is 9 10 On Value Additivity called the value of the firm. The firm is held by a number of different creditors, such as equityholders, bondholders, banks and so forth.

The value of the creditor's holdings, called liabilities, must by value additivity and no free lunches add up to the value of the firm. We define the value of a firm as the price for which one could sell the stream of cash flows that the assets of the firm generates for the traditional creditors.

By the value additivity axiom and no free lunches, The value of the firm is always equal to the sum of the values of the company's liabilities to the traditional creditors. For example, if the company's liabilities consist of equity with value E and debt with value B, and the firm value equals V, then: The summation on the RHS is justified by value additivity, the equality by "no free lunches. The value of the firm is not necessarily equal to the price of the assets of the company. This because some of the cash flows go to third parties, in the form of taxes, lawyers and accountants fees and also because it may be advantageous to keep the company alive rather than selling it.

The "live firm" may be more valuable e. Example The assets of a firm generate a perpetual, riskfree, after-tax cash flow of SIO per year. The reader should return to these after having gone through the book and see if she can understand why these may not be violations after all. Is this a violation of value additivity and no free lunch? Options give the right to purchase call or sell put an underlying asset at a pre-determined price the strike price during a particular period.

Merton proved in that the value of an option on a portfolio of assets think of it as equity in a firm with multiple subsidiaries can never exceed, and will usually be lower than, the sum of the values of options on the component assets of the portfolio think of it as the combined equity in each of the subsidiaries of the firm. How can it ever be lower? That seems to violate value additivity and no free lunch, but it does not really. Merton's result has been used to explain why the stock price of companies that disintegrate or spin off subsidiaries increases.

Warren Buffett claims to have made a living out of violations of value additivity and no free lunch. Does 2 ketchup bottles of 0. What does this tell you about value additivity in financial markets? The basic intuition of efficient markets hypothesis is that to a large degree, 13 14 On the Efficient Markets Hypothesis one cannot predict security prices.

In fact, if there is "too much" predictability, speculators move in and profit. Their actions will make the market more "efficient," that is, less predictable. But what is "too much" predictability? Fama, "Information is always correctly reflected in securities prices" The problem with this statement is that it is ambiguous. What "information" are we talking about? What does "correctly reflected" mean?

The way we specify information makes clear we are really thinking about how we can empirically test the efficient markets hypothesis. To make concrete the notion of information, we usually use three different "information sets," illustrated in figure 4. If knowing only this much information we are able to predict future prices, this is said to be a violation of weak form efficiency.

This is also how one can test this form of market efficiency. Example The presence of autocorrelation negative price movements are more likely to be followed by positive than negative price movements in stock market returns will violate some formulations of weak form efficiency. The second information set we consider is the set of all publicly available information. Note that past prices is part of this, so past prices is a subset of all publicly available information.

If financial prices reflect all publicly available information, prices are said to satisfy semi-strong form efficiency. Here it is easy to see how one would go about testing this particular version of the efficient markets hypothesis: Do market prices react correctly to new information when it becomes public?

Example When a corporation issues its quarterly earnings announcement and the earnings are twice what the market expected, the corporation's stock price should rise. The final information set is the set of all information. If markets were perfectly informed, and prices reflected all this information, prices would be strong form efficient. Under strong form efficiency, prices also reflect private "insider" information. The reasoning is that if an an insider attempts to trade on her information, prices will move against her, and, because of this, reflect her information.

Trading will reveal even inside information. There is lots of evidence in favor of weak form and semi-strong form efficiency. There is solid evidence against strong form efficiency, markets can not perfectly read the information in trades. It means that markets use the information available to them in the best possible way to generate its assessment of the current value of a financial security.

Lectures on Corporate Finance - PDF Free Download

Here is how this is traditionally interpreted. For simplicity, this security does not pay dividends. Information is the information available at time t, and is used in generating the conditional expectation above. Notice how finance really takes an anthropomorphic view of the market. The previous formal statements of the EMH are too simple. They fail to take into account that for anybody to be willing to hold a financial security, they must be offered an inducement to holding the security until the payoff is realized.

We therefore refine the basic statement above. If a security was risk free, we could discount its next period price at the risk free interest rate rftt to arrive at todays price. We term this a risk premium, and use pitt for the risk premium for security i in period t.

Note the implications of this. Price changes cannot be predicted beyond the compensation pi 4. Surveys can be found in Fama and Leroy You could try to get short-term bridge financing. But that will carry a high opportunity cost, reflected in the high short-term interest rate. Of course, the market may not know that coupon rates will decrease soon.

But that very possibility contradicts market efficiency. If such news has continuously been positive lately, would you not expect the average realized return to be exceptionally high as well? Fama revisited the EMH 20 years later in Fama For the risk premium formulation of efficient markets see Lucas Long term interest rates are at record highs. Most companies therefore find it cheaper to finance with common stock or relatively inexpensive short-term bank loans.

What does the Efficient Market Hypothesis have to say about the correctness of this? UPC has been found guilty of discriminatory practices in hiring. She cites an empirical study where 17 well-managed firms and a control group of 17 average firms were followed for 8 years after the former were reported in the press to be "excelling" as far as management is concerned. Is this evidence that the stock market does not recognize good management?

It states that it changed how it accounts for inventory. There is no other surprises in the earnings report. Would the stock price now jump on the release of this earnings report? The Japanese economy has deep structural problems, which the Japanese seem reluctant to overcome. We do not see any major change in this situation over the next two to three years. Hence, we advise against investing in the Tokyo stock market, because we expect returns to be below average for the next two to three years. There are two important properties of these cash flows. Two, the dates at which the amounts are paid.

This is the typical example of a risk free security, one with no uncertainty as to both the amount of and timing of cash flow. As another example, consider an oil company about to start drilling for oil in a new area. The oil company is facing uncertainty at several levels. There is uncertainty about 23 24 Present Value whether the oil company actually locates oil. Even if they locate oil, there is a lot of uncertainty to what price they can sell the oil.

In this case it is very hard to find the future cash flows, about the best one can do is to estimate the expected future cash flows. Alternatively one can consider contingent future cash flows. We will for the rest of the chapter concentrate on the valuation of a sequence of certain future cash flows.

The valuation of future risky cash flows is the topic of later chapters. We use the symbol Xt for the amount X to be paid at a future date t, and we want to value a set of future cash flows: More precisely, it is the cost to obtain the same stream of cash flows in the market. Evaluating the PV is simplified by using the axiom of value additivity, since we can then split the problem into summing the values of the individual dated cash flows. The problem is then reduced to finding the value today of a cash flow Xt at some future date t.

To do this we use the set of prices Pt today of receiving one dollar at time t in the future. The PV of the entire stream is then: Today's value of one dollar received days from now is USD 0. For example, we can split the evaluation of the present value into several steps. They are typically estimated from actual prices in financial markets. Since most people are impatient, and would put more value today on receiving a dollar tomorrow than one year from now, you would expect the following property to hold: For each price Pt there is a corresponding interest rate rt.

We will therefore need to spend some time on transformations involving interest rates. Generally, the rate of return on an asset is Rate of return: The bank offers you a given interest on the balance of your account. How much would you have to invest now at the per period interest rate rt to get one dollar at time t? To complicate matters, there is another facet of interest rates, the frequency of compounding. We will return to this, for now we will use discrete compounding. One thing to note about the expressions 5.

If you know interest rates you also know prices, and vice versa. For most purposes, such as calculating present values, it is the prices that are of interest, not returns. Ever seen a grocery store that quotes the prices of its apples as the — t'th root of its dollar price? There are however also cases where interest rates may be more meaningful.

The interest rate rt denotes the percentage return on investing one dollar in a security that promises one dollar at time t. The return is normalized to percent per period, so returns on securities with different maturity different t's can be compared. Many people will also compare investment opportunities by calculating an implied return on the project. Such comparisons are however full of pitfalls, many mistakes continue to be made by people who only use interest rates to compare investment opportunities. The term structure can take a multitude of shapes. Typically, it is rising, but it can also be decreasing, or even "humped.

A one-year T bill, with a face value of , and no coupons, sells for Given these market prices, we can find Pi and Pz that gives the securities the correct prices: The Net Present Value NPV of an investment project is the difference between the Present Value and how much it costs you to generate the same cash flows with your project.

If the Net Present Value of a project is positive, it is obviously a valuable project: This goes for any kind of investment project. The basic decision rule is to Invest in any project with a positive Net Present Value Positive NPV occurs when your cost to generate a stream of cash flows is less than the price that the market charges. Example Suppose you know how to generate a particular risky stream of cash flows by building new computer chips and selling them. The investment is cheap. You would have to pay a lot more to get the a stream of cash flows with the same properties distribution over time and type of randomness by combining equity, futures, bonds, etc.

Positive NPV reflects the presence of economic rents. You own or know something that nobody else does. One example is a the calculation of a perpetuity. A perpetuity is a sequence of payments each period into indefinite future. You receive 10 next year. How much are you willing to pay the bank today for this set of cash flows? The present value of an annuity that last T periods is found as. Such annuity factors are tabulated in a lot of places. Most financial calculators will also provide them.

The choice should be based on the alternative with the highest present value. In discrete compounding, you receive a dividend only once or a number of times n, depending on the case every period. The continuous-time case is really the limit of discrete compounding, whereby the rate r is paid and reinvested faster and faster: We can also find the present values for respectively discrete and continuous compounding as: The typical fixed income security is a bond.

Valuing bonds should by now be straightforward. We need to find the present value of the promised sequence of payments, using either prices Pt or interest rates rt. Example A bond promises the following sequence of payments: Collectively, the equity owners are the residual claimants on the value of the firm, net of all the firm's liabilities.

For one stock, however, this is not the relevant startmg point for valuation. What counts is the cash flows accruing to the stock. For the company the cash flows are the dividends paid. The individual owner of a stock has an alternative source of cash flow, though: Example You currently own one stock in the XYZ company. XYZ will pay dividend one year from now of one dollar. You also know that the price of one XYZ share one year from now, just after the dividend payments, will be for sure.

What is the current value of the stock? This is just the present value: Clearly the buyer of the share must believe that the value of one XYZ share at that time is The source of the value must be cash flow from the XYZ share at some point in the future. We are in other words in the following situation: To value a stock it is not necessary to estimate some future stock price.

One can also concentrate on the cash flows from the company, namely the dividends. The price of a stock is the present value of all future dividends. How to estimate all these future dividends? In general this is clearly impossible. Usually one unit equals one dollar of expected cash flow. That is, if Xt now denotes the cash flow itself a random variable , and E[Xt] its expected value, then one measures the risky cash flow in terms of number of expected dollars, and writes: In fact, you will want to use different prices depending on the level of risk.

We will be returning to this risky case. References Textbook References Any basic text book on corporate finance, such as Brealey and Myers or Ross, Westerfield, and Jaffe covers this material in much more detail. Calculate the implied interest rates and graph the term structure of interest rates. Calculate the present value of the following cash flows: BankThree is offering personal loans at Which is the better offer?

How would you make a lot of money? Three banks have offered loans. The first bank offers 4. The second bank offers 4. The third bank offers 4. Determine which is is the best offer. Determine todays stock price. Present Value 36 5. What is the current value of receiving one dollar at time 3? Consider now the bond D, with the following characteristics Bond Cashflow in period 1 2 3 20 20 2. What is the current price of bond D? Consider next bond E, which last for four periods. Bond E has the following characteristics: Bond Cashflow in period 1 2 3 4 10 10 10 3.

If the market does not allow any free lunches arbitrage , what is the maximal price that bond E can have? The company is paying a dividend of 5 next period. In an efficient market, can these numbers be sustained? After that, the payments grows at a rate of g per year for the next T years. The present value of the annuity is Can you find a simplified expression for this present value? How much must she invest today to have that amount at graduation?

If she invested once a year for four years beginning today until the end of the 4 years how much must she invest? Given the opportunity to invest in one of the three bonds listed below, which would you buy? We will first look at some problems t h a t t e n d to appear when calculating net 39 Capital Budgeting 40 present values, such as the mistaking of accounting numbers for cash flows, how to treat sunk cost, repeated projects. We will also discuss a number of alternative approaches that have been suggested to evaluate projects. While the most important is clearly NPV, it is necessary to be aware of the alternatives, and how they can be and are misused.

Note that the whole discussion in this chapter is in terms of risk free cash flows. This is for simplicity. As will become clear in later chapters, all what we are doing is also relevant for risky cashflows. It is merely a matter of adjusting prices, or discount factors. To find the present value we need either prices Pt for cash flows at a future date t, or alternatively, an interest rate rt for cash flows at t. In capital budgeting, the interest rates implicit in prices of future cash flows are referred to as discount rates.

Discount rates reduce the cash flow towards their present value. One should be very careful here. With the risk of angering accountants, keep the following in mind: Only to be used to reduce tax payments. These are different from the ones used in finance. In particular, finance obtains values from market prices and the axioms of chapter 2. Earnings numbers would be irrelevant, were it not that their calculation determine taxes, and taxes are cash flows and therefore affect value.

Normally, taxes are paid on the accounting profits from an investment. You can reduce these by depreciating the investment. This is called the depreciation tax shield. Note though that if you have no profits, you cannot use the depreciation to reduce your taxes. Depreciation can be used to reduce taxes only to the extent that you have positive profits. Cash outflows costs that occurred in the past are sunk costs and are irrelevant for the computation of NPV. The market does not reward stupid past investments, so they do not enter in the computation of value.

The fact that you already sunk so many billion dollars in the super conductor-collider does not make the project more attractive. There are some relevant cash flows that are often forgotten, namely opportunity costs. When you make an investment, you may have to forgo money that you would otherwise have made automatically. Example The opportunity cost of getting a college education is the money you could have made flipping burgers at MacDonald's.

This opportunity cost should be added in as cash outflow in the computation of the NPV of a college education. Capital Budgeting 42 Alternatively, opportunity costs should be acknowledged as cash flows in a separate project whose NPV has to be computed as well. Example In the college education case, think of it as two possible projects: Compute the NPV of both projects and compare.

Most projects have some cashflows that will be adjusted as a result of inflation. Sales is a typical example of this, the prices you can sell goods for is easily adjustable. There are however some cashflows that are fixed in nominal terms, such as debt payments. Conceptually dealing with inflation should be straightforward. All that is needed is consistency.

When using nominal cash flows, discount using the interest rates that apply to cash flows which are expressed in nominal terms. When adjusting cash flows for inflation "constant dollars" , use "real" interest rates. People often compute the latter as the nominal rate less expected inflation. That is not necessarily correct, so maybe you want to forget about real cash flows entirely. Besides, economists don't really agree on what inflation is. Example A project has the following projected cash flows in real terms. If these projects are one shot projects, they should be compared using NPV.

But when the projects are to be repeated e. One way of doing this is to find the periodicity of matching cycles and compute cash flows over this period. Example Consider two projects with life lengths 2 and 3 years. Make them into a comparable project by repeating the first one 3 times and the second one 2 times. This way you will have two projects with a 6 year life length to compare. An equivalent procedure is called the Equivalent discussed at length in standard textbooks. These range from methods that will agree with NPV "most of the time" to methods that will only agree with NPV by accident.

While we advocate the use of NPV to make all decisions, it is sometimes useful to understand the alternatives and when they will lead to "wrong" decisions. The decision rule involving payback is to accept projects with a payback period shorter than some given period of time. Example A company is considering two projects with the following cash flows. Project 1 2 Cost Xi 50 X2 50 X3 X4 44 Capital Budgeting The payback period for project 1 is 2 years, whereas the payback period for project 2 is 3 years. If the company only accepts projects with a payback less than two years, only project 1 will be acceptable.

Project 1 is however the worst of these two project, it only returns its initial cost, and must therefore have a negative NPV. Let us calculate the IMPV for the two projects: The example illustrates some of the weaknesses of using payback period as a decision criterion: There is no discounting of future payoffs. Payments beyond the payback period are ignored. There is no economic rationale for choosing a "cutoff" payback period.

There are attempts to "rescue" the payback period by discounting the future cashflows before calculating the payback period. While this would show that project 1 in the previous example is clearly undesirable, it will still not take account of cash flows beyond the discounted payback period. The IRR is a popular summary measure of the return from an investment. It is primarily useful because it is a relative measure, it is easy to compare the IRR of two investment projects.

The IRR has a number of weaknesses, though, that should make one be very careful in using it. One assumption one is making when using the IRR is that the interest rate is constant. As we have seen it is far from obvious that this is the case. What if the term structure of interest rates is not "flat"?

First, it is hard to solve higher order equations, it must be done numerically. Also, most such equations have multiple solutions, some of which may be imaginary. Example A project has cashflows: For all interest rates between these two the NPV of the project is positive. Sometimes you want the IRR on your project to be as low as possible!

When you borrow you want the lowest possible interest rate! If you look at the NPV profiles for these two projects, it is immediately obvious what the problem is: As yet another area full of pitfalls in the application of IRR, the ranking of projects using IRR should be avoided, in particular when the projects are mutually exclusive. Example Consider the following two projects Project 1 2 t- 0 1 2 50 50 3 50 IRR The projects are mutually exclusive.

Which one to choose? Based on IRR, project 1 seems like the better one. But this would be a mistake. Project 2 is the one that adds most value, and it is the one that should be chosen. The problem stems from the fact that the scale of the two projects are different. Project 1 earns a higher return, but only on a tenth of the investment of project 2. If you have available for your investment, what do you earn on the remaining if you invest in project 1? The internal rate of return is widely used in fixed-income analysis, where it is referred to as "bond yield," or "yield to maturity," but is very dangerous in that context.

Example Two bonds with the same maturity and principal, but different coupons, can have different yields - this does not imply an arbitrage opportunity. The decision rule used is to accept projects with a profitability index larger than one. This is equivalent to choosing projects with positive net present values. The rationale behind the use of the profitability index is that it is an attempt to get a relative measure of the desirability of a given project.

But the fact that it is a relative measure already points to a problem using the profitabihty index, it ignores scale, and we therefore have the same problems as we had using the IRR rule to rank mutually exclusive projects. To use this measure for project valuation one would estimate the future accounting numbers for the project, calculate the accounting earnings for the future and compare the resulting estimate with some "hurdle" accounting rate of return. The problem with the accounting rate of return is that it uses accounting numbers, which usually have very little to do with cash flow.

Accounting numbers are easily manipulated, and do not really reflect prices in a marketplace. Despite their shortcomings, it's important to understand these and other new ideas , because shrewd managers and consultants abuse them. Example To increase bonuses, which often depend on return to equity, managers can take one-time charges for future costs. They can for example be justified as "re-organization costs. As a result of the charge return on equity accounting profit divided by book value of equity increases. As will the managers' bonuses References Any basic text book on corporate finance, such as Brealey and Myers or Ross et al.

Find the Payback periods for the two projects. Which project has the shortest payback period? Calculate the Internal Rate of Return on the two projects. Which project has the higher IRR?

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How would the NPV rule rank these two projects? The project has positive cash flows of in years one and two. At the end of the life of the project there are large environmental costs resulting in a negative cash flow in year 3 of — Determine the internal rate s of return for the project. All three bonds have a face value of 1, at maturity. Find the time zero prices, P i , P2, and P3, of one dollar to be delivered in years 1, 2, and 3, respectively.

At a cost of 47 they can make some small repairs on their current machine which will make it last 6. At a higher cost of 90 they can make some more extensive repairs on their current machine which will make it last for 4 more years.

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A new machine costs and will last for 8 years. Determine the best action. It will have a life of 3 years. The cost will be depreciated straight-line to a zero salvage value, and is worth 40 at that time.


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Cash sales will be per year and cash costs will run per year. The firm will also need to invest 60 in working capital at year 0. What is the project's NPV? The machine's initial cost is , and can be depreciated on a straight-line basis to a zero salvage in 5 years. The machine's per year fixed cost is , and its variable cost is 0. The selling price per unit is 1. Calculate the machine's accounting break-even point on the new machine i. Calculate the machine's present value break-even point i.

It can follow one of two strategies. The first is to manufacture a medication aimed at relieving headache pain. The second strategy is to make a pill designed to relieve headache and arthritis pain. Both products would be introduced at a price of 4 per package in real terms. The broader remedy would probably sell 10 million packages a year.

This is twice the sales rate for the headacheonly medication. Cash costs of production in the first year are expected to be 1. Production costs are expected to be 1. Either strategy would require further investment in plant. The headache-only pill could be produced using equipment that would cost The machinery required to produce the broader remedy would cost 12 million and last three years. At this time the firm would be able to sell it for 1 million in real terms. The production machinery would need to be replaced every three years, at constant real costs.

For both projects the firm will use straight-line depreciation. What pain reliever should the firm produce? Chapter 7 Valuation under Uncertainty: That involves two things: Recognizing risk categories, the ability to classify risky cash flows in "bins" of equal "risk. Pricing each risk category, finding a price per dollar of expected cash flow. The intuitive implication of risk is that the higher the risk, the lower the current price for the future risky cash flows. It will turn out that our intuition is correct. We will in this chapter mainly work with returns.

Returns is the transformation of prices into equivalent interest rates, as discussed in chapter 5, defined by For the price to decrease when risk increases move to a higher "risk category" it must be the case that the interest rate increases when risk increases. The "asset pricing question" is then to find what is the relevant measure of risk.

To summarize what we will be showing in this chapter: For a given asset, the asset's volatility is not an appropriate measure of risk. Rather, the relevant measure of risk depends on how much this particular asset contributes to a measure of overall risk. The central result of asset pricing theory we will show is therefore that Only the covariance with some diversified "benchmark portfolio" is priced.

To do so, we must first cover some preliminaries on portfolios and returns. The intuition for the CAPM is based on per period returns for a portfolio of assets. An investor's preferences are supposed to be such that he prefers higher expected portfolio returns, but dislikes variability, measured by the variance, or equivalently standard deviation, of the portfolio returns. Let fj be the return on asset i. There is no indexing by time, for now we only deal with one period.

The return is random, and E[fi] is its expected value, of is the variance of returns on asset i. The standard deviation of returns is then 7. Your total wealth is the sum of these values, The fraction of your wealth invested in the each share, or portfolio weights, is a half for each asset: The point is usually made the following way.

Repeat this for increasing numbers of n. A picture like figure 7. With only one or two assets in the portfolio, the standard deviation is large. By increasing the number of assets the standard deviation decreases, but only up Valuation under Uncertainty: This is the finance way of telling you the folly of putting all your eggs in one basket.

Should one care about the diversifiable risk? Not if you aren't rewarded for holding it. That is precisely what happens in the CAPM. What are the feasible portfolios for such an investor? The investor takes the set of assets, expected returns, variances and covariances as given. The only thing the investor can vary are the weights. By varying the weights the investor can generate a feasible set of portfolio combinations.

If we accept that investors wants high expected portfolio return but dislike portfolio volatility, clearly each investors optimal portfolio is one that maximizes portfolio expected return for a given level of portfolio standard deviation. The set of such portfolios is called the efficient set, and it is the upward sloping part of the curve in figure 7. Let Tf be the return on such a security. A typical example of a risk free security is a short term government Treasury bill. A risk free security has standard deviation of zero.

If we combine a risky security with the risk free security, the combination maps as a line in the mean-standard deviation plot, as shown in figure 7. As is obvious from the figure, the only risky asset that it makes sense for the investor to use is m, the tangency portfolio. Assets on the line from rf through m dominates all other feasible assets. With a risk free asset, the efficient set is therefore the line from rf through the tangency portfolio m. The arguments above tells us that any investor will want to hold as his portfolio a combination of the risk free asset and the portfolio m of risky asset.

But then the total demand by investors of securities will also be some combination of the risk free asset and the portfolio m. In an economic equilibrium, demand has to equal supply. A logical consequence is then that the portfolio m has to equal the total supply of risky assets, the market portfolio. To 58 Valuation under Uncertainty: Risk Free Asset combined with portfolios p and m. E[fp] Feasible Set Tj crP make demand equal supply, the prices of risky assets will have to adjust, changing expected returns and return variances. In the end, each investors portfolio consists of a combination of the risk free asset and the market portfolio m of all risky assets.

Hence, in equilibrium, the market portfolio must be mean-variance efficient, which really means that The market portfolio has the highest possible reward-to-risk ratio "Sharpe ratio" Mathematically, this is equivalent to: From the CAPM equation, we see that the only thing that affects an asset's risk is the assets covariance with the market portfolio. The CAPM relation 7. To implement this in practice we need to estimate the beta, the covariance between the return on the cash flows and the market portfolio.

If for example the asset in question is a stock, we can easily estimate this from historical data of stock returns. The only problem is the choice of the market portfolio. In practice we usually chose a broad based stock index as a representative market portfolio, and calculate beta from the covariance of these returns. The beta of stock J is HJ 0. It turns out that the statement in the previous section, "Average returns are determined solely by covariation with returns on some benchmark portfolio," is vacuous without qualification about what that benchmark portfolio is Roll, The CAPM identifies the market portfolio as the benchmark portfolio.

But is it observable? If not, the CAPM may not have much empirical content. This point is very relevant for an extremely mundane practice, namely, mutual fund performance evaluation. Discussion has to be delegated to an investments text, though. The CAPM 60 7. The CAPM gives us a tool to adjust for the riskiness of a project, one that is remarkably simple to apply. Risk is measured by the beta of a project, the covariance of the project returns and the market returns. Given the project beta, use the CAPM equation 7.

Use this discount rate to discount the expected cash flows from the project. Variance of r m where rm denotes the return on the market portfolio. We next use this "beta" to compute the required discount rate: Our principle of valuation has been to compare the cost X0 of a project to the price P it takes to buy the same cash flow Xi in the marketplace. We will not elaborate here, because we would like to move on and introduce a far more powerful valuation procedure under uncertainty.

The interested reader is referred to Ekern Example A project costs today and has expected cashflows Spfi] of next period. The covariance between the project return and the market portfolio is 0. The variance of the market portfolio is 0. Determine the NPV of the project. Use this discount rate to discount the future cash flows: The EMH implies that average returns are determined solely by risk. Asset pricing theory determines the content of "risk," namely covariance with the returns on a diversified benchmark portfolio.

Historical data support this proposition across broad categories of assets. However, a closer look reveals ample violations. These violations can be "worked away" by a clever choice of benchmark portfolio, but that does not advance our understanding of financial markets. This is the state of the art The empirical evidence is only weak at best. We want to reflect on this a little bit, because the CAPM is the main model on which actual corporate finance decisions including litigation are based. The model is theoretically compelling beautiful logic , but that is not enough.

There are lots of things that can go wrong. Besides picking the wrong "market portfolio," investor beliefs may be wrong, there may be sample selection bias, investors may care about more than volatility for example downside risk, which manifests itself as skewness , there are taxes, the markets may not continuously be in equilibrium, etc.

Experimental financial markets were set up at Caltech in a very simple way. If the CAPM did not emerge in our simple setting, what hope would there be that it does in the far more 62 Valuation under Uncertainty: The results of the experiment shows that, yes, the CAPM emerges, but very slowly. The "price discovery" process is painstakingly slow. The conclusion to draw from the experimental evidence: Many view this simple relationship as too simplistic, both on theoretical and empirical grounds. To implement the APT one would need to choose a set of "factors," or observable market variables that one thinks influences stock returns.

Given this one estimates the coefficients in a regression of stock returns on the "factors. For the same volatility, a security with higher skewness may be perceived to be more risky. In fact, many financial securities have highly skewed payoffs. The prototype of such a security is the option. Many securities that are issued by corporations have option-like payoffs. The CAPM is not designed to price such securities. Hence, we will have to consider another approach.

That will be done in Part III. The Arbitrage Pricing Theory is primarily due to Ross The experiments mentioned are discussed in Bossaerts and Plott A B E[r] What does this tell you about how diversification possibilities varies with covariances? What is the expected return of a stock with a beta value of 0.

What is the beta of stock A? What is the highest cost that makes this project worth investing in? The correlation between the two is shares is 0. What is the current expected market return? The other is an asset with expected return E[ri] and standard deviation a-i- Show that combinations of these two assets map as a straight line in a mean-standard deviation plot. Stock 1 has a standard deviation of 0. The correlation between the stocks is —0. Calculate both the variance and the standard deviation of the portfolio.

Suppose you desire to invest in any one of the stocks listed above singly. Can any be recommended? Now suppose you diversify into two securities. Given all choices, can any portfolio be eliminated? The beta for the overall firm is 1. The firm is considering the following capital expenditures: Which projects would it accept if it uses the opportunity cost of capital for the entire company? Which projects would it accept if it estimates the cost of capital separately for each division?