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.

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.

NPV = 2582

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

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:

(millions)
• 1. Revenue ( $3750x100,000) $375
• 2. Variable Cost ($3000x100,000) 300
• 3. Noncapital Fixed Costs 30
• 4. Depreciation (S/L) 15
• 5. PreTax Profits 30
• 6. Tax @ 50% 15
• 7. Net Profit 15
• 8. Net Cash Flow ( 7 + 4 ) 30

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:

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

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

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

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

Error		Proportion 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

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:

• 1. The present value of future dividends
• 2. The present value of future earnings
• 3. The present value of future cash flows
• 4. The capitalized value of current earnings plus
  the present value of growth opportunities (PVGO).

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

• 1. The present value of future net dividends
  dividends minus sale of new stock
• 2. The present value of future net earnings
  earnings minus imputed interest on accumulated capital
• 3. The present value of future free cash flows
  cash flow minus investment outlay
• 4. The capitalized value of current earnings plus
  the present value of growth opportunities (PVGO).
• 5. The sum of the net present values of all worthwhile
  projects of the corporation

A Mathematical Derivation of M&M's Results

M&M analysis starts from the proposition that the stock price at any time t, p_t, 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) p_t = (d_t+1 + p_t+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 N_t be the number of shares outstanding at time t. The value of the equity in the corporation at time, P_t, is just N_tp_t. Thus if (1) is multiplied by N_t one obtains

(2) N_tpt = P_t = (N_td_t+1 + N_tp_t+1)/(1+r) = (D_t+1 + N_t p_t+1)/(1+r).

The quantity N_tp_t+1 can be expressed as

N_tp_t+1 = N_tp_t+1 -N_t+1p_t+1 +N_t+1p_t+1 = -p_t+1(N_t+1-N_t)+P_t+1.

The term p_t+1(N_t+1-N_t) 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, S_t +1 . Thus equation (2) reduces to

(3) P_t = ((D_t+1 - S_t+1) + P_t+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 e_t is earnings per share then E_t=N_te_t is total earnings. If K_t is the sum of all past investment as of time t then the imputed interest on investment is rK_t. M&M defined Net Earnings as E_t-rK_t 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, P_t, is equal to

E_t/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:

r_port = (x_A)r_A + (x_B)r_B

Var(r_port) = (x_A)²Var(r_A) + (x_B)²Var(r_B)
+ 2(x_A)(x_B )Cov(r_A,r_B)

You will also need these formulas:

Cov(r_A,r_B) = Correlation(r_A,r_B)xStDev(r_A)xStDev (r_B)

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

Example:

Let r_A=E(r_A)=20%,
r_B =E(r_B)=10%,
σ_A=stdev(r_A)=15%,
σ_B =stdev(r_B)=5% and
σ_AB=Cov(r_A,r_B) =10.

This means σ_A²= Var(r_A)=225, σ_B ²=Var(r_B)=25,
and the correlation of r_A and r_B is 0.2.

Exercise 32. Plot r_port versus StDev(r_port)

A dollar slot machine has reels with the following symbols:

The probability distribution of the payoff is

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 r_A, r_B, and r_P be the rates of return on the two stocks and the portfolo. The proportion of the portfolio in A and B are denoted as x_A and x_B. The share in A, x_A, is the value of the A stock as a proportion of the total value of the portfolio.

Then

r_P = x_Ar_A + x_Br_B,

and E{r_P} = x_AE{r_A} + x_BE{r_B}.

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 r_P - E{r_P}, square it and take the expected value of the squared deviations. From the equations for r_P and E{r_P} we have

r_P - E{r_P} = x_Ar_A + x_Br_B - x_AE{r_A} - x_BE{r_B} = x_A(r_A - E{r_A}) + x_B(r_B - E{r_B}).

When this expression is squared we get:

( r_P - E{r_P})= (x_A(r_A - E{r_A}) ² + (x>>BB(r_B - E{r_B}) ² +2(x_A(r_A - E{r_A}) (x_B(r_B - E{r_B}) . x_A ²(r_A - E{r_A})+ x_B²(r_B - E{r_B}) ² +2x_AxB(r_A - E{r_A})(r_B - E{r_B})

When we take the expected values we get

Var(r_P) = x_A ²Var(r_A) + x_B ²Var(r_B) + 2x_Ax_BE{(r_A - E{r_A})(r_B - E{r_B})}.

The last term involves the expression

E{(r_A - E{r_A})(r_B - E{r_B})},

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

It represent the extent to which r_A and r_B vary together.

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

Var(r_P) = x_A²Var(r_A) + x_B²Var(r_B) + 2x_A x_BCov(r_A,r_B).

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:

• 1. Two statistics, the mean and variance of the rates of return, are sufficient to describe investor preferences over the distribution of future returns on a portfolio.
• 2. Investors prefer higher expected returns to lower expected returns for a given level of portfolio variance and prefer lower variance to higher variance of portfolio returns for a given levelof expected returns.
• Additional assumptions made be Sharpe (1964) for the Capital Assets Pricing Model (CAPM):
  • 1. All investors have the same expectations about means, variances, and covariances of security returns.
  • 2. All investors have a common time horizon for investment decisions.
  • 3. All investors can borrow and lend at the risk-free rate of interest.
  • 4. Taxes are assumed to have no effect on asset pricing.
  • 5. All assets are sold in complete and perfect markets (with zero transactions costs).

Economic Determinants of β and Variance

• A. Financial Leverage
• C. Operating Leverage
• C. Lines of Business
  • 1. Business risks
    • a. volume of sales
    • b. unit price
    • c. unit variable cost
    • d. tax rate
    • e. availability of raw materials

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, r_m)/Variance(r_m).

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:

• 1. Finance leverage (the debt/equity ratio)
• 2. Operating leverage (fixed costs as a ratio to the value of the project)
• 3. The mix of products produced by the project.

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 β_eq and β_d and the β for total assets be

β_assets. Then

β_assets = (E/V)β_eq + (D/V)β_d.

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

β_assets = (E/V)β_eq.

Solving this equation for β_eq in terms of β_assets gives:

β_eq = (V/E)β_assets = [(E+D)/E]β_assets

= (1 + D/E)β_assets = (1+L)β_assets,

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)

β_revenue =β_{fixed cost}(PV(fixed cost)/ (PV(revenue)) +β_{variable cost} (PV(variable cost)/PV(revenue)) +β_asset (PV(asset)/PV(revenue))

β_{fixed cost} = 0

Both revenue and variable costs are determined by sales so

β_revenue =β_{variable 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

β_revenue = β_{fixed cost} [PV(fixed cost)/PV(revenue) + β_{variable cost}[PV(variable cost)/PV(revenue)] + β_asset[NPV/PV(revenue)]

If fixed cost is not subject to variation then
β_{fixed cost} = 0.

Also revenue and variable cost vary because of variation in sales volume. Therefore we expect
β_revenue = β_{variable cost}.

Therefore

β_revenue [ 1 - PV(variable cost)/PV(revenue)] = β_assetNPV/PV(revenue).

Multiplying both sides by PV(revenue) gives

β_revenue [ PV(revenue)-PV(variable cost)] = β_assetNPV

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

so β_revenue[ NPV + PV(fixed cost)] = β_assetNPV.

Dividing through by NPV gives:

β_asset = β_revenue [ 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

• A. Standard Approach -- Risk Premium
  • 1. Estimate the expected income of project year by year.
  • 2. Estimate risk of project incomes
  • 3. Determine the risk premium appropriate for the project
  • 4. Add risk premium to the risk-free rate
  • 5. Use the risk adjusted discount rate to find the net present value of the expected net incomes (free cash flows)
  • 6. If the expected NPV is positive, accept the project; otherwise reject it.
• B. Certainty Equivalent
  • 1. Determine the amount of certain income that would be just as good as the opportunity to receive the uncertain income of the project in each year
  • 2. Discount the certainty equivalent net income at the risk-free rate
  • 3. Accept or reject the project on the basis of the NPV of the certainty equivalent income
• C. Tobin's q value
  • 1. Estimate expected present value of stream of profits from project
  • 2. Determine the purchase price or replacement costs of capital requirements of the project
  • 3. Find the q ratio of expected present value to capital costs
  • 4. Undertake the project if q is greater than 1.0, reject it otherwise
• D. The Option Value Approach
  • 1. Estimate the present value of the expected net income of the project
  • 2. Determine the present value of the costs
  • 3. Determine the option value of keeping the investment opportunity alive
  • 4. Undertake the project if the expected net present value is greater than the option value of the 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{r_A} = r_f + (r_M - r_f)b_A

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

r_A = r_f + (r_M - r_f)b _A + u_A.

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: β_asset = β_revenue[ 1 + (PV(fixed cost)/NPV ].

For this example β_asset = β_revenue[ 1 + 334.35/34.35 ] β_asset = β_revenue[ 1 + 334.35/34.35 ] β_asset = β_revenue[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

β_asset = β_revenue[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:

• Joint probability P(y,x) = probability that both y and x will occur.
• Conditional probability P(y:x) = probability that y will occur given that x has already occurred.
• Unconditional probabilities P(x) = probability that x occurs.

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. Policy	Expansionary		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

• 1. Preparation of the capital budget
• 2. Project Authorizations
• 3. Monitoring of project under construction
• 4. Postaudit checks of completed projects.

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

Types of Investment Projects:

• 1. Mandated safety and environmental outlays (must meet technical requirements)
• 2. Maintenance and cost reduction (must have positive NPV)
• 3. Expansion of capacity in existing businesses (must have postive expected NPV, forecasts crucial)
• 4. Investment for new product lines and/or regions (must have positive expected NPV, forecasts crucial but there are many uncertainties.

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:

• 1. Insure that forecasts are consistent among projects.
• 2. Eliminate conflicts of interest between the success indicators applied to investment decision makers and the criterion of investment. "This problem may also reflect the way that the manager's performance is measured and rewarded. A good measurement system should have some tolerance for mistakes and should be able to discriminate between good decisions and lucky ones. Ideally, managers would be rewrded for good decisions thwarted by bad luck and penalized for bad decisions rescued by good luck."
• 3. Reduce forecast bias. If cost estimates systematically underestimate the final cost take that into account when reviewing proposals.
• 4. Get senior management the information it needs.

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:

• Accounting depreciation versus economic depreciation
• Accounting income versus economic income.

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.

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

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

σ_t = t^1/2σ₁

where σ_t is the standard deviation of price changes over a period of length t, t^1/2 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.,

r_equity = r_assets + (r_assets - r_debt)D/E,

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

β_equity = β_assets + (β_assets - β_debt)D/E.

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

β_equity = β_assets(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.

Corporate Value Independent of Capital Structure

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.

Corporate Value, Capital Structure and Taxes

Because the interest paid on debt is deductible when coporate taxes are computed the value of a corporation will depend upon its financial structure. Let T be the corporate profit tax rate and E be the earnings of two identical corporations which goes on forever. To keep things simple suppose all earnings after taxes are paid out as dividends.

For an unleveraged corporations then its after tax earings are (1-T)E. Suppose the capitalized value of those after tax earnings is V_U=(1-T)E/r. Now consider a leveraged corporation with debt D and interest payments rD. Its taxable profits are then E−rD and its profit taxes are T[E−rD}. Its after-tax earnings are then

E − T[E−rD] = (1-T)E + rTD
the capitalized value of this
is the value of the leveraged coporation
V_L = (1-T)E/r + rTD/r
which reduces to
V_L = V_U + TD

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

Thatcher Glass

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:

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 C_t be the payment in year t. Then value V is given by V = C₁/(1+r) + C₂/(1+r) ² + ... + C_t/(1+r) ^t. Differentiating V with respect to r gives dV/dr = -C₁/(1+r)² - 2C₂/ (1+r) ³ + ... - tC_t/(1+r) ^t+1. = -[C₁/(1+r) + 2C₂/(1+r)+ ... + tC_t/ (1+r) ^t]/(1+r). If we denote the present value of C_i by PV_i and note that V = PV₁ + PV₂ + ... + PV_t and dV/dr = -[1PV₁ + 2PV₂ + ... + tPV_t]/(1+r), then the division of the second equation by the first gives (1/V)dV/dr= -[(1PV₁+2PV₂+...+tPV_t)/ (PV₁+PV₂+...+PV_t)]/(1+r). But (1PV₁+2PV₂+...+tPV_t)/(PV₁+ PV₂+...+PV_t) is simply what we have defined as Duration.

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

A Hypothetical Leveraged Buyout

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.

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

• V = Replacement value of capital
• D = Debt
• E₁ = Value of equity before LBO
• E₂ = Value of equity after LBO
• r_D = Interest rate on debt
• r_V = Rate of return on assets
• d = Depreciation rate
• Dep = Depreciation
• CF = Cash Flow
• t = Profit tax rate
• S = Surplus = Revenue - Operating Cost
• BTP = Before-tax profit
• ATP = After-tax profit

The relationships that exists between the variables are:

• BTP = S - Dep - Int
• Int = r_D
• PTax = tBTP
• ATP = BTP - PTax = (1-t)BTP
• CF = ATP + Dep
• Inv = gV
• FCF = CF - Inv
• S = r_aV
• Ret = Inv - Dep
• Div = ATP - Ret
• Val = Div/(r_eq-g)
• r_eq = r_f +β_eq(r_m-r_f)
• β_eq = (1+L)β_asset

(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 r_L and the expected rates of inflation are E{inf_$} and E{inf_L}, 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 r_A and r_B = (Covariance (r_A,r_B) /(Stdev_A)(Stdev_B)

β of A = Covariance(r_A,r_m)/Variance(r_m)

r_A=alpha_A+β_Ar_m

r_A=r_F+β_A(r_m-r_F)

r_assets=(E/(E+D))r_equity+(D/(E+D))r_debt

r_equity=r_assets+(D/E)[r_assets -r_debt]

β_assets=(E/(E+D))β_equity +(D/(E+D))β_debt

β_equity=β_assets[1+(D/E)] -(D/E)β_debt

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

r = r_f + β(r_m - r_f)

β_equity = (1+D/E)β_assets

r_equity = r_assets + (D/E)(r_assets - r_debt)

PV = payment/(1+r)a^time

β_assets = β_revenue (1 + (PV of fixed costs)/NPV) expected value = sum of probabilities times payoffs W.A.C.C. = (E/(E+D))r_equity + (D/(E+D))r_debt(1-tax) standard deviation over periods of length t = t^1/2(standard deviation periods of length 1)

Price = Div/(r-g)

The Black and Scholes formula for call value:

Relevant Variables:

S = current stock price

X = exercise price

r_f = risk-free interest rate

σ = instanteous standard deviation of the rate of return of the stock

T = time until expiration of the option

C = value of call

PVX = present value of the exercise price X
= X/(1+r_f)^T

C = S[N(d₁)-N(d₂)(PVX/S)]

where N(d₁) and N(d₂) are the probabilities of a standard normal variable having a value less than or equal to d₁ and d₂.

The values of d₁ and d₂ are given by:

d₁ = [natural logarithm of (S/PVofX) + T(stdv)²/2]

and

d₂=d₁ - stdv x (square root of T)

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

• 1. Determine the time until expiration T in years or a fraction of a year
• 2. Find the present value of the exercise price using the risk-free rate of interest
• 3. Find the ratio of the current stock price to the present value of the exercise price
• 4. Find the product of the square root of time to expiration and the volatility of the stock price (standard deviation)
• 5. Using the ratio found in #3 and the product found in #4 read from the table the ratio of call option value to the current stock price expressed as a percent.
• 6. Multiply the ratio found in #5 time the current stock price to get the value of the 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.

• a. What is the present value of the exercise price?
• b. What is the ratio of the current stock price to the PV of the exercise price?
• c. What is the value of the parameter
  SQRT(time to expiration)xvolatility?
• d. What is the value read from the table of Black and Scholes values?
• e. What is the value of the call option?

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

• r_debt = r_f = 5 percent
• β_asset = 0.8
• r_m = 13 percent
• income of corporation = $2 million
• number of shares = 500,000

Intitial Case: No debt (L=0)

• β_equity = β_asset = 0.8
• required rate of return on equity
• r_equity = r_f + β_equity
• (r_m - r_f)
• = 5 + 0.8(13-5) = 5 + 0.8(8)
• = 11.4 percent

This is also the rate of return on assets r_assets = r_equity = 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.

β_equity = (1+L)β_asset = 4(0.8) = 3.2

r_equity = r_assets + L(r_assets - r_debt)

= 11.4 + 3(11.4-5.0)

= 11.4 + 3(6.4) = 30.6 percent

required rate of return on equity = r_f + β_equity

(r_m-r_f) = 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:

• Suppose the company sells debt to buy up the 75 percent of the common stock. This requires issuing debt equal to (.75)($10.526)= $7.895 million and leaves 125,000 shares in the market. The leverage ratio increase to 3.0.
• The required rate of return is the same as in the not tax case = 0.306
• The interest on the debt is (0.05)(7.895)= $0.395 million
  The taxable profit is (2 -0.395)= $1.605 million
  Aftertax profit is (0.6)(1.605)= $0.963 million
  Income for equity holders = $0.963 million
  
• Valuation of equity = (0.963 million)/(0.306) = $3.147 million
• Price of share = ($4.386 million)/(125,000) = $25.176
• Valuation of company = Value of debt + Value of equity
  = $7.895 million + $3.147 million
  = $11.042 million
• Thus the value of the company rose from $10.526 million to $11.042 million as a result of the leveraging.
• When there is a tax on corporate profits leveraging of the company will increase its value.