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Thayer Watkins
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ECONOMICS 137A
NOTES FOR:
FOUNDATIONS OF CORPORATE FINANCE
by
Thayer Watkins

No. CONTENTS:PAGE
0. MAJOR POINTS IN CORPORATE FINANCE1
1. HOW ECONOMIC DECISIONS SHOULD BE MADE2
2. THE TIME VALUE OF MONEY 4
3. EXERCISES ON NET PRESENT VALUE AND
THE NET PRESENT VALUE RULE FOR MAKING INVESTMENT DECISIONS
6
4. THE USE OF THE ANNUITY TABLE TO MAKE THE COMPUTATION OF PRESENT VALUE EASIER 7
5. PERPETUITIES 8
6. THE DEFINITIVE ANSWER TO THE QUESTION OF HOW INVESTMENT DECISIONS SHOULD BE MADE: THE NET PRESENT VALUE RULE 9
7. A FORMULA FOR THE PRESENT VALUE OF A GROWING ANNUITY 11
8. THE DISTRIBUTION OF THE RATES OF RETURN OF SECURITIES 12
9. THE PRESENT VALUE OF GROWTH OPPORTUNITIES 13
10. RESULTS OF INVESTIGATIONS OF THE ABILITY OF STOCK MARKET PROFESSIONALS TO PREDICT THE EARNINGS OF COMPANIES 14
11. CORPORATE FINANCE 15
12. MODIGLIANI AND MILLER'S PROPOSITONNS CONCERNING THE EQUITY VALUE OF A CORPORATION17
13. RISKY PROJECTS 20
14. PORTFOLIO ANALYSIS 22
15. THE CAPITAL ASSETS PRICING MODEL 25
16. BETA THEORY 26
17. TESTS OF THE CAPITAL ASSET PRICING MODEL 29
18. THEOREM: CHANGES IN CONDITIONAL EXPECTATIONS FOR ONE TIME PERIOD ARE UNCORRELATED WITH CHANGES IN THE CONDITIONAL EXPECTATIONS FOR A PREVIOUS TIME PERIOD 32
19. THE SOURCE OF POSITIVE NET PRESENT VALUE FOR PROJECTS 35
20. THE INVESTMENT-OPPORTUNITIES LINE36
21. THE FOUR STAGES OF CAPITAL BUDGETING 37
22. THE LIMITED LIABILITY CORPORATION 39
23. A RIGHTS ISSUE OF STOCK 41
24. THE EFFICIENT MARKETS HYPOTHESIS 43
25. CHAOS AND ORDER IN CAPITAL MARKETS 45
26.MODIGlIANI AND MILLER'S PROPOSITION THAT, IN THE ABSENCE OF DIFFERENTIAL TAXES ON INTEREST AND DIVIDENDS, THE VALUE OF A CORPORATION IS INDEPENDENT OF ITS CAPITAL STRUCTURE 46
27. LEVERAGED BUYOUTS (LBO'S) 47
28. THE CONCEPT OF THE "DURATION" OF AN INVESTMENT 49
29. A HYPOTHETICAL LEVERAGED BUYOUT 51
30. INTEREST RATES, INFLATION RATES AND EXCHANGE RATES 53
31. FORMULAS 55
32. NOTES ON BENJAMIN STEIN'S "LICENSE TO STEAL" 56
33. THE SAVINGS AND LOAN INDUSTRY 64
34. NOTES ON "THE PREDATOR'S BALL" BY CONNIE BRUCK 65
35. NOTES ON "FUNNY MONEY" BY MARK SINGER 69

Major Points in Corporate Finance


HOW ECONOMIC DECISIONS SHOULD BE MADE

When we have a choice of actions often there will be some advantages and some disadvantages for different alternatives. The advantages are also called benefits and the disadvantages are called costs. A worthwhile project is one where the advantages are worth more than the disadvantages. If the disadvantages outweigh the advantages then the project should be turned down. Cost-benefit analysis is just this process, formalized. Usually the advantages of a project are that it will produce additional goods and services and the disadvantages are the resources it uses up. Consider the building of a hydroelectric project. The dam will reduce flood damage during the rainy season and retain water for irrigation during the dry season. It may also allow increased use of the river for transportation by maintaining the water level sufficiently high throughout the year. It may also produce electricity. The reservoir behind the dam may also increase the production of lake fish. On the disadvantage side there are the cement, gravel, machines and labor that are used up in building the dam. The dam may prevent flooding of property downstream, but the reservoir of the dam usually floods some homes and farmland. It also is an impediment in the river preventing the navigation by riverboats and stream fish. In tropical areas dam reservoirs promote the spread of parasite-carrying snails and flies that cause blindness. In Egypt the Aswan High Dam stopped the annual flooding of the Nile that each brought nutrient-rich soil to farms. Now Egyptian farmers have to buy fertilizers to replace the benefits of the annual Nile flooding. So the question is whether the value of the advantages was greater or less than the value of the disadvantages. In the case of the Aswan Dam in Egypt the government was convinced that the benefits outweighed the costs but now many Egyptians doubt that that was true.

Exercise 1: List the benefits and costs of switching from doing your laundry in a laundromat to buying your own washer and dryer to do laundry at home.

Exercise 2: List the types of benefits and costs that may result from building a new section of highway which reduces travel times and accidents.

One cannot go very far in cost-benefit analysis without needing to reduce the various benefits and costs to a common unit of measure. The natural procedure for doing this is to use market prices to reduce everything to money values. This is basically the correct solution but one should have a proof that it is correct. Consider whether the price that prevails for some commodity is a true market price or an artificial price administered by the government. It is appropriate to measure the costs and benefits using prices only if those prices are ones at which people can buy as much or as little as they want.

Exercise 3: Suppose some commodity is rationed so people are not able to buy as much as they would like at that ration price. For example, suppose that gasoline is rationed so that consumers cannot buy more than ten gallons per week. If the control price is $1.00 per gallon would the value of a gift of two gallons per week be equal to $2 per week?

THE TIME VALUE OF MONEY

A typical investment project involves the costs occuring in the immediate future but the benefits coming it over an extended period of time beyond the immediate future. The question is how to establish an equivalency between money now and money in the future.

Exercise 4: Suppose you put $10,000 in a bank account at an interest rate of 12 percent. How much will you have next year? How much will you have after two years? After five years?

Exercise 5: Suppose you are due to inherit $10,000 next year. How much could you borrow at an interest rate of 8 percent and still pay the loan off after one year? Suppose your inheritance is due in two years. How much could you get now when the interest rate is 8 percent?

Exercise 6: Use the information in Exercise 5 and create a table showing the present value of $10,000 various numbers of years in the future.

Number of years Present Value of $10,000
in the futureat 8 percentat 10 percent
0 10,000 10,000
1 _______ ______
2 ______ ______
3 ______ ______
4 ______ ______
5 ______ ______

Exercise 7: Suppose you have $40,000 which you can spend all this year or save some part of it, drawing 10 percent interest, to spend next year. Draw a graph showing all of the combinations of expenditures this year and expenditures next year that you can achieve with your $40,000.

Exercise 8: About 350 years ago an Indian tribe sold Manhattan Island to the Dutch for twenty four dollars. How much would that $24 be worth now if it had been invested at 8 percent interest?

Exercise 9: A new truck will cost $50,000 and bring in additional revenues over operating costs of $4,000 per year over a twenty year period. Is it a good investment?

Exercise 10: If the interest rate is 8 percent is the truck in Exercise 9 a good investment?

Exercise 11: Compute the NPV of the truck in Exercise 9 at a number of different interest rates. Try to find a pair of interest rates such that the NPV is positive at the lower rate and negative at the higher rate. Draw a simple graph and estimate the interest rate at which the NPV is zero. What is the definition of the Internal Rate of Return (IRR) of a project? What is the approximate IRR of the truck in Exercise 9.

Exercise 12: Suppose the truck in Exercise 9 were to last 25 years instead of 20. Estimate the IRR for this case.

Exercise 13: Suppose the government takes 50 percent of the profits of the truck in Exercise 9 but gives a tax credit for the initial investment in the truck. What would be the effect (increase, decrease, or leave unchanged) of such a tax on the IRR of the truck?

EXERCISES ON NET PRESENT VALUE AND THE NET PRESENT VALUE RULE FOR MAKING INVESTMENT DECISIONS

Example of the Computation of the NPV at Various Interest Rates and the Determination of its IRR

A company is considering investing in a mill that costs $400,000 and will last six years. Each year the mill will produce $90,000 in profits (revenues in excess of operating costs). However the motors of the mill will wear out after 3 years and have to be replaced at a cost of $100,000. Therefore the cash flow for the third year will be -$10,000. Several values for the discount rate (interest rate) will be taken.

TIMECASH FLOWDISCOUNT
FACTOR@20%
PV OF CF
0 -$400,000 1.000 -$400,000
1 +90,000 0.833 75,000
2 +90,000 0.694 62,500
3 -10,000 0.579 -5,787
4 +90,000 0.482 43,403
5 +90,000 0.402 36,169
6 +90,000 0.335 30,141

The net present value at a discount rate of 20 percent is -$158,574. The mill is not a good investttmmmeent. But at other discount rates the NPV is positive and the mill is a good investment. Compute the NPV for the mill at 5, 10, and 15 percent discount rates.

DISCOUNT
RATE
NET PRESENT
VALUE
0 +$40,000
5 -------
10 _______
15 _______
20 -$158,574

Net Present Value Calculator for Internet Explorer.

A Present Value Calculator for Netscape and Internet Explorer.

The Use of the Annuity Table to Make the Computation of Present Value Easier

If there is a cash flow of $2000 per year from year 1 to year 10 you don't have to discount each $2000 individually. The annuity factor for year n is the sum of the discount factors for years 1 to n. Therefore to get the present value of $2000 per year for years 1 to 10 when the interest rate is 12 percent you only have to multiply $2000 by the annuity factor for ten years at 12 percent; i.e., 2000x5.650 = $11,300. The annuity factors can also be used for other cases. Suppose there will be a cash flow of $45000 per year for years 8 through 13. If the cash flow were for years 1 through 13 you would use the annuity factor for 13 years but that would include cash flow for years 1 throught 7 that do not exist. Therefore the annuity factor for 7 years should be subtracted from the annuity factor for 13 years. If the interest rate is 8 percent then the calculation would be 7.904 - 5.206 = 2.698 and then 45,000x2.698 which gives the present value of the cash flow as $121,410. Another way to evaluate this annuity is to consider the annuity from the point of view of year 7; then it is an annuity that runs from year 1 to year 6. The value of this annuity discounted to year 7 is the annuity factor for 6 years (4.623)times $45,000 or $208,035. But we want the value as of time 0 so we must apply the discount factor for 7 years to the value of the annuity as of year 7; i.e., (0.583)($208,035)=121,284. The difference between the two values, $121,410 versus $121,284, is due to rounding errors. If the computation used unrounded numbers the results would be exactly the same.

Exercise 14: What is the present value of $1 per year for years 1 through 10 when the interest rate is 16 percent? What is the present value of $1 per year for years 1 through 5 when the interest rate is 16 percent?

Exercise 15: What is the present value of $120,000 per year for years 6 through 10 when the interest rate is 16 percent?

Exercise 16: What is the present value of $90,000 per year for years 2 through 8 when the interest rate is 10 percent?

Exercise 17: If the interest rate is 9 percent, what is the present value of $25,000 per year for years 1 through 40 and then $15,000 for years 41 through 60?

Exercise 18: What is the present value of $15,000 per year for years 1 through 60 and $10,000 per year for years 41 through 60 when the interest rate is 9 percent?

THE DEVELOPMENT OF THE CAPITAL ASSET PRICING MODEL

THE BETA FOR THE MARKET PORTFOLIO ITSELF IS EXACTLY ONE.

THE DEVELOPMENT OF THE CAPITAL ASSET PRICING MODEL FROM PORTFPOLIO ANALYSIS

Results and Conclusions of Portfolio Analysis

The Development of the Statistical (Stochastic) Approach to Securities

THE COST OF CAPITAL

WHEN EQUIPMENT IS PURCHASED AND USED THERE ARE SEVERAL COSTS:

THE INTEREST COST SHOULD BE BASED UPON THE MARKET VALUE OF THE EQUIPMENT, WHICH DECLINES OVER ITS ECONOMIC LIFE.

THE TRUE ECONOMIC DEPRECIATION (AS OPPOSED TO THE ACCOUNTING DEPRECIATION FOR TAX PURPOSES) IS THE DECREASE IN MARKET VALUE FROM ONE YEAR TO ANOTHER. THIS ALSO MAY VARY OVER THE ECONOMIC LIFE OF THE EQUIPMENT.

ALTHOUGH BOTH THE ECONOMIC DEPRECIATION AND INTEREST COSTVARY OVER THE LIFE OF THE EQUIPMENT THE COMBINED TOTAL IS CONSTANT. THIS COMBINED TOTAL OF DEPRECIATION AND INTEREST COST CAN BE FOUND BY DETERMINING THE PAYMENT THAT WOULD HAVE TO BE MADE TO PAY OFF A LOAN FOR THE PURCHASE PRICE IN EXACTLY THE ECONOMIC LIFE OF THE EQUIPMENT. THIS PAYMENT IS CALLED THE ANNUALIZED COST OF THE EQUIPMENT, OR THE EQUIVALENT ANNUAL COST.

TIMING OF AN INVESTMENT

SUPPOSE A COMPANY HAS A STAND OF TIMBER WHICH CAN BE HARVESTED NOW OR ALLOWED TO GROW. IN ADDITION TO THERE BEING MORE WOOD THE PRICE OF LUMBER MAY BE HIGHER IN THE FUTURE. BUT BEYOND A CERTAIN AGE THE RATE OF GROWTH OF THE TREES SLOWS. THE PROBLEM IS WHEN SHOULD THE TREES BE HARVESTED FOR MAXIMUM ADVANTAGE TO THE COMPANY.

YEAR OF HARVEST 0 1 2 3 4 5
NET REVENUE ($000) 50 64.4 77.5 89.4 100 109.4

THE INTEREST RATE IS 10 PERCENT.

THE SOLUTION TO THE PROBLEM CAN BE DETERMINED BY COMPUTING THE NET PRESENT VALUES.

YEAR OF HARVEST 0 1 2 3 4 5
NET PRESENT VALUE 50 58.5 64.0 67.2 68.3 67.9

THE MAXIMUM NPV IS ACHIEVED IF THE TIMBER IS ALLOWED TO GROW FOUR MORE YEARS AND THEN HARVESTED

THIS ANSWER COULD ALSO BE OBTAINED BY COMPARING THE RATE OF INCREASE IN NET REVENUE FROM YEAR TO YEAR WITH THE INTEREST RATE. IF THE COMPANY GETS A GREATER RATE OF RETURN BY HOLDING THE TIMBER ANOTHER YEAR THAN IT WOULD GET BY HARVESTING THE TIMBER AND INVESTING THE PROCEEDS AT THE CURRENT INTEREST RATE IT SHOULD POSTPONE HARVESTING THE TIMBER.

YEAR OF HARVEST 0 1 2 3 4 5
NET REVENUE ($000) 50 64.4 77.5 89.4 100 109.4
CHANGE FROM
PREVIOUS YEAR
28.8 20.3 15.4 11.9 9.4

FOR YEARS 1 THROUGH 4 HOLDING THE TIMBER GIVES A BETTER RETURN THAN 10 PERCENT BUT NOT FOR HOLDING FROM YEAR 4 TO YEAR 5.

INVESTMENT PROJECT

PURCHASE OF A TRUCK FOR $100,000 WHICH WILL LAST THREE YEARS AND BRING IN PROFITS OF $40,000 THE FIRST YEAR AND $50,000 IN EACH OF THE SECOND AND THIRD YEARS. AT THE END OF THREE YEARS IT HAS A SCRAP VALUE OF $5,000.

THE INTEREST RATE IS 16 PERCENT.

COMPUTATION OF NET PRESENT VALUE

TIMEFREE
CASH FLOW
DISCOUNT
FACTOR @ 16%
PRESENT VALUE
0 -$100,000 1.000 -$100,000
1 40,000 0.862 34,480
2 50,000 0.743 37,150
3 55,000 0.641 35,255

NET PRESENT VALUE = $6,885

THE NET PRESENT VALUE IS POSITIVE SO IT IS A WORTHWHILE INVESTMENT.

ALTERNATIVE METHODS OF EVALUATING THE PROJECT

COMPARISON OF INVESTMENT PROJECT WITH KEEPING FUNDS IN BANK EARNING INTEREST

TIMEBANK ACCOUNT PROJECT
0 $100,000 0
1 116,000 40,000
2 134,560 96,400
3 156,090 166,824

NET GAIN = 166,824 - 156,090 = 10,734

PV OF NET GAIN = 6,885

TIME FREE CASH FLOW NET INTEREST
@16%
VALUE
0     -100,000
1 40,000 $16,000 -76,000
2 50,000 12,160 -38,160
3 55,000 6,106 +10,734

CASH FLOW EQUALS AFTER-TAX PROFIT PLUS DEPRECIATION

FREE CASH FLOW EQUALS CASH FLOW MINUS INVESTMENT OUTLAY

EXAMPLE OF COMPUTATION OF CASH FLOW

REVENUE 60,000
COST 10,000
SURPLUS 50,000
-DEPRECIATION -30,000
TAXABLE PROFIT 20,000
TAX @ 40% 8,000
AFTER-TAX PROFIT 12,000
DEPRECIATION +30,000
CASH FLOW 42,000

Pricing of Assets

THE PRICE OF AN ASSET, SUCH AS A BOND, SHARE OF STOCK, OR RENTAL PROPERTY, SHOULD EQUAL THE PRESENT VALUE OF THE FUTURE INCOME TO BE RECEIVED FROM IT.

THIS RESULT MAY BE DERIVED AS FOLLOWS:

LET Pt BE THE PRICE AT TIME t AND LET Yt BE THE INCOME (INTEREST, DIVIDEND, OR RENT) TO BE RECEIVED AT TIME t.

CAPITAL GAIN AT TIME t = Pt - Pt-1

REWARD FOR HOLDING THE ASSET FROM TIME t-1 TO TIME t

INCOME + CAPITAL GAIN

Yt + Pt - Pt-1

RATE OF RETURN =

REWARD DIVIDED INVESTMENT AT t-1 = (Yt + Pt - Pt-1)/Pt-1

IF THIS RATE OF RETURN EQUALS THE INTEREST RATE r THEN

(Yt + Pt - Pt-1)/Pt-1 = r

(Yt + Pt)/Pt-1 -1 = r

(Yt + Pt)/Pt-1 = 1 + r

Pt-1 = (Yt + Pt)/(1 + r)

THIS HOLD FOR t=1, 2, 3, ....

SO P0 = (Y1 + P1)/(1 + r)

P1 = (Y2 + P2)/(1 + r)

SUBSTITUTING THE EXPRESSION FOR P1 INTO THE EQUATION FOR P0 GIVES

P0 = (Y1 + (Y2 + P2) /(1 + r))/(1 + r)

P0 = Y1/(1 + r) + Y2/(1 + r) + P2/(1 + r)2

CONTINUING THIS PROCESS YIELDS

P0 = Y1/(1 + r) + Y2/(1 + r)2

+ Y3/(1 + r)3 + ...

THIS PROPOSITION ALSO MEANS THAT IF PRICES ARE EQUAL TO THE FUTURE INCOME DISCOUNTED AT A RATE r, THEN THE RATE OF RETURN, TAKING INTO ACCOUNT NET CAPITAL GAINS AS WELL AS INCOME, WILL ALSO BE r.

Investment Criteria

THE CORRECT CRITERION FOR MAKING INVESTMENT DECISIONS IS THE NET PRESENT VALUE RULE: AN INVESTMENT IS WORTHWHILE ONLY IF THE PRESENT VALUE OF THE FREE CASH FLOWS IS POSITIVE.

THE INTERNAL RATE OF RETURN OF A PROJECT IS THE DISCOUNT RATE THAT MAKES THE NET PRESENT VALUE OF A PROJECT EQUAL TO ZERO.

THE INTERNAL RATE OF RETURN RULE IS THAT AN INVESTMENT PROJECT SHOULD BE APPROVED IF ITS INTERNAL RATE OF RETURN IS GREATER THAN THE COST OF CAPITAL.

GENERALLY THE I.R.R. RULE GIVES THE SAME DECISIONS FOR INVESTMENT PROJECTS AS DOES THE NET PRESENT VALUE RULE, BUT IN SOME CASES THE I.R.R. RULE MAY GIVE THE WRONG DECISION, OR BE MISLEADING. THE PROBLEMS WITH THE I.R.R. CRITERION ARE:

IN APPLYING THE NPV RULE:

BONDS AND DEBENTURES PAY A FIXED INTEREST BASED UPON THEIR FACE VALUE (THE PAYMENT TO BE MADE AT MATURITY) AND THEIR COUPON RATE. THE MARKET INTEREST RATE MAY BE DIFFERENT FROM THE COUPON RATE. THE PRICE OF A SECURITY SHOULD BE EQUAL TO THE PRESENT VALUE OF THE INTEREST AND FACE VALUE PAYMENTS DISCOUNTED AT THE MARKET RATE OF INTEREST.

Asset Valuation

BONDS AND DEBENTURES PAY A FIXED INTEREST BASED UPON THEIR FACE VALUE (THE PAYMENT TO BE MADE AT MATURITY) AND THEIR COUPON RATE. THE MARKET INTEREST RATE MAY BE DIFFERENT FROM THE COUPON RATE. THE PRICE OF A SECURITY SHOULD BE EQUAL TO THE PRESENT VALUE OF THE INTEREST AND FACE VALUE PAYMENTS DISCOUNTED AT THE MARKET RATE OF INTEREST.

EXAMPLE: CONSIDER A FIVE YEAR BOND ISSUED ISSUED ON JANUARY 1ST IN 2000 WITH A FACE VALUE OF $1000 AND A COUPON RATE OF 5 PERCENT. THE MARKET INTEREST RATE IS 8 PERCENT. IN 2000 THE SCHEDULE OF PAYMENTS IS:

TIME PAYMENTDISCOUNT FACTOR
@8%
PRESENT VALUE
1/1 2001 $50 0.926 $46.30
1/1 2002 $50 0.857 $42.85
1/1 2003 $50 0.794 $39.70
1/1 2004 $50 0.735 $36.75
1/1 2005 $1050 0.681 $715.05

PRESENT VALUE = $880.65

A MORE EFFICIENT METHOD OF COMPUTATION IS:

PRESENT VALUE =

(SUM OF DISCOUNT FACTORS)x$50 + 0.681x$1000

= 3.993x$50 + 0.681x$1000 = $880.65

ONE YEAR LATER ON 1/1 2001 AFTER THE FIRST YEAR'S INTEREST HAS BEEN PAID THE BOND HOLDER FACES THIS SCHEDULE OF PAYMENTS:

TIME PAYMENTDISCOUNT FACTOR
@8%
PRESENT VALUE
1/1 2002 $50 0.926 $46.30
1/1 2003 $50 0.857 $42.85
1/1 2004 $50 0.794 $39.70
1/1 2005 $1050 0.735 $771.75

PRESENT VALUE = $900.60

OR 3.312x50 + 0.735x1000 = $900.60

AFTER TWO YEARS, ON 1/1 2002 THE SCHEDULE OF PAYMENTS IS:

TIME PAYMENTDISCOUNT FACTOR
@8%
PRESENT VALUE
1/1 2003 $50 0.926 $46.30
1/1 2004 $50 0.857 $42.85
1/1 2005 $1050 0.794 $833.70

PRESENT VALUE = 2.577x50 + 0.794x1000 = $922.85

ON 1/1 2003 AFTER THE PAYMENT OF THE INTEREST THE VALUE OF THE BOND IS $946.15

ON 1/1 2004 AFTER THE PAYMENT OF THE INTEREST THE VALUE OF THE BOND IS $972.30 ON 1/1 2005 AFTER THE PAYMENT OF THE INTEREST THE VALUE OF THE BOND IS $1000

CONSIDER NOW THE RATE OF RETURN THAT THE BOND HOLDER MAKES. ON 1/1 2000 THE BONDBUYER PAYS $880.65 FOR THE BOND. ON 1/1 2001 THE BONDHOLDER GETS A $50 INTEREST PAYMENT AND HAS A BOND THAT IS WORTH $900.60. THE RETURN TO THE BONDHOLDER IS $50 IN INTEREST PLUS A CAPITAL GAIN OF ($900.60-$880.65) $19.95 FOR A TOTAL RETURN OF $69.95 ON AN INVESTMENT OF $880.65. THE RATE OF RETURN IS THUS 69.95/880.65 = 0.08 OR 8 PERCENT.

LIKEWISE SOMEONE PURCHASING THE BOND ON 1/1 2001 FOR $900.60 WILL ON 1/1 2002 HAVE $50 IN INTEREST PLUS A BOND THAT IS WORTH $922.85. THE TRETURN ON THE BOND AS OF 1/1 2002 IS $50 IN INTEREST AND A CAPITAL GAIN OF ($922.85-$900.60)=$22.25 FOR A TOTAL RETURN OF $72.25 ON A $900.60 INVESTMENT. THE RATE OF RETURN IS 72.25/900.60 = 0.08. AGAIN THE RATE OF RETURN IS 8 PERCENT.

THE FINANCIAL MARKET ESTABLISHES ASSET PRICES SO THAT THE HOLDER OF AN ASSET MAKES A RATE OF RETURN EQUAL TO THE MARKET INTEREST RATE.

THE ABOVE EXAMPLE WAS OF A RISK-FREE BOND. LET US CONSIDER HOW AN ASSET WOULD BE PRICED IF THERE IS A SUBSTANTIAL RISK OF DEFAULT. SUPPOSE THE PROBABILITY OF DEFAULT IS 1/3. THE HOLDER OF THE BOND ON 1/1 2004 FACES THE PROSPECT THAT ON 1/1 2005 THERE IS A 2/3 PROBABILITY OF RECEIVING ON 1/1 2005 A PAYMENT OF $50 FOR INTEREST AND $1000 PAYMENT FOR THE FACE VALUE OF THE BOND, BUT A 1/3 PROBABILITY OF GETTING NOTHING. THE PRESENT VALUE OF THE $1050 PAYMENT ON 1/1 2005 IS $972.22 AS OF 1/1 2004. THE PRESENT VALUE OF A ZERO PAYMENT IS OF COURSE ZERO. ON AVERAGE THEN THE RETURN ON HOLDING THE BOND IS (2/3)972.22 + (1/3)0 = $648.15. THUS THE RISKY BOND WOULD SELL FOR $648.15 ON 1/1 2004. TWO THIRDS OF THE TIME SOMEONE MAKING A PURCHASE OF SUCH A RISKY BOND WOULD MAKE A RATE OF RETURN OF ($1050-$648.15)/$648.15 = 0.62 OR 62 PERCENT. BUT THIS HIGH RATE OF RETURN IS COUNTERBALANCED BY THE FACT THAT ONE THIRD OF THE TIME SUCH A BOND WOULD GIVE A RATE OF RETURN OF -100 PERCENT. THUS THE AVERAGE RATE OF RETURN WOULD BE (2/3)(62%) + (1/3)(-100%) = 8%. IF ONE LOOKED ONLY AT THE CASES WHERE THE BOND PAID OFF IT WOULD LOOK LIKE A FABULOUSLY PROFITABLE INVESTMENT BUT CONSIDERING CASES OF THE LOSES AS WELL AS THE GAINS THE RATE OF RETURN IS ON AVERAGE SIMPLY EQUAL TO THE MARKET RATE OF RETURN.

THE ABOVE CONSIDERED THE VALUATION OF THE BOND ONLY AT ONE YEAR PRIOR TO MATURITY BUT THE ANALYSIS COULD BE EXTENDED BACKWARDS. THE RESULT WOULD BE THE SAME. THE AVERAGE RATE OF RETURN ON AN ASSET IS EQUAL TO THE MARKET RATE OF RETURN (ADJUSTED FOR THE DEGREE OF RISK).

COMPUTATION OF NET PRESENT VALUE

TIMEFREE CASH FLOWDISCOUNT FACTOR PRESENT VALUE
0      
1      
2      
3      

A Proposal for a Bold Investment Project

Charles Owen and the Institute of Design of the Illinois Institute of Technology in Chicago have developed a design for a very large airship to carry cargo and passengers. Although airships have been considered obsolete since the destruction of the Hindenberg in 1937, this may be based upon irrational fear rather than the technology or economics of airships.

Before the Hindenberg disaster airships had decades of success in carrying freight and passengers between many cities of Europe. In 1928 the airship Graf Zeppelin made the first cross-Atlantic trip carrying 12 tons of freight and 60 passengers and crew. In contrast, Charles Lindbergh's cross-Atlantic flight in an airplane in 1927 carried only Lindbergh. In 1929, the Graf Zeppelin flew around the world in 21 days, making only three stops. Altogether the Graf Zeppelin made 650 flights (144 of them across the Atlantic) for about 1 million miles and carried more than 18 thousand passengers. Over a period of 40 years Germany built 118 dirigibles which flew without an accident due to fire until the Hindenberg disaster even though Germany had to use flammable hydrogen instead of nonflammable helium as a lifting agent. The Hindenberg disaster itself was caused by sabotage with an incendiary bomb, a fact withheld from the general public. Nevertheless, due to the public's perception of airships as being dangerous, airships disappeared. But economically there may be a role for airships as carriers of freight. Charles Owen's design calls for an ellipsoidal airship 2400 meters long (almost 1.5 miles), 1000 meters wide (5/8 of a mile), and 640 meters high (2/5 of mile). It is made up of a three meter (ten foot) thick shell of composite, honeycombed "space age" material which has a high strength for its weight. The construction of the airship would require a large, dry canyon such as is found in the Grand Canyon area. Lift is provided by plastic cells or bags filled with helium. These cells are approximately 100 meters by 100 meters in cross section with heights varying from 250 meters to 540 meters depending upon location within the airship. The lifting cells are fastened by ropes and nets to the lower surface of the airship and the internal structure is hung with ropes from the upper surface. Filled with helium these cells will provide a lift of about three quarters of a million metric tons or 827 thousand regular tons. The shell of the airship and the ropes and nets will weigh 578 thousand tons leaving about 250 thousand tons for equipment, cargo and other elements. The design cargo capacity of the airship is 35 thousand tons.

The design calls for a maximum speed of 100 miles per hour. The propulsion is provided by 160 electric motors turning six-bladed propellers which can pivot to any angle. At lower speeds far fewer engines are needed. At 50 miles per hour only 40 engines are required.

There are two sources of electricity for the engines. Almost forty percent of the top of the airship will be covered with photovoltaic cells which will generate 156 megawatts of power. There will also be turbines driven by compressed air. The air is compressed through heating either by sunlight on the surface of the ship or by burning hydrogen. When the photovoltaic cells produce more power than is needed the excess is used to electrolize water into hydrogen and oxygen.

The airship keeps reservoirs of water as ballast as well as other purposes. During emergencies the ballast can be jettisoned for quick lift. The runoff of rain from the top of the airship is channeled into tanks and when the water supply runs low the airship can just run through a rain storm. The airship will remain permanently aloft. Loading and unloading are achieved by means of six doughnut shaped shuttle pods, each capable of transferring 825 tons of cargo. The outer diameter is 240 meteres and the diameter of the hole is 30 meters. These shuttles are as large or larger than the airships of the 1930's. In addition to the shuttles, there will be aircraft which can land and takeoff on the 1600 meter runway on the top of the airship.

One special feature of the Owen design is that the underside of the airship functions as a communication screen. The dots or "pixels" of this screen will be over two feet by two feet. It would take 16,840 kilowatts of power to light this screen.

The living, working and cargo handling facilities are located in an elliptical, keel-like structure on the bottom of the airship. This structure is 1600 meters long, 650 meters wide, and 120 meters high. The airship would have 13 times the space of the ocean liner Queen Mary which carried 3400 passengers and crew. The design calls for 2000 passengers and a crew of 1500. The airship can carry four times as much cargo as the largest container ship.

Summary of Charles Owen's Airship Design
CHARACTERISTICMAGNITUDE
Length1.5 miles
Width5/8 mile
Height2/5 mile
Gross Lifting Power827 thousand tons
Cargo Capacity35 thousand tons
Maximum Speed100 miles per hour
Source of Electricity 156 Megawatts of Photovoltaic Cells
Compressed Air Turbines
Source of WaterRain Runoff
Transfer of CargoShuttle Pods Carrying 825 tons
Crew1500
Passengers2000
Cabin1600 meters long
650 meters wide
120 meters high
Communicaation Screen

The question of whether Owen's airship should be built depends not just on technical feasibility, but also on economics. The gargantuan size of the airship is essentially a matter of economics. If the scale of a structure, such as an oil tanker ship, is doubled the volume and hence its productive capacity goes up by a factor of eight. But the construction costs for the structure are related to the surface area of the structure rather than its volume so these increase by only a factor of four. Thus the bigger the structure the more economical it is. This is why oil tanker size was doubled and redoubled in the 1950's.

The Institute of Design has not provided estimates of the costs and revenues that might result from such an airship, but we can make some guestimates and use them as a practice problem in investment analysis.

PERPETUITIES

Surprisingly, it is possible to compute the present value of $1 per year from year 1 to infinity. Such a cash flow stream is called a perpetuity. One can create such a perpetuity simply by putting money in the bank. If t he interest rate is 10 per cent the way to generate $1 per year forever is simply to put $10 in the bank. Therefore the present value of $1 per year forever is simply $10. If the interest rate were 5 per cent you would have to put $20 in the bank to get $1 per year forever. Therefore the present value of $1 per year forever when the interest is 5 percent is $20.

Exercise 19: What is the present value of $1 per year forever when the interest rate is 8 percent?

Exercise 20: What is the present value of $150 per year forever when the interest rate is 15 percent?

Exercise 21: What is the present value of $52,000 per year starting in year 10 and going forever when the interest rate is 20 percent?

THE DEFINITIVE ANSWER TO THE QUESTION OF HOW INVESTMENT DECISIONS SHOULD BE MADE: THE NET PRESENT VALUE RULE

When practitioners and theorists in the 1930's formulated the net present value rule as the proper way to decide whether an investment project was worthwhile the result perhaps seemed obvious. However it took a long time to reach a resolution of this issue and there were several plausible alternatives. We will go into those alternatives later, along with the question of how the market economy could have functioned for so many centuries without people knowing exactly how an investment decision should be made.

The Net Present Value (NPV) Rule says that an investment is worthwhile if its NPV is positive and not worthwhile if its value is negative. If the NPV is zero then it does not matter if the project is accepted or rejected. So the rule is utterly simple to apply, the only problem is the computation of the NPV.

Example of the Computation of NPV:

Suppose a company has a tiremaking machine that is still functioning but requires $2000 per year in repairs. It will last five more years. A new machine that will also last five years costs $8000. The old machine has a scrap value of $3000. Both machines will do the same job. It is important to realize that the proper decision generally depends upon the level of the interest rate.

Let us first look at how cash flows will be affected by the decision.

Keep Buy
Time Old Machine New Machine Net Change
0 0 $5000=8000-3000 -$5000
1 $2000 0 $2000
2 $2000 0 $2000
3 $2000 0 $2000
4 $2000 0 $2000
5 $2000 0 $2000

The question of whether the saving of $2000 a year in years 1 to 5 outweighs the $5000 outlay at time zero depends upon the interest rate. How much is a dollar one year from now worth compaired to a dollar now? If the interest rate is 10 percent then a dollar put in the bank now at 10% will grow to be $1.10 next year. Therefore since $1.10 one year from now equals $1.00 now, $1.00 one year from now equals 1/1.1 = .9091 dollars now. That is to say, the present value of one dollar one year from now when the interest rate is ten percent is 0.9091. Likewise a dollar in the bank now will grow to be $1.21 two years from now so the present value of a dollar two years from now is 0.8264.

Time Net ChangeDiscount Factor
@10%
Present Value
0 -5000 1.0000 -5000
1 2000 .9091 1818
2 2000 .8264 1653
3 2000 .7513 1503
4 2000 .6830 1366
5 2000 .6209 1242

NPV = 2582

If the interest rate were 30 percent instead of 10 percent the figures would look like this:

Time Net ChangeDiscount Factor
@10%
Present Value
0 -5000 1.0000 -5000
1 2000 .7692 1538
2 2000 .5917 1183
3 2000 .4552 910
4 2000 .3501 700
5 2000 .2693 539

NPV = -130

Thus if the interest rate is 30 percent the project is not worthwhile. In other words, it would be better not to buy the replacement machine and put the $5000 it would cost in the bank and earn 30 percent interest.

We might like to know at what interest rate the project switched from being a good investment to being a bad investment. If we tried a number of different interest rates we might by chance find one at which the NPV is zero.

Consider a project for producing electric cars. It is expected that the sales will be 100,000 per year. The cost for the construction of the factory is $150 million. The factory is expected to last for ten years. In addition to the cost of the depreciation and interest on the factory there are other fixed costs of $30 million per year, such as for administrative personnel. The unit variable costs are $3000 and the cars are expected to sell for $3750. There is a 50 percent profit tax and the interest rate is 10 percent.

Computation of the cash flow for years 1 through 10:

Net Present Value (NPV) = - 150 + 30x(A.F.10yrs@10%) = - 150 + 30x6.145 = 34.35

The PV of the fixed costs is 150 + 30x6.145 = 334.35

A Formula for the Present Value of a Growing Annuity

The annuity factors are quite useful but they are limited to the case in which a payment is the same over a number of years. A more general situation is that in which the payment grows at a constant rate for a number of years. Suppose the payment is y1 at the end of year 1 and grows at an annual rate of g until the end of the n-th year.

At an interest rate of r the present value of this stream of payments is:

PV = y1 (1 - [(1+g)/(1+r)]n) /(r - g)

When g is zero this reduces to the formula for a simple annuity.

Illustration: Suppose a project earns $10,000 the first year and the earnings grow five percent a year for ten years. After ten years the payments end. If the interest rate is eight percent, what is the present value of the payments.

Since g=0.05 and r=0.08, (1+g)/(1+r)=1.05/1.08=0.97222. This ratio raised to the tenth power is 0.7545, so

PV = 10,000 (1-0.7545)/(0.08-0.05)= 10000 (0.2455/0.03)

PV = 10000x8.1836 = $81,836

The formula also works if the growth rate is greater than the interest rate but some of the intermediate computations may look peculiar.

Suppose y1=1, n=12, g=0.1 and r=0.6.

Then (1+g)/(1+r)=1.1/1.06=1.0377 and so [(1+g)/(1+r)]12 =1.5597.

Thus

PV = 1 (1-1.5597)/(0.06-0.1)=(-.5597)/(-0.04)=13.9926.

Exercise 22:

A factory will generate a growing net income over the 16 years of its life. In the first year it will earn $50,000 and this will grow seven percent a year. The factory costs $800,000 and the interest rate is nine percent. Is the factory a worthwhile investment?

Exercise 23: Estimate your salary in the first year after you graduate and the number of years you will work until retirement. Also estimate the average growth rate of your salary over your working life. Estimate also the interest rate over the same period. Now determine the present value of your expected salary over your working life.

Exercise 24: Suppose a payment starts at $5 and goes on forever growing at 4 percent a year. If the interest rate is 10 percent what is the present value.

Rates of Return on Securities, 1926-1988

Security Average Annual
Rate of Return
Average Annual Real
Rate of Return
Average Annual
Risk Premium
Standard Deviation
of Returns
Treasury Bills 3.6% 0.5% 0.0 3.3
Government Bonds 4.7 1.7 1.1 8.5
Corporate Bonds 5.3 2.4 1.7 8.4
Common Stocks 12.1 8.8 8.4 20.9

The Distribution of the Rates of Return of Securities

The accompanying table gives the real rates of return of four types of securities.

Exercise 25: Tabulate the frequencies of the real rates of return for each type of security using the following categories:

For this exercise use the table of rates of return given at this site: http://www.sjsu.edu/faculty/watkins/returns0.htm

Rate of Return Frequency
 Large Company
Common Stock
Small Company
Common Stock
Long Term
Corporate Bonds
Long Term
Government Bonds
Intermediate
Government Bonds
Treasury Bills
below -40% _____ _____ _____ _____ _____ _____ _____
-40 to -30.1 _____ _____ _____ _____ _____ _____ _____
-30 to -20.1 _____ _____ _____ _____ _____ _____ _____
-20 to -10.1 _____ _____ _____ _____ _____ _____ _____
-10 to -0.1 _____ _____ _____ _____ _____ _____ _____
0 to 9.9 _____ _____ _____ _____ _____ _____ _____
10 to 19.9 _____ _____ _____ _____ _____ _____ _____
20 to 29.9 _____ _____ _____ _____ _____ _____ _____
30 to 39.9 _____ _____ _____ _____ _____ _____ _____
40 and above _____ _____ _____ _____ _____ _____ _____
Total _____ _____ _____ _____ _____

Exercise 26: Use the above frequencies to compute the proportions of the time the rates of return fall into the various categories.

Exercise 27: Plot graphs of the frequencies in Exercise 25.

Exercise 28: Calculate the average real rate of return for the four types of securities.

Exercise 29: Find the maximum and minimum for each type of security. Compute the range (max-min) for each type.

The Present Value of Growth Oppportunities

An Exercise on the Present Value of Growth Opportunities of a Corporation Price = Capitalized Value of Current Earnings Plus the Present Value of Growth Opportunities Capitalized Value of Current Earnings is the present value of a perpetuity equal to the current earnings; i.e., (Current Earnings)/(Capitalization Rate) The capitalization rate is the discount rate that is appropriate for a security of a particular risk.

Exercise 30: From the stock market page of a newspaper get information on price and the P/E ratio for six stocks (two you consider high risk, two you consider low risk, and any other two). Estimate the current earnings by dividing the price by the P/E ratio. (This is just an approximation because the P/E is rounded off and perhaps out-of-date.)

We do not have the method yet for getting an appropriate capitalization rate for different risk stocks so just use 12% for all of them.

Compute the PVGO for each stock and what proportion of price is accounted for by PVGO.

Results of Investigations of the Ability of Stock Market Professionals to Predict the Earnings of Companies

Study I. Brown and Rozeff (1978)

Forecaster: Major investment advisory service Set of Firms: Fifty companies which were customers of the servic Time frame: 1971-1975

ErrorProportion of Forecasts
With a Smaller Error
  Forecast Error
as % of actual
Earnings
  Analysts Extrapolation
of past growth
5% 18.0 15.0
10 32.0 26.5
25 63.5 54.5
50 86.5 81.0
75 90.5 87.5
100 92.0 89.5

Study 2: (Michael B. O'Higgins)

Forecaster: Institutional Brokerage Estimate System Set of Firms: Thirty companies in the Dow-Jones Industrials Time frame: 1974-1983

Year Year
Average Error in
Earnings Forecast
Year Current Coming
1974 9.5% 36.6%
1975 16.5 54.1
1976 24.9 46.7
1977 9.7 25.3
1978 12.4 24.2
1979 10.1 74.8
1980 16.3 42.5
1981 9.6 38.4
1982 14.8 153.3
1983 56.2 42.1
average 18.0% 53.8%

Study 3 (Niederhaffer and Regan)

Forecaster: Consensus forecasts reported by Standard and Poor:

Set of Firms: 150 Stocks (50 stocks with the highest price gains, 50 stocks with greatest price loss, 50 stocks selected at random)

Time frame: 1970

Group Median Price
Change
Median Actual
Earnings Change
Median Forecasted
Earnings Change
Top 50 +48.4% +21.4% +7.7%
Bottom 50 -56.7 -83.0 +15.3
Random 50 -3.2 -10.5 +5.8

Corporate Finance

A corporation can raise new capital by selling debt or selling equity. If it chooses to sell equity (common stock) it has a choice of a public issue or a private placement. A public issue of common stock requires compliance with a complex procedure specified by the Securities and Exchange Commission (SEC).

Private placement involves offering the stock to a small number of buyers, a dozen or less, such as institutional investors. The stock sold through private placement is called "letter stock" and there are limitations on the resale of it by its buyers.

A public issue starts with the filing of a registration statement with the SEC. A preliminary prospectus giving the history, current state, and future prospects of the company is prepared for review by the SEC. This preliminary prospectus is printed in red ink and is an advertisement of the issue but not an actual offer of the stock for sale. Because it is in red and not an actual offer it is known as a "red herring." After any changes required by the SEC are incorporated, a final prospectus including the offer price is distributed. The corporation then appoints a registrar and transfer agent to handle the legalities of the sale of the new stock.

The public issue can take to forms, a cash offer and a rights issue. A rights issue is when the new stock is offered only to the current stockholders in the company. The current stockholders receive the right to buy shares of the new stock in proportion to their current holdings. These rights usually can be sold. Often the price of the new stock is significantly below the market price of the old stock and therefore the rights are valuable.

In a cash offer the corporation usually selects underwriters to market the new stock. The offering and list of underwriters are advertised in a special type of display called a tombstone advertisement. The underwriters are listed in order of importance and great prestige is attached to the positions of the underwriters' names in the tombstone advertisement.

The underwriters are compensated for their services by being able to buy the stock at a level below the offer price. This spread overall averages about six percent of the capital raised, but the there are major differences for different sizes of offerings. For issues of about one million dollars the spread is about 15 percent whereas for issues of about $100 million the spread is about 4 percent.

Preferred stock does not involve the voting rights of common stock but does not have the same assurance of a payment that debt securities have. On the other hand the dividend payment on preferred stock is not tax deductible. T he most important issuers of preferred stock are regulated public utilities, who can pass the cost disadvantages of preferred stock on to their rate payers. Because dividends paid by one corporation to another are 70 percent tax free most preferred stock is owned by corporations.

LIFETIME EARNINGS EXPECTATIONS

  1. In how many semesters do you expect to graduate? __________
  2. Will you be getting a bachelor's degree or a master's degree? ________
  3. What starting salary (in $ 000's) do you expect when you go to work after graduation? _____________
  4. What salary would you expect if you dropped out of college and went to work? __________
  5. How many years do you expect to work after graduation before retirement? ______
  6. What rate of increase in salary do you expect over your working life if you complete your degree? _________
  7. What rate of increase in salary do you expect over your working life if you do not complete your degree? _________
  8. What rate of inflation do you expect over your working life? _____
  9. Are you working now? _________
  10. If you are working now, are you working part-time or full-time? _______
  11. What proportion of your salary do you expect to receive after taxes? ____________
  12. What are your costs per year for attending college; i.e., costs such as for fees and books that you would not have if you did not attend college? _________

Modigliani and Miller's Propositions Concerning the Equity Value of a Corporation

There are four schools of thought
as to what determines
the value of a corporation:

Modigliani and Miller (M&M) established that, properly interpreted, each of these approaches will give the same total equity value for the corporation. "Properly interpreted" means that dividends must be taken as "net dividends;" i.e., dividends paid out minus funds raised from the sale of new stock. Earnings must be taken as net earnings, earnings minus an imputed interest on cumulative investment. Cash flow must take into account the cash outflow of investment (free cash flow). It is important to note that their result is for the total equity in the corporation, as opposed to the value of a single share.

The cash flow of a corporation, which is defined as

Aftertax profits plus Depreciation,

can go for dividends or for investment. Investment could also be covered by the sale of new stock. This means that

Cash Flow + Sale of New Stock = Dividends + Investment.

Therefore

Dividends - Sale of New Stock = Cash Flow - Investment.

Cash Flow minus Investment is called Free Cash Flow, and Dividends - Sale of New Stock is called Net Dividends.

Thus Free Cash Flow is the same as Net Dividends and therefore the value of the equity in a corporation is equal to the present value of future free cash flows and also the present value of net dividends.

M&M established that if one computes the cumulative investment of a corporation and deducts an interest from earnings based on this cumulative investment the result, which they label Net Earnings, is equal to both Net Dividends and Free Cash Flow. Therefore the equity value of a corporation is equal to the present value of all future Net Earnings.

M&M also established that if the present value of growth opportunities is calculated as the present value of of the earnings of investment projects which are in excess of the rate of discount r then the capitalized value of the current earnings plus the present value of the growth opportunities is equal to the other three methods of determining the equity value of a corporation. In the form of an equation, this says that the equity value of the corporation, P, is equal to

P = E/r + PVGO.

So M&M's analysis revealed that there is no conflict between the four schools of thought on the valuation of the equity in a corporation. The relationship also applies for a single share of the corporation.

In addition to the four approaches that M&M brought together, there is a fifth more fundamental approach. The value of a corporations is the sum of the net present values of all its worthwhile projects. The present value of the free cash flows is the totalling up of the cash flows for the separate projects and then computing the present values. Since net dividends are identically equal to free cash flows, it follows that the present value of the net dividends must be equal to the value of the corporation.

Thus, the total value of the equity in a corporation should be

A Mathematical Derivation of M&M's Results

M&M analysis starts from the proposition that the stock price at any time t, pt, is equal to the present value of the next dividend payment and the price of the stock at the time of that payment; i.e.,

(1) pt = (dt+1 + pt+1)/(1+r)

This is equivalent to the condition that the price of the stock is equal to the present value of all future dividends.

In this analysis a lower case letter will denote the quantity per share and the upper case letter the quantity for the whole corporation. Let Nt be the number of shares outstanding at time t. The value of the equity in the corporation at time, Pt, is just Ntpt. Thus if (1) is multiplied by Nt one obtains

(2) Ntpt = Pt = (Ntdt+1 + Ntpt+1)/(1+r) = (Dt+1 + Nt pt+1)/(1+r).

The quantity Ntpt+1 can be expressed as

Ntpt+1 = Ntpt+1 -Nt+1pt+1 +Nt+1pt+1 = -pt+1(Nt+1-Nt)+Pt+1.

The term pt+1(Nt+1-Nt) is the number of new shares sold between t and t+1 valued at the price of the stock at t+1. This may be taken to be the value of new stock sold in year t+1, St +1 . Thus equation (2) reduces to

(3) Pt = ((Dt+1 - St+1) + Pt+1)/(1+r)

This means that the value of the corporation is equal to the present value of all future Net Dividends, dividends paid out less the funds brought in by the sale of new stock. The cash flow of a corporation, which is defined as

Aftertax profits plus Depreciation,

can go for dividends or for investment. Investment could also be covered by the sale of new stock. The relationship is

Cash Flow + Sale of New Stock = Dividends + Investment.

Therefore

Dividends - Sale of New Stock = Cash Flow - Investment.

Cash Flow minus Investment is called Free Cash Flow.

Thus Free Cash Flow is the same as Net Dividends and therefore the value of the equity in a corporation is equal to the present value of future free cash flows. M&M established that if one computes the cumulative investment of a corporation and deducts an interest from earnings based on this cumulative investment the result, which they label Net Earnings, is equal to Net Dividends and Free Cash Flow. Therefore the equity value of a corporation is equal to the present value of all Net Earnings. If et is earnings per share then Et=Ntet is total earnings. If Kt is the sum of all past investment as of time t then the imputed interest on investment is rKt. M&M defined Net Earnings as Et-rKt and found its present value is the same as the equity value of the corporation. M&M also established that if the present value of growth opportunities is calculated as the present value of of the earnings of investment projects which is in excess of the rate of discount r the capitalized value of the current earnings plus the present value of the growth opportunities is equal to the other three methods of determining the equity value of a corporation. In the form of an equation, this says that the equity value of the corporation, Pt, is equal to

Et/r + PVGO.

So M&M's analysis revealed that there is no conflict between the four schools of thought on the valuation of the equity in a corporation.

RISKY PROJECTS

For Risky Project we would like to know the probability distribution. We summarize the information in the probability distribution with the mean and standard deviation. The skewness and kurtosis might also be of interest. Later the expected payoffs of a slot machine, parlay cards, craps, and roulette will be computed. Definition of variance. Notation mu and E(), sigma and Var. Computation of variance for some simple cases. Rate of return of a portfolio. Expected rate of return of a portfolio. Derivation of the formula. The concept of covariance. The concept of correlation. Covariance as CinderElla. The variance of the rate of return of a portfolio of n stocks.

Portfolio Analysis

Exercise 31.

Compute the risk and return for various portfolios of two stocks using the formulas:

rport = (xA)rA + (xB)rB

Var(rport) = (xA)2Var(rA) + (xB)2Var(rB)
+ 2(xA)(xB )Cov(rA,rB)

You will also need these formulas:

Cov(rA,rB) = Correlation(rA,rB)xStDev(rA)xStDev (rB)

StDev(r) = Square Root (Var(r))

Example:

Let rA=E(rA)=20%,
rB =E(rB)=10%,
σA=stdev(rA)=15%,
σB =stdev(rB)=5% and
σAB=Cov(rA,rB) =10.

This means σA2= Var(rA)=225, σB 2=Var(rB)=25,
and the correlation of rA and rB is 0.2.

xA xB rport Var(rport ) σport
_____ ____ ____ _______ ______
1.00 0.00 20% 225.0 15.0
.90 0.10 19 184.3 13.6
.80 0.20 18 _____ ____
.70 ____ ___ _____ ____
.60 ____ ___ _____ ____
.50 ____ ___ _____ ____
.40 ____ ___ _____ ____
.30 ____ ___ _____ ____
.20 ____ ___ _____ ____
.10 ____ ___ _____ ____
.00 ____ ___ _____ ____

Exercise 32. Plot rport versus StDev(rport)

A dollar slot machine has reels with the following symbols:

Symbol Reel 1 Reel 2 Reel 3
Bar 1 3 1
Bell 1 3 3
Plum 5 1 5
Orange 3 6 7
Lemon 3 0 4
Cherries 7 7 0

The probability distribution of the payoff is

Event
PayoffProbability
bar bar bar $85 3/8000
bell bell bell $18 9/8000
bell bell bar $18 3/8000
plum plum plum $14 25/8000
plum plum bar $14 5/8000
orange orange orange $10 126/8000
orange orange bar $10 18/8000
cherries cherries lemon $ 5 196/8000
cherries cherries bell $ 5 147/8000
cherries cherries anything else $ 3 637/8000

Portfolio Analysis

Consider a portfolio made up of stock from Company A and Company B. The rate of return for the portfolio will depend upon the rates of return for the two companies and the shares of the portfolio in each of the stocks. Likewise the risk (i.e., variability of the rate of return of the portfolio) depends upon the risks of the two stocks and the shares, but there is another factor related to the correlation between the rates of return of the two stocks. In order to establish these relationships it is necessary to use some mathematics.

If z is any random variable let E{z} be the expected value of z. The rates of return on the stock of the two companies are random variables. The rate of return on a portfolio is also a random variable. Let rA, rB, and rP be the rates of return on the two stocks and the portfolo. The proportion of the portfolio in A and B are denoted as xA and xB. The share in A, xA, is the value of the A stock as a proportion of the total value of the portfolio.

Then

rP = xArA + xBrB,

and E{rP} = xAE{rA} + xBE{rB}.

These formulas mean that the rate of return of the portfolio is the weighted average of the rates of return of the securities in the portfolio.

The variance of any random variable is defined as the expected value of the squared deviation of that random variable from its expected value; i.e.,

Var(z) = E{ (z - E{z})2 }.

We use the square root of the variance, called the standard deviation, as the appropriate measure of risk in Portfolio Analysis.

To find the variance of the rate of return on a portfolio we have to find rP - E{rP}, square it and take the expected value of the squared deviations. From the equations for rP and E{rP} we have

rP - E{rP} = xArA + xBrB - xAE{rA} - xBE{rB} = xA(rA - E{rA}) + xB(rB - E{rB}).

When this expression is squared we get:

( rP - E{rP})= (xA(rA - E{rA}) 2 + (x>>BB(rB - E{rB}) 2 +2(xA(rA - E{rA}) (xB(rB - E{rB}) . xA 2(rA - E{rA})+ xB2(rB - E{rB}) 2 +2xAxB(rA - E{rA})(rB - E{rB})

When we take the expected values we get

Var(rP) = xA 2Var(rA) + xB 2Var(rB) + 2xAxBE{(rA - E{rA})(rB - E{rB})}.

The last term involves the expression

E{(rA - E{rA})(rB - E{rB})},

which is important enough to be given a special name, covariance.

It represent the extent to which rA and rB vary together.

The formula for the variance of the rate of return for the portfolio is then

Var(rP) = xA2Var(rA) + xB2Var(rB) + 2xA xBCov(rA,rB).

Another concept that is important and closely related to covariance is correlation. The correlation coefficient is the ratio of covariance to the product of the standard deviations of the two variable. It is a number that must be between -1 and +1. A correlation of +1 indicates that the variable move exactly together; up together, down together. A correlation of zero indicates there is no relation whatsoever between the changes in one variable and changes in the other variable. A correlation coefficient of -1 indicates that the variables are closely related but increases in one correspond to decreases in the other.

Table 1 gives the computations of the expected rate of return and risk (standard deviation) when Company A has an expected return of 20% and Company B 10%. The standard deviations of the returns are 15% and 5%, respectively. Company A is of realtively higher risk than B and also has a relatively higher rate of return. In Table 1 the correlation between the rates of return is taken to be -0.8 so the covariance is -60.

Exercise 33: Plot a graph of the return and risk for the various portfolios in which the expected rate of return is on the vertical axis and the standard deviation of the return is on the horizontal axis.

Exercise 34: Compute the return and risk for a portfolio which is 75% in Company A's stock and 25% in Company B's stock.

Exercise 35: Plot a graph using Table 2, which shows the results when the correlation between the two returns is zero.

More on Portfolio Analysis

Below are given the combinations of risk and return that can be achieved with portfolios composed of stocks in two companies.

The data are presented in the format of setting the proportion of Company A's stock and then varying the proportions for B and C. Plot up the data on a graph. To help you visualize the pattern connect the dots for all those portfolios containing a fixed proportion of A's stock. Label these lines by the proportion of A's stock they contain. This helps you identify a particular portfolio. Afterwards connect all of the points containing a fixed proportion of B's stock. You may find it helpful to use dotted lines or a colored pencil in this case. Do the same for the proportions of C's stock. Sketch the envelope of the combinations of risk and return that can be achieved by all possible portfolios.

Capital Assets Pricing Model

(based upon Financial Statement Analysis by George Foster)

Assumptions of Portfolio Analysis:

Economic Determinants of β and Variance

Beta Theory

The primary use of β is for computing the expected rate of return for an investment. This is done using the market line,

expected rate of return = risk-free rate + (risk premium for the market portfolio) times β.

The risk premium for the market portfolio is the difference between the expected rate of return for the market portfolio and the risk-free interest rate. The expected rate of return may also be looked upon as the required rate of return for a project; i.e. its cost of capital.

The procedure for deciding whether a risky project is worthwhile is compute its expected Net Present Value based upon the expected value of its cash flows and using a discount rate that includes a risk premium for the riskiness of the project. According to the Capital Asset Pricing Model (CAPM) the risk premium depends upon the β.

The β for a company that is already in business is measured by correlating its rate of return with the rate of return on the market portfolio; i.e.,

β = Covariance(r, rm)/Variance(rm).

For a proposed project the β would have to be estimated. This is difficult to do, but there are some insights that may help. Some of the factors which influence β are:

These factors will be covered in reverse order. A company, division, or project may produce not just one product but a variety of products. The company or division or project may be thought of as a portfolio of products. Just as the β for a financial portfolio is a weighted average of the βs of the securities in the portfolio, the β for a company etc. is the weighted average of the industry βs for the products produced. Below are some estimates of industry βs.

Exercise 36: Propose a company producing four different types of products and specify the proportions of company resources devoted to each type and calculate the asset β for your proposed company.

Financial Leverage and β

Just as a company may be thought of as a portfolio of different industry investments it can be thought of as a portfolio of equity and debt capital. The β for the total is a weighted average of the βs of the different components of the portfolio.

Let E be equity capital, D debt capital and V(=E+D) be total assets. Let the βs for equity and debt be βeq and βd and the β for total assets be

βassets. Then

βassets = (E/V)βeq + (D/V)βd.

If the corporation debt is risk-free then βd=0 and

βassets = (E/V)βeq.

Solving this equation for βeq in terms of βassets gives:

βeq = (V/E)βassets = [(E+D)/E]βassets

= (1 + D/E)βassets = (1+L)βassets,

where L is the debt/equity ratio D/E.

Operating Leverage and β

Cash Flow = Revenue - Fixed Cost - Variable Cost

Value of Asset = PV(Cash Flow)

PV(asset) = PV(revenue) - PV(fixed cost) - PV(variable cost)

PV(revenue) = PV(fixed cost) + PV(variable cost) + PV(asset)

βrevenuefixed cost(PV(fixed cost)/ (PV(revenue)) +βvariable cost (PV(variable cost)/PV(revenue)) +βasset (PV(asset)/PV(revenue))

βfixed cost = 0

Both revenue and variable costs are determined by sales so

βrevenuevariable cost

Derivation of the influence of operating leverage on β:

cash flow = revenue - fixed cost - variable cost

PV(cash flow) = PV(revenue) - PV(fixed cost) - PV(variable cost)

The present value of the cash flow is simply the NPV of the operation. The above equation can be rearranged to give

PV(revenue) = PV(fixed cost) + PV(variable cost) + NPV.

This suggests that we think of revenue being divided into three parts, each going to a different group and having a different degree of security. Those who receive the fixed costs are like debt-holders. They receive a fixed payment and the receivers of cash flow get what ever is left over after the payment of fixed costs and variable costs.

This means that

βrevenue = βfixed cost [PV(fixed cost)/PV(revenue) + βvariable cost[PV(variable cost)/PV(revenue)] + βasset[NPV/PV(revenue)]

If fixed cost is not subject to variation then
βfixed cost = 0.

Also revenue and variable cost vary because of variation in sales volume. Therefore we expect
βrevenue = βvariable cost.

Therefore

βrevenue [ 1 - PV(variable cost)/PV(revenue)] = βassetNPV/PV(revenue).

Multiplying both sides by PV(revenue) gives

βrevenue [ PV(revenue)-PV(variable cost)] = βassetNPV

But PV(revenue) - PV(variable cost) = PV(fixed cost) + NPV,

so βrevenue[ NPV + PV(fixed cost)] = βassetNPV.

Dividing through by NPV gives:

βasset = βrevenue [ 1 + (PV(fixed cost)/NPV ].

Exercise 37.

Suppose a project has a NPV of $1 million and the initial investment (which is the same as the present value of the fixed costs) is $4 million. If the revenue β is 0.3 what is the asset β for the project?

Exercise 38. Suppose there is another method of production for the project in #37 having PV(fixed costs)= $2.5 million and the same NPV. What would the asset β be for this case?

Exercise 39

Suppose the asset β of a company is 1.2 and its leverage ratio is 2.0. What is the equity β of the company? If it increases its leverage ratio to 3.0 what is its equity β?

Exercise 40.

Suppose the equity β of a company has been determined by the statistical relationship between its rate of return and that of the market portfolio and that its equity β is 2.4. If its leverage ratio is 3.0 what is its asset β? If the company were to increase its leverage ratio to 4.0 what would its asset β be? What would its equity β be?

Approaches to Making Decisions
About A Risky Investment Project

Tests of the Capital Asset Pricing Model

The standard version of the model says that the expected rate of return of a company is a linear function of its β and nothing else; i.e.,

E{rA} = rf + (rM - rf)bA

The actual rate of return is assumed be the expected rate plus some random variable. Thus,

rA = rf + (rM - rf)b A + uA.

The CAPM theory is tested by doing a statistical analysis (called regression analysis) that finds the linear function that best explains the variation in actual rate of return in terms of variation in β and some other variable or variables. The presumption is that if the CAPM is correct then the other variables will no value in explaining variation in r once the influence of β has been taken into account. Empirical tests (Banz 1981) found that other variables do have an explanatory value. The size of the company, as measured by the market value of equity ME, influences the rate of return. Smaller companies have a higher value than could be explained on β alone and larger companies a smaller rate of return. According to the CAPM leverage affects the equity β (the β which is measured by the performance of common stock is the equity β). Therefore, according to the CAPM, leverage affects risk and risk affects the expected rate of return. Presumably, if you know the β of a company knowing the leverage does not add any relevant information. But Bhandari (1988) found that the leverage helps explain the rate of return even in statistical analyses where the size (ME) is known in addition to the β. Several studies have found that the P/E ratio is valuable, in addition to β, in explaining variations in the rate of return of stocks. Generally low P/E ratio stocks have a higher rate of return than high P/E ratio stocks with the same βs. Studies have also found that the standard deviation of the rates of return of stocks is useful in explaining variation in rates of return even when β is included.

Effect of "Operating Leverage" on Risk

The formula for the asset β is: βasset = βrevenue[ 1 + (PV(fixed cost)/NPV ].

For this example βasset = βrevenue[ 1 + 334.35/34.35 ] βasset = βrevenue[ 1 + 334.35/34.35 ] βasset = βrevenue[10.734]

The revenue β would be derived from correlating the proportional changes in the revenues for the project with the rate of return on the market. Suppose the project were modified so that some of the labor force is converted from hourly wages to salaries so their costs become part of fixed costs instead of variable cost. Let us say that $10 million per year are involved in this switch. The profit and NPV is not affected. But the PV of the fixed costs is now $61.45 million larger so the operating leverage is (334.35+61.45)/34.35 and hence

βasset = βrevenue[11.523].

The project is riskier as a result of the change.

Theorem: Changes in conditional expectations for one time period are uncorrelated with changes in the conditional expectations for a previous time period.

Definitions:

If the joint probabilities are know then the conditional probabilities and unconditional probabilities may be calculated from the formulas

P(x) = yP(y,x)

P(y:x) = P(y,x)/P(x)

Conditional expected value of a variable y E{y:x} = the expected value of y given that x has already occurred; i.e.,

E{y:x} = yP(y:x)y.

ILLUSTRATION OF THE THEOREM

Suppose the probabilities for some variable w depends upon three events. Let us suppose the price of gold on January 1, 2002 depends upon the rate of inflation in 2001, which can either be high (10%) or low (3%). The rate of inflation may be affected by the fiscal and monetary policy which can be either expansionary or tight. Economic policy may be affected by who wins the next election. Suppose the next election is a contest between Hillary Clinton, Jack Kemp, and Ross Perot.

Election Outcome:

 >
Clinton Wins Kemp Wins Perot Wins No Decision
Probability 0.4 0.4 0.1 0.1

Econ. Policy: Expansive or Tight

 >
Clinton Wins Kemp Wins Perot Wins No Decision
 Exp Tight Exp Tight Exp Tight Exp Tight
Conditional
Probability
0.8 0.2 0.5 0.5 0.3 0.7 0.5 0.5

Econ. PolicyExpansionary Tight
Inflation Rate High Low High Low
Conditional
Probability
0.6 0.4 0.3 0.7

Inflation Rate High Low
Price of Gold($/oz) 300 400 500 300 400 500
Conditional
Probability
0.1 0.4 0.5 0.7 0.2 0.1

Conditional Expectations for the 1/1/02 Price of Gold G:

Expected Price of Gold given the 2001 Inflation Rate

E{G:H} = (0.1)300+(0.4)400+(0.5)500 = 440

E{G:L} = (0.7)300+(0.2)400+(0.1)500 = 340

Expected Price of Gold given the 2001 Economic Policy

E{G:Exp} = (0.6)440 + (0.4)340 = 400

E{G:Tight} = (0.3)440 + (0.7)340 = 370

Expected Price of Gold given the outcome of the 2000 Election

E{G:C} = (0.8)400 + (0.2)370 = 394

E{G:B} = (0.5)400 + (0.5)370 = 385

E{G:P} = (0.3)400 + (0.7)370 = 379

E{G:N} = (0.5)400 + (0.5)370 = 385

Expected Price of Gold before the 2000 Election

E{G} = (0.4)394 + (0.4)385 + (0.1)379 + (0.1)385 = 388

Change in Expected
1/1/02 Price of Gold
Probability
+6 0.4
-3 0.5
-9 0.1

Expected Price Change = (0.4)(+6)+(0.5)(-3)+(0.1)(-9) = 0

Changes in the 1/1/02 Price of Gold after economic policy is chosen:

 
Change Probability
(a) if Clinton wins  
Exp 400-394 = +6 0.8
Tight 370-394 = -24 0.2
Expected Change = (0.8)(+6)+(0.2)(-24) = 0  
(b) if Kemp wins or no decision   
Exp 400-385 = +15 0.5
Tight 370-385 = -15 0.5
Expected Change = (0.5)(+15)+
(0.5)(-15) = 0
 
(c) if Perot wins   
Exp 400-379 = +21 0.3
Tight 370-379 = -9 0.7
Expected Change = (0.3)(+21)+
(0.7)(-9) = 0
 

Correlation of the change in the expected price of gold after economic policy is chosen with the change in the expected price of gold as a result of the election equals zero.

The expected future price of precious metals (and other assets not paying a dividend) is the current price scaled up by the required rate of return; the present value of the future price is just the current price.

THE SOURCE OF POSITIVE NET PRESENT VALUE FOR PROJECTS

The source of positive NPV projects for a company is in those areas where it has a special advantage and can produce some good or service cheaper than its actual or potential competitors. In such situations it is said to earn economic rents. The Marvin Enterprises example of Brealey and Myers shows how price is determined by the condition for equilibrium in the market; i.e. equilibrium means that no one has an incentive to enter or leave the industry and that condition means the NPV of an investment in the industry is zero. For a company considering leaving the industry the NPV is calculated using the salvage value of the equipment whereas for entering the industry the NPV is based upon the purchase price of equipment.

The investment-opportunities line

Suppose a company has $10 million that it can use either for dividends or investment in the following projects

         Project    investment    return     rate of return
            A         3 mill        4 mill       33%
            B         2 mill        3 mill       50%
            C         1 mill        2 mill      100%
            D         4 mill        7 mill       75%
  
    Discount rate = 2/3 = 67%   Discount factor = 0.6


         Situation   investment  Payout now     Payout 1 yr PV
            1          none        10 mill         0 mill   10.0
            2           C           9              2        10.2  
            3          C, D         5              9        10.4
            4          C, D, B      3             12        10.2
            5          all          0             16         9.6

The Four Stages of Capital Budgeting

The Capital Budget: the list of worthwhile investment projects and the sources of financing.

Types of Investment Projects:

Flow of information:

Plant -> Division Managers -> Senior Management

Plant managers generally generate projects of types #1, #2, division management would generate projects of type #3, and senior management would be the source of projects of type #4.

Division managers can approve smaller projects. Typically small means less than 0.1 of 1 percent of the company's capital budget. Larger projects are filtered up to senior management for final approval. There is often a separate approval and appropriation of funds actions.

Problem of investment criterion to use in capital budgeting and the matter of success indicators for decision makers. NPV is the theoretically correct criterion but others such as payback period and return on book value are also very common because they are easier to understand.

Goals to be achieved in designing an investment decision framework:

Brealey and Myers Second Law: The proportion of proposed projects having a positive NPV is independent of top management's estimate of the opportunity cost of capital.

In evaluating performance take into account:

THE LIMITED LIABILITY CORPORATION

The limited liability corporation arose to become the overwhelmingly dominant economic institution of our times because it facilitated raising capital. Proprietors and partners in business find it relatively difficult to raise new capital because any participant in these forms of business is liable for all of the debts of the business. Because it is risky to enter such organization it is difficult to get out thus creating a further impediment to securing financing. Corporations, on the other hand, involve the limited risk of the initial investment and so can secure more investors. Corporations are financed by a great variety of means. The major categories are: internal (retained earnings and depreciation credits), equity (common stock, preferred stock), long term debt (bonds, debentures), short term debt (commercial paper, trade credit). There are other odd types including warrants and convertible debt. Common stock involves the right to share in the dividends and usually a right to vote. There are some cases of common stock that does not have a voting right. For example, when Henry Ford died the Ford family dealt with the tax problem by creating the Ford Foundation and donating non-voting shares in the Ford Motor Company to Foundation. The Ford family retained the voting common stock and thus control of the company.

Preferred stock is a peculiar case. Generally preferred stock is issued by regulated public utilities where the disadvantage of the non-deductibility of the dividend is not important. Preferred stock is purchased by corporations, for whom the dividend is 70 percent tax free.

In recent years U.S. corporation have met about 80 percent of their capital needs from internal sources of funds (retained earnings and depreciation allowances). The vast majority of the external funds coming from increases in debt. There has been a small amount created by increases in accounts payable but no new capital coming from new stock issues. In fact, corporation have been buying back their own stock with funds raised by issuing debt. Obviously the trend for corporation is higher debt ratios. Rather steadily the debt/asset ratio for all corporations has risen from about 35 percent in 1954 to about 60 percent in 1990. However the extent of debt financing in the U.S. is about the same now as that of most other industrial nations and significantly below that of Japan.

Corporations may issue new securities through a private placement or a public offering. A private placement avoids the costly process of registering the issue with the Securities and Exchange Commission. The private placements are traditionally for the small, risky, and unusual issue.

The first stage of the registration process is the distribution of a prospectus, sometimes called a red herring, because of the warning printed in red informing the reader that the company is trying to sell securities before the registration is effective.

A public offering may be either a general cash offer or a rights issue. The general cash offering make use of underwriters. The underwriters fees will typically be less than five percent for a large issue but could be as much as ten percent for a small offering. A "tombstone" advertisement is published for an underwritten issue listing all the underwriters involved.

A RIGHTS ISSUE OF STOCK

In a rights issue of new stock a lower offer price for the stock does not make the stockholders any better off than a higher price because what they gain in terms of the increase in the price of the new stock is offset by the decrease in the price of the stock they already own.

When a corporation is raising money by the sale of new stock they are doing it to fund a worthwhile investment project. The only real gain for the stockholders is from the positive net present value of the investment project. All the rest of the operation is "smoke and mirrors." To see what this mean consider the following example.

Suppose a corporation has a project that costs $1 million but will return $3 million. Its NPV is thus $2 million. The corporation can raise the $1 million by selling 20,000 shares at $50 each or 100,000 at $10 each. Or it can use any other combination of price and shares that amounts to $1 million.

Suppose the shares are selling at $60 before the project and that there are 500,000 shares outstanding. It thus has $30 million in equity capital. Consider the results after the sale of the new shares and the undertaking of the investment project.

New Shares New Shares
Sell for $50 Sell for $10
Old Shares 500,000 500,000
New Shares 20,000 100,000
Total Shares 520,000 600,000
Shares Required
to Buy One Share
25 5
Equity
Before Project $30 million $30 million
After Project $33 million $33 million
Share Price
Before Project $60.00 $60.00
After Project $63.46 $55.00

Consider an owner of 25 shares. Before the project his/her holdings are worth 25(60)=$1500. Under the $50 per share offering he/she can buy one share and so owns 26 after the financing of the project. Each of the shares is worth $63.46 so the total value of his/her holdings is 26(63.46)=$1650. Since he/she paid out $50 for the new share the net gain is $1650-1500-50=$100. This $100 net gain is just the shareholders share of the NPV of the project. The $100 net gain came as a result of a capital gain of $63.46-50=$13.46 on the one new share and capital gains of $3.46 on each of the 25 old shares. This amounts to a total of 86.54 on the old shares and $13.46 on the new share for a total of $100.

Under the $10 per share offering the owner of 25 shares can buy 5 shares. After the financing each of the 30 shares is worth $55 so his/her holdings are worth $1650 the same as in the $50 per share offering and their the same net gain of $100. The $100 is the net gain of $55-$10=$45 per share on the 5 new shares and a capital loss of $5 per share on the old shares. The net is thus 5(45)-25(5)=225-125=$100.

The Efficient Market Hypothesis

The Efficient Market Hypothesis (EMH) can be stated in several different ways. In addition, there are some simplified versions of the EMH which have an intuitive appeal, but are not strictly accurate. These simplified versions are useful while learning the concept but ultimately must be abandoned for the more rigorous formulations. The EMH was initially stated in terms of changes in prices. It was said that the probability of the price increasing was always equal to the probability of it dereasing. A more accurate statement is that the expected value of the price change is always zero. Thus, if there is a 90 percent probability of the price going up by 10 cents and a 10 percent probability of it going down by 90 cents the expected price change is still zero, even though the price is nine times as likely to go up as it is to go down. This may be expressed mathematically as E{(pt - ps)} = 0. This says that the expected change in price at time t from what it was at time s is zero. Actually, we should be more explicit. The expected value is calculated using some set of probabilities. The probabilities involved above are the conditional probabilities, the probabilities given the information that was available at time s. If we let Is represent the information that was available at time s then the proper way to state the EMH is: E{(pt-ps)|Is} = 0. This expression is equivalent to: E{pt|Is} = ps. This says that the best estimate of what a price will be equal to a some future time t is the price now at time s. Another way of expressing this is to say that all of the information available at time s that has relevance for what the future price will be at time t is incorporated into the price at time s. The formulation in terms of price changes is called the additive model. An alternate model, and one that is better, says that it is not the price change that is random but the rate of return. If the security pays no dividend and the rate of return for holding the security is r then the price after one period is the original price times (1+r). Thus, E{pt |Is} = ps (1+E{rs+1|Is})(1+E{rs+2|Is}).... (1+E{rt-1|Is}) If E{r} is not zero but instead is positive then the price drifts upward so this model is called a random walk with drift. If the security pays a dividend d1 then since r = [d1 + (p1-p0)] /p0 p1 = (1+r)p0 - d1.

Thus the more the dividend the less the capital gain for a given rate of return and the slower the drift in prices. An informationally efficient market is one in which the current price reflects all relevant and ascertainable information. This means that the best estimate of the price in the future; i.e., its expected value; is easily determined from the current price. Every one that has access to the current price has the best forecast possible. In an efficient market the price basically follows a random walk (possibly with drift). There may be small deviations from randomness so long as it is not great enough to allow anyone to make a profit from the nonrandomness. In an efficient market the purchase or sale of a security is never a positive NPV transaction. Any gains from the nonrandomness are not enought to cover the transaction costs. In other words, trust market prices because markets never lie (very much).

Harry Roberts defined three forms of the efficient market hypothesis. The weak form is where the current market price reflects an information contained in past prices. If a market is weak form efficient then charting is a waste of time. The past changes in prices is of no value in predicting the future changes; there is no correlation between past price changes and current price changes. There is no momentum or resilience in the market. Bull markets and bear markets are figments of investors imaginations.

Roberts defined a semi-strong form of market efficiency where the current price reflect no only all information in past prices but also all information in published information. The strong form of efficiency reflect not only public information but all information that can be acquired by analysis of the company and the economy. A super strong form of efficiency would be where the price reflects all information including insider information. If insiders act on their information it may affect prices.

If a market is efficient in the strong form sense then there are no financial illusions. Investors would not be fooled by purely cosmetic changes such as stock splits or changes in accounting conventions. This also means mergers should not change the combined value of the companies involved because investor could have created the equivalent of the merged company by creating a portfolio made up of those companies.

It often happens that a stock split is associated with an increase in the price of a stock but that is because the stock split is precipitated by some good fortune for the company. The stock split is not the cause of the increase in the value of the stock but a side effect of the same things that caused the value of the stock to go up. When companies do a stock split with out there being any real improvement the value may go up and it may go down; on average there is no change.

CHAOS AND ORDER IN CAPITAL MARKETS

Notes on Chaos and Order in the Capital Markets by Edgar Peters 1991

The discoverers of the random walk character of market prices, such as Maurice Kendall and M. Osborne, were looking for linear dependence of current price changes on past price changes. They calculated correlation coefficients that measure linear dependence and found no significant correlations. This still leaves open the possibility of some more sophisticated nonlinear dependence.

Statistical work makes use of random numbers. But the random numbers are not truly random. They are generated by some complicated by totally deterministic process. One of the early methods for generating "random" numbers was to take a four digit number, square it, take the middle four digits and this number to generate the next random number. For example, take the number 9633 as the starting point. This number squared is 92794689 so the first random number is 7946. The next one is 1389 because 7946 squared is 63 138916. A sequence of number generated in this manner passes tests for randomness. So passing linear tests for randomness does not mean the variable cannot be predicted.

A random walk implies zero correlation between changes in one time period and changes in a previous time period. But there are other implications of a random walk. Suppose we look at the variance of month-to-month price changes and the variance of price over a year. How should the variances be related? To answer that question let us consider the variance of price changes over a two month period. Suppose x1 is the change over the first month and x2 the change over the second of the two month period. Then the two month price change is x1+x2. What is the variance of x1+x2? It is Var(x1) +Var(x2)+2Cov(x1,x2). But for a random walk Cov(x1,x2) is zero and Var(x1)=Var(x2)=Var(x) so Var(x1 +x2)=2Var(x). But this means the standard deviation of the price changes over a two month period is the square root of 2 times the standard deviation of the price changes over aone month period. The variance of the price changes over a twelve month period is twelve times the variance of the price changes over a one month period. So the standard deviation of price changes over a year is the square root of twelve times the standard deviation of monthly price changes. Generalized this rule is

σt = t1/2σ1

where σt is the standard deviation of price changes over a period of length t, t1/2 is the square root of t and σ1 is the standard deviation over a time period of length 1. This is called the t-one half power rule. It was published by Albert Einstein in 1905 in reference to Brownian motion.

Hurst's Rescaled Range Analysis

Raymond Hurst developed a test for randomness in time series based upon the t-one half power rule while serving as a hydrologist for the Nile River Dam Project. Below is the graph that Hurst used showing how the data should look for a random walk.

On the vertical axis is shown the logarithm of the ratio of the range (maximum - minimum) divided by the standard deviation of the original observations, the so-called rescaled range R/S. The horizontal axis has the logarithm of t or the logarithm of the number of observations (which is proportional to 1/t). For contrast the R/S results are shown for sunspot activity that has a cyclical pattern. Also shown are the Hurst diagrams for several financial markets.


Coca Cola


IBM


Yen/Dollar Exchange Rate

The results seem to indicate a deviation of the Efficient Market Hypothesis but the nature and significance of that deviation.


Proof of Modigliani and Miller's Proposition That, in the Absence of Differential Taxes on Interest and Dividends, the Value of a Corporation is Independent of Its Capital Structure

Assume for now there are no taxes on profits. The rate of return on equity is a function of leverage; i.e.,

requity = rassets + (rassets - rdebt)D/E,

where D/E is the debt-equity ratio. If rassets < rdebt then an increase in leverage raises the rate of return on equity. But risk also increases with leverage according to the formula

βequity = βassets + (βassets - βdebt)D/E.

If debt is risk free (βdebt=0) then this formula reduces to

βequity = βassets(1 + D/E).

The higher rate of return on equity is offset by the higher discount rate due to the higher risk. It is not clear that the offset is exact. Modigliani and Miller develop another argument to demonstrate that the higher leverage leaves the value of the corporation unchanged.

Imagine two corporation identical in every respect except that one is unleveraged and the other is leveraged. Both have the same earnings. The unleveraged corporation distributes all of its earnings as dividends. The leveraged corporation pays out some of its earnings as interest on its debt and distributes the remainder as dividends.

Now imagine an investor who is considering two portfolios. One portfolio is made up of one percent of the stock of the unleveraged corporation. This portfolio pays dividends equal to one percent of its earnings. The second portfolio contains one percent of the stock of the leveraged corporation and one percent of its debt. This portfolio pays dividends equal to one percent of the earnings less the interest on the debt plus one percent of the interest on the debt. This is equal to exactly one percent of the earnings since the interest on the debt exactly recapture the interest that was deducted from the earnings. Since, by assumption the two corporation have the same earnings, the two portfolios provide the exactly the same income. But the value of the two portfolios would have to be exactly the same if they provide exactly the same income. But both portfolios contain one percent of the value of their corporations so the value of the corporations (debt + equity) must be exactly the same.

This proof worked because it was possible to undo the leveraging of the one corporation by buying up an equal share of its debt. Clearly the same construction would work if the leveraged corporation had a more complicated capital structure involving unsecured debt (debentures) as well as secured debt (bonds). It would still work if there were preferred stock involved as well as common stock. No matter how complicated the structure; e.g. involving warrants, convertible bonds, promissory notes or whatever; the same argument works.

Examples of Leveraged Buyouts (LBO's)

Metromedia

John W. Kluge (kloo gy), an immigrant from Germany, was for many years a food broker in Washington, D.C. In 1959 he and some friends bought control of the Metropolitan Broadcasting Corporation (MBC) which was a television network (far smaller than NBC, CBS, and ABC) and owned two radio stations. Kluge changed the name to Metromedia and proceeded to build up over 25 years its holdings to seven television stations and 14 radio stations. Metromedia also owned the Ice Capades and the Harlem Globetrotters.

Kluge was always open to new ventures for making a profit. For example, he bought the depreciation rights to the $100 billion of New York's buses and subways. Because of his sometimes unorthodox ventures he felt constrained by having to report to public stockholders even though he held 25 percent of Metromedia. In 1983 he decided to take Metromedia private through a leveraged buyout.

Kluge and his backers had to raise $1.45 billion to purchase 28.6 million shares, refinance Metromedia's existing debt, purchase employee stock options, and provide working capital for the transition. Ten banks led by Manufacturers Hanover Trust provided an eight year loan of $1.3 billion taking Metromedia's assets as collateral. The rate of interest was 1.5 percent above the prime rate and there was a covenant that required Metromedia to maintain a minimum net worth of $100 million.

Kluge made a per share offer to the shareholders of Metromedia of $30 in cash and subordinated debentures having a face value of $22.50. The market value of these fifteen year debentures which would not pay interest until the sixth year was estimated to be $9.50 to $10.00. A threatened stockholder suit brought an increase in the offer by $1 per share. The offer was accepted and Kluge ended up with 93 percent of the voting stock of Metromedia.

Metromedia's ratio of long-term debt to equity was 3.5 and Moody's and Standard & Poors lowered the rating of Metromedia's debentures. In June of 1984 Metromedia went private. Six months later Kluge had Drexel Burnham Lambert sell $1.3 billion of junk bonds to replace his bank loan. It was the biggest junk bond issue up to that point but it sold out within two hours.

In 1985 Kluge sold six of Metromedia's seven television stations to a group headed by Rupert Murdock for $1.5 billion and the remaining station (in Boston) was sold to the Hearst Corporation for $450 million. In 1986 he sold off a billboard operation for $710 million and the Harlem Globetrotters and Ice Capades for $30 million. In 1986 the chain of radio stations was sold for $285 million. This left telecommunications as the main element of Metromedia's operations. But a few months later Kluge sold Metromedia's cellular telephone and paging operations to Southwestern Bell for $1.65 billion. Altogether Kluge sold $4.6 billion of Metromedia's assets.

Gibson Greeting Cards

This company was founded in 1850 and became a corporation in 1895. In 1964 CIT Financial Corporation acquired Gibson Greeting Cards, but in 1980 CIT Financial was acquired by RCA and Gibson became a subsidiary of RCA. Gibson was doing well. In 1984 it was the third largest greeting card company with sales of $304 million. RCA was however implementing a policy of concentrating on its core business of NBC, Hertz, electronics, and communications and decided to sell Gibson Greeting Cards. It was sold to Wesray Corporation, a creation of former Secretary of the Treasury William Simon. The price was $81 million. Wesray gave Gibson management 20 percent of the company. The new Gibson stockholders, including Simon, invested $1 million. The funds came in part from loans from General Electric Credit Corporation ($40 million), Barclays American Business Credit ($13 million). The rest came from Gibson selling and leasing back its three major manufacturing and distribution facilities. Thus the price was actually only $54 million. General Electric Credit Corporation got warrants to purchase 2.3 million shares at 14 cents per share and additional interest in proportion to any dividends paid by Gibson Greeting Cards. The interest rate in 1982 on Gibson's debt averaged 19 percent. Eighteen months after Wesray bought Gibson it cashed in by a public offering of 10 million shares at $27.50. William Simon realized a payoff of $66 million on an investment of about one third of a million dollars. Later the Gibson stock price fell to $18 but rebounded to $28 when a takeover battle developed between Walt Disney Production and Saul Steinberg.

Thatcher Glass

In 1981 Dart & Craft, a major consumer goods manufacturer, decided to sell off its subsidiary Thatcher Glass. Dart & Craft felt the glass container industry had little growth potential. Thatcher Glass was the third largest bottle manufacturer in the nation and had sales of about $350 million and profits of about $30 million. A new company, Dominick International, was formed to buy Thatcher. This new company included the president of Thatcher Glass. Dominick International offered $120 million in cash and $18 million in subordinated debentures and preferred stock. A group of banks, including Manufacturers Hanover Trust and Chase Manhattan provided $110 million at an interest rate of 22.5 percent. There was $4 million of equity and $3 million of Dominick preferred stock. The rest was raised by other high interest loans. The buyout was completed in 1983 and Thatcher had a debt ratio of 95 percent. This means the debt/equity ratio was about 19. By the end of 1984 Thatcher's sales had dropped sharply due to increased competition from plastic and aluminum containers. Thatcher reacted by cutting prices and laying off 80 percent of its 4,000 employees. In 1985 Thatcher filed for bankruptcy and attempted to reorganize under a new president. Dominick International sold off three of Thatcher's six plants and realized $40 million on the sale. Thus the third largest bottle manufacturer in the U.S. folded as a result of the leveraged buyout.

The Concept of the Duration of an Investment

A security often involves revenues (and perhaps costs) that occur at different times. For example, a five year $1000 corporate bond with a coupon rate of 5 percent has five interest payments of $50 at the end of years 1 through 5 and a payment of $1000 at the end of year 5. If the market interest rate is 8 percent and is expected to stay at that level for the life of the bond then the bond would have a value equal to $880, which is the sum of the present values of the interest payment and the final payment of $1000 at maturity. Some of that value is realized in one year, some in two years and so on. The following table gives a breakdown of the value of the bond:

Time Payment Present
Value
PVxTime
1 $50 $46.30 46.30
2 $50 $42.87 85.74
3 $50 $39.69 119.07
4 $50 $36.75 147.00
5 $1050 $714.61 3573.06
Total $880.22 3971.17
Duration = 3971.17/880.22 =4.51 years

The duration of the bond is the average time until receipt of the present values. In other words, the duration is the weighted average of the times until receipt of the payments, where the weights are the present value of the payments. The discount rate that make the present value of the payments equal to the price of a bond is called its yield. The relationship of the change in the value of a security to changes in its yield rate is called its volatility; i.e. Volatility= proportional change in value per unit change in yield = (dV/V)/dr = (1/V)dV/dr The significance of duration, in addition to its being a convenient way to summarize the time scale of a security, is that the following relationship exists: Volatility = -Duration/(1+r) This relationship can be proven using calculus. Let Ct be the payment in year t. Then value V is given by V = C1/(1+r) + C2/(1+r) 2 + ... + Ct/(1+r) t. Differentiating V with respect to r gives dV/dr = -C1/(1+r)2 - 2C2/ (1+r) 3 + ... - tCt/(1+r) t+1. = -[C1/(1+r) + 2C2/(1+r)+ ... + tCt/ (1+r) t]/(1+r). If we denote the present value of Ci by PVi and note that V = PV1 + PV2 + ... + PVt and dV/dr = -[1PV1 + 2PV2 + ... + tPVt]/(1+r), then the division of the second equation by the first gives (1/V)dV/dr= -[(1PV1+2PV2+...+tPVt)/ (PV1+PV2+...+PVt)]/(1+r). But (1PV1+2PV2+...+tPVt)/(PV1+ PV2+...+PVt) is simply what we have defined as Duration.

Thus Volatility = - Duration/(1+r).

A Hypothetical Leveraged Buyout

Existing conditions
of corporation
  Capital
Equity $20 million
Debt $10 million
Total $30 million
   
Revenues $32 million
Operating Cost $20 million
Surplus $12 million
   
Interest $0.9 million
Depreciation $3.0 million
   
Taxable Profit $8.1 million
Tax @ 40% $3.24 million
   
After-Tax Profit $4.86 million
Cash Flow $7.86 million
   
Investment $3.00 million
Free Cash Flow $4.86 million
   
Dividends $4.86 million
   
Capitalized Value
of Dividends
$20.0 million

  Cost of Capital After tax
Equity 24.3% 24.3%
Debt 9.0 5.4
WACC 19.2 18.0

After LBO
  Capital
Equity $0 million
Debt $30 million
Total $30 million
   
Revenues $32 million
Operating Cost $20 million
Surplus $12 million
   
Interest $2.7 million
Depreciation $3.0 million
   
Taxable Profit $6.3 million
Tax @ 40% $2.52 million
   
After-Tax Profit $3.78 million
Cash Flow $6.78 million
   
Investment $3.00 million
Free Cash Flow $3.78 million
   
Dividends $3.78 million
   
Capitalized Value
of Dividends
$15.56 million

The organizers of the LBO thus gain $15.56 million with no investment of equity capital. The market value of the corporation is after the leveraged buyout equal to $30 million + $15.56 million or $45.56 million compared to the market value of $30 million before the LBO.

The market value of the corporation after the leveraged buyout is equal to: $30 million + $15.56 million or $45.56 million compared to the market value of $30 million before the LBO.

It is instructive to determine where the additional $15.56 million of value comes from. We know the $15.56 million is the capitalized value of the $3.78 million in annual dividends that are paid to the owners after the LBO. So the question is, "Where does the $3.78 million come from. Some comes from the reduction of taxes paid as a result of the tax deductibility of interest. Comparing the before and after LBO figures we see that the tax payment is reduced from $3.24 million to $2.52 million, a savings of $0.72 million. But this is just a small part of the total.

The majority of the gain comes from shifting from a high cost source of capital, equity for which the cost is 24.3%, to a lower cost source of capital, debt for which the cost is 9%. Before the LBO the providers of the $20 million of equity capital were paid dividends of $4.86 million. After the LBO the providers of that $20 million of capital, the new bondholders, are paid $1.8 million. The savings on the cost of the $20 million of capital is thus 4.86 - 1.8 or $3.06 million. If we add together the savings on capital cost of $3.06 to the $0.72 saved on taxes we get the $3.78 million that goes to the new owners after the LBO. Thus most of the gain from the LBO comes from the switch from higher cost sources of capital to lower cost sources of capital.

The switch from equity to debt comes at the expense of increasing the risk for equity holders of the corporation. The switch also increases the risk for debt holders because it increases the chance that the corporation will not be able to meet the interest payments in a bad year. When there is substantial equity capital there is a cushion of earnings that can be diverted from dividends to the payment of interest. The example did not reflect the fact that the debt after the LBO is riskier and must pay a higher interest rate. In the example suppose the interest rate on debt rose from 9% before the LBO to 15% after the LBO. The interest cost after the LBO would then be $4.5 million and the effect on taxes and so forth are as shown below.

After LBO
with increase
in the cost of debt
  Capital
Equity $0 million
Debt $30 million
Total $30 million
   
Revenues $32 million
Operating Cost $20 million
Surplus $12 million
   
Interest $4.5 million
Depreciation $3.0 million
   
Taxable Profit $4.5 million
Tax @ 40% $1.8 million
   
After-Tax Profit $2.7 million
Cash Flow $5.7 million
   
Investment $3.00 million
Free Cash Flow $2.7 million
   
Dividends $2.7 million
   
Capitalized Value
of Dividends
$11.11 million

There is still an increase in value from the LBO because even at 15% the cost of debt capital is less than the cost of capital from equity. Again in this version of the example the effect of leverage on the capitalization rate was ignored. The following provides a more rigorous analysis of the problem.

The Gains From a Leveraged Buyout

Definition of variables

The relationships that exists between the variables are:

(Analysis to be completed later)

Interest rates, inflation rates, and exchange rates

There are relationships among the nominal interest rates and inflation rates of two countries and the exchange rates, spot and forward, between their currencies. These relationships are based upon the notion that for financial equilibrium to exist between two countries the real rates of interest must be the same. This is true only if there is no differences in the tax rates and levels of risk in the two countries. Suppose the two countries are the United States and the United Kingdom. If the nominal rates of interest are r$ and rL and the expected rates of inflation are E{inf$} and E{infL}, then the real buying powers of a unit of money of invested in the two countries are:

                1 + r$         1 + rL 
               __________ and  ____________.
               1 + E{inf$}     1 + E{infL/} 

Equating these two ratios and rearranging gives:

1+rL E{1+infL} __________ = ___________ 1+r$ E{1+inf$}

An investor could exchange domestic currency for foreign currency at the spot exchange rate, invest in the foreign bond market, and sell the anticipated foreign currency now at the forward exchange rate. Or the investor could simply invest in the domestic bond market. For equilibrium the investors in both countries would have to be indifferent between these two investments. This requires that:

forexchL/$ _______________ be equal to spotL/$ 1+rL __________ 1+r$

Also for equilibrium the forward exchange rate must be an unbiased estimate of what the spot rate will be in the future; i.e.,

forexchL/$ = E{spotL/$}.

This means that:

forexchL/$ E{spotL/$} _______________ = ____________ spotL/$ spotL/$

These relationships can be combined into one diagram:

1+rL E{1+infL} __________ equals ___________ 1+r$ E{1+inf$} equals equals forexchL/$ E{spotL/$} _______________ equals ____________ spotL/$ spotL/$

Thus if one knows three of the quantities involved in any two of these ratios one can solve for the fourth.

FORMULAS:

PV= [1-1/(1+r)t]/r

PV= 1/(r-g)

Annual Cost= PV(costs)/(Annuity Factor for life of asset)

Var(r)= Expected value of (r-expected r) 2

Stand.Dev.=(Var)1/2

Correlation between rA and rB = (Covariance (rA,rB) /(StdevA)(StdevB)

β of A = Covariance(rA,rm)/Variance(rm)

rA=alphaAArm

rA=rFA(rm-rF)

rassets=(E/(E+D))requity+(D/(E+D))rdebt

requity=rassets+(D/E)[rassets -rdebt]

βassets=(E/(E+D))βequity +(D/(E+D))βdebt

βequityassets[1+(D/E)] -(D/E)βdebt

W.A.Cost of Capital= (E/(E+D))requity +(D/(E+D))rdebt(1-taxrate)

r = rf + β(rm - rf)

βequity = (1+D/E)βassets

requity = rassets + (D/E)(rassets - rdebt)

PV = payment/(1+r)atime

βassets = βrevenue (1 + (PV of fixed costs)/NPV) expected value = sum of probabilities times payoffs W.A.C.C. = (E/(E+D))requity + (D/(E+D))rdebt(1-tax) standard deviation over periods of length t = t1/2(standard deviation periods of length 1)

Price = Div/(r-g)

The Black and Scholes formula for call value:

Relevant Variables:

Steps in using the table for the Black and Scholes value of a call option:

Exercises on Stock Option Valuation

Exercise 41. Find the value of a European call option for a share of Dole, Inc. with an exercise price of $60 with 1 year to expiration when the current risk-free interest rate is 12 percent. The current market price of the stock is $58.93 and its volatility is 20.25 percent. Dole will not be paying a dividend with in the next year.

Exercise 42. What is the value of an American call option for a share of Clinton & Co. with an exercise price of $96 and which which expires in November 2003 (one year from now). The current risk-free interest rate is 8 percent and the market price of the stock is $95.47. Clinton & Co.'s volatility is 21.21 percent. As is the case with most stocks in its industry Clinton & Co. will not pay a dividend in the next year.

Exercise 43. What is the value of a European put option for the stock in #41 when the exercise price is the same?

Exercise 45. What is the value of a put option on Kennedy & Co. for the next six months when the exercise price is $0.25 and the current market price is $0.10. The current interest rate is zero and the volatility of Kennedy is 10.6 percent. Kennedy has never paid a dividend and is not expected to do so in the next six months.

The Informational Efficiency of Markets and Accounting Manipulation of Earnings

One of the implication of the efficient market hypothesis in its semi-strong form is that cosmetic changes in accounting for a corporation will not affect the price of its stock.

Example showing the effect of leveraging:

Case of No Taxes on Corporate Profits

Intitial Case: No debt (L=0)

This is also the rate of return on assets rassets = requity = 11.4%

Value of assets = Value of equity

= (2 million)/(0.114) = $17.544 million

Price of a share = ($17.544 million)/(500,000) = $35.09

Case of leverage: Suppose the company sells debt to buy up the 75 percent of the common stock. This requires issuing debt equal to (.75)($17.544) = $13.158 million and leaves 125,000 shares in the market. The leverage ratio increase to 3.0.

βequity = (1+L)βasset = 4(0.8) = 3.2

requity = rassets + L(rassets - rdebt)

= 11.4 + 3(11.4-5.0)

= 11.4 + 3(6.4) = 30.6 percent

required rate of return on equity = rf + βequity

(rm-rf) = 5 + 3.2(8) = 30.6 percent

interest on debt = (0.05)(13.158) = $657,900

income for equity holders = 2,000,000 - 657,900 = $1,342,100

Valuation of equity = (1.342 million)/(0.306) = $4.386 million

Price of share = ($4.386 million)/(125,000) = $35.09

Valuation of company = Value of debt + Value of equity

= $13.158 million + $4.386 million

= $17.544 million

Thus the value of the company is the same as in the initial case.

The leveraging of the company left its value unchanged.

Case of Taxation of Corporate Profits Tax rate on corporate profits= 40 percent Tax rate on dividends and interest income = 0 percent No debt case (L=0) The required rate of return is the same as the no tax case = 0.114 Profits after taxes = $2 million - 0.4( 2 million) = 0.6(2 million) = $1.2 million Value of assets = Value of equity = (1.2 million)/(0.114) = $10.526 million Price of a share = ($10.526 million)/(500,000) = $21.05 Case of leverage:

The relationships can be expressed algebraically.