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8/11/2019 An Alternative Model for Risk-Adjusted Returns: Effect of Capital Structure on Returns, Derivative Pricing and Volat

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Pablo Carbonell

June 21, 2013

An Alternative Model for Risk-Adjusted Returns:

Effect of Capital Structure on Returns, Derivative Pricing and Volatility

Last updated 10/20/2013

Abstract

The purpose of this study is to assess how capital structure modifies the probability

distribution of returns in stocks and its consequences on derivative pricing and volatility. Under

the Black-Scholes-Merton model and commonly in financial literature, a lognormal distribution

of returns is assumed for stocks. We show that a lognormal distribution cannot be assumed for

stocks as arbitrage opportunities would arise when comparable assets are traded at different

levels of leverage. Consequently we propose a new distribution of returns to adjust for non-

arbitrage conditions. The goal is not to find a definitive probability distribution, as all pricing

mechanisms would have to be factored in for that objective, but rather to adjust our benchmark

lognormal distribution for considerations on capital structure and study the consequences of such

adjustment. Hence we explore those consequences along four areas. First, we show that under

the proposed distribution we predict a higher frequency of larger market declines along with

increases in volatility as the market falls. Second, we derive an adjusted calculation for the

Black-Scholes equation where leverage is introduced as an additional parameter, and show that

this adjustment constitutes a better fit of market prices given that it reduces the dispersion in

implied volatilities of options. Third, we derive an alternative model to CAPM for stocks, where

we show that leverage pulls stocks downwards in relation to a theoretical average Market

Security Line. Fourth, we assess the properties of the Fama-French portfolios under the light of

our alternative model to CAPM. We verify, after adjusting for differences in leverage, that the

market premiums for small and high book to price stocks are consistent with our model.


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Part 1: Capital Structure and Distribution of Returns

Lets consider the scenario of two almost identical companies,

and

, which own the

same type of assets. These firms keep minimum cash to operate and give back to investors all

excess of it, and do not expand or shrink. For illustration purposes lets assume that they are two

farms next to each other. However, companyhas no debt, while has its assets partiallyfinanced through debt.

Now, we could assume either that the value ofis lower than the value of because of tax shields, or that the value ofalready includes the opportunity value of the assetsdebt absorption capacity. The reasoning for the latter is that companycould be part of a largerportfolio: while not levered itself, it would be providing additional debt capacity for the overall

portfolioand correspondingly tax savings would be made somewhere else in the portfolio as a

consequence of. Furthermore, the highest bidder for would include this debt capacitypremium, effectively pricing

in the market at such level. Thus we have

.

If the two firms remain very similarand therefore, their assets interchangeableduring

a period of time where interest rates do not change we will observe:

(1.1)Where is the yield on the debt, and, are the respective prices of and

adjusted for returns at time. Unless something structurally changes, the value of and needto keep this relationship; deviations from such would imply that the same asset has twoprices, opening a window for arbitrage opportunities. In our example of the farms this equationmeans that the two farms maintain equal prices.


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The following graph illustrates a possible path for the adjusted prices ofand:

0.00

20.00

40.00

60.00

80.00

100.00

120.00

Fig. 1.1

A E

What follows is that at maximum only one of the two firms can present returns that fit a

lognormal distribution. Lets call and the respective prices ofand observed afterregular intervals i. If we assume that has lognormal returns, then the distributions of returns forand are given by:

+ + + + + , The parameters and are respectively mean and volatility per interval of time, and

is the yield of the debt per interval of time. It follows that:

+ (1.2)We define R as the returns on E, such that +/. The Cumulative DistributionFunction (CDF) for R is given by:

(1.3)


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We introduce a change of variable /, we have: 1 (1.4)

1 1

Given that X is ,we get: 1 ; , (1.5)

Substituting in the CDF for the normal distribution of we get the CDF for:

12 12

1

2 (1.6)Deriving over the previous expression we get the probability distribution function of R:

;, ,, 1 12 + (1.7)

We have defined a new distribution probability for the returns R, with

parameters, , , . The parameters and are respectively the drift and volatility of theunderlying asset; represents the leverage and the yield on the debt. In the specific casewhen 0 we verify that 1 and the CDF formula becomes the same as in the lognormaldistribution. One way to calculate this CDF is by using the Excel function:

., , ,, where + The expected value of R is given by:

1 1 + 1 (1.8)The variance on is given by:

R [] (1.9)


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From equation (1.4) we have that 1, whereis, . Tosimplify the notation we set , , , .

We introduce a change of variables 1 , and calculateR:R 2 2 Applying properties of the lognormal distribution:

+ 2+ (1.10)We calculate[], using a previous result from (1.8):

[] [+ ] + 2+ (1.11)

Substituting (1.10) and (1.11) into (1.9):

+( 1) (1.12)The standard deviation on is thus given by:

+/ 1 (1.13)


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The following graph illustrates the p.d.f. of ,, , for different levels of. Thevalues chosen are 0.1, 0.2, 0.05. For the case when 1the distribution reducesto the lognormal distribution.

We can see that as the leverage increases, the distribution becomes heavier on the tails,effectively predicting a higher frequency or larger declines in the market.

When the market declines the variance of

will increase. This is because after a market

decline our leverage increases, given that equity is reduced yet debt keeps its value. Fromequation (1.13) we see that the standard deviation of is proportional to the leverageand thusshould increase along with.

In the scenario described we assumed that and were two almost identical companiesand as such the value of their assets had to be tied together. We can generalize this result for

comparing very different types of companies. From a valuation perspective, the value of an asset

is the net present value of the cash flows associated with it. Then we only need to think of any

two firms at different levels of leverage where the cash flows derived from their respective assets

are interchangeable given the markets risk preferences, and the result still applies.

0.3 0.5 0.6 0.8 0.9 1.1 1.2 1.4 1.5 1.7 1.8 2.0 2.1 2.3 2.4 2.6

Fig. 1.9

= 0.8

= 1.0

= 1.4


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Part 2: Implications in Derivative Pricing

Geske (1979) has shown that the value of call option on a stock can be calculated as a

compounded call option, where the underlying stock is itself a call option on its underlying

assets. The stock becomes a call on the firms assets: thestrike price is the value of the debt at

maturity, and the maturity date of the call option is the maturity date of the debt. This model

introduces leverage as a parameter for assessing derivative pricing. However, a company

normally has debt with different maturities that are renewed on a rolling basis. In the case of

amortized loans, each payment date is the maturing of a portion of the principal and carries the

risk of default. Even if we extend this model for multiple maturities (Chen and Yeh, 2006), we

still need to solve for the fact that debt is renewed. An option is an instrument such that as time

goes by the time to maturity decreases (if maturity is in five years, after one year maturity is in

four years). But maturing debt in a firm is renewed with new debt issued, and average time to

maturity of overall debt keeps being pushed forward. Hence in this model a maturity date for the

debt loses meaning; arguably it does not correspond to a relevant assumption for valuation, but

rather to a parameter that needs to be filled-in in order to treat stock as an option. The approach

in this paper is to modify the Black-Scholes formula itselfto adjust for the reasons it does not

represent the effect of leverage in the first place.


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Lets consider the scenario of a stock which is a levered position on asset, asdescribed in the previous section. The price of

follows a lognormal distribution of returns or

Geometric Brownian Motion. Let be the spot price of, and the debt minus excess cash inper share terms.

It follows that:

(2.1)Where 0,1. Letbe a derivative on. Applying the multi-variable extension ofIts Lemma where , we get:

12 We define the portfolio

with -1 derivative, and

shares of

. Most likely there will

be no shares ofavailable to trade, so we build these shares synthetically using ,where is a comparable long position in debt that yields.

The value of portfolio is therefore given by:

Consequently:

12 The term cancels out, thus the portfolio is riskless and yields the risk-free rate:


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Substituting in the previous equations:

12 So that: 12 (2.2)The equation above is an adjusted version of the Black-Scholes-Merton general

differential equation, where is a derivative on ,and . We verify that for 0theequation reduces to the general Black-Scholes-Merton equation.

The specific case of a European call is a function such that, when : max , 0 Replacing for we get:

max , 0We verify in the equation that a call option on with strike is equivalent to a call

option on

with strike price

, where

. As both call options have the

same payoff in any possible scenario, their value has to be the same. Given thatfollows thelognormal distribution we can directly apply Black-Scholes equation on. Thus the solution fora price is:

1

/2

(2.3)

In this equation the parameter is the standard deviation of .


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Part 3: Experimental Results with Adjusted Black-Scholes formula

In this section we assess how well the proposed model fits actual implied volatilities on

S&P 500. To do so, first we need to estimate the leverage parameter for the aggregate of

companies in the index. Our initial assumption is that the underlying productive assets present

lognormal returns. We define these underlying assets as the following:

A = Cash balance needed to operate + Non-cash net working capital + Fixed assets and intangibles

To estimate the value of the unlevered assets, we start with market capitalization, add the

liabilities tied to those assets, and subtract excess cash. We decide to include along with long

term debt other long term liabilities, given that the market value of equity is discounting these

liabilities from the underlying assets. To estimate excess cash we assume that on average, an

unlevered operation needs to have access to cash to cover three months of SG&A expenses. Thus

we define overall net debt as follows:

Net Debt = Debt Excess Cash

Where:

Debt = Current/Short Term Debt + Long Term Debt + Other Long Term Liabilities

Excess Cash = Cash and Equivalents + Short Term InvestmentsS&GA * 3/12


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The following table shows aggregate financial data from stocks in S&P 500:

Fig. 2.1

Portfolio Overall

# Tickers 500

Market Cap 14,978,890,000

Interest Expense 199,699,175

Short/Current Debt 1,997,490,702

Long Term Debt 4,298,836,037

Other Non-Current Liabilities 4,904,666,364

Cash and Equivalents 4,615,930,453

Short Term Investments 519,325,825

SG&A 2,015,029,063

Source: Financial statements from Yahoo! Finance, 06/21/2013.

The result of our calculation places Net Debt at 6,569,494,091or 44% of market

capitalizationwith prices as of June 21st2013.

To assess implied volatilities we look at prices for call options on the ETF ticker SPY for

December 2014 and 2015. We assume a Risk-free rate of 0.5% for all maturities, dividend yield

of 1.9%, and an average interest on net debt of 1.65% which corresponds to 5-year AAA.

Current price of SPY is $159.59, thus becomes $69.99.


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Figures 2.1a and 2.1b show the implied volatilities of options using respectively the

standard and adjusted calculations of Black-Scholes:

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250 300

ImpliedVolatility

Strike Price

Fig. 2.1a - Standard Black-Scholes

2014 2015

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250 300

ImpliedVolatility

Strike Price

Fig. 2.1b - Adjusted Black-Scholes

2014 2015

However we may still be underestimating the full effect of leverage. There are situations

still not accounted for, such as operational leases, which are one type of financial commitment

that does not appear on balance sheets. If we plug a value of 2we get a flatter fit for prices:

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250 300

ImpliedVolatility

Strike Price

Fig. 2.1a - Standard Black-Scholes

2014 2015

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250 300

ImpliedVolatility

Strike Price

Fig. 2.2b - Adjusted Black-Scholes, Lambda=2

2014 2015


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Comparing graphs 2.1a with 2.1b, 2.2b we see that our adjusted calculation for Black-

Scholes reduces the dispersion in implied volatilities, and is therefore a better fit for real prices.

The following chart further illustrates how dispersion in implied volatilities is reduced:

Fig. 2.3

Option Method Strike=50 Strike=260 Ratio

December 2014 Standard B.S. 0.76 0.15 1 : 4.94

Adjusted, D=70 (44% of S) 0.39 0.11 1 : 3.50

Adjusted, D=159.59 (100% of S) 0.24 0.08 1 : 2.86

Option Method Strike=30 Strike=250 Ratio

December 2015 Standard B.S. 1.01 0.16 1 : 6.50

Adjusted, D=70 (44% of S) 0.43 0.11 1 : 3.87

Adjusted, D=159.59 (100% of S) 0.25 0.08 1 : 3.06

The fact that the implied volatility lines are not completely flattened means that this

distribution leaves room for other causes of accelerated market declines, whether systemic (such

as stochastic volatility and interest rates), or non-systemic (such as market drops that are due to

external shocks or single discrete events).


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Part 4: An alternative to CAPM

Let

, , , be the distribution of returns for the overall market, and

, , , the distribution of returns for a stock, where is defined as our distributionof returns. Let be the risk-free rate in continuous compounding terms.

The current value of the market is given by, where represents themarkets aggregate capital structureas defined previously. The stock price is .

The unlevered assetsandfit the following equations:

Lets consider a portfolio with -1 shares ofand shares of. Thevalue of is given by:

It follows that:

Now, the terms and follow the same distribution0,1and are positivelycorrelated. They do not cancel out directly, but tend to do so in a portfolio that has many

for

different stocks:

lim = 0


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In other words, the terms and are diversified away. Thus becomes a risk-freeportfolio. Consequently, this risk-free portfolio should yield the risk-free rate:

Combining the previous equations:

Dividing byand rearranging, we get to our model alternative to CAPM:

(4.1)Notice how this equation is similar to CAPM, yet here the parameters belong to our

distribution,, ,. To compare this model against CAPM we need first to express theexpected return and standard deviation in CAPMs terms, where return is defined as 1.The expected return on the stock for 1is the expected value of 1.

From equation (1.8):

1 1 +

1 1 (4.2)

From equation (1.13):

1 +/ 1 (4.3)


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Using equations (4.1), (4.2) and (4.3), the following table (4.9) illustrates expected

returns and volatility expressed in CAPMs terms. We show the results for values of

0.1, 0.2, 0.03, 0.05.Table 5.9

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.065 0.0825 0.1 0.1175 0.135 0.1525 0.17 0.1875 0.205For =0.7:

E(R-1) 7% 8% 10% 13% 15% 12% 21% 25% 29%

Std. Dev. 0.08 0.12 0.16 0.21 0.26 0.31 0.37 0.44 0.52

For =1.0:

E(R-1) 7% 10% 13% 16% 20% 24% 28% 33% 39%Std. Dev. 0.11 0.17 0.23 0.29 0.37 0.45 0.53 0.63 0.74

For =1.3:

E(R-1) 8% 11% 15% 19% 24% 29% 35% 42% 49%

Std. Dev. 0.14 0.22 0.30 0.38 0.48 0.58 0.70 0.82 0.96

Graphically:

0%

10%

20%

30%

40%

50%

60%

0.00 0.20 0.40 0.60 0.80 1.00 1.20

ExpectedValueof(R-1)

Standard Deviation of (R-1)

Fig. 5.9

=0.7

=1

=1.3


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Results differ from CAPM since it would predict one straight line independent of. Theassumption from CAPM that conflicts with our model is that an investor seeks to optimize the

tradeoff between the expected return

1and volatility

1. This is not the case

in our model, where tradeoffs between and are instead consequences of non-arbitrageconditions on a risk-free portfolio. The key reason for a difference is that the volatility 1does not fully characterize the shape of the distribution of returns. It turns out thattwo distributions of returns with same variance may still have a different shapeand

consequently different expected returns. Graphically, as the CAPM investor trades its portfolio to

move along the CAPMs Security Market Line, the volatility 1is kept constant. Butthe probability distribution is nevertheless changed, and accordingly expected return is modified.


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Part 5: Effect of Capital Structure in the CAPM model

Penman, Richardson and Tuna (2005) have shown that leverage is negatively associated

with returns, after adjusting for differences in Enterprise Book-to-Price value. The purpose of

this section is to derive analytically that leverage is negatively correlated with CAPM risk-

adjusted returns.

In the previous section we showed that the tradeoff between returns and volatility is

governed by equation 4.1, and using equations 4.3 and 4.8 we show that our model contradicts

CAPM. A graphical interpretation for this finding is that different stocks will show a different

slope for the line that connects the stock with the risk free rate:

0%

1%

2%

3%

4%

5%

6%

7%

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

ExpectedValueofR-1

Standard Deviation of (R-1)

Fig. 5.1

Stock


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Under CAPM we would expect only one value for the slope across stocks. Graphically,

increasing the slopemakes the expected return escape CAPMs line upwards. The value of theslope in CAPM is given by definition as:

1 1 1 (5.2)Using equations (4.3), (4.8) and (5.2) we calculate the slope for the points in table 4.9:

Table 5.3

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50=0.7 0.474 0.463 0.465 0.471 0.478 0.485 0.491 0.495 0.498

=1.0 0.391 0.409 0.426 0.441 0.454 0.465 0.474 0.481 0.486

=1.3 0.346 0.380 0.405 0.425 0.441 0.454 0.465 0.474 0.480

Table 5.3 shows higher slope values for stocks with lower leverage, and an apparent

general tendency to increase the slope as underlying asset volatility increases. In the steps that

follow we derive analytically the impact of in the slope.Let

,

,

,

be the distribution of returns for the market and

, , , the distribution of returns for a stock. We substitute in (5.2) with (4.2) and (4.3): + 1 1 1+/ 1 (5.3)Rearranging, and substituting using equation (4.1):

1

1 1

++/

1

(5.4)

Deriving over: 1

(5.5)


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The sign of the derivative on in equation (5.5) is given by the sign of , whereis the risk-free rate and is the interest rate on net debt. Unless the firm piles cash in excess atno or very little interest,

will be higher than the risk-free rate, and thus the derivative will be

negative. Yet normally we would expect companies to place excess cash in short term securities

that yield at least the risk-free rate. As this derivative is negative, it means that decreasing

leverage increases the slope, making the stock escape the CAPMs security line upwards. If a firm could borrow at the risk-free rate the derivative of would cancel, in that case

the slope would not depend on leverage. To illustrate why changes when the risk-free rateand the yield of debt are different, lets imagine that we are managing a levered company. We

have $100 that we can use for either invest in risk-free to add to our treasury, or pay off some of

our debt. An outsider would prefer risk-free debt than lending to us; but from our perspective, we

could consider both as risk-free investments as neither will add volatility to our holding. Yet

paying off the debt offers a return instead of. Consequently, doing so would improve ourposition in terms of the tradeoff between risk and expected returns, measured respectively as

standard deviation and expected value of 1.


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Part 6: Experimental results with the Fama-French portfolios

The Fama-French three-factor model provides an example of portfolios that escape the

CAPMs Security Market Line.Empirical results show that portfolios with higher book-to-price

value or smaller companies present higher average returns. The notation for the Fama-French

model is as follows:

( ) . . The graphical interpretation of this formula is that stocks with SMB or HML

characteristic will show a higher slope, as defined in equation 5.2.In the following experiment we download the financial statements and historical prices ofthe stocks in S&P 500 and in Russell 2000, and estimate the slope for the four quadrants ofSmall/Big and High/Low.

For each stock, the return on the underlying assets in given by:

+ + +

(6.1)

For small intervals of time, we can assume:

+ (6.2)Rearranging equation 6.10 and substituting using 6.11 and 1.2 we get:

1 1 (6.3)The return is directly available from the historical adjusted closing prices. Usingformula (6.3) we can estimate the corresponding return on the underlying asset. The volatility

parameter of each stock is given by the standard deviation of. We calculate the of theaggregate of a portfolio as the average of its stocks weighted by market capitalization.


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Table (6.4) shows the data for the aggregate portfolios:

Fig. 6.4

Portfolio Big High Big Low Small High Small Low Overall

# Tickers 172 328 998 841 2,339

Market Cap 4,611,000,000 10,367,890,000 660,101,190 879,072,320 16,518,063,510

Interest Expense 133,145,964 66,553,211 20,603,877 16,143,088 236,446,140

Short/Current Debt 1,725,755,524 271,735,178 153,757,480 37,154,904 2,188,403,086

Long Term Debt 2,710,081,840 1,588,754,197 261,597,705 227,977,876 4,788,411,618

Other Non-Current Liabilities 4,154,758,958 749,907,406 325,524,621 67,972,100 5,298,163,085

Cash and Equivalents 3,920,606,669 695,323,784 365,284,372 94,447,737 5,075,662,562

Short Term Investments 157,144,811 362,181,014 23,760,318 19,900,860 562,987,003

Total Assets 21,491,280,311 7,491,721,196 2,231,830,147 746,861,137 31,961,692,791

Total Liabilities 17,834,624,946 4,826,549,577 1,678,600,083 530,115,824 24,869,890,430

Weekly Sigma * Market Cap 106,880,762 253,887,186 22,815,951 36,878,053 420,461,952

SG&A 763,621,264 1,251,407,799 176,909,871 157,601,511 2,349,540,445

Source: Financial statements from Yahoo! Finance, 06/21/2013.

We estimate Debt, Net Debt and in the same way as in section 3. To estimate ourparameterwe look directly at the ratio between interest expense and Debt.

Table (6.5) shows the estimates, all values per annum:

Fig. 6.5

Portfolio Big High Big Low Small High Small Low Overall Big Small High Low 0.1671 0.1766 0.2492 0.3025 0.1836 0.1737 0.2797 0.1774 0.1864 0.0154 0.0252 0.0274 0.0473 0.0191 0.0177 0.0336 0.0163 0.0277 2.0201 1.1800 1.6000 1.2937 1.4373 1.4386 1.4250 1.9675 1.1888


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We assume a Risk-free rate of 0.50% and =0.04. Table (6.6) shows the expectedreturn, standard deviation, and the resulting alpha on a CAPM regression. To calculate the

expected return and standard deviation we use formulas (4.2) and (4.3).

Fig. 6.6

Portfolio Big High Big Low Small High Small Low Overall

Expected return (R-1) 0.0895 0.0612 0.1228 0.1340 0.0756

Std. Dev. of (R-1) 0.3578 0.2217 0.4404 0.4464 0.2816

Alpha -0.52% 0.06% 0.73% 1.70% 0.00%

Table (6.6) shows a premium for small over big companies. The premium becomes

negative for Value companies, those with high book-to-price. However, from table (6.5) we see

that these high book-to-price portfolios are more highly levered than the others, and as shown in

equation (5.6) there is a negative premium brought by leverage.

We can adjust the results to leave out the effect of leverage. For this we calculate the

expected returns and volatility of the portfolios using 1.44, the leverage of the overallmarket. Table (6.7) shows the result comparing Big versus Small, and High versus Low:

Fig. 6.7

Portfolio Adj. Big Adj. Small Adj. High Adj. Low

Expected return (R-1) 0.0707 0.1321 0.0734 0.0734

Std. Dev. Of (R-1) 0.2653 0.4519 0.2715 0.2864

Alpha -0.08% 1.38% 0.03% -0.34%

Once adjusting for leverage, table 6.7 shows the premiums for Small versus Big, and for

High versus Low. The differences in these portfolios are given by the parameters

,,that

characterize the shape of the distributions. From table (6.5) we see that smaller firms present

higher volatility of their unlevered assets, and high book-to-price firms have lower costs of debt.

These characteristics modify risk in a way that is not fully captured by CAPM, translating into

different slopes on CAPM as we apply equation (5.4).


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Modigliani, Franco; Miller, Merton. The Cost of Capital, Corporation Finance and the Theory of

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Economy 81 (3): 637654.

Merton, Robert. Theory of Rational Option Pricing. Bell Journal of Economics and Management Science

4 (1): 141183. doi: 10.2307/3003143.

Fama, Eugene F.; French, Kenneth R. (1993). Common Risk Factors in the Returns on Stocks and Bonds.

Journal of Financial Economics 33 (1): 356. doi:10.1016/0304-405X(93)90023-5

John C. Cox, Stephen A. Ross, and Mark Rubinstein. 1979. Option Pricing: A Simplified Approach.

Journal of Financial Economics 7: 229-263

Robert Geske. 1978. The Valuation of Compound Options. Journal of Financial Economics 7: 63 -81.

Stephen H. Penman, Scott A. Richardson, Irem Tuna. 2005. The Book-to-Price Effect in Stock Returns:

Accounting for Leverage. The Rodney L. White Center for Financial Research at Wharton

School.

Hull, John C. (2008). Options, Futures and Other Derivatives(7th Ed.). Prentice Hall. ISBN 0-13-

505283-1.

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