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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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